**1. Introduction**

Delay differential equations are used to model biological, physical, system engineering, and sociological processes as well as naturally occurring oscillatory systems (see, for examples, [1–9]).

It is known that in differential and difference equations, the involvement of the delay term causes deep difficulties in the analysis of these equations. Lu [10] studies modified iterative schemes by combing the method of upper-lower solutions and the Jacobi method or the Gauss–Seidel method for finite-difference solutions of reaction-diffusion systems with time delays. Ashyralyev and Sobolevskii [11] study the initial-value problem for linear delay parabolic equations in a Banach space and present a sufficient condition for the stability of the solution of this initial-value problem. The stability estimates in Hölder norms for the solutions of the initial-boundary value problem for delay parabolic equations were established.

Ashyralyev and Agirseven [12–18] studied some initial-boundary value problems for linear delay parabolic differential equations. Theorems on stability and convergence of difference schemes for the

numerical solution of initial and boundary value problems for linear parabolic equations with time delay were proved. Moreover, Ashyralyev, Agirseven, and Ceylan [19] investigated finding sufficient conditions for the existence and uniqueness of a bounded solution of the initial value problem for the semilinear delay parabolic equation in a Banach space. The main theorem on the existence and uniqueness of a bounded solution of this problem was established. In applications, existence and uniqueness of a bounded solution of problems for four semilinear delay parabolic equations were obtained. Numerical results were given.

Henriquez, Cuevas, and Caicedo [20] investigated the existence of almost periodic solutions for linear retarded functional differential equations with finite delay. The existence of almost periodic solutions with the stabilization of distributed control systems was obtained.

Hao, Fan, Cao, and Sun [21] proposed a linearized quasi-compact finite difference scheme for semilinear space-fractional diffusion equations with a fixed time delay. Under the local Lipschitz conditions, they proved the solvability and convergence of the scheme in the discrete maximum norm by the energy method.

Liang [22] studied the convergence and asymptotic stability of semidiscrete and full discrete schemes for linear parabolic equations with time delay. She proved that the semidiscrete scheme, backward Euler and Crank–Nicolson full discrete schemes can unconditionally preserve the delay-independent asymptotic stability with some additional restrictions on time and spatial stepsizes of the forward Euler full discrete scheme.

Bhrawy, Abdelkawy, and Mallawi [23] investigated the Chebyshev Gauss–Lobatto pseudospectral scheme in spatial directions for the approximate solution of one-dimensional, coupled, and two-dimensional parabolic equations with time delays. They also develop an efficient numerical algorithm based on the Chebyshev pseudospectral algorithm to obtain the two spatial variables in approximate solving the two-dimensional parabolic equations with time delay.

Applying Vishik's results and methods of operator tools, Ismailov, Guler, and Ipek [24] described all solvable extensions of a minimal operator generated by linear delay differential-operator expression of first order in the Hilbert space of vector-functions in finite interval. Sharp formulas for the spectrums of these solvable extensions were obtained. Theoretical results have been supported by applications.

Piriadarshani and Sengadir [25] obtained an existence theorem for a semilinear partial differential equation with infinite delay employing a phase space in which discretizations can naturally be performed. For linear partial differential equations with infinite delay they show that the solutions of the ordinary differential equation with infinite delay obtained by the semi-discretization converge to the original solution.

Castro, Rodriguez, Cabrera, and Martin [26] developed an explicit finite difference scheme for a model with coefficients variable in time and studied their properties of convergence and stability.

Hyperbolic equations without time delay arise in many branches of science and engineering, for example, electrodynamics, thermodynamics, hydrodynamics, fluid dynamics, wave propagation, hyperbolic geometry, and discrete mathematics (see, e.g., [8,9,27–34], and the references given therein). The geometry of complex networks is closely related with their structure and function. Shang Yilun [34] investigated the Gromov-hyperbolicity of the Newman–Watts model of small-world networks. It is known that asymptotic Erd˝os–Rényi random graphs are not hyperbolic. We show that the Newman–Watts ones built on top of them by adding lattice-induced clustering are not hyperbolic as the network size goes to infinity. Numerical simulations are provided to illustrate the effects of various parameters on hyperbolicity in this model. The geometry of complex networks has a close relationship with their structure and function. Shang Yilun [33] investigated Gromov-hyperbolicity of inhomogeneous random networks modeled by the Chung-Lu model *G*(*w*). His numerical simulations illustrate this non-hyperbolicity of *G*(*w*) for power law degree distributions among others.

