*Article* **A Remark for the Hyers–Ulam Stabilities on** *n***-Banach Spaces**

**Jaeyoo Choy 1, Hahng-Yun Chu 2,\* and Ahyoung Kim <sup>2</sup>**


**Abstract:** In this article, we deal with stabilities of several functional equations in *n*-Banach spaces. For a surjective mapping *f* into a *n*-Banach space, we prove the generalized Hyers–Ulam stabilities of the cubic functional equation and the quartic functional equation for *f* in *n*-Banach spaces.

**Keywords:** *n*-Banach space; cubic mappings; quartic mappings; the generalized Hyers–Ulam stability

### **1. Introduction**

A question of the stability of functional equations concerning group homomorphisms was first raised by S. M. Ulam in 1940 [1]. In the next year, a partial affirmative answer to the question of Ulam was given by D. H. Hyers [2] for additive mappings on Banach spaces. Hyers' theorem was generalized by T. Aoki [3] for additive mapping. In 1978, Rassias [4] provided a generalization of the theorem for linear mappings by allowing the Cauchy differences to be unbounded. Subsequently, the result of Rassias' theorem was generalized by P. G˘avruta [5], allowing the Cauchy difference controlled by a general unbounded function which is called *the generalized Hyers–Ulam stability*. On the other hand, Rassias and Šemrl found an example of a continuous real-valued function from R for which the Hyers–Ulam stability does not occur. See [6].

Let *X* and *Y* be real vector spaces and *f* : *X* → *Y* a mapping. For a cubic function *<sup>f</sup>*(*x*) = *cx*<sup>3</sup> (*<sup>c</sup>* <sup>∈</sup> <sup>R</sup>, *<sup>X</sup>* <sup>=</sup> *<sup>Y</sup>* <sup>=</sup> <sup>R</sup>), *<sup>f</sup>* clearly satisfies the following functional equation

$$f(\mathbf{x} + 2\mathbf{y}) + 3f(\mathbf{x}) = 3f(\mathbf{x} + \mathbf{y}) + f(\mathbf{x} - \mathbf{y}) + 6f(\mathbf{y}).\tag{1}$$

For this reason, it is natural that Equation (1) is called a cubic functional equation and every solution of Equation (1) is also called a cubic function. The general solution for Equation (1) was solved by J. M. Rassias [7] for a mapping from a real normed space to a Banach space. Jun et al. [8] proved that the cubic functional Equation (1) is equivalent to the following functional equation

$$f(2\mathbf{x} + \mathbf{y}) + f(2\mathbf{x} - \mathbf{y}) = 2f(\mathbf{x} + \mathbf{y}) + 2f(\mathbf{x} - \mathbf{y}) + 12f(\mathbf{x}).\tag{2}$$

In [9], Chu et al. extended the cubic functional equation to the following generalized form

$$\begin{aligned} f(\sum\_{j=1}^{n-1} \mathbf{x}\_j + 2\mathbf{x}\_n) + f(\sum\_{j=1}^{n-1} \mathbf{x}\_j - 2\mathbf{x}\_n) + \sum\_{j=1}^{n-1} f(2\mathbf{x}\_j), \\ = 2f(\sum\_{j=1}^{n-1} \mathbf{x}\_j) + 4\sum\_{j=1}^{n-1} (f(\mathbf{x}\_j + \mathbf{x}\_n) + f(\mathbf{x}\_j - \mathbf{x}\_n)), \end{aligned}$$

where *n* ≥ 2 is an integer, and they also proved the generalized Hyers–Ulam stability. The stability problem for cubic functional equations has been extensively investigated by many mathematicians (see [10–12].)

**Citation:** Choy, J.; Chu, H.-Y.; Kim, A. A Remark for the Hyers–Ulam Stabilities on *n*-Banach Spaces. *Axioms* **2021**, *10*, 2. https://doi.org/ 10.3390/axioms10010002

Received: 6 November 2020 Accepted: 17 December 2020 Published: 29 December 2020

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In [13], J. M. Rassias introduced the functional equation as follows:

$$f(2\mathbf{x} + y) + f(2\mathbf{x} - y) = 4f(\mathbf{x} + y) + 4f(\mathbf{x} - y) + 24f(\mathbf{x}) - 6f(y) \tag{3}$$

It is obvious that *f*(*x*) = *x*<sup>4</sup> is a solution of Equation (3), so we call Equation (3) a quartic functional equation. Chung and Sahoo [14] investigated the general solution of (3) and A. Najati [15] proved the generalized Hyers–Ulam stability for the quartic functional Equation (3) using the idea of G˘avruta [5]. The stability results of quartic functional equations can be found in several other papers (see [16–18].) There are a number of papers and research monographs regarding various generalizations and applications of the generalized Hyers–Ulam stability of several functional equations. See [19–22]. Park investigated the generalized Hyers–Ulam stability for additive mappings, Jensen mappings and quadratic mappings in 2-Banach spaces in [23,24].

Misiak [25,26] introduced the notion of *n*-normed spaces which is one of the generalizations of normed spaces and 2-normed spaces. For more information of the phase spaces, we refer to the papers [27–30]. Recently, Chu et al. [31] studied the generalized Hyers–Ulam stabilities of the Cauchy functional equations, the Jensen functional equations and the quadratic functional equations on *n*-Banach spaces. In [32], Brzde¸k and Ciepli ´nski proved a fixed point theorem for operator acting on a class of functions with values in an *n*-Banach space. For study of the Hyers–Ulam stability, the extension to *n*-Banach spaces is valuable in terms of development of the field of functional equations.

Motivated by results in [31,32], we focus on the generalized Hyers–Ulam stabilities of several functional equationss—in detail, the cubic functional equation expressed as Equation (2) and the quartic functional equation expressed as Equation (3) on *n*-Banach spaces. We prove the generalized Hyers–Ulam stabilities of the functional equations on the spaces.

*The contents of paper*: In Section 2, we recall definitions and lemma in *n*-Banach spaces to investigate the generalized Hyers–Ulam stabilities on the spaces. In Section 3, we investigate the generalized Hyers–Ulam stability problem in *n*-Banach spaces. The problems for the generalized Hyers–Ulam stability related on the cubic functional equation in *n*-Banach spaces are studied in Section 3.1. We also deal with applications of the stabilities for the functional equations on the spaces. In Section 3.2, we focus on the the quartic functional equation and prove the generalized stability on the *n*-Banach spaces.

### **2. Preliminaries**

In this section, we recall definitions and lemma in *n*-Banach spaces as a preliminary step toward the main theorems.

**Definition 1** ([25,26])**.** *Let <sup>X</sup> be a real linear space with* dim *<sup>X</sup>* <sup>≥</sup> *<sup>n</sup> and* ·, ··· , · : *<sup>X</sup><sup>n</sup>* <sup>→</sup> <sup>R</sup> *be a function. Then* (*X*, ·, ··· , ·) *is called a linear n-normed space if*


*for all <sup>α</sup>* <sup>∈</sup> <sup>R</sup> *and all x*, *<sup>y</sup>*, *<sup>x</sup>*1, ··· , *xn* <sup>∈</sup> *X. The function* ·, ··· , · *is called an n-norm on X.*

**Definition 2** ([31])**.** *Let* {*x*-} *be a sequence in a linear n-normed space X. The sequence* {*x*-} *is said to be n-convergent in X if there exists an element x* ∈ *X such that*

$$\lim\_{\ell \to \infty} \|\mathbf{x}\_{\ell} - \mathbf{x}\_{\prime}\mathbf{y}\_{2\prime} \cdot \cdot \cdot \prime y\_{n}\| = 0$$

*for all y*2, ··· , *yn* ∈ *X. In this case, we say that a sequence* {*x*-} *converges to the limit x, simply denoted by* lim-<sup>→</sup><sup>∞</sup> *x*-= *x with a slight abuse of notation.*

**Definition 3** ([31])**.** *A sequence* {*x*-} *in a linear n-normed space X is called an n-Cauchy sequence if for any ε* > 0*, there exists N* ∈ **N** *such that for all s*, *t* ≥ *N, xs* − *xt*, *y*2, ··· , *yn* < *ε for all y*2, ··· , *yn* ∈ *X. For convenience, we will write* lim*s*,*t*→<sup>∞</sup> *xs* − *xt*, *y*2, ··· , *yn* = 0 *for an n-Cauchy sequence* {*x*-}*. An n-Banach space is defined to be a linear n-normed space in which every n-Cauchy sequence is n-convergent.*

The following lemma is a useful toolbox for a linear n-normed space.

**Lemma 1** ([31])**.** *Let* (*X*, ·, ··· , ·) *be a linear n-normed space and x* ∈ *X. Then*


$$\lim\_{m \to \infty} ||x\_{m'} y\_{2'} \dots y\_n|| = ||\lim\_{m \to \infty} x\_{m'} y\_{2'} \dots y\_n||$$

*for all y*2,..., *yn* ∈ *X*.

From now on, let *X* be a real linear space and let (*Z*, ·, ··· , ·) be an *n*-Banach space unless otherwise stated.

### **3. Main Results**

In this section, we present the generalized Hyers–Ulam stabilities for the several functional equations in *n*-Banach spaces. We solve the problems for the stabilities and consider applications of the results in *n*-Banach spaces.

### *3.1. Stability of the Cubic Functional Equation*

We start this subsection by investigating the generalized Hyers–Ulam stability for the cubic functional Equation (2) in *n*-Banach spaces. For convenience, we use the the notation *Df*(*x*, *y*) as follows:

$$D\_f(\mathbf{x}, y) := f(2\mathbf{x} + y) + f(2\mathbf{x} - y) - 2f(\mathbf{x} + y) - 2f(\mathbf{x} - y) - 12f(\mathbf{x})$$

for all *x*, *y* ∈ *X*. If *Df*(*x*, *y*) = 0, then the function *f* is a solution of the cubic functional equation. Thus, *Df*(*x*, *y*) is an approximate remainder of the functional Equation (2) and acts as a perturbation of the equation. We use this approximate remainder to solve the generalized Hyers–Ulam stability for the cubic functional equation in *n*-Banach spaces.

Now, in the following theorem, we present a solution of stability for the cubic funtional equation in the spaces.

**Theorem 1.** *Let <sup>ϕ</sup>* : *<sup>X</sup>n*+<sup>1</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> *be a function such that*

$$\sum\_{i=0}^{\infty} \frac{\varrho(2^i \ge 0, \mathbf{x}\_{2^i} \cdot \cdots, \mathbf{x}\_n)}{8^i} < \infty, \lim\_{\mathfrak{n} \to \infty} \frac{\varrho(2^n \ge 2^n \mathfrak{y}, \mathbf{x}\_{2^i}, \dots, \mathbf{x}\_n)}{8^n} = 0$$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*. *Suppose that a function f* : *X* → *Z be a surjective mapping satisfying*

$$\|\|D\_f(\mathbf{x}, \mathbf{y}), z\_{2\prime}, \dots, z\_n\|\| \le q(\mathbf{x}, \mathbf{y}, \mathbf{x}\_{2\prime}, \dots, \mathbf{x}\_n) \tag{4}$$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, ... , *n. Then there is a unique cubic mapping C* : *X* → *Z such that*

$$\|f(\mathbf{x}) - \mathbb{C}(\mathbf{x}), z\_2, \dots, z\_n\| \le \frac{1}{16} \sum\_{i=0}^{\infty} \frac{\varrho(2^i \mathbf{x}, 0, \mathbf{x}\_{2^i}, \dots, \mathbf{x}\_n)}{8^i} \tag{5}$$

*for all x*, *x*2,..., *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, . . . , *n.*

We call the function *f* the pseudo-cubic function for the error function *ϕ*, and the solution function *C* is the cubic function induced from the pseudo-cubic function *f* .

**Proof.** Let *zi* = *f*(*xi*)(*i* = 2, 3, ..., *n*). First, take *y* = 0 in (4) to have

$$\|\frac{f(2\mathbf{x})}{8} - f(\mathbf{x}), z\_2, \dots, z\_n\| \le \frac{1}{16} \rho(\mathbf{x}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \tag{6}$$

for all *x*, *x*2,..., *xn* ∈ *X*. Replacing *x* by 2*x* in (6) and dividing by 8, we obtain

$$\|\frac{f(2^2\mathbf{x})}{8^2} - f(\mathbf{x}), z\_{2^\prime}, \dots, z\_n\| \le \frac{1}{16} [\varphi(\mathbf{x}, 0, \mathbf{x}\_{2\prime} \cdot \dots \cdot, \mathbf{x}\_n) + \frac{\varphi(2\mathbf{x}, 0, \mathbf{x}\_{2\prime}, \dots, \mathbf{x}\_n)}{8}] \tag{7}$$

for all *x*, *x*2,..., *xn* ∈ *X*. Using the induction on *n*, we get that

$$\left| \left| \frac{f(2^n \mathbf{x})}{8^n} - f(\mathbf{x}), z\_{2^n} \cdot \dots \cdot z\_n \right| \right| \le \frac{1}{16} \sum\_{i=0}^{n-1} \frac{\varrho(2^i \mathbf{x}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n)}{8^i} \tag{8}$$

for all *<sup>x</sup>*, *<sup>x</sup>*2, ... , *xn* <sup>∈</sup> *<sup>X</sup>*. For 0 <sup>≤</sup> *<sup>m</sup>* <sup>&</sup>lt; *<sup>n</sup>*, divide inequality (8) by 8*<sup>m</sup>* and also replace *<sup>x</sup>* by 2*mx* to find that

$$\begin{aligned} \|\frac{f(2^n 2^m \mathbf{x})}{8^{n+m}} - \frac{f(2^m \mathbf{x})}{8^m}, z\_{2^1, \dots, z\_n} \|\| &= \frac{1}{8^m} \|\frac{f(2^n 2^m \mathbf{x})}{8^n} - f(2^m \mathbf{x}), z\_{2^1, \dots, z\_n} \|\| \\ &\leq \frac{1}{16 \cdot 8^m} \sum\_{i=0}^{n-1} \frac{\varrho(2^i 2^m \mathbf{x}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n)}{8^i} \\ &\leq \frac{1}{16} \sum\_{i=m}^{n-1} \frac{\varrho(2^i \mathbf{x}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n)}{8^i} \end{aligned}$$

for all *x*, *x*2,..., *xn* ∈ *X*. We then obtain

$$\lim\_{m,n \to \infty} \parallel \frac{f(2^n 2^m \mathbf{x})}{8^{n+m}} - \frac{f(2^m \mathbf{x})}{8^m}, z\_{2'}, \dots, z\_{\mathbb{N}} \parallel = 0$$

for all *<sup>x</sup>*2, ... , *xn* <sup>∈</sup> *<sup>X</sup>*. Since *<sup>f</sup>* is surjective, by Lemma 1, the sequence { <sup>1</sup> <sup>8</sup>*<sup>n</sup> <sup>f</sup>*(2*nx*)} is an *n*-Cauchy sequence in *Z*. Therefore, we may define a mapping *C* : *X* → *Z* by

$$\mathcal{C}(\mathfrak{x}) := \lim\_{n \to \infty} \frac{1}{8^n} f(2^n \mathfrak{x})$$

for all *x* ∈ *X*. By letting *n* → ∞ in (8), we arrive at the formula (5). To show that the mapping *<sup>C</sup>* : *<sup>X</sup>* <sup>→</sup> *<sup>Z</sup>* satisfies Equation (2), replace *<sup>x</sup>*, *<sup>y</sup>* with 2*nx*, 2*ny*, respectively, in (4) and divide by 8*n*; then it follows that

$$\begin{aligned} 8^{-n} \| f(2^n(2\mathbf{x} + \mathbf{y}) + f(2^n(2\mathbf{x} - \mathbf{y}) - 2f(2^n(\mathbf{x} + \mathbf{y})) - 2f(2^n(\mathbf{x} - \mathbf{y}) - 12f(2^n\mathbf{x}), z\_2, \dots, z\_n) \| &\tag{9} \\ &\le 8^{-n} \varphi(2^n\mathbf{x}, 2^ny, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \end{aligned}$$

for all *x*, *x*2, ... , *xn* ∈ *X*. Taking the limit as *n* → ∞ in (9), we immediately obtain that the mapping *C* satisfies (2).

