**Exact Solutions for a Class of Wick-Type Stochastic (3+1)-Dimensional Modified Benjamin–Bona–Mahony Equations**

### **Praveen Agarwal 1, Abd-Allah Hyder 2,3, M. Zakarya 2,4, Ghada AlNemer 5, Clemente Cesarano 6,\* and Dario Assante <sup>7</sup>**


Received: 9 October 2019; Accepted: 18 November 2019; Published: 3 December 2019

**Abstract:** In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions.

**Keywords:** modified BBM equations; (3+1)-dimensional equations; white noise; Brownian motion; travelling wave solutions; wick-type stochastic

**MSC:** 60H15; 60H35; 35C07; 60H40

### **1. Introduction**

In this paper, with the help of white noise theory, Hermite transform and a generalised, modified tanh–coth method, we deduce stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations as the forms:

$$\mathcal{U}\mathcal{U}\_t + \mathcal{R}\_1(t) \diamond \mathcal{U}\_z + \mathcal{R}\_2(t) \diamond \mathcal{U}^{\diamond 2} \diamond \mathcal{U}\_x + \mathcal{R}\_3(t) \diamond \mathcal{U}\_{xyt} = \mathbf{0},\tag{1}$$

$$\left(V\_t + R\_4(t) \diamond V\_x + R\_5(t) \diamond V^{\diamond 2} \diamond V\_y + R\_6(t) \diamond V\_{xxx} = 0,\tag{2}$$

and

$$\mathcal{W}\_t + R\_7(t) \diamond \mathcal{W}\_y + R\_8(t) \diamond \mathcal{W}^{\diamond 2} \diamond \mathcal{W}\_z + R\_9(t) \diamond \mathcal{W}\_{\text{xxt}} = 0,\tag{3}$$

where (*x*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*) <sup>∈</sup> <sup>R</sup><sup>3</sup> <sup>×</sup> <sup>R</sup>+, *Ri*(*<sup>i</sup>* <sup>=</sup> 1, 2, ...9) are non-zero integrable functions from <sup>R</sup><sup>+</sup> to the Kondrative distribution space (S)−1, which was defined by Holden et al. in [1] as a Banach algebra with the Wick-product!Equations (1)–(3) are the perturbations of the (3+1)-dimensional modified BBM equations with variable coefficients:

$$u\_t + r\_1(t)u\_z + r\_2(t)u^2u\_x + r\_3(t)u\_{xyt} = 0,\tag{4}$$

$$v v\_t + r\_4(t)v\_x + r\_5(t)v^2 v\_y + r\_6(t)v\_{xxx} = 0,\tag{5}$$

and

$$w\_t + r\_7(t)w\_y + r\_8(t)w^2w\_z + r\_9(t)w\_{xxt} = 0,\tag{6}$$

where *ri*(*i* = 1, 2, ...9) are non-zero integrable functions o nR+. The modified BBM equation:

$$
\mu\_t + k(t)\mu\_x + l(t)\mu^2\mu\_x + m(t)\mu\_{\text{xxx}} = 0. \tag{7}
$$

which describes the surface long waves in nonlinear dispersive media. It is also used as a character to acoustic-gravity waves in compressible fluids, hydromagnetic waves in cold plasma, and acoustic waves in anharmonic crystals [2]. The study of (3+1)-dimensional nonlinear equations is promising because these equations model the real features in a wide assortment of science, technology, fluid mechanics, wave propagations, electrodynamics, and engineering fields [3–6]. For this reason, Hereman [4,5] proposed the (3+1)-dimensional nonlinear modified KdV equation. Analogously, and by the same sense, Wazwaz [7] introduced Equations (4)–(6). Moreover, if Equations (4)–(6) are considered in a random environment, we have random (3+1)-dimensional modified BBM equations. In order to obtain the exact solutions of random (3+1)-dimensional modified BBM equations, we only consider them in a white noise environment; that is, we will discuss the Wick-type, stochastic, (3+1)-dimensional modified BBM Equations (1)–(3).

Recently, the study of solutions to nonlinear partial differential equations (PDEs) is prospering [8–10]. Many authors have researched the subject of the random travelling wave, which is a significant subject of stochastic partial differential equations (SPDEs). Wadati [11] first proposed and discussed the stochastic KdV equation and gave the propagation of soliton of the KdV equation under the effect of Gaussian noise. Furthermore, Ghany and Hyder [12–15], Ghany, Hyder and Zakarya [16,17], Chen and Xie [18–20], Hyder and Zakarya [21,22], Hyder [23,24], and Agarwal, Hyder and Zakarya [25] investigated a wide class of Wick-type stochastic evolution equations by using different extension methods and white noise analysis.

There are many methods to obtain travelling wave solutions to nonlinear PDEs, such as the inverse scattering method [26], the Newton's method [27], the tanh method [28], the Sinc–Galerkin method [29], the residual power series method [30], the semi-inverse variational principle and the first integral method [31], and the Daftardar-Gejji and Jafari method [32]. The tanh method, established by Malfliet [33], pursues a specially straightforward and effective algorithm to obtain exact solutions for a wide class of nonlinear PDEs. Moreover, a variety of research papers have focused on the different applications and extensions of the tanh method. Fan [34] has introduced an extended tanh method and gave new travelling wave solutions that cannot be obtained by the tanh method. Also, Wazwaz extended the tanh method and named it the tanh–coth method [35]. Furthermore, El-Wakil [36] and Soliman [37] modified the tanh–coth method and presented new, exact solutions for some nonlinear PDEs.

Our aim in this work was to obtain new stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Firstly, we give a generalised version of the modified tanh–coth method to make it convenient for the nonlinear (3+1)-dimensional and multi dimensional PDEs. Secondly, we use the generalised, modified tanh–coth method, white noise theory, and Hermite transform to produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations, this set includes solutions of exponential, hyperbolic, and trigonometric types. Finally, we use the inverse Hermite transform to obtain stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Moreover, by an application example, we show how the stochastic solutions can be given as white noise functional solutions. In our work, the modified BBM equation describes the surface long waves in nonlinear dispersive media. It is also used as a character of acoustic gravity waves in compressible fluids, hydromagnetic waves in cold plasma, and acoustic waves in harmonic crystals [2]. The study of (3+1)-dimensional nonlinear equations is prospering because these equations model the real features in a wide assortment of science, technology, fluid mechanics, wave propagations, electrodynamics, and engineering fields [3–6]. The origin and references of Equation (1) are given in Holden [1]. Ghany and Fathallah studied white-noise functional solutions for wick-type stochastic time-fractional Benjamin–Bona–Mahony (BBM) equation in [38]. Recently, Sahoo and Saha Ray studied by other methods the stochastic solutions of wick-type stochastic time-fractional BBM equation for modeling long surface gravity waves of small amplitude, in [39]. The PDE of Benjamin et al. [2] is now often called the BBM equation, although it is also known as the regularised long wave (RLW) equation. Morrison et al proposed the one-dimensional PDE, as an equally valid and accurate model for the same wave phenomena simulated by the KdV and RLW equations [40]. Random waves are an important subject of random PDEs. In essence, to investigate the exact solutions of random BBM equation, we restricted our attention to consider this problem in white noise environment [38].

This paper is organized as follows: In Section 2, we recall some requisites from Gaussian white noise analysis. In Section 3, we give a generalisation to the modified tanh–coth method to make it convenient for the nonlinear (3+1)-dimensional equations. In Section 4, we employ the generalised, modified tanh–coth method, white noise theory, and Hermite transform to obtain a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. In Section 5, we apply the inverse Hermite transform to explore stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. In Section 5, we give some examples to show that the stochastic solutions can be given as Brownian motion functional solutions and white noise functional solutions. In Section 6, we give a summary and discussion.

### **2. Requisites from Gaussian White Noise Analysis**

The Gaussian white noise analysis starts with the rigging <sup>S</sup>(R*d*) <sup>⊂</sup> *<sup>L</sup>*2(R*d*) ⊂ S∗(R*d*), where <sup>S</sup>(R*d*) is the Schwartz space of rapidly decreasing, infinite differentiable functions on <sup>R</sup>*d*, and <sup>S</sup>∗(R*d*) is the space of tempered distributions. From the Bochner–Minlos theorem [1], we have a unique white noise measure *<sup>μ</sup>*, on <sup>S</sup>∗(R*d*), *<sup>β</sup>* <sup>S</sup>∗(R*d*) . Assume that *<sup>ξ</sup>n*(*x*) = *<sup>π</sup>*−1/4((*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)!)−1/2*e*−*x*2/2*hn*−1( <sup>√</sup>2*x*), *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> are the Hermite functions, where *hn*(*x*) denotes the Hermite polynomials. It is well known that the collection (*ξn*)*n*∈<sup>N</sup> forms an orthonormal basis for *<sup>L</sup>*2(R). Let *<sup>α</sup>* = (*α*1, ..., *<sup>α</sup>d*) be a *<sup>d</sup>*-dimensional multi-indices with *<sup>α</sup>*1, ..., *<sup>α</sup><sup>d</sup>* <sup>∈</sup> <sup>N</sup>; then, the family of tensor products *ξα* :<sup>=</sup> *<sup>ξ</sup>*(*α*1,...,*αd*) <sup>=</sup> *ξα*<sup>1</sup> <sup>⊗</sup> ... <sup>⊗</sup> *ξα<sup>d</sup>* , *<sup>α</sup>* <sup>∈</sup> <sup>N</sup>*<sup>d</sup>* constitutes an orthonormal basis for *<sup>L</sup>*2(R*d*). Now, introduce an ordering in N*<sup>d</sup>* by

$$\text{if } i < j \Rightarrow \sum\_{k=1}^{d} a\_k^{(i)} \le \sum\_{k=1}^{d} a\_k^{(j)}, \text{ where } a^{(i)} = \left(a\_k^{(i)}\right)\_{k=1}^{d}, a^{(j)} = \left(a\_k^{(j)}\right)\_{k=1}^{d} \in \mathbb{N}^d. \tag{8}$$

Using this ordering, we define *η<sup>i</sup>* := *ξα*(*i*) = *ξ <sup>α</sup>*(*i*) 1 ⊗ ... ⊗ *ξ <sup>α</sup>*(*i*) *d* , *<sup>i</sup>* <sup>∈</sup> <sup>N</sup>. Let <sup>J</sup> <sup>=</sup> N<sup>N</sup> 0 *<sup>c</sup>* be the set of all sequences *<sup>α</sup>* = (*αi*)*i*∈<sup>N</sup> with *<sup>α</sup><sup>i</sup>* <sup>∈</sup> <sup>N</sup><sup>0</sup> and with compact support. For *<sup>α</sup>* <sup>∈</sup> <sup>J</sup>, we define

$$H\_{\mathfrak{a}}(\omega) = \prod\_{i=1}^{\infty} h\_{\mathfrak{a}\_i}(\langle \omega\_\prime \eta\_i \rangle), \ \omega \in \mathcal{S}^\*(\mathbb{R}^d). \tag{9}$$

Let *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>, the Kondrative space of stochastic test functions (S)*<sup>n</sup>* <sup>1</sup> is defined by:

$$(\mathcal{S})\_1^n = \left\{ f = \sum\_{\mathfrak{a}} c\_{\mathfrak{a}} H\_{\mathfrak{a}} \in \bigoplus\_{k=1}^n L^2(\mathfrak{a}) \, : \, c\_{\mathfrak{a}} \in \mathbb{R}^n \text{ and } \|f\|\_{1,k}^2 := \sum\_{\mathfrak{a}} c\_{\mathfrak{a}}^2 (a!)^2 (2! \mathbb{N})^{\mathbb{N}} < \infty \,\forall k \in \mathbb{N} \right\}, \tag{10}$$

and the Kondrative space of stochastic distributions (S)*<sup>n</sup>* <sup>−</sup><sup>1</sup> is defined by:

$$\left(\left(\mathcal{S}\right)\_{-1}^{n} = \left\{ F = \sum\_{\mathfrak{a}} b\_{\mathfrak{a}} H\_{\mathfrak{a}} \, : \, b\_{\mathfrak{a}} \in \mathbb{R}^{n} \text{ and } \left\| F \right\|\_{-1, \mathfrak{a}}^{2} := \sum\_{\mathfrak{a}} b\_{\mathfrak{a}}^{2} (2\mathbb{N})^{-\mathfrak{q}\mathfrak{a}} < \infty \text{ for some } q \in \mathbb{N} \right\}.\tag{11}$$

The family of seminorms *<sup>f</sup>* 1,*k*, *<sup>k</sup>* <sup>∈</sup> <sup>N</sup> produces a topology on (S)*<sup>n</sup>* <sup>1</sup> and (S)*<sup>n</sup>* <sup>−</sup><sup>1</sup> can be represented as the dual of (S)*<sup>n</sup>* <sup>1</sup> under the action *F*, *<sup>f</sup>* <sup>=</sup> <sup>∑</sup>*α*(*bα*, *<sup>c</sup>α*)*α*!, where *<sup>F</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup> <sup>b</sup>αH<sup>α</sup>* <sup>∈</sup> (S)*<sup>n</sup>* <sup>−</sup>1, *<sup>f</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup> <sup>c</sup>αH<sup>α</sup>* <sup>∈</sup> (S)*<sup>n</sup>* <sup>1</sup> and (*bα*, *<sup>c</sup>α*) is the usual scalar product on <sup>R</sup>*n*.