In numerical methods for solving hyperbolic equations, the problem of stability has received a great deal of importance and attention. The method of operators as a tool for the study of the stability of the solution of local and nonlocal problems to hyperbolic differential and difference equations in Hilbert and Banach spaces has been systematically developed by many authors (see, e.g., [27–32,35–37]).

A large cycle of works on difference schemes for hyperbolic equations (see, e.g., [38–42] and the references given therein), in which stability was established under the assumption that the magnitude of the grid steps *τ* and *h* with respect to the time and space variables, were connected. In abstract terms that means that the condition *τ*||*Ah*|| → 0 when *τ* → 0 holds.

Of course there is great interest in the study of absolute stable difference schemes of a high order of accuracy for hyperbolic equations, in which stability was established without any assumptions in respect to the grid steps *τ* and *h*. Such type of stability inequalities for the solutions of the first order of accuracy difference scheme for the abstract hyperbolic equation in Hilbert spaces were established for the first time in [43]. The first and second order of accuracy difference schemes generated by integer power of space operator of approximate solutions of the abstract initial value problem for the abstract hyperbolic equation in Hilbert spaces were presented in [11]. The stability estimates for the solution of these difference schemes were established.

The survey paper [44] contains the recent results on the local and nonlocal well-posed problems for second order differential and difference equations. Stability of differential problems for hyperbolic equations and of difference schemes for approximate solution of these problems were presented.

However, the stability theory of problems for a hyperbolic equation with unbounded time delay term is not well-investigated. A few researchers are interested in these kinds of problems. Bounded solutions of semilinear one dimensional hyperbolic differential equations with time delay term have been investigated in earlier papers [45–48]. In the paper [49] the existence and uniqueness of a bounded solution of nonlinear hyperbolic differential equations with bounded time delay term were established. The generality of the operator approach allows for treating a wider class of delay nonlinear hyperbolic differential equations with bounded time delay term. In general, hyperbolic differential equations with unbounded time delay term are blown up [7]. Therefore, the boundedness solution of problems for hyperbolic equations with unbounded time delay term is not well-investigated.

Our goal in the present paper is to investigate the boundedness solution of problems for semilinear hyperbolic equations with unbounded time delay term. We study the initial value problem for the semilinear hyperbolic differential equation with time delay

$$\begin{cases} \frac{d^2u(t)}{dt^2} + Au(t) = f(t, u(t), u\_t(t - w), u(t - w)), \; t > 0, \\\\ u(t) = \varphi(t), \; -\omega \le t \le 0 \end{cases} \tag{1}$$

in a Hilbert space *H* with a self-adjoint positive definite operator *A*. Here *ϕ*(*t*) is a continuously differentiable abstract function defined on the interval [−*ω*, 0] with values in *H* and *f*(*t*) is continuous abstract function defined on the interval [0, ∞) with values in *H*. Assume that *A* is unbounded operator and (*Ax*, *x*) > *δ*(*x*, *x*), for *x* = 0, *x* ∈ *H* and *δ* > 0.

A function *u*(*t*) is called a solution of problem (1), if the following conditions are fulfilled:


In the present paper, the main theorem on the existence and uniqueness of a bounded solution of the differential problem (1) is established. In applications, the existence and uniqueness of a bounded solution of four problems for semilinear hyperbolic equations with time delay are obtained. A first order of accuracy difference scheme for the numerical solution of this problem is presented. The theorem on the existence and uniqueness of a uniformly bounded solution of this difference scheme

with respect to *τ* is established. In applications, existence and uniqueness of a bounded solution of a problem for four semilinear delay parabolic equations were established. Numerical results for the solution of difference schemes for one and two dimensional nonlinear hyperbolic equation with time delay are presented.