Now, let *D* : *X* → *Z* be another cubic mapping satisfying (5). Then we have

$$\begin{aligned} \|\mathbb{C}(\mathbf{x}) - D(\mathbf{x}), z\_2, \dots, z\_n\|\| &= \|\mathbb{S}^{-n}\|\mathbb{C}(2^n \mathbf{x}) - D(2^n \mathbf{x}), z\_2, \dots, z\_n\|\\ &\le \|\mathbb{S}^{-n}(\|\mathbb{C}(2^n \mathbf{x}) - f(2^n \mathbf{x}), z\_2, \dots, z\_n\|\| + \|f(2^n \mathbf{x}) - D(2^n \mathbf{x}), z\_2, \dots, z\_n\|\|) \\ &\le \frac{1}{8} \sum\_{i=0}^{\infty} \frac{\rho(2^i 2^n \mathbf{x}, 0, x\_2, \dots, x\_n)}{8^{n+i}} \end{aligned}$$

which tends to zero as *k* → ∞ for all *x*, *z*2, ... , *zn* ∈ *X*. By Lemma 1, we conclude that *C*(*x*) = *D*(*x*) for all *x* ∈ *X*. This completes the proof of the theorem.

As an application of Theorem 1, we obtain a stability of Equation (2) in the following corollary.

**Corollary 1.** *Assume that* (*X*, ·) *is a real normed space and that* (*Z*, ·, ··· , ·) *is a linear n-normed space. Let θ* ∈ [0, ∞), *p*, *q*,*r* ∈ (0, ∞) *and p*, *q* < 3 *and let f* : *X* → *Z be a surjective mapping satisfying*

$$\|D\_f(\mathbf{x}, y), z\_{2\prime}, \dots, z\_n\| \le \theta (||\mathbf{x}||^p + ||y||^q) \|\mathbf{x}\_2\|^r \cdot \dots \cdot \|\mathbf{x}\_n\|^r$$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, ... , *n. Then there is a unique cubic mapping C* : *X* → *Z such that*

$$\left\| f(\mathbf{x}) - \mathbf{C}(\mathbf{x}), z\_2, \dots, z\_n \right\| \le \frac{\theta}{16 - 2^{p+1}} \left\| \mathbf{x} \right\|^p \left\| \mathbf{x}\_2 \right\|^r \cdot \dots \cdot \left\| \mathbf{x}\_n \right\|^r$$

*for all x*, *x*2,..., *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, . . . , *n.*

**Proof.** The assertion follows from Theorem 1 by setting

$$\varphi(x, y, \mathbf{x\_2}, \dots, \mathbf{x\_n}) = \theta(||\mathbf{x}||^p + ||y||^q) ||\mathbf{x\_2}||^r \cdot \dots ||\mathbf{x\_n}||^r$$

for all *x*, *y*, *x*2,..., *xn* ∈ *X*.

In the next theorem we also deal with a solution of the cubic functional equation in *n*-Banach spaces under different conditions. Next, we investigate the change of conditions for the pseudo-cubic function *f* and the error function *ϕ*, and also obtain a stability of Equation (2) in the following theorem (compare with Theorem 1).

**Theorem 2.** *Let <sup>ϕ</sup>* : *<sup>X</sup>n*+<sup>1</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> *be a function such that*

$$\sum\_{i=0}^{\infty} 8^i \varphi(\frac{\chi}{2^{i+1}}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n) < \infty,\\ \lim\_{n \to \infty} 8^n \varphi(\frac{\chi}{2^n}, \frac{\chi}{2^n}, \mathbf{x}\_2, \dots, \mathbf{x}\_n) = 0.$$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, ... , *n. Suppose that a function f* : *X* → *Z be a surjective mapping satisfying* (4)*. Then there is a unique cubic mapping C* : *X* → *Z such that*

$$\|f(\mathbf{x}) - \mathbf{C}(\mathbf{x}), z\_2, \dots, z\_n\| \le \frac{1}{2} \sum\_{i=0}^{\infty} 8^i \varphi(\frac{\mathbf{x}}{2^{i+1}}, 0, \mathbf{x}\_{2^i}, \dots, \mathbf{x}\_n) \tag{10}$$

*for all x*, *x*2,..., *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, . . . , *n.*

**Proof.** Let *zi* = *f*(*xi*) for each *i* = 2, ... , *n*. Take *x* = *<sup>x</sup>* <sup>2</sup> in (6) and multiply by eight to have

$$\|f(\mathbf{x}) - 8f(\frac{\mathbf{x}}{2}), z\_{2\prime}, \dots, z\_n\| \le \frac{1}{2} \varphi(\frac{\mathbf{x}}{2}, 0, \mathbf{x}\_{2\prime}, \dots, \mathbf{x}\_n) \tag{11}$$

for all *<sup>x</sup>*, *<sup>x</sup>*2,..., *xn* <sup>∈</sup> *<sup>X</sup>*. By replacing *<sup>x</sup>* with *<sup>x</sup>* <sup>2</sup> in (11) and multiplying by eight, we get

$$\|f(\mathbf{x}) - 8^2 f(\frac{\mathbf{x}}{2^2}), z\_2, \dots, z\_n\| \le \frac{1}{2} \varrho(\frac{\mathbf{x}}{2}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n) + 4 \varrho(\frac{\mathbf{x}}{2^2}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n)$$

for all *x*, *x*2,..., *xn* ∈ *X*. Then we can find a unique cubic mapping *C* : *X* → *Z* defined by

$$\mathcal{C}(x) := \lim\_{n \to \infty} 8^n f(\frac{x}{2^n})$$

for all *x* ∈ *X*, as in the proof of Theorem 4. This completes the proof.

Using the above theorem, we immediately get the following corollary.

**Corollary 2.** *Assume that* (*X*, ·) *is a real normed space and that* (*Z*, ·, ··· , ·) *is a linear n-normed space. Let θ* ∈ [0, ∞), *p*, *q*,*r* ∈ (0, ∞)*, and p*, *q* > 3*. Let f* : *X* → *Z be a surjective mapping satisfying*

$$\|\|D\_f(\mathbf{x}, y), z\_{2^r}, \dots, z\_n\|\| \le \theta (||\mathbf{x}||^p + ||y||^q) \|\|\mathbf{x}\_2\|\|^r \cdot \dots \cdot \|\|\mathbf{x}\_{\text{H}}\|\|^r$$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, ... , *n. Then there is a unique cubic mapping C* : *X* → *Z such that*

$$\left| \| f(\mathbf{x}) - \mathbb{C}(\mathbf{x})\_{\prime} z\_{2\prime} \dots z\_{n} \| \right| \le \frac{\theta}{2^{p+1} - 16} \| |\mathbf{x}| |^{p} \| |\mathbf{x}\_{2}| |^{r} \cdot \dots | |\mathbf{x}\_{n}| |^{r}$$

*for all x*, *x*2,..., *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, . . . , *n.*

**Proof.** The proof follows from Theorem 2 with

$$\varrho(\mathbf{x}, \mathbf{y}, \mathbf{x}\_2, \cdots, \mathbf{x}\_n) = \theta(||\mathbf{x}||^p + ||\mathbf{y}||^q) ||\mathbf{x}\_2||^r \cdots ||\mathbf{x}\_n||^r$$

for all *x*, *y*, *x*2,..., *xn* ∈ *X*.

### *3.2. Stability of the Quartic Functional Equation*

In this subsection, we discuss the generalized Hyers–Ulam stability of the quartic functional Equation (3) in *n*-Banach spaces. For *x*, *y* ∈ *X*, we define *Ef*(*x*, *y*) given by

$$E\_f(\mathbf{x}, y) := f(2\mathbf{x} + y) + f(2\mathbf{x} - y) - 4f(\mathbf{x} + y) - 4f(\mathbf{x} - y) - 24f(\mathbf{x}) + 6f(y).$$

The difference *Ef*(*x*, *y*) means an approximate remainder of the functional Equation (3).

Now we provide important consequences for the stability of the quartic functional equation.

**Theorem 3.** *Let <sup>ϕ</sup>* : *<sup>X</sup>n*+<sup>1</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> *be a function such that*

$$\sum\_{i=0}^{\infty} 2^{-4k} \varphi(2^i \mathbf{x}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_{\mathbb{N}}) < \infty \text{ and } \lim\_{n \to \infty} 2^{-4n} \varphi(2^n \mathbf{x}, 2^n y, \mathbf{x}\_2, \dots, \mathbf{x}\_{\mathbb{N}}) = 0, \text{ } $$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*. *Suppose that a function f* : *X* → *Z be a surjective mapping satisfying*

$$\|\|E\_f(\mathbf{x}, y), z\_{2\prime}, \dots, z\_n\|\| \le \delta + \varrho(\mathbf{x}, y, \mathbf{x\_{2\prime}}, \dots, \mathbf{x\_n})\tag{12}$$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, ... , *n, where δ* ≥ 0*. Then there is a unique quartic mapping T* : *X* → *Z such that*

$$\|f(\mathbf{x}) - T(\mathbf{x}), z\_2, \dots, z\_n\| \le \frac{1}{30}\delta + \frac{1}{32} \sum\_{i=0}^{\infty} 2^{-4k} \varphi(2^i \mathbf{x}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n) + \frac{1}{5} \|f(0), z\_2, \dots, z\_n\| \tag{13}$$

*for all x*, *x*2,..., *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, . . . , *n.*

From now on we call the function *f* the pseudo-quartic function for *ϕ*, and the solution function *T* is the quartic function induced from the pseudo-quartic function *f* .

**Proof.** Let *zi* = *f*(*xi*) for each *i* = 2, . . . , *n*. By letting *y* = 0 in (12), we get

$$\|2^{-4}f(2\mathbf{x}) - f(\mathbf{x}) + \frac{3}{16}f(0), \mathbf{z}\_{2\prime}, \dots, \mathbf{z}\_{n}\| \le \frac{1}{32}\delta + \frac{1}{32}\rho(\mathbf{x}, 0, \mathbf{z}\_{2\prime}, \dots, \mathbf{x}\_{n}),\tag{14}$$

for all *<sup>x</sup>*, *<sup>x</sup>*2, ... , *xn* <sup>∈</sup> *<sup>X</sup>*. If we replace *<sup>x</sup>* by 2*n*−1*<sup>x</sup>* in (14) and divide both sides of (14) by 24*n*−4, we have that

$$\left\| 2^{-4n} f(2^n \mathbf{x}) - 2^{4-4n} f(2^{n-1x}) + 3 \cdot 2^{-4n} f(0), z\_2, \dots, z\_n \right\| \le \frac{\delta}{2^{4n+1}} + \frac{1}{2^{4n+1}} q(2^{n-1} \mathbf{x}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n) $$

for all *x*, *x*2, ... , *xn* ∈ *X* and integers *n* ≥ 1. Therefore, for all integers 0 ≤ *m* < *n*, we obtain

$$\begin{aligned} &\|\sum\_{k=m+1}^n [2^{-4k}f(2^kx) - 2^{4-4k}f(2^{k-1}x) + 3 \cdot 2^{-4k}f(0)], z\_{2^\prime}, \dots, z\_n\| \\ &\le \sum\_{k=m+1}^n \|[2^{-4k}f(2^kx) - 2^{4-4k}f(2^{k-1}x) + 3 \cdot 2^{-4k}f(0)], z\_{2^\prime}, \dots, z\_n\| \\ &\le \delta \sum\_{k=m+1}^n 2^{-4k-1} + \sum\_{k=m+1}^n 2^{-4k-1} \varphi(2^{k-1}x, 0, \mathbf{x}\_{2^\prime}, \dots, \mathbf{x}\_n) \end{aligned}$$

and so

$$\begin{aligned} & \| 2^{-4n} f(2^n \mathbf{x}) - 2^{-4m} f(2^m \mathbf{x}), z\_{2^\*}, \dots, z\_{\mathbb{R}} \| \\ & \le 3 \cdot \| f(0), z\_{2^\*}, \dots, z\_{\mathbb{R}} \| \sum\_{k=m+1}^n 2^{-4k} + \delta \sum\_{k=m+1}^n 2^{-4k-1} + \sum\_{k=m+1}^n 2^{-4k-1} \varphi(2^{k-1} \mathbf{x}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_{\mathbb{R}}) \end{aligned} \tag{15}$$

for all *x*, *x*2, ... , *xn* ∈ *X*. In a similar way as in the proof of Theorem 1, we can show that the sequence {2−4*<sup>n</sup> <sup>f</sup>*(2*nx*)} is an *<sup>n</sup>*-Cauchy sequence in *<sup>Z</sup>* for all *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*. Define a mapping *T* : *X* → *Z* by

$$T(\mathbf{x}) := \lim\_{n \to \infty} 2^{-4n} f(2^n \mathbf{x})$$

for all *x* ∈ *X*. By (15), we have the inequality (13). It follows from (12) that

$$\begin{aligned} &\|T(2\mathbf{x}+\mathbf{y}) + T(2\mathbf{x}-\mathbf{y}) - 4T(\mathbf{x}+\mathbf{y}) - 4T(\mathbf{x}-\mathbf{y}) - 24T(\mathbf{x}) + 6T(\mathbf{y}), \; z\_2, \dots, z\_n\|\_2 \\ &= \lim\_{\mathbf{n}\to\infty} 2^{-4n} \|f(2^n(2\mathbf{x}+\mathbf{y})) + f(2^n(2\mathbf{x}-\mathbf{y})) - 4f(2^n(\mathbf{x}+\mathbf{y})) - 4f(2^n(\mathbf{x}-\mathbf{y})) \\ &- 24f(2^nx) + 6f(2^ny), \; z\_2, \dots, z\_n\| \\ &\le \lim\_{\mathbf{n}\to\infty} 2^{-4n} \varphi(2^nx, 2^ny, \mathbf{x}\_2, \dots, \mathbf{x}\_n) = 0 \end{aligned}$$

for all *x*, *y*, *x*2, ... , *xn* ∈ *X*. This implies that *T* : *X* → *Z* is a quartic mapping. Let *Q* : *X* → *Z* be another quartic mapping satisfying (13). Therefore we have

$$\begin{aligned} \|T(\mathbf{x}) - Q(\mathbf{x}), z\_2, \dots, z\_n\| &= \lim\_{n \to \infty} 2^{-4n} \|f(2^n \mathbf{x}) - Q(2^n \mathbf{x}), z\_2, \dots, z\_n\| \\ &\le \lim\_{n \to \infty} 2^{-4n} \left(\frac{1}{30}\delta + \frac{1}{32}\tilde{\rho}(2^n \mathbf{x}) + \frac{1}{5} \|f(0), z\_2, \dots, z\_n\|\right) \\ &= \frac{1}{32} \lim\_{n \to \infty} \sum\_{k=n}^{\infty} 2^{-4k} \rho(2^k \mathbf{x}, 0, \mathbf{x\_2}, \dots, \mathbf{x\_n}) = 0 \end{aligned}$$

for all *x*, *y*, *x*2, ... , *xn* ∈ *X*. It follows from Lemma 1 that *T*(*x*) = *Q*(*x*) for all *x* ∈ *X*. This proves the uniqueness of *T*.