The Wick product of two distributions *<sup>F</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup> <sup>a</sup>αHα*, *<sup>G</sup>* <sup>=</sup> <sup>∑</sup>*<sup>β</sup> <sup>b</sup>βH<sup>β</sup>* <sup>∈</sup> (S)*<sup>n</sup>* <sup>−</sup><sup>1</sup> with *<sup>a</sup>α*, *<sup>b</sup><sup>β</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* is defined by:

$$F \diamond G = \sum\_{a,\emptyset} (a\_{a\prime} b\_{\not\otimes}) H\_{a+\not\otimes} \,. \tag{12}$$

Let *<sup>F</sup>* <sup>=</sup> <sup>∑</sup>*<sup>α</sup> <sup>a</sup>αH<sup>α</sup>* <sup>∈</sup> (S)*<sup>n</sup>* <sup>−</sup><sup>1</sup> with *<sup>a</sup><sup>α</sup>* <sup>∈</sup> <sup>R</sup>*n*. The Hermite transform of *<sup>F</sup>* is defined by:

$$\mathcal{H}F(w) = \tilde{F}(w) = \sum\_{a} a\_{a} w^{a} \in \mathbb{C}^{n} \quad \text{(when convergent)},\tag{13}$$

where *<sup>w</sup>* = (*w*1, *<sup>w</sup>*2, ...) <sup>∈</sup> <sup>C</sup><sup>N</sup> and *<sup>w</sup><sup>α</sup>* <sup>=</sup> <sup>Π</sup><sup>∞</sup> *i*=1*wα<sup>i</sup> <sup>i</sup>* , with *<sup>α</sup>* = (*α*1, *<sup>α</sup>*2, ...) <sup>∈</sup> <sup>J</sup> and *<sup>w</sup>*<sup>0</sup> *<sup>i</sup>* = 1.

For *<sup>F</sup>*, *<sup>G</sup>* <sup>∈</sup> (*S*)*<sup>n</sup>* <sup>−</sup>1, by the definition of Hermite transform, we get:

$$
\widehat{F \diamond G}(w) = \check{F}(w)\check{G}(w) \,, \tag{14}
$$

for all *w* such that *F*(*w*) and *G*(*w*) exist. The multiplication on the right hand side of the above equality is the complex bilinear multiplication in C*<sup>n</sup>* which is defined by (*w*<sup>1</sup> 1, ...*w*<sup>1</sup> *n*)(*w*<sup>2</sup> <sup>1</sup>, ..., *<sup>w</sup>*<sup>2</sup> *<sup>n</sup>*) = ∑*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> *w*<sup>1</sup> *<sup>i</sup> <sup>w</sup>*<sup>2</sup> *i* , where *w<sup>k</sup> <sup>i</sup>* <sup>∈</sup> <sup>C</sup>. Hence, The Hermite transform converts the Wick product into the usual product and convergence in (S)*<sup>n</sup>* <sup>−</sup><sup>1</sup> into pointwise and bounded convergence in a specific neighbourhood of zero in C*n*. For more details about stochastic Kondrative spaces, Wick product, and Hermite transform we refer the reader to [1].

### **3. Generalization of the Modified Tanh–Coth Method**

Consider a multi dimensional, nonlinear PDE of wave propagation:

$$P\left(\mu, \mu\_{t\prime}, \mu\_{\mathbf{x}\_{i}\prime}, \mu\_{\mathbf{x}\_{i}\mathbf{x}\_{j\prime}\prime}, \mu\_{\mathbf{x}\_{i}\mathbf{x}\_{j}\mathbf{x}\_{k\prime}\prime}\ldots\right) = 0\,,\tag{15}$$

where *u* is the dependent variable and *t* = *x*0, *x*1, *x*2, ..., *xm* are the independent variables. Introduce the wave transformation:

$$
\mu = \mu(\xi), \qquad \xi = \sum\_{i=0}^{m} a\_i x\_i. \tag{16}
$$

where *ai*(*i* = 0, 1, 2, ..., *m*) are unknown constants. Therefore, Equation (15) can be transformed into a nonlinear ordinary differential equation (NODE):

$$\mathbb{Q}(\mathfrak{u}, \mathfrak{u}', \mathfrak{u}'', \mathfrak{u}''', \dots) = 0. \tag{17}$$

For simplicity, we integrate the NODE (17), provided that all terms include derivatives, and set the integration constants to be zero. Subsequently, the transformed Equation (17) can be solved by expanding its general solution in finite series as follows:

$$\mu(\xi) = \sum\_{k=0}^{N} A\_k \Phi^k(\xi) + \sum\_{k=1}^{N} B\_k \Phi^{-k}(\xi), \tag{18}$$

where Φ solves the first order Riccati equation [41]:

$$\Phi'(\xi) = \mathfrak{w}\mathfrak{o} + \mathfrak{a}\_1 \Phi(\xi) + \mathfrak{a}\_2 \Phi^2(\xi),\tag{19}$$

where *α*0,*α*1, and *α*<sup>2</sup> are constants to be determined. The positive constant *N* can be specified by balancing the linear and nonlinear terms of highest order in Equation (17). Inserting Equations (18) and (19) into Equation (17), yields an algebraic equation in Φ and its powers. Equating the coefficients of Φ*<sup>k</sup>* to zero, gives an algebraic system of equations in *Ak* and *Bk*. With the help of the computer symbolic system *Mathematica*, we can obtain *Ak* and *Bk*. The Riccati Equation (19) has the following particular solutions [42]:

$$\begin{cases} \Phi(\underline{\zeta}) = e^{\overline{\zeta}} - 1, & \mathfrak{a}\_{0} = 1, \mathfrak{a}\_{1} = 1, \mathfrak{a}\_{2} = 0, \\ \Phi(\underline{\zeta}) = \coth(\underline{\zeta}) \pm (\underline{\zeta}), \ \tanh(\underline{\zeta}) \pm i \left( \underline{\zeta} \right), & \mathfrak{a}\_{0} = \frac{1}{2}, \mathfrak{a}\_{1} = 0, \mathfrak{a}\_{2} = -\frac{1}{2}, \\ \Phi(\underline{\zeta}) = \tan(\underline{\zeta}), \ -\cot(\underline{\zeta}), & \mathfrak{a}\_{0} = 1, \mathfrak{a}\_{1} = 0, \mathfrak{a}\_{2} = 1, \\ \Phi(\underline{\zeta}) = \frac{1}{2}\cot(2\underline{\zeta}), \ \frac{1}{2}\tan(2\underline{\zeta}), & \mathfrak{a}\_{0} = 1, \mathfrak{a}\_{1} = 0, \mathfrak{a}\_{2} = 4. \end{cases} \tag{20}$$

### **4. The Wick-Type, Stochastic, (3+1)-Dimensional Modified BBM Equations**

We first investigate the model (1) of the Wick-Type, stochastic, (3+1)-dimensional modified BBM equations. Applying Hermite transform to Equation (1), gets the deterministic equation:

$$\begin{split} \hat{\mathcal{U}}\_{1}(\mathbf{x},\mathbf{y},z,t,w) + \check{\mathcal{R}}\_{1}(t,w)\check{\mathcal{U}}\_{2}(\mathbf{x},\mathbf{y},z,t,w) + \check{\mathcal{R}}\_{2}(t,w)\check{\mathcal{U}}^{2}(\mathbf{x},\mathbf{y},z,t,w)\mathcal{U}\_{\mathbf{x}}(\mathbf{x},\mathbf{y},z,t,w) + \\ + \check{\mathcal{R}}\_{3}(t,w)\check{\mathcal{U}}\_{\mathbf{x}\mathbf{y}t}(\mathbf{x},\mathbf{y},z,t,w) = \mathbf{0}, \end{split} \tag{21}$$

where *<sup>w</sup>* = (*w*1, *<sup>w</sup>*2, ...) <sup>∈</sup> CN *c* . To obtain travelling wave solutions to Equation (21), we introduce the transformations *R* ;1(*t*, *w*) = *r*1(*t*, *w*),*R* ;2(*t*, *w*) = *r*2(*t*, *w*),*R* ;3(*t*, *w*) = *r*3(*t*, *w*), and *U*(*x*, *y*, *z*, *t*, *w*) = *u*(*x*, *y*, *z*, *t*, *w*) = *u*(*ξ*(*x*, *y*, *z*, *t*, *w*)) with

$$\xi(x, y, z, t, w) = a\_1 x + a\_2 y + a\_3 z + b \int\_0^t \chi(\tau, w) d\tau,\tag{22}$$

where *ai* (*i* = 1, 2, 3), *b*, and *c* are arbitrary constants satisfying *aib* = 0 and *χ* is a non-zero function to be determined. Hence, Equation (21) can be converted to the following NODE:

$$a\_1(b\chi + a\_3r\_1)u + \frac{1}{3}a\_1r\_2u^3 + a\_1a\_2b\chi r\_2u^{\prime\prime} = 0.\tag{23}$$

Balancing *u*<sup>3</sup> with *u* , gives *N* = 1. Therefore, we put the solution of Equation (21) in the form:

$$u(x, y, z, t, w) = A\_0(t, w) + A\_1(t, w) \Phi(\xi) + \frac{B\_1(t, w)}{\Phi(\xi)},\tag{24}$$

where Φ is the solution of Equation (19). Substituting Equations (24) and (19) into Equation (23), collecting the coefficients of <sup>Φ</sup>*<sup>k</sup>* (*<sup>k</sup>* <sup>=</sup> <sup>−</sup>3, <sup>−</sup>2, <sup>−</sup>1, 0, 1, 2, 3), and equating them to zero, gives the following system of seven algebraic equations in *A*0, *A*1, *B*1, and *χ*.

$$\begin{cases} (b\chi + a\_3 r\_1)A\_0 + \frac{1}{3} a\_1 r\_2 I\_0 + a\_1 a\_2 b \chi r\_3 E\_0 = 0, \\ (b\chi + a\_3 r\_1)A\_1 + \frac{1}{3} a\_1 r\_2 I\_1 + a\_1 a\_2 b \chi r\_3 E\_1 = 0, \\ (b\chi + a\_3 r\_1)B\_1 + \frac{1}{3} a\_1 r\_2 I\_1 + a\_1 a\_2 b \chi r\_3 F\_1 = 0, \\ \frac{1}{3} a\_1 r\_2 I\_2 + a\_1 a\_2 b \chi r\_3 E\_2 = 0, \\ \frac{1}{3} a\_1 r\_2 I\_3 + a\_1 a\_2 b \chi r\_3 E\_3 = 0, \\ \frac{1}{3} a\_1 r\_2 I\_2 + a\_1 a\_2 b \chi r\_3 F\_2 = 0, \\ \frac{1}{3} a\_1 r\_2 I\_3 + a\_1 a\_2 b \chi r\_3 F\_3 = 0. \end{cases} \tag{25}$$

where *I*<sup>0</sup> = *A*0*G*<sup>0</sup> + *A*1*H*<sup>1</sup> + *B*1*G*1, *I*<sup>1</sup> = *A*0*G*<sup>1</sup> + *A*1*G*<sup>0</sup> + *B*1*G*2, *I*<sup>2</sup> = *A*0*G*<sup>2</sup> + *A*1*G*1, *I*<sup>3</sup> = *A*1*G*2, *J*<sup>1</sup> = *A*0*H*<sup>1</sup> + *A*1*H*<sup>2</sup> + *B*1*G*0, *J*<sup>2</sup> = *A*0*H*<sup>2</sup> + *B*1*H*1, *J*<sup>3</sup> = *B*1*H*2, *G*<sup>0</sup> = *A*<sup>2</sup> <sup>0</sup> + 2*A*1*B*1, *G*<sup>1</sup> = 2*A*0*A*1, *G*<sup>2</sup> = *A*<sup>2</sup> <sup>1</sup>, *<sup>H</sup>*<sup>1</sup> = <sup>2</sup>*A*0*B*1, *<sup>H</sup>*<sup>2</sup> = *<sup>B</sup>*<sup>2</sup> <sup>1</sup>, *E*<sup>0</sup> = *α*0*C*<sup>1</sup> − *α*2*D*1, *E*<sup>1</sup> = *α*1*C*<sup>1</sup> + 2*α*0*C*2, *E*<sup>2</sup> = *α*2*C*<sup>1</sup> + 2*α*1*C*2, *E*<sup>3</sup> = 2*α*2*C*2, *F*<sup>1</sup> = −*α*1*D*<sup>1</sup> − 2*α*2*D*2, *F*<sup>2</sup> = −*α*0*D*<sup>1</sup> − 2*α*1*D*2, *F*<sup>3</sup> = −2*α*0*D*2, *C*<sup>0</sup> = *α*0*A*<sup>1</sup> − *α*2*B*1, *C*<sup>1</sup> = *α*1*A*1, *C*<sup>2</sup> = *α*2*A*1, *D*<sup>1</sup> = −*α*1*B*1, *D*<sup>2</sup> = −*α*0*B*1.

Now, we solve the system (25) for some cases relating to the Riccati equation (19).