#### **2. Main Existence and Uniqueness Theorem of the Differential Problem**

Throughout this paper, *c*(*t*) and *s*(*t*) are operator-functions defined by formulas

$$c(t)u = \frac{e^{iA^{\frac{1}{2}}t} + e^{-iA^{\frac{1}{2}}t}}{2}u,\ \ s(t)u = \int\_0^t c(y)u dy. \tag{2}$$

We will give some auxiliary statements which will be useful in the sequel.

Recall that the norm *AH*→*<sup>H</sup>* of a bounded operator *A* : *H* → *H* is by definition the smallest number *C* for which estimate

$$\|Au\|\_{H} \le \mathbb{C} \|u\|\_{H}$$

holds. Equivalently,

$$\|\|A\|\|\_{H \to H} = \sup\_{\|\|u\|\|\_{H} = 1} \|Au\|\_{H}.$$

**Lemma 1.** *For t* ≥ 0*, the following estimates hold:*

$$\|\|A^{-\frac{1}{2}}\|\|\_{H\to H} \le \delta^{-\frac{1}{2}}, \quad \|c(t)\|\|\_{H\to H} \le 1, \ \|As(t)\|\|\_{H\to H} \le 1. \tag{3}$$

**Proof.** Applying formulas (2) and the spectral representation of the self-adjoint positive definite operator *A* in a Hilbert space *H*, we can write (see [50])

$$\left\|A^{-\frac{1}{2}}\right\|\_{H\to H}\leq\sup\_{\delta\leq\lambda<\infty}\left|\lambda^{-\frac{1}{2}}\right|\leq\delta^{-\frac{1}{2}},$$

$$\left\|c(t)\right\|\_{H\to H}\leq\sup\_{\delta\leq\lambda<\infty}\left|\frac{e^{i\lambda^{\frac{1}{2}}t}+e^{-i\lambda^{\frac{1}{2}}t}}{2}\right|=\sup\_{\delta\leq\lambda<\infty}\left|\cos\left(\lambda^{\frac{1}{2}}t\right)\right|\leq 1,$$

$$\left\|A^{\frac{1}{2}}s(t)\right\|\_{H\to H}\leq\sup\_{\delta\leq\lambda<\infty}\left|\frac{e^{i\lambda^{\frac{1}{2}}t}-e^{-i\lambda^{\frac{1}{2}}t}}{2i}\right|=\sup\_{\delta\leq\lambda<\infty}\left|\sin\left(\lambda^{\frac{1}{2}}t\right)\right|\leq 1,$$

for any *t* ≥ 0. Lemma 1 is proved.

The approach of proof of main theorem is based on reducing problem (1) to an integral equation of Volterra type

$$\begin{aligned} u(t) &= c \left( t - (m-1)w \right) u((m-1)w) + s \left( t - (m-1)w \right) \frac{du((m-1)w)}{dt} \\ &+ \int\_{(m-1)w}^t s \left( t - y \right) f(y, u(y), u\_y(y-w), u(y-w)) dy, \\ &(m-1)w \le t \le mw, \ m = 1, 2, \ldots, \ u(t) = \varrho(t), \ -w \le t \le 0 \end{aligned}$$

in [0, ∞) × *H* and the application of successive approximations. Note that on (*m* − 1)*w* ≤ *t* ≤ *mw*, *m* = 1,2,..., *ut*(*t* − *w*) and *u*(*t* − *w*) are given. Therefore, the recursive formula for the solution of

*Mathematics* **2019**, *7*, 1163

problem (1) is

$$\begin{array}{ll} u\_i(t) = & c\left(t - (m-1)w\right)u\left((m-1)w\right) + s\left(t - (m-1)w\right)\frac{du\left((m-1)w\right)}{dt} \\ & + \int\_{(m-1)w}^t s\left(t - y\right)f(y, u\_{i-1}(y), u\_y(y-w), u(y-w)) dy,\end{array} \tag{4}$$

$$u\_0(t) = c\left(t - (m-1)w\right)u\left((m-1)w\right) + s\left(t - (m-1)w\right)\frac{du\left((m-1)w\right)}{dt},$$