Now we show a simple application of Theorem 3 to obtain a stability of Equation (3).

**Corollary 3.** *Assume that* (*X*, ·) *is a real normed space and that* (*Z*, ·, ··· , ·) *is a linear n-normed space. Let ε*, *δ*, *θ*, ∈ [0, ∞), *p*, *q*,*r* ∈ (0, ∞) *and p*, *q* < 4*. Let f* : *X* → *Z be a surjective mapping satisfying*

$$||E\_f(\mathbf{x}, y)\_r z\_{2^r} \dots\_r z\_n|| \le \delta + (\varepsilon ||\mathbf{x}||^p + \theta ||y||^q) ||\mathbf{x}\_2||^r \cdot \dots ||\mathbf{x}\_n||^r$$

*for all x*, *y*, *z*2, ... , *zn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, ... , *n. Then there is a unique quartic mapping T* : *X* → *Z such that*

$$\|f(\mathbf{x}) - T(\mathbf{x}), z\_{2^{p}}, \dots, z\_{n}\| \le \frac{\delta}{24} + \frac{\varepsilon}{32 - 2^{p+1}} ||\mathbf{x}||^{p} ||\mathbf{x}\_{2}||^{r} \cdot \dots ||\mathbf{x}\_{n}||^{r}$$

*for all x*, *x*2,..., *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, . . . , *n.*

**Proof.** It is a consequence of Theorem 3 with

$$\varrho(\mathbf{x}, \mathbf{y}, \mathbf{x}\_2, \dots, \mathbf{x}\_n) = (\varepsilon ||\mathbf{x}||^p + \theta ||\mathbf{y}||^q) ||\mathbf{x}\_2||^r \cdot \dots \cdot ||\mathbf{x}\_n||^r$$

for all *x*, *y*, *x*2,..., *xn* ∈ *X*.

Next we consider the changes of conditions of the pseudo-quartic function and the error function, and we find a solution of the quartic functional equation in *n*-Banach spaces. We prove the existence of a solution of Equation (3) in *n*-Banach spaces.

**Theorem 4.** *Let <sup>ϕ</sup>* : *<sup>X</sup>n*+<sup>1</sup> <sup>→</sup> <sup>R</sup><sup>+</sup> *be a function such that*

$$\sum\_{i=0}^{\infty} 2^{4i} \varrho(\frac{\mathbf{x}}{2^{i+1}}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n) < \infty,\\ \lim\_{n \to \infty} 2^{4n} \varrho(\frac{\mathbf{x}}{2^n}, \frac{\mathbf{y}}{2^n}, \mathbf{x}\_2, \dots, \mathbf{x}\_n) = 0,\tag{16}$$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*. *Suppose that a function f* : *X* → *Z be a surjective mapping satisfying*

$$\|E\_f(\mathbf{x}, y), z\_{2\prime}, \dots, z\_n\| \le \varphi(\mathbf{x}, y, \mathbf{x}\_{2\prime}, \dots, \mathbf{x}\_n) \tag{17}$$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, ... , *n. Then there is a unique quartic mapping T* : *X* → *Z such that*

$$\|f(\mathbf{x}) - T(\mathbf{x}), z\_2, \dots, z\_n\| \le \frac{1}{2} \sum\_{i=0}^{\infty} 2^{4i} \varphi(\frac{\mathbf{x}}{2^{i+1}}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \tag{18}$$

*for all x*, *x*2,..., *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, . . . , *n.*

**Proof.** It follows from (16) that *ϕ*(0, 0) = 0. Thus, we have *f*(0) = 0 from (17). By letting *y* = 0 in (17), we get

$$\|2^{-4}f(2\mathbf{x}) - f(\mathbf{x}), z\_{2\prime}, \dots, z\_n\| \le \frac{1}{32} \varrho(\mathbf{x}, 0, \mathbf{x}\_{2\prime}, \dots, \mathbf{x}\_n) \tag{19}$$

for all *<sup>x</sup>*, *<sup>x</sup>*2, ... , *xn* <sup>∈</sup> *<sup>X</sup>*, where *zi* <sup>=</sup> *<sup>f</sup>*(*xi*) for each *<sup>i</sup>* <sup>=</sup> 2, ... , *<sup>n</sup>*. Through replacing *<sup>x</sup>* by *<sup>x</sup>* <sup>2</sup> in (19) and multiplying by 24, we obtain

$$\|f(\mathbf{x}) - 2^4 f(\frac{\mathbf{x}}{2}), z\_2, \dots, z\_n\| \le \frac{1}{2} \rho(\frac{\mathbf{x}}{2}, 0, \mathbf{x}\_{2\prime}, \dots, \mathbf{x}\_n) \tag{20}$$

for all *x*, *x*2,..., *xn* ∈ *X*, where *zi* = *f*(*xi*) for each *i* = 2, . . . , *n*. By (20), we have

$$\|f(\mathbf{x}) - 2^8 f(\frac{\mathbf{x}}{2^2}), z\_2, \dots, z\_n\| \le \frac{1}{2} \varrho(\frac{\mathbf{x}}{2}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n) + 2^3 \varrho(\frac{\mathbf{x}}{2^2}, 0, \mathbf{x}\_2, \dots, \mathbf{x}\_n)$$

for all *x*, *x*2, ... , *xn* ∈ *X*, where *zi* = *f*(*xi*) for each *i* = 2, ... , *n*. As in the proof of Theorem 3, we can find a unique quartic mapping *T* : *X* → *Z* defined by

$$T(\mathfrak{x}) := \lim\_{n \to \infty} \mathfrak{2}^{4n} f(\frac{\mathfrak{x}}{2^n})$$

for all *x* ∈ *X*. This completes the proof.

As an application of Theorem 1, we obtain a stability of Equation (2) in the following corollary.

**Corollary 4.** *Assume that* (*X*, ·) *is a real normed space and that* (*Z*, ·, ··· , ·) *is a linear n-normed space. Let ε*, *θ* ∈ [0, ∞), *p*, *q*,*r* ∈ (0, ∞) *and p*, *q* > 4*. Let f* : *X* → *Z be a surjective mapping satisfying*

$$\left| \| E\_f(\mathbf{x}, y), z\_{2\prime}, \dots, z\_n \| \right| \le (\varepsilon \| \| \mathbf{x} \| \| ^p + \theta \| \| y \| ^q) \| \| \mathbf{x}\_2 \| \| ^r \cdot \dots \cdot \| \| \mathbf{x}\_n \| ^r$$

*for all x*, *y*, *x*2, ... , *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, ... , *n. Then there is a unique quartic mapping T* : *X* → *Z such that*

$$\|f(\mathbf{x}) - T(\mathbf{x}), z\_{2\prime}, \dots, z\_n\| \le \frac{\varepsilon}{2^{p+1} - 32} \|\mathbf{x}\|^p \|\mathbf{x}\_2\|^r \cdot \dots \cdot \|\mathbf{x}\_n\|^r$$

*for all x*, *x*2,..., *xn* ∈ *X*, *where zi* = *f*(*xi*) *for each i* = 2, . . . , *n.*

**Proof.** It is a direct consequence of Theorem 4 with

$$\varrho(\mathbf{x}, y, \mathbf{x\_{2}}, \dots, \mathbf{x\_{n}}) = (\varepsilon ||\mathbf{x}||^{p} + \theta ||y||^{q}) ||\mathbf{x\_{2}}||^{r} \cdot \dots \cdot ||\mathbf{x\_{n}}||^{r}$$

for all *x*, *y*, *x*2,..., *xn* ∈ *X*.

### **4. Conclusions**

In this paper, we considered the cubic functional equation and quartic functional equation in *n*-Banach spaces. We dealt with stabilities of the functional equations in *n*-Banach spaces. For a surjective mapping *f* into an *n*-Banach space, called a pseudocubic function or a pseudo-quartic function, we solved the stability problem for the cubic functional equations and the quartic functional equations for *f* , as we demonstrated the existence of the solutions of the functional equations. As applications, we got the solutions of the generalized Hyers–Ulam stabilities under the changes of conditions of the pseudofunctions and the error functions. Our results about the equations in *n*-Banach spaces are a new approach and are key extensions for the study of functional equations, where the novelty of our results lies.

**Author Contributions:** Conceptualization, Investigation and Methodology: J.C., H.-Y.C. and A.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by research fund of Chungnam National University.

**Acknowledgments:** The authors are deeply grateful to the referees whose remarks helped to improve the manuscript.

**Conflicts of Interest:** The authors declare no conflicts of interest.

### **References**


### *Article* **An Inertial Generalized Viscosity Approximation Method for Solving Multiple-Sets Split Feasibility Problems and Common Fixed Point of Strictly Pseudo-Nonspreading Mappings**

**Hammed Anuoluwapo Abass 1,2 and Lateef Olakunle Jolaoso 3,***<sup>∗</sup>*


**Abstract:** In this paper, we propose a generalized viscosity iterative algorithm which includes a sequence of contractions and a self adaptive step size for approximating a common solution of a multiple-set split feasibility problem and fixed point problem for countable families of *k*-strictly pseudononspeading mappings in the framework of real Hilbert spaces. The advantage of the step size introduced in our algorithm is that it does not require the computation of the Lipschitz constant of the gradient operator which is very difficult in practice. We also introduce an inertial process version of the generalize viscosity approximation method with self adaptive step size. We prove strong convergence results for the sequences generated by the algorithms for solving the aforementioned problems and present some numerical examples to show the efficiency and accuracy of our algorithm. The results presented in this paper extends and complements many recent results in the literature.

**Citation:** Abass, H.A.; Jolaoso, L.O. An Inertial Generalized Viscosity Approximation Method for Solving Multiple-Sets Split Feasibility Problems and Common Fixed Point of Strictly Pseudo-Nonspreading Mappings. *Axioms* **2021**, *10*, 1. https://dx.doi.org/ 10.3390/axioms10010001

Received: 30 November 2020 Accepted: 14 December 2020 Published: 24 December 2020

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/).

**Keywords:** multiple-sets split feasibility problem; strictly pseudocontractive mappings; nonexpansive mappings; viscossity iterative scheme; fixed point problem

### **1. Introduction**

The problem of finding a point in the intersection of closed and convex subsets in real Hilbert spaces has appeared severally in diverse areas of mathematics and physical sciences. This problem is commonly referred to as the Convex Feasibility Problem (shortly, CFP), and finds its applications in various disciplines such as image restoration, computer tomograph and radiation therapy treatment planning, see [1]. A generalization of the CFP is the Split Feasibility Problem (SFP) which was introduced by Censor and Elfving [2] and defined as finding a point in a nonempty closed convex set, whose image under a bounded operator is in another set. Mathematically, the SFP can be formulated as:

$$\text{find} \quad \mathbf{x}^\* \in \mathbb{C} \quad \text{such that} \quad Ax^\* \in Q\_{\prime} \tag{1}$$

where *C* and *Q* are nonempty closed convex subsets of R*<sup>N</sup>* and R*<sup>M</sup>* respectively, and *A* is a given matrix of dimension *N* × *M*. The SFP also models inverse problems arising from phase retrieval and intensity modulated radiation therapy [2]. Censor et al. [3] further introduced another generalization of the CFP and SFP called the Multiple Set Split Feasibility Problem (MSSFP) which is formulated as

$$\text{find} \quad \mathbf{x}^\* \in \mathbb{C} := \cap\_{i=1}^k \mathbb{C}\_i \quad \text{such that} \quad A\mathbf{x}^\* \in \mathbb{Q} := \cap\_{j=1}^t Q\_{j\prime} \tag{2}$$

where *k* ≥ 1 and *r* ≥ 1 are given integers, *A* is a given *M* × *N* real matrix with *A*<sup>∗</sup> its transpose, {*Ci*}*<sup>k</sup> <sup>i</sup>*=<sup>1</sup> and {*Qj*}*<sup>t</sup> <sup>j</sup>*=<sup>1</sup> are nonempty closed convex subsets of <sup>R</sup>*<sup>N</sup>* and <sup>R</sup>*M*, respectively. Observe that when *k* = *r* = 1, the MSSFP reduces to SFP (1). In this paper, we focus on the MSSFP in a unified framework. We denote the set of solutions of (2) by Ω and assume that Ω is consistent (i.e., nonempty). It is well known that the MSSFP is equivalent to the following minimization problem:

$$\min \left\{ \frac{1}{2} ||\mathbf{x} - P\_C(\mathbf{x})||^2 + \frac{1}{2} ||A\mathbf{x} - P\_Q(A\mathbf{x})||^2 \right\},\tag{3}$$

where *PC* and *PQ* are the orthogonal projections onto *C* and *Q* respectively. For solving (3), Censor et al. [3] defined a proximity function *p*(*x*) for measuring the distance of a point to all sets as follows:

$$p(\mathbf{x}) := \frac{1}{2} \sum\_{i=1}^{k} a\_i ||\mathbf{x} - P\_{\mathbb{C}\_i}(\mathbf{x})||^2 + \frac{1}{2} \sum\_{j=1}^{t} \beta\_j ||Ax - P\_{\mathbb{Q}\_j}(Ax)||^2 \tag{4}$$

where *<sup>α</sup><sup>i</sup>* <sup>&</sup>gt; 0, *<sup>β</sup><sup>j</sup>* <sup>&</sup>gt; <sup>0</sup> <sup>∀</sup>*<sup>i</sup>* and *<sup>j</sup>* respectively, and <sup>∑</sup>*<sup>k</sup> <sup>i</sup>*=<sup>1</sup> *<sup>α</sup><sup>i</sup>* + <sup>∑</sup>*<sup>t</sup> <sup>j</sup>*=<sup>1</sup> *β<sup>j</sup>* = 1. It is easy to see that

$$\bigtriangledown p(\mathbf{x}) := \sum\_{i=1}^{k} \alpha\_i (\mathbf{x} - P\_{\mathbf{C}\_i}(\mathbf{x})) + \sum\_{j=1}^{t} \beta\_j A^\*(I - P\_Q) A \mathbf{x}.$$