### *4.1. Case I*

We reduce the system (25) by using *α*<sup>0</sup> = *α*<sup>1</sup> = 1 and *α*<sup>2</sup> = 0. By using *Mathematica*, we can find a set of solutions for the reduced system as follows:

$$A\_0 = \pm i \sqrt{\frac{3a\_3 r\_1}{a\_1 r\_2}}, \quad A\_1 = 0, \quad B\_1 = \pm \sqrt{\frac{3a\_2 a\_3 r\_1}{a\_1 a\_2 r\_2 r\_3 - 2}}, \quad \chi = \frac{2a\_3 r\_1}{b(a\_1 a\_2 r\_3 - 2)}.\tag{26}$$

Substituting the values (26) in Equation (24) and using (20), yields a travelling wave solution of Equation (21) of exponential type:

$$\begin{split} \mu\_{1}(\mathbf{x}, y, z, t, w) &= \\ &= \frac{\pm i \sqrt{3 a\_{3} r\_{1}(t, w) (a\_{1} a\_{2} r\_{2}(t, w) r\_{3}(t, w) - \mathfrak{Z})} (\exp(\mathfrak{z}\_{1}(\mathbf{x}, y, z, t, w)) - 1) \pm \sqrt{3 a\_{1} a\_{2} a\_{3} r\_{1}(t, w) r\_{2}(t, w)}, \\ & \qquad \sqrt{a\_{1} r\_{2}(t, w) (a\_{1} a\_{2} r\_{2}(t, w) r\_{3}(t, w) - \mathfrak{Z})} (\exp(\mathfrak{z}\_{1}(\mathbf{x}, y, z, t, w)) - 1) \end{split} \tag{27}$$

where

$$\xi\_1(\mathbf{x}, y, z, t, w) = a\_1 \mathbf{x} + a\_2 y + a\_3 z + 2a\_3 \int\_0^t \frac{r\_1(\mathbf{r}, w)}{a\_1 a\_2 r\_3(\mathbf{r}, w) - 2} d\mathbf{r} \,. \tag{28}$$

### *4.2. Case II*

We reduce the system (25) by using *α*<sup>0</sup> = <sup>1</sup> <sup>2</sup> , *<sup>α</sup>*<sup>1</sup> <sup>=</sup> 0, and *<sup>α</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup><sup>1</sup> <sup>2</sup> . By using *Mathematica*, we can find a set of solutions for the reduced system as follows:

$$A\_0 = 0, \quad A\_1 = \pm \sqrt{\frac{3a\_2 a\_3 r\_1 r\_3}{r\_2 (2 - a\_1 a\_2 r\_3)}}, \quad B\_1 = \pm i \sqrt{\frac{3a\_1 a\_2 r\_1}{2r\_2 r\_3}}, \quad \chi = -\frac{4a\_3 r\_1}{b (4 + a\_1 a\_2 r\_3)}.\tag{29}$$

*Axioms* **2019**, *8*, 134

Substituting the values (29) in Equation (24) and using (20), yields travelling wave solutions of Equation (21) of hyperbolic type:

$$\begin{split} u\_{2}(\mathbf{x},y,z,t,w) &= \ \pm \quad \sqrt{\frac{3a\_{2}a\_{3}r\_{1}(t,w)r\_{3}(t,w)}{r\_{2}(t,w)(2-a\_{1}a\_{2}r\_{3}(t,w))}} \big(\coth(\not\!\!\_{2}(\mathbf{x},y,z,t,w)) \pm (\not\!\!\_{2}(\mathbf{x},y,z,t,w))\big) \\ &\pm \quad i\frac{\sqrt{3b u\_{2}r\_{1}(t,w)}}{\sqrt{2r\_{2}(t,w)r\_{3}(t,w)}\left(\coth(\not\!\!\_{2}(\mathbf{x},y,z,t,w)) \pm (\not\!\!\_{2}(\mathbf{x},y,z,t,w))\right)}, \end{split} \tag{30}$$

$$\begin{split} u\_3(\mathbf{x}, y, z, t, w) &= \quad \pm \quad \sqrt{\frac{3a\_2 a\_3 r\_1(t, w) r\_3(t, w)}{r\_2(t, w) (2 - a\_1 a\_2 r\_3(t, w))}} \left( \tanh(\not\!\!\_2 (\mathbf{x}, y, z, t, w)) \pm i \left( \not\!\!\!\_2 (\mathbf{x}, y, z, t, w) \right) \right)} \\ &\quad \pm \quad i \frac{\sqrt{3b a\_2 r\_1(t, w)}}{\sqrt{2r\_2(t, w) r\_3(t, w)} \left( \tanh(\not\!\!\!\_2 (\mathbf{x}, y, z, t, w)) \pm i \left( \not\!\!\!\_2 (\mathbf{x}, y, z, t, w) \right) \right)}, \end{split} \tag{31}$$

where

$$\xi\_2(\mathbf{x}, y, z, t, w) = a\_1 \mathbf{x} + a\_2 \mathbf{y} + a\_3 z - 4a\_3 \int\_0^t \frac{r\_1(\mathbf{r}, w)}{4 + a\_1 a\_2 r\_3(\mathbf{r}, w)} d\mathbf{r} \,. \tag{32}$$

### *4.3. Case III*

We reduce the system (25) by putting *α*<sup>0</sup> = *α*<sup>2</sup> = 1 and *α*<sup>1</sup> = 0. By using *Mathematica*, we can find a set of solutions for the reduced system as follows:

$$A\_0 = \pm \sqrt{\frac{3a\_2 a\_3 r\_1}{1 - a\_1 a\_2 r\_2 r\_3}}, \quad A\_1 = B\_1 = \pm \sqrt{\frac{6a\_2 a\_3 r\_1 r\_3}{1 + 2a\_1 a\_2 r\_2 r\_3}}, \quad \chi = \frac{-a\_3 r\_1}{b(1 + 2a\_1 a\_2 r\_3)}.\tag{33}$$

Substituting the values (33) in Equation (24) and using (20), yields travelling wave solutions of Equation (21) of trigonometric type:

$$\begin{array}{l} \mu\_{4}(\mathbf{x},y,z,t,w) = \mathbf{u}\_{5}(\mathbf{x},y,z,t,w) = \\ = \pm \sqrt{\frac{3a\_{2}a\_{3}r\_{1}(t,w)}{1 - a\_{1}a\_{2}r\_{2}(t,w)r\_{3}(t,w)}} \pm \sqrt{\frac{6a\_{2}a\_{3}r\_{1}(t,w)r\_{3}(t,w)}{1 + 2a\_{1}a\_{2}r\_{2}(t,w)r\_{3}(t,w)}} \left(\sec(\underline{\xi}\_{3}(\mathbf{x},y,z,t,w)) \cdot \csc(\underline{\xi}\_{3}(\mathbf{x},y,z,t,w))\right) \,, \end{array} \tag{34}$$

where

$$f\_3^\mathbf{r}(\mathbf{x}, y, z, t, w) = a\_1 \mathbf{x} + a\_2 \mathbf{y} + a\_3 z - a\_3 \int\_0^t \frac{r\_1(\mathbf{r}, w)}{1 + 2a\_1 a\_2 r\_3(\mathbf{r}, w)} d\mathbf{r} \,. \tag{35}$$

### *4.4. Case IV*

We reduce the system (25) by putting *α*<sup>0</sup> = 1, *α*<sup>1</sup> = 0 and *α*<sup>2</sup> = 4. By using *Mathematica*, we can find a set of solutions for the reduced system as follows:

$$A\_0 = \pm 3\sqrt{\frac{3a\_2 a\_3 r\_1}{9a\_1 a\_2 r\_2 r\_3 - 2}}, \quad A\_1 = \pm 8\sqrt{\frac{3a\_2 a\_3 r\_1 r\_3}{2 + 15a\_1 a\_2 r\_2 r\_3}}, \quad B\_1 = \pm i\sqrt{\frac{3a\_3 r\_1}{a\_1 r\_2 r\_3}}, \quad \chi = -\frac{2a\_3 r\_1}{b(9a\_1 a\_2 r\_2 r\_3 - 2)}.\tag{36}$$

Substituting the values (36) in Equation (24) and using (20), yields travelling wave solutions of Equation (21) of trigonometric type:

$$\begin{split} \mu\_{6}(\mathbf{x},y,z,t,w) &= \pm 3 \sqrt{\frac{3a\_{2}a\_{3}r\_{1}(t,w)}{9a\_{1}a\_{2}r\_{2}(t,w)r\_{3}(t,w)-2}} \pm 4 \sqrt{\frac{3a\_{2}a\_{3}r\_{1}(t,w)r\_{3}(t,w)}{2+15a\_{1}a\_{2}r\_{2}(t,w)r\_{3}(t,w)}} \left(\cot(2\xi\_{4}(\mathbf{x},y,z,t,w))\right) \\ &\pm i \frac{\sqrt{3a\_{3}r\_{1}(t,w)}}{2\sqrt{a\_{1}r\_{2}(t,w)r\_{3}(t,w)}(\cot(2\xi\_{4}(\mathbf{x},y,z,t,w)))} \end{split} \tag{37}$$

*Axioms* **2019**, *8*, 134

$$\begin{split} \text{var}(\mathbf{x}, y, z, t, w) &= \pm 3 \sqrt{\frac{3a\_2 a\_3 r\_1(t, w)}{9a\_1 a\_2 r\_2(t, w) r\_3(t, w) - 2}} \pm 4 \sqrt{\frac{3a\_2 a\_3 r\_1(t, w) r\_3(t, w)}{2 + 15a\_1 a\_2 r\_2(t, w) r\_3(t, w)}} \left( \tan(2 \underline{x}\_4(\mathbf{x}, y, z, t, w)) \right) \\ &+ i \underline{i} \underline{\qquad} \end{split} \tag{28}$$

$$\pm i \frac{\pm i}{2\sqrt{a\_1 r\_2(t, w) r\_3(t, w)} (\tan(2\xi\_4(x, y, z, t, w)))} \; , \tag{38}$$

where

$$f\_4(x, y, z, t, w) = a\_1 x + a\_2 y + a\_3 z + 2a\_3 \int\_0^t \frac{r\_1(\tau, w)}{9a\_1 a\_2 r\_3(\tau, w) - 2} d\tau. \tag{39}$$

Obviously, there are several particular solutions for the system (25) with the Riccati equation (19), coming from many different cases. In the above cases we just clarified how far our technique is applicable.

Now, for *q* < ∞, *r* > 0, consider the infinite dimensional neighbourhoods *Kq*(*r*) = {(*w*1, *w*2, ...) ∈ <sup>C</sup><sup>N</sup> : <sup>∑</sup>*α*=<sup>0</sup> <sup>|</sup>*wα*<sup>|</sup> <sup>2</sup>(2N)*q<sup>α</sup>* <sup>&</sup>lt; *<sup>r</sup>*2} of zero in <sup>C</sup><sup>N</sup> [1]. The properties of exponential, hyperbolic, and trigonometric functions yield that there exists a bounded open set **<sup>D</sup>** <sup>⊂</sup> <sup>R</sup><sup>3</sup> <sup>×</sup>R+, *<sup>q</sup>* <sup>&</sup>lt; <sup>∞</sup>,*<sup>r</sup>* <sup>&</sup>gt; 0, such that the solution *u*(*x*, *y*, *z*, *t*, *w*) of Equation (21) and all its derivatives which are involved in Equation (21) are uniformly bounded for (*x*, *y*, *z*, *t*, *w*) ∈ **D** × *Kq*(*r*), continuous with respect to (*x*, *y*, *z*, *t*) ∈ **D** for all *w* ∈ *Kq*(*r*) and analytic with respect to *w* ∈ *Kq*(*r*), for all (*x*, *y*, *z*, *t*) ∈ **D**. From Theorem 4.1.1 in [1], there exists *U*(*x*, *y*, *z*, *t*) ∈ (S)−<sup>1</sup> such that *u*(*x*, *y*, *z*, *t*, *w*) = *U*(*x*, *y*, *z*)(*w*) for all (*x*, *y*, *z*, *t*, *w*) ∈ **D** × *Kq*(*r*) and *U*(*x*, *y*, *z*, *t*) solves Equation (1) in (S)−1. Hence, by applying the inverse Hermite transform to Equations (27), (30), (31), (34), (37), and (38), we obtain the solutions of Equation (1) as follows:

**(I)** Stochastic Travelling Wave Solution of Exponential Type:

$$\begin{split} \mathcal{U}l\_{1}(\mathbf{x},y,z,t) &= \\ \frac{\pm i \sqrt{3a\_{3}R\_{1}(t) \diamond (a\_{1}a\_{2}R\_{2}(t) \diamond R\_{3}(t) - 2)} \diamond (\exp^{\diamond}(\Xi\_{1}(\mathbf{x},y,z,t)) - 1) \pm \sqrt{3a\_{1}a\_{2}a\_{3}R\_{1}(t) \diamond R\_{2}(t)}}{\sqrt{a\_{1}R\_{2}(t) \diamond (a\_{1}a\_{2}R\_{2}(t) \diamond R\_{3}(t) - 2)} (\exp^{\diamond}(\Xi\_{1}(\mathbf{x},y,z,t)) - 1)} \end{split} \tag{40}$$

with

$$\Xi\_1(x, y, z, t) = a\_1 x + a\_2 y + a\_3 z + 2a\_3 \int\_0^t \frac{R\_1(\tau)}{a\_1 a\_2 R\_3(\tau) - 2} d\tau. \tag{41}$$

**(II)** Stochastic Travelling Wave Solutions of Hyperbolic Type:

$$\begin{split} \mathcal{U}\_{2}(\mathbf{x},y,z,t) &= \pm \sqrt{\frac{3a\_{2}a\_{3}R\_{1}(t)\diamond R\_{3}(t)}{R\_{2}(t)\diamond (2-a\_{1}a\_{2}R\_{3}(t))}} \diamond \left(\coth^{\diamond}(\Xi\_{2}(\mathbf{x},y,z,t)) \pm^{\diamond} (\Xi\_{2}(\mathbf{x},y,z,t))\right) \\ &\pm i \frac{\sqrt{3ba\_{2}R\_{1}(t)}}{\sqrt{2R\_{2}(t)\diamond R\_{3}(t)} \diamond \left(\coth^{\diamond}(\Xi\_{2}(\mathbf{x},y,z,t)) \pm^{\diamond} (\Xi\_{2}(\mathbf{x},y,z,t))\right)} , \end{split} \tag{42}$$

$$\begin{split} lL3(\mathbf{x},y,z,t) &= \pm \sqrt{\frac{3a\_2a\_3R\_1(t) \diamond R\_3(t)}{R\_2(t) \diamond (2 - a\_1a\_2R\_3(t))}} \diamond \left(\tanh^{\diamond}(\Xi\_2(\mathbf{x},y,z,t)) \pm i^{\diamond}(\Xi\_2(\mathbf{x},y,z,t))\right) \\ &\pm i \frac{\sqrt{3ba\_2R\_1(t)}}{\sqrt{2R\_2(t) \diamond R\_3(t)} \diamond \left(\tanh^{\diamond}(\Xi\_2(\mathbf{x},y,z,t)) \pm i^{\diamond}(\Xi\_2(\mathbf{x},y,z,t))\right)} ,\end{split} \tag{43}$$

with

$$\Xi\_2(x, y, z, t) = a\_1 x + a\_2 y + a\_3 z - 4a\_3 \int\_0^t \frac{R\_1(\tau)}{4 + a\_1 a\_2 R\_3(\tau)} d\tau. \tag{44}$$