$$(m-1)w \le t \le mw,\quad m = 1,2,\ldots,$$

$$u\_i(t) = q(t),\; i = 1,2,\ldots, \quad -w \le t \le 0.$$

**Theorem 1.** *Suppose that ϕ*(*t*) *is a continuously differentiable function on* [−*ω*, 0] *and ϕ*(*t*) ∈ *D*(*A*)*, ϕ* (*t*) <sup>∈</sup> *<sup>D</sup>*(*A*<sup>1</sup> <sup>2</sup> ) *and*

$$\|A\varphi(t)\|\_{H} \le M, \quad \|A^{1/2}\varphi'(t)\|\_{H} \le \tilde{M}.\tag{5}$$

*Besides let f* : [0, ∞) × *H* × *H* × *H* −→ *H be continuous and bounded function, that is*

$$\|f(t, u, v, z)\|\_{H} \le \bar{M} \tag{6}$$

*in* [0, ∞) × *H* × *H* × *H and Lipschitz condition holds uniformly with respect to t*, *v and z*

$$\|\|f(t,u,v,z) - f(t,w,v,z)\|\|\_{H} \le L\|u - w\|\_{H}.\tag{7}$$

*Here, <sup>M</sup>*, *<sup>M</sup>*:, *<sup>M</sup>*¯ , Ł *are positive constants. Then there exists a unique solution to problem (1) which is bounded in C*(*H*)*. Here, C*(*H*) = *C*([0, ∞), *H*) *stands for the Banach space of the abstract continuous and bounded functions v*(*t*) *defined on* [0, ∞) *with values in H, equipped with the norm*

$$\|\|v\|\|\_{C(H)} = \sup\_{0 \le t < \infty} \|v(t)\|\_H.$$

**Proof.** Let 0 ≤ *t* ≤ *ω*. Then, according to Equation (4), we get

$$u\_i(t) = c\left(t\right)\varrho(0) + s\left(t\right)\varrho'(0) + \int\_0^t s\left(t - y\right)f(y, u\_{i-1}(y), \varrho\_y(y - w), \varrho(y - w))dy,\tag{8}$$

$$u\_i'(t) = -A\mathbf{s}\left(t\right)\boldsymbol{\varrho}(0) + \mathbf{c}\left(t\right)\boldsymbol{\varrho}'(0) + \int\_0^t \mathbf{c}\left(t - \boldsymbol{y}\right)f(\boldsymbol{y}, \boldsymbol{u}\_{i-1}(\boldsymbol{y}), \boldsymbol{\varrho}\_{\boldsymbol{y}}(\boldsymbol{y} - \boldsymbol{w}), \boldsymbol{\varrho}(\boldsymbol{y} - \boldsymbol{w}))d\boldsymbol{y} \tag{9}$$

for all *i* = 1, 2, .... Therefore,

$$u(t) = u\_0(t) + \sum\_{i=0}^{\infty} (u\_{i+1}(t) - u\_i(t)),\tag{10}$$

$$u'(t) = u\_0'(t) + \sum\_{i=0}^{\infty} (u\_{i+1}'(t) - u\_i'(t)),\tag{11}$$

where

$$
\mu\_0(t) = \mathfrak{c}\left(t\right)\varrho(0) + \mathfrak{s}\left(t\right)\varrho'(0), \\
\mathfrak{u}\_0'(t) = -A\mathfrak{s}\left(t\right)\varrho(0) + \mathfrak{c}\left(t\right)\varrho'(0).
$$

Applying the triangle inequality and estimates (3) and (5), we get

$$\|u\_0(t)\|\_{H} \le \|A^{-1}\|\_{H \to H}$$

$$\delta \times \left[ \|\mathbf{c}\left(t\right)\|\_{H \to H} \|A\mathbf{q}(0)\|\_{H} + \|A^{1/2}\mathbf{s}\left(t\right)\|\_{H \to H} \|\|A^{1/2}\mathbf{q}'(0)\|\|\_{H} \right]$$

$$\le \delta^{-1} \left[ \|A\mathbf{q}(0)\|\|\_{H} + \|A^{1/2}\mathbf{q}'(0)\|\|\_{H} \right] \le \delta^{-1} \left[M + \tilde{M}\right],$$