Censor et al. [3] also introduced the following projection method for solving the MSSFP:

$$\mathbf{x}\_{n+1} = P\_{\Omega}(\mathbf{x}\_n - \mathbf{s} \oslash p(\mathbf{x}\_n)),\tag{5}$$

where *s* is a positive scalar. They further proved the weak convergence of (5) under the condition that the stepsize *s* satisfies

$$0 < s\_L \le s \le s\_\mu < \frac{2}{L'} $$

where *L* = ∑*<sup>k</sup> <sup>i</sup>*=<sup>1</sup> *<sup>α</sup><sup>i</sup>* + *<sup>ρ</sup>*(*A*∗*A*) <sup>∑</sup>*<sup>t</sup> <sup>j</sup>*=<sup>1</sup> *β<sup>j</sup>* is the Lipschitz constant of *p*. A major setback of (5) is the fact that the algorithm used a fixed stepsize which is restricted by the Lipschitz constant (this depends on the largest eigenvalue of the matrix *A*∗*A*). Computing the largest eigenvalue of *A*∗*A* is usually difficult and its conservation results in slow convergence. More so, note that the projection onto the sets *C* and *Q* are often difficult to calculate when the sets are not simple. This can also result in the complication of (5). Several efforts have been made in order to find best appropriate modifications of (5) without the setbacks in infinite dimensional real Hilbert spaces. For instance, Zhao and Yang [4] introduced a new projection method such that the stepsize *s* is selected via an Armijo line search technique for solving the MSSFP. However, this line search process required extra inner iteration for obtaining a suitable stepsize. The authors in [5] also introduced a self-adaptive projection method which requires the computation of the stepsize directly without any inner iteration. More so, López et al. [6] introduced a relaxed projection method with a fixed stepsize and proved a weak convergence result for solving the MSSFP. He et al. [7] further combined a Halpern iterative scheme with the relaxed projection method and proved a strong convergence result for solving the MSSFP. Recently, Suantai et al. [8] introduced an inertial relaxed projection method with a self-adaptive stepsize for solving the MSSFP. Also, Wen et al. [9] introduced a cyclic-simultaneous projection method and proved weak convergence result for solving the MSSFP.

Constructing iterative schemes with a faster rate of convergence are usually of great interest. The inertial-type algorithm which originated from the equation for an oscillator with damping and conservative restoring force has been an important tool employed in improving the performance of algorithms and has some nice convergence characteristics. In general, the main feature of the inertial-type algorithms is that we can use the previous

iterates to construct the next one. Since the introduction of the inertial-like algorithm, many authors combined the inertial term [*θn*(*xn* − *xn*−1)] together with different kinds of iterative algorithms, including Mann, Kranoselski, Halpern, Viscosity, to mention a few, to approximate solutions of fixed point problems and optimization problems. Most authors were able to prove weak convergence results while few proved strong convergence results. Polyak [10] was the first author to propose the heavy ball method, Alvarez and Attouch [11] employed this to the setting of a general maximal monotone operator using the Proximal Point Algorithm (PPA), which is called the inertial PPA, and is of the form:

$$\begin{cases} y\_n = \mathbf{x}\_n + \theta\_n (\mathbf{x}\_n - \mathbf{x}\_{n-1}), \\ \mathbf{x}\_{n+1} = (I + r\_n B)^{-1} y\_n, n > 1. \end{cases} \tag{6}$$

They proved that if {*rn*} is non-decreasing and {*θn*} ⊂ [0, 1) with

$$\sum\_{n=1}^{+\infty} \theta\_{\mathcal{U}} \left|| \mathbf{x}\_{\mathcal{U}} - \mathbf{x}\_{n-1} \right||^{2} < +\infty,\tag{7}$$

then the Algorithm (6) converges weakly to a zero of a maximal monotone operator *B*. More precisely, condition (7) is true for *θ<sup>n</sup>* < <sup>1</sup> <sup>3</sup> . Here *θ<sup>n</sup>* is an extrapolation factor. Other initial-type algorithms can be found in, for instance [12–17].

Motivated by the works of Wen et al. [9] and López et al. [6], in this paper, we introduce a general viscosity relaxed projection method with inertial process for solving the MSSFP with the fixed point of strictly pseudo-nonspreading mappings in real Hilbert spaces. The stepsize of our algorithm is selected self-adaptively in each iteration and its convergence does not involve prior estimate of the matrix *A*∗*A*. More so, we define some sublevel sets whose projections can be calculated explicitly using the formula in [18]. The general viscosity approximation method guarantees strong convergence of the sequences generated by the algorithm. This improves the weak convergence results proved in [6,9,19]. We further provide some numerical experiments to illustrate the performance and accuracy of our algorithm. Our results improve and complement the results of [6–9,19–24] and many other results in this direction.

### **2. Preliminaries**

We state some known and useful results which will be needed in the proof of our main theorem. In the sequel, we denote strong and weak convergence by "→" and "", respectively.

Let *C* be a nonempty closed convex subset of a real Hilbert space *H* with inner product ., . and norm ||.||. Let *S* : *C* → *C* be a nonlinear mapping and *F*(*S*) = {*x* ∈ *C* : *Sx* = *x*} be the set of all fixed points of *S*.

A mapping *S* : *C* → *C* is called

1. nonexpansive, if

$$||\mathbf{Sx} - \mathbf{Sy}|| \le ||\mathbf{x} - \mathbf{y}||\_\prime \,\forall \, \mathbf{x}, \mathbf{y} \in \mathbf{C};$$

2. quasi-nonexpansive, if *F*(*S*) is nonempty, and

$$||\mathcal{S}\mathbf{x} - p|| \le ||\mathbf{x} - p||\_\prime \,\forall \, p \in F(\mathcal{S});$$

3. nonspreading [25], if

$$2||\mathbf{S}\mathbf{x} - \mathbf{S}y||^2 \le ||\mathbf{S}\mathbf{x} - \mathbf{y}||^2 + ||\mathbf{S}\mathbf{y} - \mathbf{x}||^2, \forall \mathbf{x}, \mathbf{y} \in \mathbb{C};$$

4. *k*-strictly pseudo-nonspreading in terms of Browder-Petryshyn [26], if there exists *k* ∈ [0, 1) such that

$$||\mathbb{S}\mathbf{x} - \mathbf{y}||^2 \le ||\mathbf{x} - \mathbf{y}||^2 + k||\mathbf{x} - \mathbf{S}\mathbf{x} - (\mathbf{y} - \mathbf{S}\mathbf{y})||^2 + 2\langle \mathbf{x} - \mathbf{S}\mathbf{x}, \mathbf{y} - \mathbf{S}\mathbf{y} \rangle, \forall \mathbf{x}, \mathbf{y} \in \mathbb{C}\dots$$

	- *(b)It is also clear that every nonspreading mapping is k-strictly pseudo-nonspreading with k=0, but the converse is not true, see example 3 in [27].*

**Lemma 1.** *[27] Let T* : *C* → *C be a k-strictly pseudo-nonspreading mapping with k* ∈ [0, 1)*. Denote T<sup>β</sup>* := *βI* + (1 − *β*)*T, where β* ∈ [*k*, 1)*, then*

*(a) F*(*T*) = *F*(*Tβ*),

*(b)the following inequality holds:*

$$||T\_{\beta}\mathbf{x} - T\_{\beta}\mathbf{y}||^{2} \le ||\mathbf{x} - \mathbf{y}||^{2} + \frac{2}{1 - \beta} \langle \mathbf{x} - T\_{\beta}\mathbf{x}, \mathbf{y} - T\_{\beta}\mathbf{y} \rangle, \forall \ \mathbf{x}, \mathbf{y} \in \mathbb{C};$$

*(c) T<sup>β</sup> is a quasi-nonexpansive mapping.*

**Lemma 2.** *[28] Let C* ⊂ *H be nonempty, closed and convex set. Then,* ∀*x*, *y* ∈ *H and z* ∈ *C*


*3.* ||*PCx* <sup>−</sup> *<sup>z</sup>*||<sup>2</sup> ≤ ||*<sup>x</sup>* <sup>−</sup> *<sup>z</sup>*||<sup>2</sup> − ||*PCx* <sup>−</sup> *<sup>x</sup>*||2.

**Lemma 3.** *[29] Let H be a real Hilbert space and* {*xi*}*i*≥<sup>1</sup> *be a bounded sequence in H. For <sup>α</sup><sup>i</sup>* <sup>∈</sup> (0, 1) *such that* <sup>∑</sup><sup>∞</sup> *<sup>i</sup>*=<sup>1</sup> *α<sup>i</sup>* = 1*, the following identity holds*

$$||\sum\_{i=1}^{\infty} \alpha\_i \mathbf{x}\_i||^2 = \sum\_{i=1}^{\infty} \alpha\_i ||\mathbf{x}\_i||^2 - \sum\_{1 \le i < j < \infty} \alpha\_i \mathbf{a}\_j ||\mathbf{x}\_i - \mathbf{x}\_j||^2.$$

More so, from Lemma 3, we get the following result.

**Lemma 4.** *[30] For all x*1, *x*2,..., *xn* ∈ *H*, *the following inequality holds:*

$$||\sum\_{i=1}^{n} \lambda\_i \mathbf{x}\_i||^2 = \sum\_{i=1}^{n} \lambda\_i ||\mathbf{x}\_i||^2 - \frac{1}{2} \sum\_{i,j=1}^{n} \lambda\_i \lambda\_j ||\mathbf{x}\_i - \mathbf{x}\_j||^2, \ n \ge 2,$$

*where <sup>λ</sup><sup>i</sup>* <sup>∈</sup> [0, 1], *<sup>i</sup>* <sup>=</sup> 1, 2, . . . , *<sup>n</sup>*, <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *λ<sup>i</sup>* = 1.

**Lemma 5.** *[27] Let C be a closed convex subset of H, T* : *C* → *C be a k-strictly pseudononspreading mapping with F*(*T*) = ∅*. If* {*xn*} *is a sequence in C which converges weakly to p and* {(*I* − *T*)*xn*} *converges strongly to q, then* (*I* − *T*)*p* = *q. In particular, if q* = 0*, then p* = *T p.*

**Lemma 6.** *[31] Let* {*an*} *be a sequence of nonegative real numbers* {*γn*} *be a sequence of real numbers in* (0, 1) *with conditions* ∑<sup>∞</sup> *<sup>n</sup>*=<sup>1</sup> *γ<sup>n</sup>* = ∞ *and* {*dn*} *be a sequence of real numbers. Assume that*

$$a\_{n+1} \le (1 - \gamma\_n)a\_n + \gamma\_n d\_{n\prime} \ n \ge 1.$$

*If* lim sup*k*→<sup>∞</sup> *dnk* ≤ <sup>0</sup> *for every subsequence* {*ank*} *of* {*an*} *satisfying the condition:* lim inf*k*→∞(*ank*<sup>+</sup><sup>1</sup> − *ank* ) ≥ 0, *then* lim*n*→<sup>∞</sup> *an* = 0.

### **3. Main Results**

In this section, we present our iterative algorithm and its convergence result.

Let *H*<sup>1</sup> and *H*<sup>2</sup> be real Hilbert spaces, *C* be a nonempty, closed and convex subset of a real Hilbert space *H* and {*gn*} be a sequence of {*σn*}-contractive self maps of *H* with lim inf*n*→<sup>∞</sup> *<sup>σ</sup><sup>n</sup>* <sup>≤</sup> lim sup*n*→<sup>∞</sup> *<sup>σ</sup><sup>n</sup>* <sup>=</sup> *σμ* <sup>&</sup>lt; 1. Suppose that {*gn*(*x*)} is uniformly convergent to {*g*(*x*)} for any *x* ∈ *D*, where *D* is a bounded subset of *C*, let *Sm* : *H*<sup>1</sup> → *H*1, be a countable family of *km*-strictly pseudo-nonspreading mapping with *<sup>k</sup>* :<sup>=</sup> sup*m*≥<sup>1</sup> *km* <sup>∈</sup> (0, 1) and *Sm*,*<sup>β</sup>* :<sup>=</sup> *<sup>β</sup><sup>I</sup>* + (<sup>1</sup> <sup>−</sup> *<sup>β</sup>*)*Sm*, where *<sup>β</sup>* <sup>∈</sup> [*k*, 1), and *<sup>m</sup>* <sup>∈</sup> <sup>N</sup>\{0}.

Before we state our algorithm, we assume that the following conditions hold:


$$\partial \mathcal{c}\_i(\mathbf{x}) = \{ \mathfrak{J}\_i \in H\_1 : \mathfrak{c}\_i(z) \ge \mathfrak{c}\_i(\mathbf{x}) + \langle \mathfrak{J}\_i, z - \mathbf{x} \rangle \quad \forall z \in H\_1 \},$$

and

$$
\partial q\_j(y) = \{ \eta\_j \in H\_2 : q\_j(u) \ge q\_j(y) + \langle \eta\_j, u - y \rangle \quad \forall u \in H\_2 \}.
$$

(A3) We set *C<sup>n</sup> <sup>i</sup>* and *<sup>Q</sup><sup>n</sup> <sup>j</sup>* as the half-spaces defined by

$$\mathbb{C}\_{i}^{\mathfrak{n}} = \{ \mathbf{x} \in H\_{1} : c\_{i}(\mathfrak{x}\_{\mathfrak{n}}) + \langle \mathfrak{z}\_{i}^{\mathfrak{n}}, \mathbf{x} - \mathfrak{x}\_{\mathfrak{n}} \rangle \le 0 \},$$

where *ξ<sup>n</sup> <sup>n</sup>* ∈ *∂ci*(*xn*) (*i* = 1, 2, . . . , *k*) and

$$\mathcal{Q}\_{\rangle}^n = \{ y \in H\_2 : q\_j(A\mathfrak{x}\_n) + \langle \eta\_{j'}^n, y - A\mathfrak{x}\_n \rangle \le 0 \},$$

where *η<sup>n</sup> <sup>j</sup>* ∈ *∂qj*(*Axn*) (*j* = 1, 2, . . . , *t*). (A4) We define the proximity function by

$$f\_n(\mathbf{x}) = \frac{1}{2} \sum\_{j=1}^t \lambda\_j ||Ax - P\_{Q\_j^n}(Ax)||^2 \mu$$

where *λ<sup>j</sup>* > 0 ∀1 ≤ *j* ≤ *t*. Then the gradient of *fn*(*x*) is given by

$$\nabla f\_n(x) = \sum\_{j=1}^t \lambda\_j A^\* \left( I - P\_Q^n \right)(Ax).$$

(A5) The control sequences {*αn*}, {*wi*}, {*γn*,*m*} and {*ρn*} are chosen such that

$$\begin{aligned} \left\{ \begin{aligned} \alpha\_{n} \right\} &\subset (0,1)\_{\prime} \lim\_{n \to +\infty} \alpha\_{n} = 0, \stackrel{+\infty}{\sum} \alpha\_{n} = +\infty; \\ \left\{ \gamma\_{n,m} \right\} &\subset (0,1)\_{\prime} \liminf\_{n \to +\infty} \gamma\_{n,0} \gamma\_{n,m} > 0, \stackrel{+\infty}{\sum} \gamma\_{n,m} = 1; \\ \left\{ w\_{i} \right\} &\subseteq [0,1] \text{ with } \sum\_{i=1}^{+\infty} w\_{i} = 1; \\ \left\{ \rho\_{n} \right\} &\subset (0,4) \text{ and } \liminf\_{n \to +\infty} \rho\_{n} (4 - \rho\_{n}) > 0. \end{aligned} \right.$$

$$\text{We now present our algorithm as follows:}$$

First we show that the sequence {*xn*} generated by Algorithm 1 is bounded.