**(III)** Stochastic Travelling Wave Solutions of Trigonometric Type:

$$\begin{split} \mathcal{U}\_{4}(\mathbf{x},\mathbf{y},z,t) = \mathcal{U}\mathfrak{z}(\mathbf{x},\mathbf{y},z,t) &= \pm \sqrt{\frac{3a\_{2}a\_{3}R\_{1}(t)}{1 - a\_{1}a\_{2}R\_{2}(t) \diamond R\_{3}(t)}} \pm \sqrt{\frac{6a\_{2}a\_{3}R\_{1}(t) \diamond R\_{3}(t)}{1 + 2a\_{1}a\_{2}R\_{2}(t) \diamond R\_{3}(t)}} \\ &\quad \diamond \left(\sec^{\diamond}(\Xi\_{3}(\mathbf{x},\mathbf{y},z,t)) \diamond \csc^{\diamond}(\Xi\_{3}(\mathbf{x},\mathbf{y},z,t))\right) \,, \end{split} \tag{45}$$

with

$$\Xi\_3(\mathbf{x}, y, z, t) = a\_1 \mathbf{x} + a\_2 y + a\_3 z - a\_3 \int\_0^t \frac{R\_1(\tau)}{1 + 2a\_1 a\_2 R\_3(\tau)} d\tau. \tag{46}$$

$$\begin{split} lL\_{b}(x,y,z,t) &= \pm 3\sqrt{\frac{3a\_{2}a\_{3}R\_{1}(t)}{9a\_{1}a\_{2}R\_{2}(t)\diamond R\_{3}(t) - 2}} \pm 4\sqrt{\frac{3a\_{2}a\_{3}R\_{1}(t)\diamond R\_{3}(t)}{2 + 15a\_{1}a\_{2}R\_{2}(t)\diamond R\_{3}(t)}} \diamond \left(\cot^{o}(2\Xi\_{4}(x,y,z,t))\right) \\ &\pm i\frac{\sqrt{3a\_{3}R\_{1}(t)}}{2\sqrt{a\_{1}R\_{2}(t)\diamond R\_{3}(t)}\diamond \left(\cot^{o}(2\Xi\_{4}(x,y,z,t))\right)} .\end{split} \tag{47}$$

$$\begin{split} \mathcal{U}\_{7}(\mathbf{x},y,z,t) &= \pm 3\sqrt{\frac{3a\_{2}a\_{3}R\_{1}(t)}{9a\_{1}a\_{2}R\_{2}(t)\diamond R\_{3}(t)-2}} \pm 4\sqrt{\frac{3a\_{2}a\_{3}R\_{1}(t)\diamond R\_{3}(t)}{2+15a\_{1}a\_{2}R\_{2}(t)\diamond R\_{3}(t)}} \diamond \left(\tan^{\diamond}(2\Xi\_{4}(\mathbf{x},y,z,t))\right) \\ &\pm i\frac{\sqrt{3a\_{3}R\_{1}(t)}}{2\sqrt{a\_{1}R\_{2}(t)\diamond R\_{3}(t)}\diamond \left(\tan^{\diamond}(2\Xi\_{4}(\mathbf{x},y,z,t))\right)} ,\end{split} \tag{48}$$

with

$$\Xi\_4(\mathbf{x}, y, z, t) = a\_1 \mathbf{x} + a\_2 \mathbf{y} + a\_3 z + 2a\_3 \int\_0^t \frac{R\_1(\tau)}{9a\_1 a\_2 R\_3(\tau) - 2} d\tau. \tag{49}$$

For the other two forms of the Wick-type, stochastic, (3+1)-dimensional modified BBM equations (2) and (3), we can follow the same technique as presented for the first form (1). Therefore, we just list the stochastic travelling wave solutions for each form. For Equation (2) one obtains the following stochastic travelling wave solution:

**(I)** Stochastic Travelling Wave Solution of Exponential Type:

$$\begin{array}{l} V\_{1}(\mathbf{x},\mathbf{y},\mathbf{z},t) = \\ \hline \pm i\sqrt{\frac{3b\_{1}R\_{4}(t)\diamond(b\_{1}b\_{2}R\mathbf{g}(t)\diamond R\_{6}(t)-2)}{2}\diamond(\exp^{\diamond}(\Lambda\_{1}(\mathbf{x},\mathbf{y},\mathbf{z},t))-1)}\times\sqrt{3b\_{1}b\_{2}b\_{3}R\_{4}(t)\diamond R\mathbf{g}(t)}{2}},\end{array} \tag{50}$$

with

$$
\Lambda\_1(\mathbf{x}, y, z, t) = b\_1 \mathbf{x} + b\_2 \mathbf{y} + b\_3 z + 2b\_3 \int\_0^t \frac{\mathcal{R}\_4(\tau)}{b\_1 b\_2 \mathcal{R}\_6(\tau) - 2} d\tau. \tag{51}
$$

**(II)** Stochastic Travelling Wave Solutions of Hyperbolic Type:

$$V\_2(x, y, z, t) = \pm \sqrt{\frac{3b\_2 b\_3 R\_4(t) \diamond R\_6(t)}{R\_5(t) \diamond (2 - b\_1 b\_2 R\_6(t))}} \diamond \left(\coth^\diamond(\Lambda\_2(x, y, z, t)) \pm^\diamond (\Lambda\_2(x, y, z, t))\right)$$

$$\pm i \frac{\sqrt{3b^\* b\_2 R\_4(t)}}{\sqrt{2R\_5(t) \diamond R\_6(t)} \diamond \left(\coth^\diamond(\Lambda\_2(x, y, z, t)) \pm^\diamond (\Lambda\_2(x, y, z, t))\right)}\,\tag{52}$$

$$V\_3(\mathbf{x}, y, z, t) = \pm \sqrt{\frac{3b\_2b\_3R\_4(t) \diamond R\_6(t)}{R\_5(t) \diamond (2 - b\_1b\_2R\_6(t))}} \diamond \left(\tanh^\diamond(\Lambda\_2(\mathbf{x}, y, z, t)) \pm i^\diamond (\Lambda\_2(\mathbf{x}, y, z, t))\right)$$

$$\pm i \frac{\sqrt{3b^\*b\_2R\_4(t)}}{\sqrt{2R\_5(t) \diamond R\_6(t)} \diamond \left(\tanh^\diamond(\Lambda\_2(\mathbf{x}, y, z, t)) \pm i^\diamond (\Lambda\_2(\mathbf{x}, y, z, t))\right)} \,\,, \tag{53}$$

with

$$
\Lambda\_2(\mathbf{x}, y, z, t) = b\_1 \mathbf{x} + b\_2 y + b\_3 z - 4b\_3 \int\_0^t \frac{\mathcal{R}\_4(\tau)}{4 + b\_1 b\_2 \mathcal{R}\_6(\tau)} d\tau. \tag{54}
$$

**(III)** Stochastic Travelling Wave Solutions of Trigonometric Type:

$$V\_4(\mathbf{x}, y, z, t) = V\_5(\mathbf{x}, y, z, t) = \pm \sqrt{\frac{3b\_2 b\_3 R\_4(t)}{1 - b\_1 b\_2 R\_5(t) \diamond R\_6(t)}} \pm \sqrt{\frac{6b\_2 b\_3 R\_4(t) \diamond R\_6(t)}{1 + 2b\_1 b\_2 R\_5(t) \diamond R\_6(t)}}$$

$$\text{o } (\sec^\diamond(\Lambda\_3(\mathbf{x}, y, z, t)) \diamond \csc^\diamond(\Lambda\_3(\mathbf{x}, y, z, t))) \text{ ,} \tag{55}$$

with

$$
\Lambda\_3(\mathbf{x}, y, z, t) = b\_1 \mathbf{x} + b\_2 y + a\_3 z - b\_3 \int\_0^t \frac{R\_4(\tau)}{1 + 2b\_1 b\_2 R\_6(\tau)} d\tau. \tag{56}
$$

$$V\_{6}(\mathbf{x},y,z,t) = \pm 3\sqrt{\frac{3b\_{2}b\_{3}R\_{4}(t)}{9b\_{1}b\_{2}R\_{5}(t)\diamond R\_{6}(t) - 2}} \pm 4\sqrt{\frac{3b\_{2}b\_{3}R\_{4}(t)\diamond R\_{6}(t)}{2 + 15b\_{1}b\_{2}R\_{5}(t)\diamond R\_{6}(t)}}\,\mathrm{(\cot^{\diamond}(2\Lambda\_{4}(\mathbf{x},y,z,t)) )}$$

$$\pm i\frac{\sqrt{3b\_{3}R\_{4}(t)}}{2\sqrt{b\_{1}R\_{5}(t)\diamond R\_{6}(t)}\diamond(\cot^{\diamond}(2\Lambda\_{4}(\mathbf{x},y,z,t)))}\,\mathrm{'} \tag{57}$$

$$V\_7(x, y, z, t) = \pm 3\sqrt{\frac{3b\_2b\_3R\_4(t)}{9b\_1b\_2R\_5(t)\diamond R\_6(t) - 2}} \pm 4\sqrt{\frac{3b\_2b\_3R\_4(t)\diamond R\_6(t)}{2 + 15b\_1b\_2R\_5(t)\diamond R\_6(t)}}\diamond(\tan^\diamond(2\Lambda\_4(x, y, z, t)))$$

$$\pm i\frac{\sqrt{3b\_3R\_4(t)}}{2\sqrt{b\_1R\_5(t)\diamond R\_6(t)}\diamond(\tan^\diamond(2\Lambda\_4(x, y, z, t)))}\,'\tag{58}$$

with

$$
\Lambda\_4(\mathbf{x}, y, z, t) = b\_1 \mathbf{x} + b\_2 y + b\_3 z + 2b\_3 \int\_0^t \frac{\mathcal{R}\_4(\tau)}{9b\_1 b\_2 \mathcal{R}\_6(\tau) - 2} d\tau,\tag{59}
$$

where *bi* (*i* = 1, 2, 3) and *b*<sup>∗</sup> are arbitrary constants satisfying *bib* = 0.

For Equation (3) one obtains the following stochastic travelling wave solution:

**(I)** Stochastic Travelling Wave Solution of Exponential Type:

$$\begin{array}{l} \mathcal{W}\_{1}(\mathbf{x}, y, z, t) = \\\\ \frac{\pm i \sqrt{3c\_{1}R\_{\mathcal{T}}(t) \circ (c\_{1}c\_{2}R\_{\theta}(t) \text{ diamond}R\_{\theta}(t) - 2)} \circ (\exp^{\diamond}(\Lambda\_{1}(\mathbf{x}, y, z, t)) - 1) \pm \sqrt{3c\_{1}^{2}c\_{2}R\_{\mathcal{T}}(t) \circ R\_{\theta}(t)}}{\sqrt{c\_{1}R\_{\theta}(t) \circ (c\_{1}c\_{2}R\_{\theta}(t) \circ R\_{\theta}(t) - 2)} (\exp^{\diamond}(\Lambda\_{1}(\mathbf{x}, y, z, t)) - 1)} \end{array} \tag{60}$$

with

$$
\Delta\_1(\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{t}) = c\_1 \mathbf{x} + c\_2 \mathbf{y} + c\_3 \mathbf{z} + 2c\_1 \int\_0^t \frac{R\_7(\mathbf{r})}{c\_1 c\_2 R \rho(\mathbf{r}) - 2} d\mathbf{r} \,. \tag{61}
$$

**(II)** Stochastic Travelling Wave Solutions of Hyperbolic Type:

$$\begin{split} \mathcal{W}\_{2}(\mathbf{x},y,z,t) &= \pm \sqrt{\frac{3c\_{1}c\_{2}R\_{7}(t)\diamond R\_{9}(t)}{R\_{8}(t)\diamond (2 - c\_{1}c\_{2}R\_{9}(t))}} \diamond \left(\coth^{\diamond}(\Lambda\_{2}(\mathbf{x},y,z,t)) \pm^{\diamond} (\Lambda\_{2}(\mathbf{x},y,z,t))\right) \\ &\pm i \frac{\sqrt{3b^{\*\*}c\_{2}R\_{7}(t)}}{\sqrt{2R\_{8}(t)\diamond R\_{9}(t)\diamond (\coth^{\diamond}(\Lambda\_{2}(\mathbf{x},y,z,t)) \pm^{\diamond} (\Lambda\_{2}(\mathbf{x},y,z,t)))}}, \end{split} \tag{62}$$

*Axioms* **2019**, *8*, 134

$$\mathcal{W}\_{8}(x,y,z,t) = \pm \sqrt{\frac{3c\_{1}c\_{2}R\_{7}(t) \diamond R\_{9}(t)}{R\_{8}(t) \diamond (2 - c\_{1}c\_{2}R\_{9}(t))}} \diamond \left(\tanh^{\diamond}(\Delta\_{2}(x,y,z,t)) \pm i \, ^{\diamond} (\Delta\_{2}(x,y,z,t))\right)$$

$$\pm i \, \frac{\sqrt{3b^{\*\*} \ast c\_{2}R\_{7}(t)}}{\sqrt{2R\_{8}(t) \diamond R\_{9}(t)} \diamond (\tanh^{\diamond}(\Delta\_{2}(x,y,z,t)) \pm i \, ^{\diamond} (\Delta\_{2}(x,y,z,t)))} \, ^{\prime} \tag{63}$$

with

$$
\Delta\_2(\mathbf{x}, y, z, t) = c\_1 \mathbf{x} + c\_2 \mathbf{y} + c\_3 z - 4c\_1 \int\_0^t \frac{R\_7(\tau)}{4 + c\_1 c\_2 R\_9(\tau)} d\tau. \tag{64}
$$