$$\|\|u\_0'(t)\|\|\_{H} \le \|A^{-\frac{1}{2}}\|\_{H \to H}$$

$$\begin{aligned} &\lambda \times \left[ \|A^{1/2}s(t)\|\_{H\to H} \|A\varphi(0)\|\_{H} + \|c\left(t\right)\|\_{H\to H} \|A^{1/2}\varphi'(0)\|\_{H} \right] \\ &\leq \delta^{-\frac{1}{2}} \left[ \|A\varphi(0)\|\_{H} + \|A^{1/2}\varphi'(0)\|\_{H} \right] \leq \delta^{-\frac{1}{2}} \left[M + \breve{M}\right]. \end{aligned}$$

Applying formulas (8) and (9) and the triangle inequality and estimates (3) and (6), we get

$$\|\|u\_1(t) - u\_0(t)\|\|\_{H} \le \|A^{-\frac{1}{2}}\|\|\_{H \to H}$$

$$\times \int\_0^t \|A^{1/2}s\left(t-y\right)\|\_{H\to H} \|f(y, u\_0(y), \varphi\_y(y-w), \varphi(y-w))\|\_{\mathbb{H}} dy \le \delta^{-\frac{1}{2}} \bar{M}t,$$

$$\|u\_1'(t) - u\_0'(t)\|\_H \le \int\_0^t \|c\left(t-y\right)\|\_{H\to H} \|f(y, u\_0(y), \varphi\_y(y-w), \varphi(y-w))\|\_{\mathbb{H}} dy \le \bar{M}t.$$

Using the triangle inequality, we get

$$\begin{aligned} \|\boldsymbol{u}\_1(t)\|\|\_{H} &\leq \delta^{-1} \left[\mathcal{M} + \tilde{\mathcal{M}}\right] + \delta^{-\frac{1}{2}} \tilde{\mathcal{M}}t, \\\\ \|\boldsymbol{u}\_1'(t)\|\|\_{H} &\leq \delta^{-\frac{1}{2}} \left[\mathcal{M} + \tilde{\mathcal{M}}\right] + \tilde{\mathcal{M}}t. \end{aligned}$$

Applying formulas (8) and (9) and estimates (3), (6), and (7), we get

$$\|\|\mu\_2(t) - \mu\_1(t)\|\|\_{H} \le \|A^{-\frac{1}{2}}\|\_{H \to H}$$

$$\begin{aligned} \|\lambda \times \int\_0^t \|A^{1/2}s(t-y)\|\_{H\to H} \|f(y, u\_1(y), \varphi\_y(y-w), \varphi(y-w)) - f(y, u\_0(y), \varphi\_y(y-w), \varphi(y-w))\|\_H dy \\ \quad \le \delta^{-\frac{1}{2}} L \int\_0^t \|u\_1(y) - u\_0(y)\|\_H dy \le \delta^{-1} L \tilde{M} \int\_0^t y dy = \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}} Lt)^2}{2!}, \\ \|u\_2'(t) - u\_1'(t)\|\_H \end{aligned}$$

$$\leq \int\_{0}^{t} \|\boldsymbol{\varepsilon}(t-y)\|\_{H\to H} \|\boldsymbol{f}(\boldsymbol{y},\boldsymbol{u}\_{1}(\boldsymbol{y}),\boldsymbol{\varrho}\_{y}(\boldsymbol{y}-\boldsymbol{w}),\boldsymbol{\varrho}(\boldsymbol{y}-\boldsymbol{w})) - \boldsymbol{f}(\boldsymbol{y},\boldsymbol{u}\_{0}(\boldsymbol{y}),\boldsymbol{\varrho}\_{y}(\boldsymbol{y}-\boldsymbol{w}),\boldsymbol{\varrho}(\boldsymbol{y}-\boldsymbol{w}))\|\_{H} d\boldsymbol{y}\|\_{H}$$

$$\leq L \int\_{0}^{t} \|\boldsymbol{u}\_{1}(\boldsymbol{y}) - \boldsymbol{u}\_{0}(\boldsymbol{y})\|\_{H} d\boldsymbol{y} \leq \delta^{-\frac{1}{2}} L\tilde{M} \int\_{0}^{t} \boldsymbol{y} d\boldsymbol{y} = \delta^{-\frac{1}{2}} \frac{\bar{M}}{L} \frac{(Lt)^{2}}{2!}.$$