**Lemma 7.** *Suppose the solution set* Γ = <sup>Ω</sup> ∩ -∞ *<sup>m</sup>*=<sup>1</sup> *F*(*Sm*) <sup>=</sup> <sup>∅</sup> *and* {*xn*} *is the sequence generated by Algorithm 1. Then* {*xn*} *is bounded.*

### **Algorithm 1:** GVA

**Step 0**: Select the initial point *x*<sup>1</sup> ∈ *H* and the sequences {*αn*}, {*wi*}, {*γn*,*m*}, {*ρn*} such that Assumption (A5) is satisfied. Set *n* = 1.

**Step 1:** Given the *n*th iterate (i.e., *xn*, *n* ≥ 0), if ∇ *fn*(*xn*) = 0, STOP. Otherwise, compute

$$y\_n = \sum\_{i=1}^k \omega\_i P\_{\mathbb{C}\_i^n}(\mathfrak{x}\_n - \mathfrak{x}\_n \nabla f\_n(\mathfrak{x}\_n))\_i$$

where the stepsize *τ<sup>n</sup>* is defined by

$$\tau\_{\mathbb{H}} = \frac{\rho\_n f\_n(\mathbf{x}\_n)}{||\nabla f\_n(\mathbf{x}\_n)||^2}.$$

**Step 2:** Compute

$$\mathbf{x}\_{n+1} = \mathbf{a}\_{\hbar} \mathbf{g}\_{\hbar}(\mathbf{x}\_{\hbar}) + (1 - \alpha\_{\hbar}) \left( \gamma\_{\hbar,0} \mathbf{y}\_{\hbar} + \sum\_{m=1}^{\infty} \gamma\_{\hbar,j} S\_{m,\hbar} \mathbf{y}\_{\hbar} \right).$$

**Step 3:** Set *n* ← *n* + 1 and return to **Step 1**.

**Proof.** Let *<sup>x</sup>*<sup>∗</sup> <sup>∈</sup> <sup>Γ</sup> and *wn* <sup>=</sup> *<sup>γ</sup>n*,0*yn* <sup>+</sup> <sup>∑</sup><sup>∞</sup> *<sup>m</sup>*=<sup>1</sup> *γn*,*mSm*,*βyn*. By applying the nonexpansivity property of the projection mapping and Lemma 4, we have

$$\begin{split} ||y\_n - \mathbf{x}^\*||^2 &= ||\sum\_{i=1}^k \omega\_i \mathbf{z}\_{\mathbb{C}\_i^n} (\mathbf{x}\_n - \mathbf{z}\_n \nabla f\_n(\mathbf{x}\_n)) - \mathbf{x}^\*||^2 \\ &\le ||\mathbf{x}\_n - \mathbf{z}\_n \nabla f\_n(\mathbf{x}\_n) - \mathbf{x}^\*||^2 \\ &= ||\mathbf{x}\_n - \mathbf{x}^\*||^2 - 2\pi\_{\mathbb{H}} \langle \nabla f\_{\mathbb{H}}(\mathbf{x}\_{\mathbb{H}}), \mathbf{x}\_{\mathbb{H}} - \mathbf{x}^\* \rangle + ||\mathbf{x}\_{\mathbb{H}} \nabla f\_{\mathbb{H}}(\mathbf{x}\_{\mathbb{H}})||^2. \end{split} \tag{8}$$

Also from Lemma 2, we obtain

$$
\begin{split}
\langle\nabla f\_{\mathbf{n}}(\mathbf{x}\_{\mathrm{n}}), \mathbf{x}\_{\mathrm{n}} - \mathbf{x}^{\*}\rangle &= \langle \sum\_{j=1}^{t} \lambda\_{j} A^{\*} \left(I - P\_{Q\_{j}^{n}}\right) A \mathbf{x}\_{\mathrm{n}}, \mathbf{x}\_{\mathrm{n}} - \mathbf{x}^{\*}\rangle \\ &= \sum\_{j=1}^{t} \lambda\_{j} \langle (I - P\_{Q\_{j}^{n}}) A \mathbf{x}\_{\mathrm{n}}, A \mathbf{x}\_{\mathrm{n}} - P\_{Q\_{j}^{n}}(A \mathbf{x}\_{\mathrm{n}}) \rangle + \sum\_{j=1}^{t} \lambda\_{j} \langle (I - P\_{Q\_{j}^{n}}) A \mathbf{x}\_{\mathrm{n}}, P\_{Q\_{j}^{n}}(A \mathbf{x}\_{\mathrm{n}}) - A \mathbf{x}^{\*}\rangle \\ &\geq \sum\_{j=1}^{t} \lambda\_{j} ||A \mathbf{x}\_{\mathrm{n}} - P\_{Q\_{j}^{n}}(A \mathbf{x}\_{\mathrm{n}})||^{2} \\ &= 2f\_{\mathrm{n}}(\mathbf{x}\_{\mathrm{n}}).
\end{split}
$$

On substituting (9) into (8), we have

$$\begin{split} \left||y\_{n} - \mathbf{x}^{\*}||^{2} \leq & ||\mathbf{x}\_{n} - \mathbf{x}^{\*}||^{2} - 4\pi\_{n}f\_{n}(\mathbf{x}\_{n}) + ||\pi\_{n}\nabla f\_{n}(\mathbf{x}\_{n})||^{2} \\ &= & ||\mathbf{x}\_{n} - \mathbf{x}^{\*}||^{2} - \rho\_{n}(4 - \rho\_{n})\frac{f\_{n}^{2}(\mathbf{x}\_{n})}{||\nabla f\_{n}(\mathbf{x}\_{n})||^{2}} \\ \leq & ||\mathbf{x}\_{n} - \mathbf{x}^{\*}||^{2}. \end{split} \tag{10}$$

More so from Lemma 1, we get

$$\begin{split} ||\boldsymbol{w}\_{n} - \mathbf{x}^{\*}|| &= ||\gamma\_{n,0}\boldsymbol{y}\_{n} + \sum\_{m=1}^{\infty} \gamma\_{n,m} S\_{m,\beta} \boldsymbol{y}\_{n} - \mathbf{x}^{\*}|| \\ &\leq \gamma\_{n,0} ||\boldsymbol{y}\_{n} - \mathbf{x}^{\*}|| + \sum\_{m=1}^{\infty} \gamma\_{n,m} ||S\_{m,\beta} \boldsymbol{y}\_{n} - \mathbf{x}^{\*}|| \\ &\leq \gamma\_{n,0} ||\boldsymbol{y}\_{n} - \mathbf{x}^{\*}|| + \sum\_{m=1}^{\infty} \gamma\_{n,m} ||\boldsymbol{y}\_{n} - \mathbf{x}^{\*}|| \\ &= ||\boldsymbol{y}\_{n} - \mathbf{x}^{\*}||. \end{split} \tag{11}$$

Therefore from (10) and (11), we have

$$\begin{split} ||\mathbf{x}\_{n+1} - \mathbf{x}^\*|| &= ||a\_n \mathbf{g}\_n(\mathbf{x}\_n) + (1 - \boldsymbol{\alpha}\_n) \mathbf{w}\_n - \mathbf{x}^\*|| \\ &\le a\_n ||\mathbf{g}\_n(\mathbf{x}\_n) - \mathbf{x}^\*|| + (1 - \boldsymbol{\alpha}\_n) ||\mathbf{w}\_n - \mathbf{x}^\*|| \\ &\le a\_n \sigma\_n ||\mathbf{x}\_n - \mathbf{x}^\*|| + (1 - \boldsymbol{\alpha}\_n) ||\mathbf{x}\_n - \mathbf{x}^\*|| + \boldsymbol{\alpha}\_n ||\mathbf{g}\_n(\mathbf{x}^\*) - \mathbf{x}^\*|| \\ &\le (1 - \boldsymbol{\alpha}\_n (1 - \sigma\_n)) ||\mathbf{x}\_n - \mathbf{x}^\*|| + \boldsymbol{\alpha}\_n (1 - \sigma\_n) \frac{||\mathbf{g}\_n(\mathbf{x}^\*) - \mathbf{x}^\*||}{1 - \sigma\_n} \\ &\vdots \\ &\le \max\left\{ ||\mathbf{x}\_n - \mathbf{x}^\*||, \frac{||\mathbf{g}\_n(\mathbf{x}^\*) - \mathbf{x}^\*||}{1 - \sigma\_n} \right\}. \end{split}$$

Since {*gn*} is uniformly convergent on *D*, it follows that {*gn*(*x*∗)} is bounded. Thus, there exists a positive constant *M*, such that ||*gn*(*x*∗) − *x*∗|| ≤ *M*. By induction, we obtain

$$||\mathbf{x}\_n - \mathbf{x}^\*|| \le \max\left\{||\mathbf{x}\_1 - \mathbf{x}^\*||, \frac{M}{1 - \sigma\_\mu}\right\}.$$

Hence, {*xn*} is bounded. Consequently {*Sm*,*βxn*}, {*gn*(*xn*)}, {*yn*} and {*wn*} are all bounded.

We now give our main convergence theorem.

**Theorem 1.** *Suppose that* Γ = <sup>Ω</sup> ∩ -∞ *<sup>m</sup>*=<sup>1</sup> *F*(*Sm*) <sup>=</sup> <sup>∅</sup> *and Assumptions (A1)–(A5) hold. Then, the sequence* {*xn*} *generated by Algorithm 1 converges strongly to point z* ∈ *P*<sup>Γ</sup> *which is a unique solution of the variational inequality*

$$
\langle \mathbf{g}(z) - z, \mathbf{x}^\* - z \rangle \le 0, \forall \; \mathbf{x}^\* \in P\_{\Gamma}.
$$

**Proof.** From Lemma 1 (c), Lemma 3 and (10), we have


Now, from (10) and (12), we have that

$$\begin{split} ||\mathbf{x}\_{n+1} - \mathbf{x}^\*||^2 &= ||a\_n \mathbf{g}\_n(\mathbf{x}\_n) + (1 - a\_n) \mathbf{w}\_n - \mathbf{x}^\*||^2 \\ &\le (1 - a\_n)^2 ||\mathbf{w}\_n - \mathbf{x}^\*||^2 + 2a\_n \langle \mathbf{x}\_{n+1} - \mathbf{x}^\*, \mathbf{g}\_n(\mathbf{x}\_n) - \mathbf{x}^\* \rangle \\ &\le (1 - a\_n)^2 ||\mathbf{x}\_n - \mathbf{x}^\*||^2 - (1 - a\_n) \rho\_n (4 - \rho\_n) \frac{f\_n^2(\mathbf{x}\_n)}{||\nabla f\_n(\mathbf{x}\_n)||} \\ &- (1 - a\_n) \sum\_{n=1}^\infty \gamma\_{n,0} \gamma\_{n,n} ||\mathbf{y}\_n - \mathbf{S}\_{n,\beta} \mathbf{y}\_n|| + 2a\_n \langle \mathbf{x}\_{n+1} - \mathbf{x}^\*, \mathbf{g}\_n(\mathbf{x}\_n) - \mathbf{x}^\* \rangle \\ &= (1 - a\_n)^2 ||\mathbf{x}\_n - \mathbf{x}^\*||^2 + a\_n \left(2 \langle \mathbf{x}\_{n+1} - \mathbf{x}^\*, \mathbf{g}\_n(\mathbf{x}\_n) - \mathbf{x}^\* \rangle \right). \end{split} \tag{13}$$

Putting *dn* = 2*xn*+<sup>1</sup> − *x*∗, *gn*(*xn*) − *x*∗, in view of Lemma 5, we need to prove that lim sup*k*→<sup>∞</sup> *dnk* ≤ 0 for every {||*xnk* − *<sup>x</sup>*∗||} of {||*xn* − *<sup>x</sup>*∗||} satisfying the condition

$$\liminf\_{k \to +\infty} \{ ||\mathbf{x}\_{\mathcal{U}\_{k+1}} - \mathbf{x}^\*|| - ||\mathbf{x}\_{\mathcal{U}\_k} - \mathbf{x}^\*|| \} \ge 0. \tag{14}$$

To show this, suppose that {||*xnk* − *x*∗||} is a subsequence of {||*xn* − *x*∗||} such that (14) holds. Then

$$\begin{aligned} &\liminf\_{k \to +\infty} (||\mathbf{x}\_{n\_{k+1}} - \mathbf{x}^\*||^2 - ||\mathbf{x}\_{n\_k} - \mathbf{x}||^2) \\ & \qquad = \liminf\_{k \to \infty} \left( (||\mathbf{x}\_{n\_{k+1}} - \mathbf{x}^\*||^2 - ||\mathbf{x}\_{n\_k} - \mathbf{x}^\*||^2)(||\mathbf{x}\_{n\_{k+1}} - \mathbf{x}^\*|| + ||\mathbf{x}\_{n\_k} - \mathbf{x}^\*||) \right) \geq 0. \end{aligned}$$

Now, using (12), we have that

$$\begin{split} \limsup\_{k \to +\infty} ((1 - a\_{\mathbb{R}\_k}) \sum\_{m=1}^{+\infty} \gamma\_{\mathbb{R}\_k} y \gamma\_{\mathbb{R}\_k m} ||y\_{\mathbb{R}\_k} - S\_{m, \mathcal{G}} y\_{\mathbb{R}\_k}||) &\leq \limsup\_{k \to +\infty} ((1 - a\_{\mathbb{R}\_k}) ||x\_{\mathbb{R}\_k} - x^\*||^2 - ||x\_{\mathbb{R}\_k + 1} - x^\*||^2 \\ &+ 2a\_{\mathbb{R}\_k} \langle x\_{\mathbb{R}\_k + 1} - x^\*, y\_{\mathbb{R}\_k} (x\_{\mathbb{R}\_k}) - x^\* \rangle) \\ &\leq \limsup\_{k \to +\infty} (||x\_{\mathbb{R}\_k} - x^\*||^2 - ||x\_{\mathbb{R}\_k + 1} - x^\*||^2) \\ &+ \limsup\_{k \to +\infty} (2a\_{\mathbb{R}\_k} (x\_{\mathbb{R}\_k + 1} - x^\*, y\_{\mathbb{R}\_k} (x\_{\mathbb{R}\_k}) - x^\*)) \\ &= - \liminf\_{k \to +\infty} (||x\_{\mathbb{R}\_k + 1} - x^\*||^2 - ||x\_{\mathbb{R}\_k} - x^\*||^2) \leq 0. \end{split} \tag{15}$$