**(III)** Stochastic Travelling Wave Solutions of Trigonometric Type:

$$\mathcal{W}\_{4}(\mathbf{x},y,z,t) = \mathcal{W}5(\mathbf{x},y,z,t) = \pm \sqrt{\frac{3c\_{1}c\_{2}R\_{7}(t)}{1 - c\_{1}c\_{2}R\_{8}(t) \diamond R\_{9}(t)}} \pm \sqrt{\frac{6c\_{1}c\_{2}R\_{7}(t) \diamond R\_{9}(t)}{1 + 2c\_{1}c\_{2}R\_{8}(t) \diamond R\_{9}(t)}}$$

$$\diamond \left(\sec^{\diamond}(\Lambda\_{3}(\mathbf{x},y,z,t)) \diamond \csc^{\diamond}(\Lambda\_{3}(\mathbf{x},y,z,t))\right),\tag{65}$$

with

$$\Lambda\_3(\mathbf{x}, \mathbf{y}, \mathbf{z}, \mathbf{t}) = c\_1 \mathbf{x} + c\_2 \mathbf{y} + c\_3 \mathbf{z} - c\_1 \int\_0^t \frac{R\_7(\tau)}{1 + 2c\_1 c\_2 R\_9(\tau)} d\tau. \tag{66}$$

$$\begin{split} W\_{\theta}(\mathbf{x},y,z,t) &= \pm 3\sqrt{\frac{3c\_{1}c\_{2}R\_{7}(t)}{9c\_{1}c\_{2}R\_{8}(t)\diamond R\_{9}(t)-2}} \pm 4\sqrt{\frac{3c\_{1}c\_{2}R\_{7}(t)\diamond R\_{9}(t)}{2+15c\_{1}c\_{2}R\_{8}(t)\diamond R\_{9}(t)}} \operatorname{oc}^{\diamond}(2\Delta\_{4}(\mathbf{x},y,z,t)) \\ &\pm i\frac{\sqrt{3c\_{1}R\_{7}(t)}}{2\sqrt{c\_{1}R\_{8}(t)\diamond R\_{9}(t)}\diamond(\cot^{\diamond}(2\Delta\_{4}(\mathbf{x},y,z,t)))} \,' \end{split} \tag{67}$$

$$\mathcal{W}(\mathbf{x},y,z,t) = \pm 3\sqrt{\frac{3c\_1c\_2R\_7(t)}{9c\_1c\_2R\_8(t)\diamond R\_9(t) - 2}} \pm 4\sqrt{\frac{3c\_1c\_2R\_7(t)\diamond R\_9(t)}{2 + 15c\_1c\_2R\_8(t)\diamond R\_9(t)}}\diamond(\tan^\diamond(2\Delta\_4(\mathbf{x},y,z,t)))$$

$$\pm i\frac{\sqrt{3c\_1R\_7(t)}}{2\sqrt{c\_1R\_8(t)\diamond R\_9(t)}\diamond(\tan^\diamond(2\Delta\_4(\mathbf{x},y,z,t)))}},\tag{68}$$

with

$$
\Delta\_4(x, y, z, t) = c\_1 x + c\_2 y + c\_3 z + 2c\_1 \int\_0^t \frac{R\_7(\tau)}{9c\_1 c\_2 R\_9(\tau) - 2} d\tau,\tag{69}
$$

where *ci* (*i* = 1, 2, 3) and *b*∗∗ are arbitrary constants satisfying *cib* = 0.

### **5. Example**

In this section, we provide a specific application example to demonstrate the effectiveness of our results and to justify the real contribution of these results. We focus our attention on Equation (1). Concerning the other two equations, Equations (2) and (3), the procedure is similar. We observe that the solutions of Equation (1) are strongly depend on the shape of the given functions *R*1(*t*) and *R*2(*t*). So, for dissimilar forms of *R*1(*t*) and *R*2(*t*), we can find dissimilar solutions of Equation (1) which come from Equations (70)–(78). We illustrate this by giving the following example.

Assume that *R*2(*t*) = *δ*1*R*1(*t*), *R*3(*t*) = *δ*2*R*1(*t*) and *R*1(*t*) = *f*(*t*) + *δ*3*Wt*, where *δ*1,*δ*2, and *δ*<sup>3</sup> are arbitrary constants, *f*(*t*) is a bounded measurable function on R+, and *Wt* is the Gaussian white noise, which is the time derivative (in the strong sense in (S)−1) of the Brownian motion *Bt*. The Hermite transform of *Wt* is given by *<sup>W</sup>t*(*w*) = <sup>∑</sup><sup>∞</sup> *<sup>i</sup>*=<sup>1</sup> *wi t* <sup>0</sup> *ηi*(*τ*)*dτ* [1]. Using the definition of *Wt*(*w*), Equations (70)–(78) yield the white noise functional solution of Equation (1) as follows:

*Axioms* **2019**, *8*, 134

$$\begin{split} \mathcal{U}\_{\mathcal{W}\_{1}}(\mathbf{x},\mathbf{y},z,t) &= \\ \frac{\pm i \sqrt{3a\_{3}(a\_{1}a\_{2}\delta\_{1}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t})^{2}-2)} (\exp(\Omega\_{1}(\mathbf{x},\mathbf{y},z,t))-1) \pm \sqrt{3a\_{1}a\_{2}a\_{3}\delta\_{1}}(f(t)+\delta\_{3}\mathcal{W}\_{t})}{\sqrt{a\_{1}\delta\_{1}(a\_{1}a\_{2}\delta\_{1}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t})^{2}-2)}(\exp(\Omega\_{1}(\mathbf{x},\mathbf{y},z,t))-1)}, \end{split} \tag{70}$$

with

$$
\Omega\_1(\mathbf{x}, y, z, t) = a\_1 \mathbf{x} + a\_2 y + a\_3 z + 2a\_3 \int\_0^t \frac{f(\tau) + \delta\_3 \mathcal{W}\_{\tau}}{a\_1 a\_2 \delta\_2 (f(\tau) + \delta\_3 \mathcal{W}\_{\tau}) - 2} d\tau,\tag{71}
$$

$$\begin{split} \mathcal{U}\_{\mathcal{W}\_{2}}(\mathbf{x},\mathbf{y},\mathbf{z},t) &= \pm \sqrt{\frac{3a\_{2}a\_{3}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t})}{\delta\_{1}(2-a\_{1}a\_{2}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t}))}} \left(\coth(\Omega\_{2}(\mathbf{x},\mathbf{y},\mathbf{z},t)) \pm (\Omega\_{2}(\mathbf{x},\mathbf{y},\mathbf{z},t))\right) \\ &\pm i \frac{\sqrt{3ba\_{2}}}{\sqrt{2\delta\_{1}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t})\left(\coth(\Omega\_{2}(\mathbf{x},\mathbf{y},\mathbf{z},t)) \pm (\Omega\_{2}(\mathbf{x},\mathbf{y},\mathbf{z},t))\right)}}, \end{split} \tag{72}$$

$$\begin{split} \mathcal{U}\_{\mathcal{W}\_{3}}(\mathbf{x},\mathbf{y},\mathbf{z},t) &= \pm \sqrt{\frac{3a\_{2}a\_{3}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t})}{\delta\_{1}(2-a\_{1}a\_{2}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t}))}} \left(\tanh(\Omega\_{2}(\mathbf{x},\mathbf{y},\mathbf{z},t)) \pm i \left(\Omega\_{2}(\mathbf{x},\mathbf{y},\mathbf{z},t)\right)\right) \\ &\pm i \frac{\sqrt{3ba\_{2}}}{\sqrt{2\delta\_{1}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t})} \left(\tanh(\Omega\_{2}(\mathbf{x},\mathbf{y},\mathbf{z},t)) \pm i \left(\Omega\_{2}(\mathbf{x},\mathbf{y},\mathbf{z},t)\right)\right)}, \end{split} \tag{73}$$

with

$$
\Omega\_2(\mathbf{x}, y, z, t) = a\_1 \mathbf{x} + a\_2 \mathbf{y} + a\_3 z - 4a\_3 \int\_0^t \frac{f(\tau) + \delta\_3 \mathcal{W}\_{\tau}}{4 + a\_1 a\_2 \delta\_2 (f(\tau) + \delta\_3 \mathcal{W}\_{\tau})} d\tau,\tag{74}
$$

$$\mathcal{U}l\_{\mathcal{W}\_{4}}(\mathbf{x},\mathbf{y},\mathbf{z},t) = \mathcal{U}\_{\mathcal{W}\_{5}}(\mathbf{x},\mathbf{y},\mathbf{z},t) = \pm \sqrt{\frac{3a\_{2}a\_{3}(f(t) + \delta\_{3}\mathcal{W}\_{t})}{1 - a\_{1}a\_{2}\delta\_{1}\delta\_{2}(f(t) + \delta\_{3}\mathcal{W}\_{t})^{2}}} \pm (f(t) + \delta\_{3}\mathcal{W}\_{t})$$

$$\times \sqrt{\frac{6a\_{2}a\_{3}\delta\_{2}}{1 + 2a\_{1}a\_{2}\delta\_{1}\delta\_{2}(f(t) + \delta\_{3}\mathcal{W}\_{t})^{2}}} \left(\mathrm{sec}(\Omega\_{3}(\mathbf{x},\mathbf{y},\mathbf{z},t))\csc(\Omega\_{3}(\mathbf{x},\mathbf{y},\mathbf{z},t))\right), \tag{75}$$

with

$$
\Omega\_3(\mathbf{x}, y, z, t) = a\_1 \mathbf{x} + a\_2 \mathbf{y} + a\_3 z - a\_3 \int\_0^t \frac{f(\mathbf{r}) + \delta\_3 \mathcal{W}\_{\mathbf{r}}}{1 + 2a\_1 a\_2 \delta\_2 (f(\mathbf{r}) + \delta\_3 \mathcal{W}\_{\mathbf{r}})} d\mathbf{r} \,\,,\tag{76}
$$

$$\begin{split} l\Omega\_{W\_{6}}(\mathbf{x},y,z,t) &= \pm 3\sqrt{\frac{3a\_{2}a\_{3}\zeta(f(t)+\delta\_{3}\mathcal{W}\_{t})}{9a\_{1}a\_{2}\delta\_{1}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t})^{2}-2}} \pm 4(f(t)+\delta\_{3}\mathcal{W}\_{t}) \\ &\times \sqrt{\frac{3a\_{2}a\_{3}\delta\_{2}}{2+15a\_{1}a\_{2}\delta\_{1}\delta\_{2}(f(t)+\delta\_{3}\mathcal{W}\_{t})^{2}}} \left(\cot(2\Omega\_{4}(\mathbf{x},y,z,t))\right) \\ &\pm i\frac{\sqrt{3a\_{3}}}{2\sqrt{a\_{1}\delta\_{1}\delta\_{2}(\cot(2\Omega\_{4}(\mathbf{x},y,z,t)))}} \,, \tag{77} \end{split}$$

$$\begin{split} l\Omega\_{W\_{\delta}}(x,y,z,t) &= \pm 3\sqrt{\frac{3a\_{2}a\_{3}(f(t) + \delta\_{3}\mathcal{W}\_{t})}{9a\_{1}a\_{2}\delta\_{1}\delta\_{2}(f(t) + \delta\_{3}\mathcal{W}\_{t})^{2} - 2}} \pm 4(f(t) + \delta\_{3}\mathcal{W}\_{t}) \\ &\times \sqrt{\frac{3a\_{2}a\_{3}\delta\_{2}}{2 + 15a\_{1}a\_{2}\delta\_{1}\delta\_{2}(f(t) + \delta\_{3}\mathcal{W}\_{t})^{2}}} \left(\tan(2\Omega\_{4}(x,y,z,t))\right) \\ &\pm i\frac{\sqrt{3a\_{3}}}{2\sqrt{a\_{1}\delta\_{1}\delta\_{2}(\tan(2\Omega\_{4}(x,y,z,t)))}},\end{split} \tag{78}$$

with

$$
\Omega\_4(x, y, z, t) = a\_1 x + a\_2 y + a\_3 z + 2a\_3 \int\_0^t \frac{f(\tau) + \delta\_3 \mathcal{W}\_{\tau}}{9a\_1 a\_2 \delta\_2 (f(\tau) + \delta\_3 \mathcal{W}\_{\tau}) - 2} d\tau \,. \tag{79}
$$

### **6. Conclusions**

Due to the fact that the stochastic models are more realistic than the deterministic models, we concentrated our study in this paper on the Wick-type, stochastic, (3+1)-dimensional modified BBM equations. Besides that, we investigated and solve the deterministic, (3+1)-dimensional modified BBM equations. In this paper, we set up a new and general version of the modified tanh–coth method to deal with the nonlinear multi dimensional PDEs. By using this generalisation of the modified tanh–coth method, Hermite transform, and white noise theory, we produced a new set of exact travelling wave solutions for the variable coefficients and (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. In [7], Wazwaz has solved the deterministic, (3+1)-dimensional modified BBM equations with constant coefficients, So, our results for this model are more general than the results obtained by him. With the aid of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type, stochastic, (3+1)-dimensional modified BBM equations. Furthermore, we showed by an example how the stochastic solutions can be given as white noise functional solutions. Note that, the schema proposed in this paper can be used for solving several nonlinear evolution equations in mathematical physics, both Wick-type stochastic and deterministic. Moreover, the Riccati equation (19) has different solutions if we chose different values of *α*0, *α*1, and *α*2. Therefore, we can find many other solutions of the Wick-type stochastic and deterministic (3+1)-dimensional modified BBM equations.

The PDE of Benjamin et al. [2] is now often called the BBM equation, although it is also known as the regularised long wave (RLW) equation. Morrison et al. proposed the one-dimensional PDE, as an equally valid and accurate model for the same wave phenomena simulated by the KdV and RLW equations [40]. Random waves are an important subject of random PDEs. In essence, to investigate the exact solutions of random Benjamin–Bona–Mahony equation, we restricted our attention to consider this problem in a white noise environment [38]. The propagation of nonlinear wave in systems with polarity symmetry can be described by the (3+1)-dimensional modified Benjamin–Bona–Mahony Equation (7). If the problem is considered in a non-Gaussian stochastic environment, we can get non-Gaussian, stochastic, (2+1)-dimensional coupled KdV equation. Obviously, the planner which we have proposed in this paper can be also applied to other non-linear PDEs in mathematical physics such as KdV-Burgers, modified KdV-Burgers, Zhiber- Shabat and Benjamin–Bona–Mahony equations. We observe that the F-expansion method we used has many other particular solutions; this in turn gives many other exact solutions for the considered stochastic, (3+1)-dimensional modified Benjamin–Bona–Mahony equations. Additionally, in this work, we discussed the solutions of SPDEs driven by non-Gaussian white noise; this discussion is less detailed than the Gaussian discussion but more general, because it deals with the dual pairing generated by integration with respect to a non-Gaussian measure. Furthermore, in future work, we will discuss the solutions of SPDEs driven by non-Gaussian white noise to get exact stochastic solutions of the non-Gaussian, stochastic, (3+1)-dimensional modified Benjamin–Bona–Mahony equations; we only considered this problem in a non-Gaussian white noise environment; that is, we investigated the variable coefficients of stochastic, (3+1)-dimensional modified Benjamin–Bona–Mahony equations. For this aim, we developed a non-Gaussian Wick calculus based on the theory of hyper-complex systems to get exact travelling wave solutions of (3+1)-dimensional modified Benjamin–Bona–Mahony equations and non-Gaussian white noise functional solutions of Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony equations.