Then

$$\begin{aligned} \|\mu\_2(t)\|\_H &\leq \delta^{-1} \left[M + \breve{M}\right] + \delta^{-\frac{1}{2}} \breve{M}t + \frac{\breve{M}}{L} \frac{(\delta^{-\frac{1}{2}}Lt)^2}{2!}, \\ \|\mu\_2'(t)\|\_H &\leq \delta^{-\frac{1}{2}} \left[M + \breve{M}\right] + \breve{M}t + \frac{\breve{M}}{L\delta^{-\frac{1}{2}}} \frac{(Lt)^2}{2!}. \end{aligned}$$

Let

$$||u\_n(t) - u\_{n-1}(t)||\_H \le \frac{\bar{M}}{L} \frac{(\delta^{-\frac{1}{2}}Lt)^n}{n!}, \quad ||u'\_n(t) - u'\_{n-1}(t)||\_H \le \frac{\bar{M}}{L\delta^{-\frac{1}{2}}} \frac{(Lt)^n}{n!}$$

and

$$\begin{aligned} \|u\_n(t)\|\_H &\leq \delta^{-1} \left[M + \tilde{M}\right] + \delta^{-\frac{1}{2}} \tilde{M}t + \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}Lt)^2}{2!} + \dots + \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}Lt)^n}{n!}, \\\|u\_n'(t)\|\_H &\leq \delta^{-\frac{1}{2}} \left[M + \tilde{M}\right] + \tilde{M}t + \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}t)^2}{2!} + \dots + \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}t)^n}{n!}. \end{aligned}$$

Then, we obtain

$$\|\|u\_{n+1}(t) - u\_n(t)\|\|\_{H} \le \|\mathcal{A}^{-\frac{1}{2}}\|\_{H \to H}$$

*Mathematics* **2019**, *7*, 1163

$$\begin{split} \lambda \times \int\_{0}^{t} \|A^{1/2}s(t-y)\|\_{H\to H} \|f(y, u\_{n}(y), \varphi(y-w), \varphi(y-w)) - f(y, u\_{n-1}(y), \varphi\_{y}(y-w), \varphi(y-w))\|\_{H} dy \\ \leq \delta^{-\frac{1}{2}} L \int\_{0}^{t} \|u\_{n}(y) - u\_{n-1}(y)\|\_{H} dy \leq \delta^{-\frac{1}{2}} L \int\_{0}^{t} \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}Ly)^{n}}{n!} y dy = \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}Lt)^{n+1}}{(n+1)!}, \\ \|u\_{n+1}'(t) - u\_{n}'(t)\|\_{H} \\ \leq \int\_{0}^{t} \|c\left(t - y\right)\|\_{H\to H} \|f(y, u\_{n}(y), \varphi\_{y}(y-w), \varphi(y-w)) - f(y, u\_{n-1}(y), \varphi(y-w), \varphi(y-w))\|\_{H} dy \end{split}$$

$$0 \le \int\_0^t L \|u\_n(y) - u\_{n-1}(y)\|\_{H} ds \le \int\_0^t L \frac{\bar{M}}{L} \frac{(\delta^{-\frac{1}{2}} L y)^n}{n!} dy = \frac{\bar{M}}{L \delta^{-\frac{1}{2}}} \frac{(L \delta^{-\frac{1}{2}} t)^{n+1}}{(n+1)!}.$$