Hence

$$\lim\_{k \to +\infty} ||y\_{n\_k} - S\_{m, \emptyset} y\_{n\_k}|| = 0.$$

Please note that

$$\begin{array}{rcl}||s\_{m,\beta}y\_n - y\_n|| &=& ||\beta y\_n + (1-\beta)S\_my\_n - y\_n||.\\ &=& (1-\beta)||S\_my\_n - y\_n||.\end{array}$$

Then it follows that

$$||S\_my\_n - y\_n|| = \frac{1}{1 - \beta} ||S\_{m,\beta}y\_n - y\_n|| \to 0. \tag{16}$$

Furthermore, using (12) and following the same approach as in (15), we also have that

$$\rho\_n(4-\rho\_n)\frac{f\_n^2(\mathbf{x}\_{n\_k})}{||\bigtriangledown f(\mathbf{x}\_{n\_k})||^2} \le \rho\_{n\_k}(4-\rho\_{n\_k})\frac{f\_n^2(\mathbf{x}\_{n\_k})}{||\bigtriangledown f(\mathbf{x}\_{n\_k})||^2} \to 0,\text{ as }k \to \infty. \tag{17}$$

This implies that

$$\sum\_{n=0}^{+\infty} \frac{f\_n^2(\mathbf{x}\_{n\_k})}{||\bigtriangledown f(\mathbf{x}\_{n\_k})||^2} < +\infty. \tag{18}$$

Since ∇ *fn* is Lipschitz continuous and {*xn*} is bounded, so {∇ *fn*(*xn*)} is also bounded. Hence from (18), we can conclude that

$$\lim\_{k \to +\infty} \frac{1}{2} \sum\_{j=1}^{t} \lambda\_j ||Ax\_{n\_k} - P\_{Q\_j^n}(Ax\_{n\_k})||^2 = 0,\tag{19}$$

which also implies that

$$\lim\_{k \to +\infty} ||Ax\_{n\_k} - P\_{Q\_j^n}(Ax\_{n\_k})|| = 0, \text{ for } j = 1, 2, \dots, t. \tag{20}$$

Since *<sup>∂</sup>qj* is bounded on bounded sets, there exists *<sup>η</sup>* such that ||*η<sup>n</sup> <sup>j</sup>* || ≤ *η* ∀*j*. Please note that *PQn j Axn* <sup>∈</sup> *<sup>Q</sup><sup>n</sup> <sup>j</sup>* , thus we get

$$\begin{aligned} q(A\mathbf{x}\_{\mathfrak{n}\_k}) &\leq \quad \langle \eta\_j^{\mathfrak{n}\_k} \, \_\prime A \mathbf{x}\_{\mathfrak{n}\_k} - P\_{Q\_j^{\mathfrak{n}\_k}} A \mathbf{x}\_{\mathfrak{n}\_k} \rangle \\ &\leq \quad ||\eta\_j^{\mathfrak{n}\_k}|| \cdot ||A \mathbf{x}\_{\mathfrak{n}\_k} - P\_{Q\_j^{\mathfrak{n}\_k}} A \mathbf{x}\_{\mathfrak{n}\_k}|| \\ &\leq \quad \eta \, ||A \mathbf{x}\_{\mathfrak{n}\_k} - P\_{Q\_j^{\mathfrak{n}\_k}} A \mathbf{x}\_{\mathfrak{n}\_k}|| \to 0 \quad \text{as} \quad k \to +\infty. \end{aligned}$$

Since {*xn*} is bounded and *C* is closed and convex, we can suppose that the subsequence {*xnk*} of {*xn*} converges weakly to *x*¯ ∈ *C*. We now show that *x*¯ ∈ Ω. By the weakly lower semicontinuity of *qj* and boundedness of *A*, we have

$$q\_j(A\mathfrak{x}) \le \liminf\_{k \to +\infty} q\_j(A\mathfrak{x}\_{n\_k}) \le 0.$$

Then *Ax*¯ ∈ *Qj*, *<sup>j</sup>* = 1, 2, ... , *<sup>t</sup>*. This implies that *Ax*¯ ∈ *t <sup>j</sup>*=<sup>1</sup> *Qj*. Next we show that *<sup>x</sup>*¯ ∈ *k <sup>i</sup>*=<sup>1</sup> *Ci*.

Let *un* = *xn* − *τn*∇ *fn*(*xn*). Since {*un*}, {*wn*} and {*yn*} are bounded, there exist subsequences {*unk*}, {*wnk*} and {*ynk*} which all converges to *x*¯. Using (10), we have that

$$||\mathbf{u}\_n - \mathbf{x}^\*||^2 \le ||\mathbf{x}\_n - \mathbf{x}^\*||^2 - \rho\_{n\_k}(4 - \rho\_{n\_k}) \frac{f\_n^2(\mathbf{x}\_{n\_k})}{||\nabla f(\mathbf{x}\_{n\_k})||^2}.\tag{21}$$

By applying Lemma 2 (iii), we have that

$$\begin{split} \sum\_{i=1}^{k} \omega\_{i} ||P\_{\mathbb{C}\_{i}^{n}}(u\_{n\_{k}}) - u\_{n\_{k}}||^{2} &\leq ||u\_{n\_{k}} - \mathbf{x}^{\*}||^{2} - \sum\_{i=1}^{k} \omega\_{i} ||P\_{\mathbb{C}\_{i}}(u\_{n\_{k}}) - \mathbf{x}^{\*}||^{2} \\ &\leq ||\mathbf{x}\_{n\_{k}} - \mathbf{x}^{\*}||^{2} - ||y\_{n\_{k}} - \mathbf{x}^{\*}||^{2} \\ &\leq ||\mathbf{x}\_{n\_{k}} - \mathbf{x}^{\*}||^{2} - ||w\_{n\_{k}} - \mathbf{x}^{\*}||^{2} \\ &= ||\mathbf{x}\_{n\_{k}} - \mathbf{x}^{\*}||^{2} - ||\mathbf{x}\_{n\_{k}+1} - \mathbf{x}^{\*}||^{2} + ||\mathbf{x}\_{n\_{k}+1} - \mathbf{x}^{\*}||^{2} - ||w\_{n\_{k}} - \mathbf{x}^{\*}||^{2} \\ &\leq ||\mathbf{x}\_{n\_{k}} - \mathbf{x}^{\*}||^{2} - ||\mathbf{x}\_{n\_{k}+1} - \mathbf{x}^{\*}||^{2} + \mathfrak{a}\_{n\_{k}} ||g\_{n\_{k}}(\mathbf{x}\_{n\_{k}}) - \mathbf{x}^{\*}||^{2} \\ &+ (1 - \mathfrak{a}\_{\text{n}}) ||w\_{n\_{k}} - \mathbf{x}^{\*}||^{2} - ||w\_{n\_{k}} - \mathbf{x}^{\*}||^{2}. \end{split} \tag{22}$$

By taking lim sup as *k* → +∞ on both sides of (22) and following the same argument as in (15), we have that

$$\lim\_{k \to +\infty} ||P\_{\mathbb{C}\_i^n}(u\_{n\_k}) - u\_{n\_k}|| = 0 = \lim\_{k \to \infty} ||y\_{n\_k} - u\_{n\_k}||.\tag{23}$$

Also, from the definition of *unk* = *xnk* − *τnk f*(*xnk* ), we have from (19) that

$$\lim\_{k \to +\infty} ||\mu\_{n\_k} - \mathbf{x}\_{n\_k}|| = 0. \tag{24}$$

Using (23) and (24), we obtain that

$$\lim\_{k \to +\infty} ||P\_{C\_i^n}(u\_{n\_k}) - \mathfrak{x}\_{n\_k}|| = 0.$$

Since *<sup>∂</sup>ci* is bounded on bounded sets, there exists *<sup>ξ</sup>* such that ||*ξ<sup>n</sup> <sup>i</sup>* || ≤ *ξ* ∀*i*. Thus,

$$\begin{aligned} c\_i(\mathbf{x}\_{n\_k}) &\leq \quad \langle \boldsymbol{\xi}\_i^{n\_k}, \mathbf{x}\_{n\_k} - \boldsymbol{P}\_{\mathbf{C}\_i}^{n\_k}(\mathbf{x}\_{n\_k}) \rangle \\ &\leq \quad \xi \left( ||\mathbf{x}\_{n\_k} - \boldsymbol{u}\_{\text{fl}\_k}|| + ||\boldsymbol{u}\_{\text{fl}\_k} - \boldsymbol{P}\_{\mathbf{C}\_i^{n\_k}}(\mathbf{x}\_{\text{fl}\_k}) \right) \to 0. \end{aligned}$$

By the lower semicontinuity of *ci*, we have

$$c\_i(\mathfrak{x}) \le \liminf\_{k \to +\infty} c\_i(\mathfrak{x}\_{n\_k}) \le 0.$$

Hence *<sup>x</sup>*¯ ∈ *Ci* for *<sup>i</sup>* = 1, 2, ... , *<sup>k</sup>*, which implies that *<sup>x</sup>*¯ ∈ *k <sup>i</sup>*=<sup>1</sup> *Ci*. Hence *x*¯ ∈ Ω. Furthermore, we have from (23) and (24) that

$$\lim\_{k \to +\infty} ||y\_{n\_k} - \mathbf{x}\_{n\_k}|| = 0. \tag{25}$$

Then, from the demiclosedness of *k*-strictly pseudo-nonspreading mappings (Lemma 5), (16) and (25), we obtain *<sup>x</sup>*¯ ∈ -∞ *<sup>m</sup>*=<sup>1</sup> *F*(*Sm*). Therefore, *x*¯ ∈ Γ.

Next is to prove that {*xn*} converges strongly to *z* ∈ Γ. Also, (16), we have

$$\lim\_{k \to +\infty} ||w\_{\mathfrak{n}\_k} - y\_{\mathfrak{n}\_k}|| = 0. \tag{26}$$

More so, from (25) and (26), we obtain

$$\lim\_{k \to +\infty} ||w\_{n\_k} - x\_{n\_k}|| = 0. \tag{27}$$

From (27), we obtain

$$||\mathbf{x}\_{n\_k+1} - \mathbf{x}\_{n\_k}|| \le \mathbf{a}\_{n\_k} ||\mathbf{g}\_{n\_k}(\mathbf{x}\_{n\_k}) - \mathbf{x}\_{n\_k}|| + (1 - \alpha\_{n\_k})||\mathbf{w}\_{n\_k} - \mathbf{x}\_{n\_k}||.\tag{28}$$

Next is to prove that the lim sup*k*→+∞*xnk*<sup>+</sup><sup>1</sup> <sup>−</sup> *<sup>x</sup>*∗, *gn*(*xn*) <sup>−</sup> *<sup>x</sup>*∗ ≤ 0. Indeed, take a subsequence {*xnk*} of {*xn*} such that *xnk z*. Hence, we have

$$\limsup\_{n \to +\infty} \langle \lg(\mathbf{x}^\*) - \mathbf{x}^\*, \mathbf{x}\_n - \mathbf{x}^\* \rangle = \lim\_{k \to +\infty} \langle \lg(\mathbf{x}^\*) - \mathbf{x}^\*, \mathbf{x}\_{n\_k} - \mathbf{x}^\* \rangle.$$

Since *gn*(*x*) is uniformly convergent on D, we have that

$$\lim\_{n \to +\infty} (g\_n(\mathbf{x}^\*) - \mathbf{x}^\*) = g(\mathbf{x}^\*) - \mathbf{x}^\*.$$

Now, from (28) and Lemma 4 (i), we obtain

$$\lim\_{k \to +\infty} \langle \mathbf{g}(\mathbf{x}^\*) - \mathbf{x}^\*, \mathbf{x}\_{\mathbb{R}\_k} - \mathbf{x}^\* \rangle = \langle \mathbf{g}(\mathbf{x}^\*) - \mathbf{x}^\*, \mathbf{z} - \mathbf{x}^\* \rangle \le 0. \tag{29}$$

Using Schwartz's inequality, we have

$$\limsup\_{k \to +\infty} \langle \mathbf{x}\_{n\_k+1} - \mathbf{x}^\*, \mathbf{g}\_{n\_k}(\mathbf{x}^\*) - \mathbf{x}^\* \rangle \le \lim\_{k \to +\infty} ||\mathbf{x}\_{n\_k+1} - \mathbf{x}^\*|| \left|| \mathbf{g}\_{\mathbf{m}}(\mathbf{x}^\*) - \mathbf{g}(\mathbf{x}^\*) || + \limsup\_{k \to +\infty} \langle \mathbf{x}\_{n\_k+1} - \mathbf{x}^\*, \mathbf{g}(\mathbf{x}^\*) - \mathbf{x}^\* \rangle \rangle + \limsup\_{k \to +\infty} \langle \mathbf{x}\_{n\_k+1} - \mathbf{x}^\*, \mathbf{g}(\mathbf{x}^\*) - \mathbf{g}(\mathbf{x}^\*) \rangle$$

By the boundedness of {*xn*}, *gn*(*x*) → *g*(*x*), then by (28) and (29), we have

$$\limsup\_{k \to +\infty} \langle \mathbf{x}\_{n\_k+1} - \mathbf{x}^\*, \mathbf{g}\_{n\_k}(\mathbf{x}^\*) - \mathbf{x}^\* \rangle \le 0. \tag{30}$$

Applying (30) and Lemma 5 in (13), we obtain that {*xn*} converges to *z*. This completes the proof.