**Author Contributions:** The authors have equally contributed to the research.

**Funding:** This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

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

### **References**


© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Best Proximity Points for Monotone Relatively Nonexpansive Mappings in Ordered Banach Spaces**

### **Karim Chaira 1,†, Mustapha Kabil 1,†, Abdessamad Kamouss 1,\*,† and Samih Lazaiz 2,†**


Received: 23 September 2019; Accepted: 25 October 2019; Published: 1 November 2019

**Abstract:** In this paper, we give sufficient conditions to ensure the existence of the best proximity point of monotone relatively nonexpansive mappings defined on partially ordered Banach spaces. An example is given to illustrate our results.

**Keywords:** best proximity point; fixed point; monotone mappings; relatively cyclic nonexpansive mappings; partially ordered Banach spaces

### **1. Introduction**

Let *X* be a Banach space and (*A*, *B*) a pair of nonempty subsets of *X*. A cyclic mapping on *A* ∪ *B* is a mapping *<sup>T</sup>* : *<sup>A</sup>* <sup>∪</sup> *<sup>B</sup>* <sup>→</sup> *<sup>A</sup>* <sup>∪</sup> *<sup>B</sup>* such that *<sup>T</sup>*(*A*) <sup>⊆</sup> *<sup>B</sup>* and *<sup>T</sup>*(*B*) <sup>⊆</sup> *<sup>A</sup>*. In case *<sup>A</sup>* <sup>∩</sup> *<sup>B</sup>* <sup>=</sup> <sup>∅</sup>, *<sup>T</sup>* does not possess a fixed point, that is, a solution to the equation *Tx* = *x*. Therefore, one can consider the following minimization problem:

$$\iota(P) \colon \begin{cases} \text{find } (\mathfrak{x}, \mathfrak{y}) \in A \times B \text{ such that} \\ \|\mathfrak{x} - T\mathfrak{x}\| = \|\mathfrak{y} - T\mathfrak{y}\| = \text{dist}(A, B). \end{cases}$$

A point *x* ∈ *A* ∪ *B* is a best proximity point of *T* if *x* is a solution of the minimization problem (*P*). The best proximity point notion can be seen as a generalization of fixed point notion since most fixed point theorems can be derived as corollaries of best proximity point theorems.

The first significant result of best proximity points was studied in [1], using the proximal normal structure, the authors proved that every cyclic relatively nonexpansive mapping from *A* ∪ *B* to itself has a best proximity point provided that *A* and *B* are weakly compact and convex. Furthermore, we find in [2] a similar result without invoking Zorn's lemma, i.e., without proximal normal structure. Recently, Chaira and Lazaiz [3] gave an extension of this last result in modular spaces. For a recent account of the theory we refer the reader to [4–6]. We can also find in ([7], pp. 27–31) an application of a best proximity point theorem to a system of differential equations.

On the other hand, the combination of metric fixed point theory and order theory allows Ran and Reurings in [8] to give a Banach Contraction Principle in partially ordered metric spaces. As consequence, they solved a matrix equation. Nieto and Rodríguez-López [9], extended the Ran–Reurings theorem in order to obtain a periodic solution for a first-order ordinary differential equation with periodic boundary conditions.

Recently, many authors studied the existence of fixed points of monotone nonexpansive mappings defined on partially ordered Banach spaces (see for example [10–15]). Recall that a self mapping *T* on *X* is said to be monotone nonexpansive if *T* is monotone and *Tx* − *Ty*≤*x* − *y*, for every comparable elements *x* and *y*. We should mention that monotone nonexpansive mappings may not be continuous. The interested reader can consult the book of Carl and Heikkilä [16] for many applications of fixed point results of monotone mappings.

In this work, motivated by the recent study of a fixed point for monotone mappings, we investigate the existence of the best proximity point of monotone relatively nonexpansive mappings in partially ordered Banach spaces.

### **2. Preliminaries and Basic Results**

Let (*X*, .) be a Banach space endowed with a partial order . Throughout, we assume that the order intervals are closed and convex. Recall that an order interval is any of the subsets

$$\{a, \rightarrow\} = \{\mathbf{x} \in X; a \preceq \mathbf{x}\} \quad , \quad (\leftarrow, a] = \{\mathbf{x} \in X; \mathbf{x} \preceq a\}$$

for any *a* ∈ *X*. As a direct consequence of this, the subset

$$\{a, b\} = \{\mathbf{x} \in \mathcal{X}; a \preceq \mathbf{x} \preceq b\} = [a, \to) \cap (\leftarrow, b]\_{\prec}$$

is also closed and convex for any *a*, *b* ∈ *X*.

We will say that *x*, *y* ∈ *X* are comparable whenever *x y* or *y x*. The linear structure of *X* is assumed to be compatible with the order structure in the following sense:


Let us recall the definition of a uniformly convex Banach space.

**Definition 1.** *Let* (*X*, .) *be a Banach space. We say that X is uniformly convex (in short, UC) if for every -* > 0 *we have δ*(*-*) > 0 *such that*

$$\delta(\epsilon) = \inf \left\{ 1 - \left\| \frac{\mathbf{x} + \mathbf{y}}{2} \right\| ; \left\| \mathbf{x} \right\| \le 1 ; \left\| \mathbf{y} \right\| \le 1 ; \left\| \mathbf{x} - \mathbf{y} \right\| \ge \epsilon \right\}. \epsilon$$

*The function δ is known as the modulus of uniform convexity of X. Note that any UC Banach space is reflexive.*

A sequence {*xn*}*n*∈<sup>N</sup> in a partially ordered set (*X*, ) is said to be


The following technical lemmas will be useful to establish the main results.

**Lemma 1.** *Let X be a Banach space endowed with a partial order . Assume that* {*xn*} *and* {*yn*} *are two sequences on X which are weakly convergent to x and* ¯ *y respectively and x* ¯ *<sup>n</sup> yn for any n* <sup>∈</sup> <sup>N</sup>*, then*

*x*¯ *y*¯.

**Proof.** Note that the positive sequence {*yn* − *xn*}*<sup>n</sup>* converges weakly to *y*¯ − *x*¯. Since closed convex subsets are also weakly closed, the positive cone is weakly closed and so we conclude that *y*¯ − *x*¯ is positive.

**Lemma 2.** *[17] Let* {*xn*} *be a bounded monotone sequence in X, and assume that X is reflexive. Then* {*xn*} *is weakly convergent.*

**Lemma 3.** *[18] Let C be a nonempty closed convex subset of a UC Banach space* (*X*, .)*. Let τ* : *C* → [0, ∞) *be a type function, i.e., there exists a bounded sequence* {*xn*} ∈ *X such that*

$$\tau\left(\mathbf{x}\right) = \limsup\_{n \to \infty} ||\mathbf{x}\_n - \mathbf{x}||\_{\prime}$$

*for every x* ∈ *C. Then τ has a unique minimum point z* ∈ *C such that*

$$\tau\left(z\right) = \inf\left\{\tau\left(\mathbf{x}\right); \mathbf{x} \in \mathbb{C}\right\} = \tau\_0.$$

*Moreover, if* {*zn*} *is a minimizing sequence in C, i.e.,* lim*n*→<sup>∞</sup> *<sup>τ</sup>*(*zn*) = *<sup>τ</sup>*0*, then* {*zn*} *converges strongly to z.*

The norm . of *X* is said to be monotone if

$$\forall \boldsymbol{u} \preceq \boldsymbol{v} \preceq \boldsymbol{w} \quad \text{implies} \quad \max \left\{ \left\| \boldsymbol{w} - \boldsymbol{v} \right\|, \left\| \boldsymbol{v} - \boldsymbol{u} \right\| \right\} \leq \left\| \boldsymbol{w} - \boldsymbol{u} \right\|, \boldsymbol{\epsilon} \right\}$$

for any *u*, *v*, *w* ∈ *X*. If the norm is monotone and {*xn*} is monotone increasing (respectively, decreasing), then the sequence {*xn* − *y*} is decreasing for any *y* such that *xn y* (respectively, *y xn* ), for any *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. In this case,

$$\liminf\_{n \to \infty} \left\| \mathbf{x}\_n - y \right\| = \lim\_{n \to \infty} \left\| \mathbf{x}\_n - y \right\| = \inf\_{n \in \mathbb{N}} \left\| \mathbf{x}\_n - y \right\|.$$

Recall that a mapping *T* : *X* → *X* is said to be


We conclude this section by extending the concept of relatively cyclic nonexpansive mapping to monotone relatively cyclic nonexpansive mapping as follows:

**Definition 2.** *Let* (*X*, ., ) *be a Banach space endowed with a partially order and* (*A*, *B*) *a pair of nonempty subset of X. The mapping T* : *A* ∪ *B* → *A* ∪ *B is said to be monotone increasing (respectively decreasing) relatively cyclic nonexpansive if*


### **3. Main Result**

Throughout we assumed that (*X*, ., ) is a Banach space endowed with a partial order for which order intervals are convex and closed and the linear structure of *X* is assumed to be compatible with the order structure.

The following result gives sufficient conditions to obtain a fixed point theorem for a monotone increasing relatively cyclic nonexpansive mapping.

**Theorem 1.** *Let* (*A*, *B*) *be a nonempty bounded closed convex pair in a partially ordered Banach space* (*X*, ., )*. Assume that* (*X*, .) *is UC. Let T* : *A* ∪ *B* → *A* ∪ *B be a monotone increasing relatively cyclic nonexpansive mapping such that x*<sup>0</sup> *Tx*<sup>0</sup> *for some x*<sup>0</sup> ∈ *A, then A* ∩ *B* = ∅ *and there exists a*<sup>∗</sup> ∈ *A* ∩ *B such that Ta*∗ = *a*∗*.*

**Proof.** We assume that *x*<sup>0</sup> *Tx*<sup>0</sup> and we define the sequence {*xn*} by *xn*+<sup>1</sup> = *Txn* for all *n* ≥ 0. By using the monotonicity of *T* we get

$$\mathbf{x}\_0 \preceq \mathbf{x}\_1 \preceq \cdots \preceq \mathbf{x}\_n \preceq \mathbf{x}\_{n+1} \preceq \cdots \prec \cdots$$

Since *A* and *B* are bounded and closed, the sequence {*xn*} is bounded increasing in the reflexive space *X*. By Lemma 2,

$$
\mathfrak{x}\_{2n} \stackrel{\underline{w}}{\longrightarrow} \mathfrak{x}\_1 \in A \quad \text{and} \quad \mathfrak{x}\_{2n+1} \stackrel{\underline{w}}{\longrightarrow} \mathfrak{x}\_2 \in B.
$$

By uniqueness of the weak limit, *x*¯ = *x*¯1 = *x*¯2. We claim that *A* ∩ *B* = ∅.

Let *<sup>K</sup>* <sup>=</sup> {*<sup>x</sup>* <sup>∈</sup> *<sup>A</sup>* <sup>∩</sup> *<sup>B</sup>*, *xn <sup>x</sup>* for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>}. It is clear that *<sup>K</sup>* is nonempty, closed and convex set. Since {*xn*} is a bounded sequence in *X*, we can define the type function as follows

$$\pi \left( \mathfrak{x} \right) = \limsup\_{n \to \infty} \| |\mathfrak{x}\_n - \mathfrak{x}| \|\_{\prime}$$

for any *x* ∈ *K*. From Lemma 3, it follows that there exists a unique *a*<sup>∗</sup> ∈ *K* such that

$$\pi\left(a^\*\right) = \inf\_{x \in K} \pi\left(x\right) \dots$$

We have

$$\pi \left( \left. T a^\* \right) = \limsup\_{n \to \infty} \left\| x\_n - T a^\* \right\| = \limsup\_{n \to \infty} \left\| T x\_{n-1} - T a^\* \right\|.$$

Since *xn*−<sup>1</sup> *a*<sup>∗</sup> and *T* is monotone relatively cyclic nonexpansive mapping,

$$\tau\left(Ta^\*\right) \le \limsup\_{n \to \infty} \left\| \mathbf{x}\_{n-1} - a^\* \right\| = \tau\left(a^\*\right).$$

Hence, *τ* (*Ta*∗) = *τ* (*a*∗). Thus *Ta*∗ = *a*∗, which completes the proof.

If *B* = *A*, we get the next result for a monotone nonexpansive mapping.