Therefore,

$$\|u\_{n+1}(t) - u\_n(t)\|\_H \le \frac{\bar{M}}{L} \frac{(\delta^{-\frac{1}{2}}Lt)^{n+1}}{(n+1)!}, \ \|u\_{n+1}'(t) - u\_n'(t)\|\_H \le \frac{\bar{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}t)^{n+1}}{(n+1)!}$$

and

$$\|u\_{n+1}(t)\|\_{H} \le \delta^{-1} \left[M + \tilde{M}\right] + \delta^{-\frac{1}{2}} \tilde{M}t + \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}Lt)^2}{2!} \cdots + \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}Lt)^{n+1}}{(n+1)!},$$

$$\|u\_{n+1}'(t)\|\_{H} \le \delta^{-\frac{1}{2}} \left[M + \tilde{M}\right] + \tilde{M}t + \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}t)^2}{2!} + \cdots + \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}t)^{n+1}}{(n+1)!}$$

are true for any *n*, *n* ≥ 1 by mathematical induction. In a similar manner, for any *n*, we can obtain

$$\|A^{\frac{1}{2}}u\_{n+1}(t) - A^{\frac{1}{2}}u\_n(t)\|\_{H} \le \frac{\bar{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}t)^{n+1}}{(n+1)!}$$

and

$$\|\|A^{\frac{1}{2}}u\_{n+1}(t)\|\|\_{H} \leq \delta^{-\frac{1}{2}} \left[M + \breve{M}\right] + \breve{M}t + \frac{\breve{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}t)^2}{2!} + \dots + \frac{\breve{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}t)^{n+1}}{(n+1)!}.$$

From that and formulas (10) and (11) it follows that

*u*(*t*)*<sup>H</sup>* ≤ *u*0(*t*)*<sup>H</sup>* + ∞ ∑ *i*=0 *ui*+1(*t*) − *ui*(*t*)*<sup>H</sup>* <sup>≤</sup> *<sup>δ</sup>*−<sup>1</sup> *M* + *M*: ! + ∞ ∑ *i*=0 *M*¯ *L* (*δ*<sup>−</sup> <sup>1</sup> <sup>2</sup> *Lt*)*i*+<sup>1</sup> (*i* + 1)! <sup>≤</sup> *<sup>δ</sup>*−<sup>1</sup> *M* + *M*: ! + *M*¯ *L e δ* − 1 <sup>2</sup> *Lt*, 0 <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *<sup>w</sup>*, *u* (*t*)*<sup>H</sup>* ≤ *u* <sup>0</sup>(*t*)*<sup>H</sup>* + ∞ ∑ *i*=0 *u <sup>i</sup>*+1(*t*) − *u <sup>i</sup>*(*t*)*<sup>H</sup>* <sup>≤</sup> *<sup>δ</sup>*<sup>−</sup> <sup>1</sup> 2 *M* + *M*: ! + ∞ ∑ *i*=0 *M*¯ *Lδ*<sup>−</sup> <sup>1</sup> 2 (*Lδ*<sup>−</sup> <sup>1</sup> <sup>2</sup> *t*)*i*+<sup>1</sup> (*i* + 1)! <sup>≤</sup> *<sup>δ</sup>*<sup>−</sup> <sup>1</sup> 2 *M* + *M*: ! + *M*¯ *Lδ*<sup>−</sup> <sup>1</sup> 2 *eLδ* − 1 2 *t* , 0 ≤ *t* ≤ *w*, *A*<sup>1</sup> <sup>2</sup> *<sup>u</sup>*(*t*)*<sup>H</sup>* ≤ *A*<sup>1</sup> <sup>2</sup> *u*0(*t*)*<sup>H</sup>* + ∞ ∑ *i*=0 *A*<sup>1</sup> <sup>2</sup> *ui*+1(*t*) <sup>−</sup> *<sup>A</sup>*<sup>1</sup> <sup>2</sup> *ui*(*t*)*<sup>H</sup>*

*Mathematics* **2019**, *7*, 1163

$$\leq \delta^{-\frac{1}{2}} \left[ M + \tilde{M} \right] + \sum\_{i=0}^{\infty} \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}t)^{i+1}}{(i+1)!}$$

$$\leq \delta^{-\frac{1}{2}} \left[ M + \tilde{M} \right] + \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} \varepsilon^{L\delta^{-\frac{1}{2}}t}, \ 0 \leq t \leq w\_{\prime t}$$

which proves the existence of a bounded solution of problem (1) in [0, *w*] × *H*.