Next, we give a generalized viscosity approximation method with inertial term which can be regard as a procedure for speeding up the convergence properties of Algorithm 1. In addition to Assumptions (A1)–(A5), we choose a sequence {*<sup>n</sup>*} ⊂ (0, *-*) with *-* ∈ [0, 1) and

$$\lim\_{n \to \infty} \frac{\varepsilon\_n}{a\_n} = 0.\tag{31}$$

**Remark 2.** *From* (31) *and Step 1, it is easy to see that* lim*n*→<sup>∞</sup> *<sup>θ</sup><sup>n</sup> <sup>α</sup><sup>n</sup>* ||*xn* − *xn*−1|| = 0. *Indeed, we have θn*||*xn* − *xn*−1|| ≤ *<sup>n</sup> for each n* ≥ 1*, which together with* (31) *implies that*

$$\liminf\_{n \to +\infty} \frac{\theta\_n}{\mathfrak{a}\_n} ||\mathfrak{x}\_n - \mathfrak{x}\_{n-1}|| = 0 \le \lim\_{n \to +\infty} \frac{\mathfrak{e}\_n}{\mathfrak{a}\_n} = 0.$$

**Lemma 8.** *Suppose the solution set* Γ = <sup>Ω</sup> ∩ -+∞ *<sup>m</sup>*=<sup>1</sup> *F*(*Sm*) <sup>=</sup> <sup>∅</sup> *and* {*xn*} *is the sequence generated by Algorithm 2. Then* {*xn*} *is bounded.*

**Proof.** Let *x*<sup>∗</sup> ∈ Γ, using Step 1, we get

$$\begin{split} ||a\_{n} - \mathbf{x}^{\*}|| &= ||\mathbf{x}\_{n} + \theta\_{n}(\mathbf{x}\_{n} - \mathbf{x}\_{n-1}) - \mathbf{x}^{\*}|| \\ &\leq ||\mathbf{x}\_{n} - \mathbf{x}^{\*}|| + \theta\_{n}||\mathbf{x}\_{n} - \mathbf{x}\_{n-1}|| \\ &= ||\mathbf{x}\_{n} - \mathbf{x}^{\*}|| + \boldsymbol{\alpha}\_{n} \cdot \frac{\theta\_{n}}{\boldsymbol{\alpha}\_{n}}||\mathbf{x}\_{n} - \boldsymbol{\alpha}\_{n\_{1}}||. \end{split} \tag{32}$$

### **Algorithm 2:** IGVA

**Step 0:** Select the initial points *x*0, *x*<sup>1</sup> ∈ *H* and the sequences {*αn*}, {*wi*}, {*γn*,*m*}, {*ρn*}, {*<sup>n</sup>*} such that Assumption (A5) and (31) are satisfied. Set *n* = 1.

**Step 1:** Given the (*n* − 1)th and *n*th iterates (i.e., *xn*−<sup>1</sup> and *xn*, *n* ≥ 1). Choose *θ<sup>n</sup>* such that 0 ≤ *θ<sup>n</sup>* ≤ *θn*, where

$$
\overline{\theta}\_n = \begin{cases}
\min\{\theta\_\prime \frac{\mathcal{E}\_n}{||\mathbf{x}\_n - \mathbf{x}\_{n-1}||}\}, & \text{if} \quad \mathbf{x}\_n \neq \mathbf{x}\_{n-1}, \\
\theta\_\prime & \text{otherwise},
\end{cases}
$$

where *θ* > 0. Compute

$$a\_n = \mathbf{x}\_n + \theta\_n (\mathbf{x}\_n - \mathbf{x}\_{n-1})\_{\prime\prime}$$

and

$$y\_n = \sum\_{i=1}^k \omega\_i P\_{\mathbb{C}\_i^n}(a\_n - \tau\_n \nabla f\_n(a\_n)),$$

where the stepsize *τ<sup>n</sup>* is defined by

$$\tau\_n = \frac{\rho\_n f\_n(a\_n)}{||\nabla f\_n(a\_n)||^2}.$$

**Step 2:** Compute

$$\mathbf{x}\_{n+1} = \mathbf{a}\_n \mathbf{g}\_n(\mathbf{x}\_n) + (1 - \mathbf{a}\_n) \left( \gamma\_{n,0} \mathbf{y}\_n + \sum\_{m=1}^{\infty} \gamma\_{n,j} S\_{m,\emptyset} \mathbf{y}\_m \right).$$

**Step 3:** Set *n* ← *n* + 1 and return to **Step 1**.

By the condition *<sup>θ</sup><sup>n</sup> <sup>α</sup><sup>n</sup>* ||*xn* − *xn*−1|| → 0, there exists a constant *M*<sup>1</sup> > 0 such that *θn <sup>α</sup><sup>n</sup>* ||*xn* − *xn*−1|| → 0 ≤ *M*1, ∀ *n* ≥ 1. Following similar argument as in the prove of (10) in Algorithm 1, we have

$$||y\_n - \mathbf{x}^\*|| \le ||a\_n - \mathbf{x}^\*||.\tag{33}$$

Also as in (11), putting *wn* = *γn*,0*yn* + ∑<sup>∞</sup> *<sup>m</sup>*=<sup>1</sup> *γn*,*mSm*,*βyn*, then we get

$$||w\_n - \mathfrak{x}^\*|| \le ||y\_n - \mathfrak{x}^\*||.\tag{34}$$

Then, it follows from (32), (33) and (34) that

$$||w\_{\mathcal{U}} - \mathbf{x}^\*|| = ||\mathbf{x}\_{\mathcal{U}} - \mathbf{x}^\*|| + a\_{\mathcal{U}}M\_1. \tag{35}$$

Thus, we have

$$\begin{split} ||\mathbf{x}\_{n+1} - \mathbf{x}^\*|| &= ||a\_n \underline{\mathbf{g}}\_n(\mathbf{x}\_n) + (1 - a\_n)\overline{\mathbf{w}}\_n - \mathbf{x}^\*|| \\ &\le a\_n ||\underline{\mathbf{g}}\_n(\mathbf{x}\_n) - \mathbf{x}^\*|| + (1 - a\_n)||\overline{\mathbf{w}}\_n - \mathbf{x}^\*|| \\ &\le a\_n r\_n ||\mathbf{x}\_n - \mathbf{x}^\*|| + (1 - a\_n) ||\|\mathbf{x}\_n - \mathbf{x}^\*|| + a\_n M\_1 | + a\_n ||\underline{\mathbf{g}}\_n(\mathbf{x}^\*) - \mathbf{x}^\*|| \\ &= (1 - a\_n (1 - \sigma\_n)) ||\mathbf{x}\_n - \mathbf{x}^\*|| + a\_n (1 - \sigma\_n) \frac{||\underline{\mathbf{g}}\_n(\mathbf{x}^\*) - \mathbf{x}^\*||}{1 - \sigma\_n} + a\_n (1 - a\_n) M\_1 \\ &= (1 - a\_n (1 - \sigma\_n)) ||\mathbf{x}\_n - \mathbf{x}^\*|| + a\_n (1 - \sigma\_n) \left[ \frac{||\underline{\mathbf{g}}\_n(\mathbf{x}^\*) - \mathbf{x}^\*||}{1 - \sigma\_n} + (1 - a\_n) \frac{M\_1}{1 - \sigma\_n} \right] \\ &\vdots \\ &\le \max\left\{ ||\mathbf{x}\_n - \mathbf{x}^\*||, \frac{||\underline{\mathbf{g}}\_n(\mathbf{x}^\*) - \mathbf{x}^\*||}{1 - \sigma\_n} \right\}. \end{split}$$

Since {*gn*} is uniformly convergence on D, it follows that {*gn*(*x*∗)} is bounded. Thus, there exists a positive constant *M*<sup>2</sup> such that ||*gn*(*x*∗) − *x*∗|| ≤ *M*2. Thus, it follows by induction that

$$||\mathbf{x}\_{\mathrm{H}} - \mathbf{x}^\*|| \le \max\left\{ ||\mathbf{x}\_1 - \mathbf{x}^\*||, \frac{M\_1 + M\_2}{1 - \sigma\_{\mu}} \right\}.$$

Therefore {*xn*} is bounded.

**Theorem 2.** *Suppose that* Γ = <sup>Ω</sup> ∩ -∞ *<sup>m</sup>*=<sup>1</sup> *F*(*Sm*) <sup>=</sup> <sup>∅</sup> *and Assumptions (A1)–(A5) with* (31) *hold. Then, the sequence* {*xn*} *generated by Algorithm 2 converges strongly to point z* ∈ *P*<sup>Γ</sup> *which is a unique solution of the variational inequality*

$$
\langle \lg(z) - z, \mathbf{x}^\* - z \rangle \le 0, \forall \; \mathbf{x}^\* \in P\_\Gamma.
$$

**Proof.** Let *x*<sup>∗</sup> ∈ Γ, then we have from Step 1 that

$$\begin{split} ||\boldsymbol{a}\_{n} - \boldsymbol{\mathsf{x}}^{\*}||^{2} &= ||\boldsymbol{\mathsf{x}}\_{n} + \theta\_{n}(\mathbf{x}\_{n} - \mathbf{x}\_{n-1}) - \boldsymbol{\mathsf{x}}^{\*}||^{2} \\ &= ||(\mathbf{x}\_{n} - \mathbf{x}^{\*}) + \theta\_{n}(\mathbf{x}\_{n} - \mathbf{x}\_{n-1})||^{2} \\ &= ||\boldsymbol{\mathsf{x}}\_{n} - \mathbf{x}^{\*}||^{2} + 2\theta\_{n}\langle\mathbf{x}\_{n} - \mathbf{x}^{\*}, \mathbf{x}\_{n} - \mathbf{x}\_{n-1}\rangle + \theta\_{n}^{2}||\mathbf{x}\_{n} - \mathbf{x}\_{n}||^{2} \\ &\leq ||\mathbf{x}\_{n} - \mathbf{x}^{\*}||^{2} + 2\theta\_{n}||\mathbf{x}\_{n} - \mathbf{x}^{\*}|| ||\boldsymbol{\mathsf{x}}\_{n} - \mathbf{x}\_{n-1}|| + \theta\_{n}^{2}||\mathbf{x}\_{n} - \mathbf{x}\_{n-1}||^{2} \\ &\leq ||\mathbf{x}\_{n} - \mathbf{x}^{\*}||^{2} + \theta\_{n}||\mathbf{x}\_{n} - \mathbf{x}\_{n-1}|| \left[ 2||\mathbf{x}\_{n} - \mathbf{x}^{\*}|| + \theta\_{n} ||\mathbf{x}\_{n} - \mathbf{x}\_{n-1}|| \right] \\ &\leq ||\mathbf{x}\_{n} - \mathbf{x}^{\*}||^{2} + \theta\_{n} ||\mathbf{x}\_{n} - \mathbf{x}\_{n-1}||M\_{3\_{\mathcal{X}}} \end{split}$$

where *<sup>M</sup>*<sup>3</sup> <sup>=</sup> sup*n*≥1{2||*xn* <sup>−</sup> *<sup>x</sup>*∗|| <sup>+</sup> *<sup>θ</sup>n*||*xn* <sup>−</sup> *xn*−1||}.

Similarly as in (12), we get

$$\begin{split} ||w\_n - \mathbf{x}^\*||^2 &\le ||a\_n - \mathbf{x}^\*||^2 \\ &= ||\mathbf{x}\_n - \mathbf{x}^\*|| + \theta\_n ||\mathbf{x}\_n - \mathbf{x}\_{n-1}|| M\_3. \end{split} \tag{36}$$

Using Step 1, we have that

$$||a\_n - x\_n|| = \alpha\_n \cdot \frac{\theta\_n}{a\_n} ||x\_n - x\_{n-1}|| \to 0, \text{ as } n \to \infty. \tag{37}$$

Now, from (36), we have that

$$\begin{aligned} ||\mathbf{x}\_{n+1} - \mathbf{x}^\*|| &= ||\mathbf{a}\_n \mathbf{g}\_n(\mathbf{x}\_n) + (1 - a\_n)\mathbf{w}\_n - \mathbf{x}^\*||^2 \\ &= (1 - a\_n)^2 ||\mathbf{w}\_n - \mathbf{x}^\*||^2 + 2a\_n (\mathbf{x}\_{n+1} - \mathbf{x}^\*, \mathbf{g}\_n(\mathbf{x}\_n) - \mathbf{x}^\*) \\ &= (1 - a\_n)^2 ||\mathbf{x}\_n - \mathbf{x}^\*||^2 + (1 - a\_n)\theta\_n ||\mathbf{x}\_n - \mathbf{x}\_{n-1}||M\_3 + 2a\_n \langle \mathbf{x}\_{n+1} - \mathbf{x}^\*, \mathbf{g}\_n(\mathbf{x}\_n) - \mathbf{x}^\* \rangle \\ &= (1 - a\_n)^2 ||\mathbf{x}\_n - \mathbf{x}^\*||^2 + a\_n (1 - a\_n) \frac{\theta\_n}{a\_n} ||\mathbf{x}\_n - \mathbf{x}\_{n-1}||M\_3 + 2a\_n \langle \mathbf{x}\_{n+1} - \mathbf{x}^\*, \mathbf{g}\_n(\mathbf{x}\_n) - \mathbf{x}^\* \rangle \\ &\leq (1 - a\_n) ||\mathbf{x}\_n - \mathbf{x}^\*||^2 + a\_n \left[ (1 - a\_n) \frac{\theta\_n}{a\_n} ||\mathbf{x}\_n - \mathbf{x}\_{n-1}||M\_3 + 2\langle \mathbf{x}\_{n+1} - \mathbf{x}^\*, \mathbf{g}\_n(\mathbf{x}\_n) - \mathbf{x}^\* \rangle \right]. \end{aligned} \tag{38}$$

Next is to show that the lim sup*k*→+∞*xnk*<sup>+</sup><sup>1</sup> <sup>−</sup> *<sup>x</sup>*∗, *gn*(*xn*) <sup>−</sup> *<sup>x</sup>*∗ ≤ 0. Indeed, take a subsequence {*xnk*} of {*xn*} such that *xnk z*. Hence, we have

$$\limsup\_{n \to +\infty} \langle \lg(\mathbf{x}^\*) - \mathbf{x}^\*, \mathbf{x}\_n - \mathbf{x}^\* \rangle = \lim\_{k \to +\infty} \langle \lg(\mathbf{x}^\*) - \mathbf{x}^\*, \mathbf{x}\_{n\_k} - \mathbf{x}^\* \rangle.$$

Since *gn*(*x*) is uniformly convergent on *D*, we have that

$$\lim\_{n \to +\infty} (g\_n(\mathbf{x}^\*) - \mathbf{x}^\*) = \mathbf{g}(\mathbf{x}^\*) - \mathbf{x}^\*.$$

Now, from (28) and Lemma 4 (i), we obtain

$$\lim\_{k \to +\infty} \langle \mathbf{g}(\mathbf{x}^\*) - \mathbf{x}^\*, \mathbf{x}\_{n\_k} - \mathbf{x}^\* \rangle = \langle \mathbf{g}(\mathbf{x}^\*) - \mathbf{x}^\*, \mathbf{z} - \mathbf{x}^\* \rangle \le 0. \tag{39}$$

By applying Schwartz's inequality, we get

$$\limsup\_{k \to +\infty} \langle \mathbf{x}\_{\mathbf{n}+1} - \mathbf{x}^\*, \mathbf{g}\_{\mathbf{n}}(\mathbf{x}^\*) - \mathbf{x}^\* \rangle \le \lim\_{k \to +\infty} ||\mathbf{x}\_{\mathbf{n}+1} - \mathbf{x}^\*|| \left| |\mathbf{g}\_{\mathbf{n}}(\mathbf{x}^\*) - \mathbf{g}(\mathbf{x}^\*)| \right| + \limsup\_{k \to +\infty} \langle \mathbf{x}\_{\mathbf{n}+1} - \mathbf{x}^\*, \mathbf{g}(\mathbf{x}^\*) - \mathbf{x}^\* \rangle.$$

By the boundedness of {*xn*}, *gn*(*x*) → *g*(*x*), then by (28) and (39), we have

$$\limsup\_{k \to +\infty} \langle \mathbf{x}\_{n\_k+1} - \mathbf{x}^\*, \mathbf{g}\_{n\_k}(\mathbf{x}^\*) - \mathbf{x}^\* \rangle \le 0. \tag{40}$$

On substituting (40) in (38), we obtain that {*xn*} converges strongly to *x*∗. This completes the proof.