**Corollary 1.** *Let A be a nonempty bounded closed convex set in a partially ordered Banach space* (*X*, ., )*. Let T* : *A* → *A be a monotone increasing nonexpansive mapping. Assume that* (*X*, .) *is UC and there exists x*<sup>0</sup> ∈ *A such that x*<sup>0</sup> *Tx*0*, then there exists a*<sup>∗</sup> ∈ *A such that Ta*<sup>∗</sup> = *a*∗*.*

Now let (A <sup>0</sup> , B <sup>0</sup> ) denotes the pair obtained from (*A*, *B*) upon setting

$$\begin{array}{rcl} \mathcal{A}\_0^\ominus &=& \{ \mathbf{x} \in A; ||\mathbf{x} - \mathbf{y}|| = \text{dist}\,(A, B) \text{ for some } \mathbf{y} \in B \cap [\mathbf{x}, \rightarrow) \} \\ \mathcal{B}\_0^\ominus &=& \{ \mathbf{y} \in B; ||\mathbf{y} - \mathbf{x}|| = \text{dist}\,(A, B) \text{ for some } \mathbf{x} \in A \cap (\leftarrow, \mathbf{y}] \} \end{array}$$

**Lemma 4.** *Let* (*A*, *B*) *be a nonempty bounded closed convex pair in a partially ordered reflexive Banach space* (*X*, ., )*. Then,*

*(i)* A <sup>0</sup> = ∅ *if and only if* B <sup>0</sup> = ∅*; (ii) dist* A <sup>0</sup> , B 0 = *dist*(*A*, *B*)*; (iii)* A <sup>0</sup> , B 0 *is a closed pair; (iv)* A <sup>0</sup> , B 0 *is a convex pair.*

**Proof.** Using the definitions of A <sup>0</sup> and B <sup>0</sup> , we can easily derive (*i*) and (*ii*).

(iii) Let {*xn*}⊂A <sup>0</sup> be a sequence which converges to some *x*¯ in *A*. Then there exists a sequence {*yn*} ⊂ *B* such that

$$||\mathfrak{x}\_{\mathbb{N}} - \mathfrak{y}\_{\mathbb{N}}|| = \text{dist}(A, B) \text{ and } \mathfrak{x}\_{\mathbb{N}} \preceq \mathfrak{y}\_{\mathbb{N}}.$$

Since *B* is closed and bounded in a reflexive Banach space, there exists a subsequence {*yϕ*(*n*)} of {*yn*} such that *yϕ*(*n*) *w y*¯ ∈ *B*. From Lemma 1, it follows that *x*¯ *y*¯. On the other hand,

$$\|\|\mathfrak{x} - \mathfrak{y}\|\| \le \liminf\_{n \to \infty} \|\|x\_{\mathfrak{q}(n)} - y\_{\mathfrak{q}(n)}\|\| = dist(A, B).$$

Therefore, we have *x*¯ ∈ A <sup>0</sup> , and hence, A <sup>0</sup> is closed. By the same arguments we get that B <sup>0</sup> is also closed.

(iv) Let *x* and *x* in A <sup>0</sup> . Then there exist *y* and *y* in *B* such that

$$\begin{cases} \quad \|\mathbf{x} - \mathbf{y}\| \quad = & \text{dist}(A, B) \quad \text{and} \quad \mathbf{x} \preceq \mathbf{y},\\ \quad \|\mathbf{x}' - \mathbf{y}'\| \quad = & \text{dist}(A, B) \quad \text{and} \quad \mathbf{x}' \preceq \mathbf{y}'. \end{cases}$$

By using the fact that the linear structure of *X* is compatible with the order structure, we get for any *t* ∈ [0, 1]

$$\begin{aligned} \|t\mathbf{x} + (1-t)\mathbf{x}' - ty - (1-t)y'\| &= \|t(\mathbf{x}-\mathbf{y}) + (1-t)(\mathbf{x}'-\mathbf{y}')\| \\ &\le \|t\|\mathbf{x} - \mathbf{y}\| + (1-t)\|\mathbf{x}' - \mathbf{y}'\| \\ &= \operatorname{dist}(A,B). \end{aligned}$$

This implies that *tx* + (1 − *t*)*x* ∈ A <sup>0</sup> . It follows that A <sup>0</sup> is convex, as claimed. Similarly we prove that B <sup>0</sup> is also convex.

**Remark 1.** *Note that if T is a monotone decreasing relatively cyclic nonexpansive mapping, we have T* A 0 ⊂ B <sup>0</sup> *and T* B 0 ⊂ A <sup>0</sup> *. Indeed, let x* ∈ A <sup>0</sup> *then there exists y* ∈ *B such that*

$$\|\|\mathbf{x} - \mathbf{y}\|\| = \text{dist}(A, B) \qquad \text{and} \quad \mathbf{x} \preceq \mathbf{y}.$$

*Thus,*

$$||Tx - Ty|| \le ||x - y|| = \text{dist}(A, B) \quad \text{and} \quad Ty \le Tx.$$

*This implies Tx* ∈ B <sup>0</sup> *. Consequently T* A 0 ⊂ B 0 *.*

For the sake of simplicity, we use the following notation

$$\mathcal{A}\_T = \left\{ (\mathbf{x}\_0, \mathbf{x}\_0') \in A \times A; \ \mathbf{x}\_0 \preceq T\mathbf{x}\_0'; \ \|\mathbf{x}\_0 - T\mathbf{x}\_0'\| = dist\left(A, B\right) \right\}.$$

The next lemma gives sufficient conditions such that A*<sup>T</sup>* is nonempty.

**Lemma 5.** *Let* (*A*, *B*) *be a nonempty bounded closed convex pair in a partially ordered Banach space* (*X*, ., ) *such that* A <sup>0</sup> *is nonempty. Let T* : *A* ∪ *B* → *A* ∪ *B be a monotone relatively cyclic nonexpansive mapping. Then* A*<sup>T</sup> is nonempty.*

**Proof.** Suppose that *T* is a monotone decreasing relatively cyclic nonexpansive mapping. Since A <sup>0</sup> = ∅, we can find a *x* <sup>0</sup> in A <sup>0</sup> such that there exists an *y* ∈ *B* ∩ *x* <sup>0</sup>, → satisfying *x* <sup>0</sup> − *y* = *dist*(*A*, *B*).

Since *x* <sup>0</sup> *<sup>y</sup>* and *<sup>T</sup>* is monotone decreasing relatively cyclic nonexpansive mapping, *Ty Tx* 0 and *Tx* <sup>0</sup> − *Ty*≤*x* <sup>0</sup> <sup>−</sup> *<sup>y</sup>* <sup>=</sup> *dist*(*A*, *<sup>B</sup>*), give that *Tx* <sup>0</sup> ∈ B 0 .

Next, for *Tx* <sup>0</sup> there exists an element *x*<sup>0</sup> ∈ A <sup>0</sup> such that

$$\|\mathbf{x}\_0 \preceq T\mathbf{x}\_0^{'} \quad \text{and} \quad \|\mathbf{x}\_0 - T\mathbf{x}\_0^{'}\| = \text{dist}\left(A, B\right).$$

Now, suppose that *T* is a monotone increasing relatively cyclic nonexpansive mapping. Since A <sup>0</sup> = ∅, we can find a *x* in A <sup>0</sup> such that there exists an *y* ∈ B <sup>0</sup> satisfying *x y* and *x* − *y* = *dist*(*A*, *B*).

Since *T* is monotone increasing, *Tx Ty* and

$$\|\|T^2\mathbf{x} - T^2\mathbf{y}\|\| \le \|\|T\mathbf{x} - T\mathbf{y}\|\| \le \|\|\mathbf{x} - \mathbf{y}\|\| = \text{dist}\left(A, B\right).$$

Take *<sup>x</sup>*<sup>0</sup> <sup>=</sup> *<sup>T</sup>*2*<sup>x</sup>* <sup>∈</sup> *<sup>A</sup>* and *<sup>x</sup>* <sup>0</sup> = *Ty* ∈ *A*. We have clearly,

$$\|\mathbf{x}\_0 \preceq T\mathbf{x}\_0^{'} \quad \text{and} \quad \|\mathbf{x}\_0 - T\mathbf{x}\_0^{'}\| = \text{dist}\left(A, B\right).$$

Thus <sup>A</sup>*<sup>T</sup>* <sup>=</sup> <sup>∅</sup>.

In the following, we give a best proximity result for monotone increasing relatively cyclic nonexpansive mapping.

**Theorem 2.** *Let* (*X*, . , ) *be a partially ordered Banach space. Assume that* (*X*, .) *is UC. Let* (*A*, *B*) *be a nonempty bounded closed convex pair in X. Let T* : *A* ∪ *B* → *A* ∪ *B be a monotone increasing relatively cyclic nonexpansive mapping. Assume that T is weakly sequentially continuous, the norm* . *of X is monotone and there exists x*0, *x* 0 ∈ A*<sup>T</sup> such that x*<sup>0</sup> *x* <sup>0</sup> *<sup>T</sup>*2*x*<sup>0</sup> *then there exist <sup>x</sup>*¯ <sup>∈</sup> *<sup>A</sup> and <sup>y</sup>*¯ <sup>∈</sup> *<sup>B</sup> such that x*¯ − *Tx*¯ = *y*¯ − *Ty*¯ = *dist*(*A*, *B*)*.*

**Proof.** Suppose that there exists *x*0, *x* 0 ∈ *A* × *A* such that

$$\left\|\mathbf{x}\_{0} - T\mathbf{x}\_{0}^{'}\right\| = dist\left(A, B\right) \quad \text{and} \quad \mathbf{x}\_{0} \preceq \mathbf{x}\_{0}^{'} \preceq T^{2}\mathbf{x}\_{0}.$$

Let {*xn*} and {*yn*} be two sequences defined as follows:

$$\begin{cases} \mathbf{x}\_n = T^{2n} \mathbf{x}\_0 \\ y\_n = T^{2n+1} \mathbf{x}\_0' \end{cases} \text{ for all } n \in \mathbb{N}.$$

Note that, since *<sup>x</sup>*<sup>0</sup> *Tx* <sup>0</sup> we get *<sup>T</sup>*<sup>2</sup>*nx*<sup>0</sup> *<sup>T</sup>*2*n*+1*<sup>x</sup>* <sup>0</sup> for all *n* ≥ 0, that is, *xn yn* for all *n* ≥ 0. Since *T* is monotone increasing relatively cyclic nonexpansive mapping, we get

$$\left\| \|\mathbf{x}\_{\mathrm{H}} - \mathbf{y}\_{\mathrm{H}}\| \right\| = \left\| \left\| T^{2u} \mathbf{x}\_{0} - T^{2u+1} \mathbf{x}\_{0}' \right\| \right\| \le \left\| \left\| \mathbf{x}\_{0} - T \mathbf{x}\_{0}' \right\| \right\| = \text{dist}\left( \mathbf{A}, \mathbf{B} \right),$$

that is, *xn* <sup>−</sup> *yn* <sup>=</sup> *dist*(*A*, *<sup>B</sup>*), for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>.

Since *<sup>x</sup>*<sup>0</sup> *<sup>T</sup>*2*x*0, *<sup>x</sup>*<sup>1</sup> <sup>=</sup> *<sup>T</sup>*2*x*<sup>0</sup> *<sup>T</sup>*4*x*<sup>0</sup> <sup>=</sup> *<sup>x</sup>*<sup>2</sup> and by induction on *<sup>n</sup>*, we can get

$$
\mathbf{x}\_n \preceq \mathbf{x}\_{n+1} \quad \text{for all} \quad n \in \mathbb{N}.
$$

In the same manner, we get

$$y\_n \preceq y\_{n+1} \quad \text{for all} \quad n \in \mathbb{N}.$$

Since {*xn*} and {*yn*} are bounded increasing sequences in reflexive space, we get from Lemma 2, *xn <sup>w</sup> <sup>x</sup>*¯ and *yn <sup>w</sup> y*¯.

Note that *<sup>x</sup>*¯ <sup>=</sup> sup {*xn*; *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>} and *<sup>y</sup>*¯ <sup>=</sup> sup {*yn*; *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>}.

Let *<sup>K</sup>* <sup>=</sup> {*<sup>y</sup>* <sup>∈</sup> *<sup>B</sup>*; *yn <sup>y</sup>*, for any *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>} and define the type function *<sup>τ</sup>* : *<sup>K</sup>* <sup>→</sup> [0, <sup>∞</sup>) generated by the sequence {*xn*}, that is,

$$\pi \left( y \right) = \limsup\_{n \to \infty} \left\| \left| \mathbf{x}\_n - y \right| \right\|\_{\ell'} $$

for *y* ∈ *K*. Using the fact that *τ* is increasing function, we get

$$\pi\left(\bar{y}\right) = \inf\_{y \in K} \pi\left(y\right). \tag{1}$$

Indeed, let *<sup>z</sup>*1, *<sup>z</sup>*<sup>2</sup> <sup>∈</sup> *<sup>K</sup>* such that *<sup>z</sup>*<sup>1</sup> *<sup>z</sup>*<sup>2</sup> then for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> we have

$$x\_n \preceq y\_n \preceq z\_1 \preceq z\_2.$$

Using the fact that the norm . is monotone, we get

$$\|\|\mathbf{x\_n} - \mathbf{z\_1}\|\| \le \|\|\mathbf{x\_n} - \mathbf{z\_2}\|\|\_{\prime}$$

hence,

$$
\tau(z\_1) \le \tau(z\_2).
$$

From Lemma 3, it follows that there exists a unique *b*<sup>∗</sup> ∈ *K* such that :

$$\pi\left(b^\*\right) = \inf\_{y \in K} \pi\left(y\right). \tag{2}$$

Since *<sup>y</sup>*¯ <sup>=</sup> sup {*yn*; *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>} and *<sup>b</sup>*<sup>∗</sup> <sup>∈</sup> *<sup>K</sup>*, *<sup>y</sup>*¯ *<sup>b</sup>*∗, that is, *<sup>τ</sup>*(*y*¯) *<sup>τ</sup>*(*b*∗). Thus, *τ*(*y*¯) = *τ*(*b*∗), i.e., *y*¯ = *b*∗.

We have also

$$\begin{aligned} \left| \tau \left( T^2 \ddot{\mathcal{Y}} \right) \right| &= \limsup\_{n \to \infty} \left\| \mathbf{x}\_n - T^2 \ddot{\mathcal{Y}} \right\| \\ &= \limsup\_{n \to \infty} \left\| T^2 \mathbf{x}\_{n-1} - T^2 \ddot{\mathcal{Y}} \right\| \\ &\leq \limsup\_{n \to \infty} \left\| \mathbf{x}\_{n-1} - \ddot{\mathcal{Y}} \right\| \\ &= \tau \left( \ddot{\mathcal{Y}} \right) \end{aligned}$$

hence, *T*2*y*¯ = *y*¯.