### **4. Numerical Example**

In this section, we give some numerical experiments to illustrate the performance of our algorithms with respect to some other algorithms in the literature. All computation are carried out using Lenovo PC with the following specification: Intel(R)core i7-600, CPU 2.48GHz, RAM 8.0GB, MATLAB version 9.5 (R2019b).

**Example 1.** *We consider the MSSFP where <sup>H</sup>*<sup>1</sup> <sup>=</sup> <sup>R</sup>*<sup>N</sup> and <sup>H</sup>*<sup>2</sup> <sup>=</sup> <sup>R</sup>*M*, *<sup>A</sup>* : <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup>*<sup>M</sup> is given by <sup>A</sup>*(*x*) = <sup>G</sup>*M*×*N*(*x*), *where* <sup>G</sup>*M*×*<sup>N</sup> is a <sup>M</sup>* <sup>×</sup> *<sup>N</sup> matrix. The closed convex sets Ci* (*<sup>i</sup>* ∈ {1, ... , *<sup>k</sup>*}) *of* R*<sup>N</sup> are given by*

$$\mathbb{C}\_{i} = \{ \mathbf{x} = (\mathbf{x}\_{1}, \dots, \mathbf{x}\_{N})^{T} \in \mathbb{R}^{N} : c\_{i}(\mathbf{x}) \le 0 \}$$

*where ci*(*x*) = *<sup>x</sup>* <sup>−</sup> *di*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *<sup>i</sup> such that pi* = *p*, *where is a positive real number and di* = (*x*1,*i*, ... , *xN*,*i*)*<sup>T</sup>* = (0, ... , 0, *<sup>i</sup>* <sup>−</sup> <sup>1</sup>)*<sup>T</sup>* <sup>∈</sup> <sup>R</sup>*<sup>N</sup> for each <sup>i</sup>* <sup>=</sup> 1, 2, ... , *<sup>k</sup>*. *Also, Qj* (*<sup>j</sup>* ∈ {1, ... , *<sup>t</sup>*}) *is defined by*

$$Q\_{\bar{\jmath}} = \{ y \in \mathbb{R}^M : q\_{\bar{\jmath}}(y) \le 0 \},$$

*where qj*(*y*) = <sup>1</sup> <sup>2</sup> *<sup>y</sup>TBjy* + *<sup>b</sup><sup>T</sup> <sup>j</sup> y* + *cj*, *j*= 1, 2, ... , *k*, *Bj is a Hessian matrix, bj and cj are vectors generated randomly. For each i* ∈ {1, . . . , *k*} *and j* ∈ {1, . . . , *t*} *the subdifferentials are given by*

$$\partial \mathfrak{c}\_i(\mathfrak{x}\_{\mathcal{U}}) = \begin{cases} \left\{ \frac{\mathfrak{x}\_n - d\_i}{||\mathfrak{x}\_n - d\_i||} \right\} & \text{if } \quad \mathfrak{x}\_n - d\_i \neq 0, \\\left\{ a\_i \in \mathbb{R}^N : ||a\_i|| \le 1 \right\} & \text{otherwise}, \end{cases}$$

*and <sup>∂</sup>qj*(*Axn*) = {(*b*1,*j*,..., *bM*,*j*)*T*}. *Please note that the projection*

$$P\_{\mathbb{C}\_i^n}(\mathfrak{x}\_{\mathbb{R}}) = \arg\min \{ ||\mathfrak{x} - \mathfrak{x}\_{\mathbb{H}}|| : \mathfrak{x} \in \mathbb{C}\_i^n \},$$

*where C<sup>n</sup> <sup>i</sup>* <sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> *<sup>H</sup>*<sup>1</sup> : *ci*(*xn*) ≤ *ξ<sup>n</sup> <sup>i</sup>* , *xn* − *x*} *which is equivalent to the following quadratic programming problem*

$$\begin{cases} \text{minimize} & \frac{1}{2} \mathbf{x}^T \vec{\mathcal{H}} \mathbf{x} + \vec{\mathcal{B}}\_{\text{tr}}^T \mathbf{x} + \vec{\mathcal{E}}\_{\text{\textquotedblleft}}\\ \text{subject to} & \vec{\mathcal{D}}\_{i,\text{\textquotedblleft}}(\mathbf{x}) \le \vec{\mathcal{F}}\_{i,\text{\textquotedblright}} \end{cases} \tag{41}$$

*where* <sup>H</sup>¯ <sup>=</sup> <sup>2</sup>*IM*×*N*, <sup>B</sup>¯ *<sup>n</sup>* <sup>=</sup> <sup>−</sup>2*xn*, *<sup>c</sup>*¯ <sup>=</sup> ||*xn*||2, <sup>D</sup>¯*i*,*<sup>n</sup>* <sup>=</sup> ¯ *ξn <sup>i</sup>* = [ ¯ *ξn <sup>i</sup>*,1, ... , ¯ *ξn <sup>i</sup>*,*N*], <sup>F</sup>¯ *<sup>i</sup>* = *p*<sup>2</sup> *<sup>i</sup>* − ||*xn* − *di*||<sup>2</sup> <sup>+</sup> ¯ *ξn <sup>i</sup>* , *xn*. *The Problem* (41) *as well as the projection onto <sup>Q</sup><sup>n</sup> <sup>j</sup> can effectively be solved using Optimization Toolbox solver '*quadprog*' in MATLAB. We defined the mapping Sm* : <sup>R</sup>*<sup>N</sup>* <sup>→</sup> <sup>R</sup>*<sup>N</sup> by*

$$S\_{\mathfrak{M}}\mathbf{x} = \begin{cases} (\mathbf{x}\_{1\prime}\mathbf{x}\_{2\prime}\dots\mathbf{x}\_{i\prime}\dots) & \text{if } \begin{array}{l} \prod\_{i=1}^{\infty} \mathbf{x}\_{i} < \mathbf{0}, \\ (-2\mathbf{x}\_{1\prime} - 2\mathbf{x}\_{2\prime}\dots, -2\mathbf{x}\_{i\prime}\dots) & \text{if } \prod\_{i=1}^{\infty} \mathbf{x}\_{i} \ge \mathbf{0}. \end{array} \end{cases}$$

*It is easy to see that Sm is* <sup>1</sup> <sup>3</sup> *-strictly pseudo-nonspreading. For each <sup>n</sup>* <sup>∈</sup> <sup>N</sup> *and <sup>m</sup>* <sup>≥</sup> 0, *let* {*γn*,*m*} *be defined by*

$$\gamma\_{n,m} = \begin{cases} \frac{1}{b^{m+1}} \left( \frac{n}{n+1} \right)\_{\prime} & n \ge m+1, \\ 1 - \frac{n}{n+1} \sum\_{k=1}^{n} \frac{1}{b^k} & n = m, \\ 0 & n < m, \end{cases}$$

*where b* > 1*. For simplicity, we consider the case for which k* = *t and compare the performance of Algorithm 1, Algorithm 2 and Algorithm (42) of Wen et al. [9] using various dimension of N*. *We choose b* = 5, *gn*(*x*) = *<sup>x</sup>* <sup>4</sup> , *<sup>β</sup>* <sup>=</sup> 0.2, *<sup>α</sup><sup>n</sup>* <sup>=</sup> <sup>√</sup> <sup>1</sup> *n*+1 , *<sup>n</sup>* = <sup>1</sup> *<sup>n</sup>*+<sup>1</sup> *, <sup>θ</sup>* <sup>=</sup> 0.01, *<sup>ρ</sup><sup>n</sup>* <sup>=</sup> *<sup>n</sup> <sup>n</sup>*+<sup>1</sup> , *wi* <sup>=</sup> <sup>1</sup> *k . Similarly, for Algorithm (42) of Wen et al. [9], we take ρ<sup>n</sup>* = *<sup>n</sup> <sup>n</sup>*+<sup>1</sup> *and wi* <sup>=</sup> <sup>1</sup> *<sup>k</sup>* . *The initial points <sup>x</sup>*0, *<sup>x</sup>*<sup>1</sup> *and the matrices* <sup>G</sup>*M*×*<sup>N</sup> are generated randomly for the following values of N and M:*

*Case I: N* = 4 *and M* = 10*;*

*Case II: N* = 10 *and M* = 5;

*Case III: N* = 10 *and M* = 10;

*Case IV: N* = 15 *and M* = 20.

*We use En* <sup>=</sup> ||*xn*+<sup>1</sup> <sup>−</sup> *xn*|| <sup>&</sup>lt; <sup>10</sup>−<sup>4</sup> *as stopping criterion and plot the graphs of En against number of iterations. The numerical results are shown in Table 1 and Figure 1.*

**Table 1.** Computation result for Example 1.


**Figure 1.** Example 1, Case I–Case IV; Top–Bottom.

Finally, we present an example in infinite dimensional Hilbert spaces.

**Example 2.** *Let <sup>H</sup>*<sup>1</sup> <sup>=</sup> *<sup>H</sup>*<sup>2</sup> <sup>=</sup> *<sup>L</sup>*2([0, 1]) *with norm* ||*x*|| <sup>=</sup> <sup>1</sup> <sup>0</sup> |*x*(*t*)| <sup>2</sup>*dt*1/2 *and the inner product x*, *<sup>y</sup>* <sup>=</sup> <sup>1</sup> <sup>0</sup> *<sup>x</sup>*(*t*)*y*(*t*)*dt*. *We defined the nonempty, closed convex sets <sup>C</sup>* <sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, 1]) : *x*(*t*), 3*t* <sup>2</sup> <sup>=</sup> <sup>0</sup>} *and <sup>Q</sup>* <sup>=</sup> {*<sup>y</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, 1]) : *y*, *<sup>t</sup>*/3≥−1}*. We defined the linear operator <sup>A</sup>* : *<sup>L</sup>*2([0, 1]) <sup>→</sup> *<sup>L</sup>*2([0, 1]) *by* (*Ax*)(*t*) = *<sup>x</sup>*(*t*). *The projection onto C and Q are given by*

$$P\_{\mathbb{C}}(\mathbf{x}(t)) = \begin{cases} \mathbf{x}(t) - \frac{\langle \mathbf{x}(t), 3t^2 \rangle}{||\mathbf{3}t^2||^2} \mathbf{3}t^2 & \text{if } \quad \langle \mathbf{x}(t), \mathbf{3}t^2 \rangle \neq 0, \\ \mathbf{x}(t), & \text{if } \quad \langle \mathbf{x}(t), \mathbf{3}t^2 \rangle = 0. \end{cases}$$

*and*

$$P\_Q(y(t)) = \begin{cases} y(t) - \frac{\langle y(t), \frac{-t}{3} \rangle}{||t-\frac{t}{3}||^2} (\frac{-t}{3}), & \text{if} \quad \langle y(t), \frac{-t}{3} \rangle < -1, \\ y(t) & \text{if} \quad \langle y(t), \frac{-t}{3} \rangle \ge -1. \end{cases}$$

*We consider the MSSFP where k* = *t* = 1*, Ci* = *C*, *Qj* = *Q*, *Sm* = *I (identity mapping) and m* = 4*. We compare our Algorithm 2 with the CQ-type algorithm (Algorithm 3.1) of Vinh et al. [20]. For Algorithm 2, we take gn*(*x*) = *<sup>x</sup>* <sup>8</sup> , *<sup>β</sup>* <sup>=</sup> 0.5*, wi* <sup>=</sup> 1, *<sup>α</sup><sup>n</sup>* <sup>=</sup> <sup>1</sup> *<sup>n</sup>*+<sup>1</sup> , *<sup>n</sup>* = <sup>1</sup> (*n*+1)<sup>2</sup> , *and <sup>γ</sup>n*,*<sup>m</sup>* <sup>=</sup> <sup>1</sup> 5 *for m* = 0, 1, ... , 4. *Also, for Vinh et al. alg, we take ρ<sup>n</sup>* = *<sup>n</sup> <sup>n</sup>*+<sup>1</sup> *and <sup>β</sup><sup>n</sup>* <sup>=</sup> <sup>1</sup> *<sup>n</sup>*+<sup>1</sup> . *We use En* = <sup>1</sup> <sup>2</sup> ||*Axn* <sup>−</sup> *PQ*(*Axn*)||<sup>2</sup> <sup>&</sup>lt; <sup>10</sup>−<sup>4</sup> *as stopping criterion and test the algorithms for the following initial points:*

*Case I: x*<sup>0</sup> = exp(−2*t*), *x*<sup>1</sup> = *t* <sup>3</sup> sin(3*t*)/3, *Case II: x*<sup>0</sup> = *t* <sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>t</sup>* <sup>−</sup> 1, *<sup>x</sup>*<sup>1</sup> = (cos(2*t*) + sin(3*t*))/5, *Case III: x*<sup>0</sup> = 2*t* cos(−3*t*), *x*<sup>1</sup> = 4 sin(2*t*), *Case IV: x*<sup>0</sup> = exp(2*t*)/2, *x*<sup>1</sup> = *t* <sup>3</sup> <sup>+</sup> <sup>3</sup>*<sup>t</sup>* <sup>−</sup> 1. *The numerical results are reported in Table 2 and Figure 2.*

**Table 2.** Computation result for Example 2.


**Figure 2.** Example 2, Case I–Case IV; Top–Bottom.

### **5. Conclusions**

In this paper, we introduce a generalized viscosity approximation method with selfadaptive stepsize for finding common solution of multiple set split feasibility problem and fixed point of a countable family of *k*-strictly pseduononspreading mappings in real Hilbert spaces. We also introduce a generalized viscosity approximation method with inertial and self-adaptive stepsize for solving the underlying problem. We prove strong convergence results for the sequences generated the algorithms under some mild conditions. We also provide some numerical example to show the performance of the proposed methods with respect to some other methods in the literature. These results improve and compliment several other results (e.g., [6–9,20]) in the literature.

**Author Contributions:** Conceptualization, H.A.A. and L.O.J.; methodology, H.A.A. and L.O.J.; software, H.A.A. and L.O.J.; validation, H.A.A. and L.O.J.; formal analysis, H.A.A. and L.O.J.; writing—original draft preparation, H.A.A. and L.O.J.; writing—review and editing, H.A.A. and L.O.J.; supervision, H.A.A. and L.O.J.; project administration, H.A.A. and L.O.J.; funding acquisition, H.A.A. and L.O.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research is funded by the Mathematical research fund at the Sefako Makgatho Health Sciences University.

**Acknowledgments:** The authors acknowledge with thanks, the Department of Mathematics and Applied Mathematics at the Sefako Makgatho Health Sciences University for making their facilities available for the research.

**Conflicts of Interest:** The authors declare no conflicts of interest.

### **References**


*Article*