Furthermore, *<sup>T</sup>* is weakly sequentially continuous then *Txn <sup>w</sup> Tx*¯ and *Tyn <sup>w</sup> Ty*¯. By the lower semi continuity of the norm, we get

$$||\mathfrak{x} - \mathfrak{y}|| \le \liminf\_{\mathfrak{n} \to \infty} ||\mathfrak{x}\_{\mathfrak{n}} - \mathfrak{y}\_{\mathfrak{n}}|| = \text{dist}\left(A, B\right).$$

Let {*x <sup>n</sup>*} be a sequence defined by *x <sup>n</sup>* = *T*<sup>2</sup>*nx* <sup>0</sup>, for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. We have

$$y\_n = T^{2n+1} \mathfrak{x}\_0' = T(T^{2n} \mathfrak{x}\_0') = T \mathfrak{x}\_n'.$$

Since *x* <sup>0</sup> *<sup>T</sup>*2*<sup>x</sup>* <sup>0</sup>, *<sup>T</sup>*<sup>2</sup>*nx* <sup>0</sup> *<sup>T</sup>*2*n*+2*<sup>x</sup>* <sup>0</sup>, that is, *x <sup>n</sup> x <sup>n</sup>*+1, for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Since {*<sup>x</sup> <sup>n</sup>*} is bounded increasing sequence in reflexive space, we get by using Lemma 2 *x n w* ¯ *x* . Since *T* is weakly sequentially continuous, *yn* = *Tx n w T* ¯ *x* . By the uniqueness of the limit, *T* ¯ *x* = *y*¯, that is,

$$\left\|\left\|\mathbf{x} - T\mathbf{x}^{\prime}\right\|\right\| = dist\left(A, B\right). \tag{3}$$

Note that *x*<sup>0</sup> *x* <sup>0</sup> *<sup>T</sup>*2*x*<sup>0</sup> *<sup>T</sup>*2*<sup>x</sup>* <sup>0</sup>, that is, *x*<sup>0</sup> *x* <sup>0</sup> *x*<sup>1</sup> *x* <sup>1</sup>. Then, by induction on *n*, we can get

$$\mathbf{x}\_n \preceq \mathbf{x}\_n' \preceq \mathbf{x}\_{n+1} \preceq \mathbf{x}\_{n+1}'.$$

Define the sequence {*zn*} as follows

$$z\_{\mathfrak{n}} = \begin{cases} \begin{array}{c} \mathfrak{x}\_{\mathfrak{n}} \quad \text{if } n \text{ is even,} \\\ x\_{\mathfrak{n}-1}^{'} \quad \text{if } n \text{ is odd.} \end{array} \end{cases}$$

Since {*zn*} is bounded increasing sequence in reflexive space, by using Lemma 2, we get *zn <sup>w</sup> z*¯. In particular, the subsequences {*z*2*n*} and {*z*2*n*+1} also converge to *z*¯, that is, *z*¯ = *x*¯ = *x*¯ . Thus, by using (3) we get *x*¯ − *Tx*¯ = *dist*(*A*, *B*).

In the following, we give a best proximity result for monotone decreasing relatively cyclic nonexpansive mapping without assuming the monotonicity of the norm ..

**Theorem 3.** *Let* (*A*, *B*) *be a nonempty bounded closed convex pair in a partially ordered Banach space* (*X*, ., )*. Let T* : *A* ∪ *B* → *A* ∪ *B be a monotone decreasing relatively cyclic nonexpansive mapping. Assume that* (*X*, .) *is UC, <sup>T</sup> is weakly sequentially continuous and there exists x*0, *x* 0 ∈ A*<sup>T</sup> such that x* <sup>0</sup> *<sup>x</sup>*<sup>0</sup> *<sup>T</sup>*2*<sup>x</sup>* <sup>0</sup>*, then there exists* (*x*¯, *y*¯) ∈ *A* × *B such that*

$$\|\|\mathfrak{x} - T\mathfrak{x}\|\| = \|\|\mathfrak{y} - T\mathfrak{y}\|\| = \operatorname{dist}\left(A, B\right) \dots$$

**Proof.** Let *x*0, *x* 0 ∈ A*<sup>T</sup>* such that

$$\mathbf{x}\_0' \preceq \mathbf{x}\_0 \preceq T^2 \mathbf{x}\_0'.$$

If *<sup>A</sup>* <sup>∩</sup> *<sup>B</sup>* <sup>=</sup> <sup>∅</sup> then *<sup>x</sup>*<sup>0</sup> <sup>=</sup> *Tx* <sup>0</sup> by Lemma 5. Since *x* <sup>0</sup> *<sup>x</sup>*<sup>0</sup> *<sup>T</sup>*2*<sup>x</sup>* <sup>0</sup> and *T* is decreasing, we get *<sup>x</sup>*<sup>0</sup> *Tx*<sup>0</sup> and *Tx*<sup>0</sup> *Tx* <sup>0</sup> = *x*0. Thus, *Tx*<sup>0</sup> = *x*0.

If *A* ∩ *B* = ∅, then we consider the sequences {*xn*} and {*zn*} ⊂ *A* defined by

$$\begin{cases} z\_0 &= \begin{array}{c} \mathbf{x}'\_0\\ \mathbf{x}\_n &= \begin{array}{c} T^{2n} \mathbf{x}\_0\\ T^{2n} \mathbf{x}'\_0 \end{array} \end{cases} \text{ for all } n \in \mathbb{N}^\*. $$

Since *x* <sup>0</sup> *<sup>x</sup>*<sup>0</sup> *<sup>T</sup>*2*<sup>x</sup>* <sup>0</sup> = *<sup>z</sup>*<sup>1</sup> and *<sup>T</sup>*<sup>2</sup> is a monotone increasing mapping, by induction on *<sup>n</sup>*, we get *T*<sup>2</sup>*nx* <sup>0</sup> *<sup>T</sup>*<sup>2</sup>*nx*<sup>0</sup> *<sup>T</sup>*2*n*+2*<sup>x</sup>* <sup>0</sup>, which implies

$$z\_n \preceq\_n \chi\_n \preceq z\_{n+1} \tag{4}$$

for all *<sup>n</sup>* <sup>≥</sup> 0. Also, since *<sup>x</sup>*<sup>0</sup> *Tx* <sup>0</sup> = *Tz*<sup>0</sup> and *<sup>T</sup>*<sup>2</sup> is a monotone increasing mapping, by induction on *<sup>n</sup>*, we get *<sup>T</sup>*<sup>2</sup>*nx*<sup>0</sup> *<sup>T</sup>*(*T*<sup>2</sup>*nx* <sup>0</sup>), which implies

$$\mathbf{x}\_{n} \preceq T \mathbf{z}\_{n} \tag{5}$$

for all *<sup>n</sup>* <sup>≥</sup> 0. The sequences {*xn*} and {*zn*} are increasing. Indeed, *<sup>x</sup>*<sup>0</sup> *<sup>T</sup>*2*<sup>x</sup>* <sup>0</sup> *<sup>T</sup>*2*x*<sup>0</sup> implies by induction on *<sup>n</sup>* that *<sup>T</sup>*<sup>2</sup>*nx*<sup>0</sup> *<sup>T</sup>*2*n*+2*x*0. Thus,

$$\mathbf{x}\_n \preceq \mathbf{x}\_{n+1},$$

for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. Since {*xn*} and {*zn*} are bounded increasing sequences in a reflexive space, we get by Lemma 2, *xn <sup>w</sup> <sup>x</sup>*¯ and *zn <sup>w</sup> z*¯. Using the fact that *T* is weakly sequentially continuous we conclude that *Tzn <sup>w</sup> Tz*¯.

Since *T* is relatively cyclic nonexpansive mapping, we get

$$\begin{aligned} ||\mathbf{x}\_n - T\mathbf{z}\_n|| &= \quad ||T^2 \mathbf{x}\_{n-1} - T^3 \mathbf{z}\_{n-1}|| \\ &\le \quad ||T\mathbf{x}\_{n-1} - T^2 \mathbf{z}\_{n-1}|| \\ &\le \quad ||\mathbf{x}\_{n-1} - T\mathbf{z}\_{n-1}|| \end{aligned}$$

for all *n* in N∗. By induction on *n*, we prove that

$$||\mathfrak{x}\_n - T\mathfrak{z}\_n|| \le ||\mathfrak{x}\_0 - T\mathfrak{x}\_0'|| = dist\left(A, B\right),$$

for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>. By the lower semi continuity of the norm, we get

$$\|\|\mathfrak{x} - T\mathfrak{z}\|\| \le \liminf\_{n \to \infty} \|\|\mathfrak{x}\_n - T\mathfrak{z}\_n\|\| = dist\left(A, B\right). \tag{6}$$

It follows from the Lemma 1 and the inequality (4) that *z*¯ *x*¯ *z*¯, and hence, *z*¯ = *x*¯. Finally, by Equation (6) it follows that

$$\|\|\| - T\|\| = \text{dist}\left(A, B\right).$$

Let *y*¯ = *Tx*¯, then by inequality (5) and Lemma 1 we have *x*¯ *y*¯ and

$$\|\|\mathbf{y} - T\mathbf{y}\|\| = \|\|T\mathbf{x} - T\mathbf{y}\|\| \le \|\|\mathbf{x} - \mathbf{y}\|\| = dist\left(A, B\right).$$

So the proof is complete.

We claim that *<sup>T</sup>*2*x*¯ <sup>=</sup> *<sup>x</sup>*¯ and *<sup>T</sup>*2*y*¯ <sup>=</sup> *<sup>y</sup>*¯. Indeed, since *xn*+<sup>1</sup> <sup>=</sup> *<sup>T</sup>*2*xn <sup>w</sup> <sup>x</sup>*¯ and *xn*+<sup>1</sup> <sup>=</sup> *<sup>T</sup>*2*xn <sup>w</sup> T*2*x*¯, the uniqueness of the weak limit implies that *T*2*x*¯ = *x*¯. Furthermore, *Tx*¯ = *y*¯ then

$$T^2\mathfrak{X} = \mathfrak{x} \Longrightarrow T(T^2\mathfrak{X}) = T\mathfrak{x} \Longrightarrow T^2(T\mathfrak{X}) = \mathfrak{y} \Longrightarrow T^2\mathfrak{y} = \mathfrak{y}.$$

The following example illustrates Theorem 3.

**Example 1.** *Consider X* = R<sup>2</sup> *with usual norm and the partially order defined by:*

$$(a,b) \preceq (c,d) \quad \text{iff} \quad (a \le c \quad \text{and} \quad b \le d)\_r$$

*for any* (*a*, *b*)*,* (*c*, *d*) *in* R2*. Suppose that*

$$A = \left\{ (\mathbf{x}, 0) \in \mathbb{R}^2 \; ; \mathbf{x} \in [0, 2] \right\} \quad \text{and}$$

$$B = \left\{ (\mathbf{x}, 1) \in \mathbb{R}^2 \; ; \mathbf{x} \in [2, 4] \right\} \; .$$

*we can show that dist*(*A*, *B*) = 1*,* A <sup>0</sup> = {(2, 0)} *and* B <sup>0</sup> = {(2, 1)}*. Suppose that a mapping T* : *A* ∪ *B* → *A* ∪ *B is defined as follows*

$$\begin{cases} \begin{array}{ll} T(\mathbf{x},0) = (2,1); & \text{for all} \quad (\mathbf{x},0) \in A\\ \begin{array}{ll} T(\mathbf{x},1) = (4-\mathbf{x},0); & \text{for all} \quad (\mathbf{x},1) \in B. \end{array} \end{cases} \end{cases}$$

*We have <sup>T</sup>*(*A*) <sup>⊂</sup> *B, <sup>T</sup>*(*B*) <sup>⊂</sup> *<sup>A</sup> and <sup>T</sup> is a decreasing mapping. Also, for any* (*x*, 0),(*x* , 1) ∈ *A* × *B we have* (*x*, 0) (*x* , 1) *and*

$$\begin{cases} \|T(\mathbf{x},0) - T(\mathbf{x}',1)\| &=& \|(2,1) - (4-\mathbf{x}',0)\| \\ &=& \sqrt{(\mathbf{x}'-2)^2 + 1} \\ \|(\mathbf{x},0) - (\mathbf{x}',1)\| &=& \frac{\|(\mathbf{x}'-\mathbf{x},1)\|}{\sqrt{(\mathbf{x}'-\mathbf{x})^2 + 1}} \\ &=& \sqrt{(\mathbf{x}'-\mathbf{x})^2 + 1} \end{cases}$$

*thus, T*(*x*, 0) − *T*(*x* , 1)≤(*x*, 0) − (*x* , 1)*. Then T is a monotone decreasing relatively cyclic nonexpansive mapping.*

*If we choose x* <sup>0</sup> = (0, 0) *and x*<sup>0</sup> = (2, 0) *in A we get*

> *<sup>x</sup>*0 *Tx* <sup>0</sup>*, x*<sup>0</sup> <sup>−</sup> *Tx* <sup>0</sup> <sup>=</sup> *dist*(*A*, *<sup>B</sup>*) *and x* <sup>0</sup> *<sup>x</sup>*<sup>0</sup> *<sup>T</sup>*2*<sup>x</sup>* 0.

*Then there exist <sup>x</sup>*¯ = (2, 0) <sup>∈</sup> *A and <sup>y</sup>*¯ = (2, 1) <sup>∈</sup> *B such that T*2*x*¯ <sup>=</sup> *x, T* ¯ <sup>2</sup>*y*¯ <sup>=</sup> *y and* ¯

$$\|\|\bar{\mathbf{x}} - T\bar{\mathbf{x}}\|\| = \|\|\bar{y} - T\bar{y}\|\| = dist\left(A, B\right) \dots$$

**Author Contributions:** The authors contributed equally to this work.

**Funding:** This research received no external funding.

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

### **References**


18. Alfuraidan, M.; Khamsi, M.A. A fixed point theorem for monotone asymptotically nonexpansive mappings. *Proc. Am. Math. Soc.* **2018**, *146*, 2451–2456. [CrossRef]

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