Let *mω* ≤ *t* ≤ (*m* + 1) *ω*, *m* = 1, 2, .... Then, according to Equation (4), we can write

$$\begin{aligned} u\_i(t) &= c \left( t - mw \right) u(mw) + s \left( t - mw \right) \frac{du(mw)}{dt} \\ + \int\_{mw}^{t} s \left( t - y \right) f(y, u\_{i-1}(y), u\_y(y - w), u(y - w)) dy, \ i &= 1, 2, \dots \end{aligned} \tag{12}$$

$$\begin{array}{c} u'\_i(t) = -As\left(t - mw\right)u(mw) + c\left(t - mw\right)\frac{du(mw)}{dt} \\ + \int\_{mw}^{t} c\left(t - y\right)f(y, u\_{i-1}(y), u\_y(y - w), u(y - w)) dy, \ i = 1, 2, \dots \end{array} \tag{13}$$

Therefore,

$$u(t) = u\_0(t) + \sum\_{i=0}^{\infty} (u\_{i+1}(t) - u\_i(t)),\tag{14}$$

$$u\_i'(t) = u\_0'(t) + \sum\_{i=0}^{\infty} (u\_{i+1}'(t) - u\_i'(t)),\tag{15}$$

where

$$u\_0(t) = c \left(t - mw\right) u(mw) + s \left(t - mw\right) \frac{du(mw)}{dt},$$

$$u\_0'(t) = -As \left(t - mw\right) u(mw) + c \left(t - mw\right) \frac{du(mw)}{dt}.$$

Assume that problem (1) has a bounded solution in [(*m* − 1)*ω*, *mw*] × *H* and

$$\|\|A^{1/2}u(t)\|\|\_{H} \le M\_{m-1}, \ \|\|u'(t)\|\|\_{H} \le \tilde{M}\_{m-1}.\tag{16}$$

2  !

Applying estimates (3) and (16), we get

$$\begin{aligned} \|\boldsymbol{\mu\_0}(t)\|\_{H} &\leq \|\boldsymbol{A}^{-\frac{1}{2}}\|\_{H\to H} \\ \|\boldsymbol{c}\left(t\right)\|\_{H\to H} \|\boldsymbol{A}^{1/2}\boldsymbol{u}(m\boldsymbol{w})\|\_{H} + \|\boldsymbol{A}^{1/2}\boldsymbol{s}\left(t\right)\|\_{H\to H} \|\boldsymbol{u}'(m\boldsymbol{w})\|\_{H} &\leq \delta^{-\frac{1}{2}} \end{aligned}$$

$$\begin{aligned} &\times \left[ \| \boldsymbol{c} \left( t \right) \| \|\_{H \to H} \| \| \boldsymbol{A}^{1/2} \boldsymbol{u} (\boldsymbol{m} \boldsymbol{w}) \| \|\_{H} + \| \boldsymbol{A}^{1/2} \boldsymbol{s} \left( t \right) \| \|\_{H \to H} \| \| \boldsymbol{u}' (\boldsymbol{m} \boldsymbol{w}) \| \|\_{H} \right] \leq \delta^{-\frac{\alpha}{2}} \left[ M\_{m-1} + M\_{m-1} \right], \\ &\| \boldsymbol{u}'\_{0}(t) \| \|\_{H} \leq \left[ \| \boldsymbol{A}^{1/2} \boldsymbol{s} \left( t \right) \| \_{H \to H} \| \| \boldsymbol{A}^{1/2} \boldsymbol{u} (\boldsymbol{m} \boldsymbol{w}) \| \|\_{H} + \| \boldsymbol{c} \left( t \right) \| \_{H \to H} \| \| \boldsymbol{q}'(0) \| \|\_{H} \right] \leq M\_{m-1} + \tilde{M}\_{m-1}. \end{aligned}$$

Applying formulas (12) and (13) and estimates (3) and (6), we get

$$\|u\_1(t) - u\_0(t)\|\_H \le \|A^{-\frac{1}{2}}\|\_{H \to H}$$

$$\times \int\_{m\omega}^t \|A^{1/2}s(t-y)\|\_{H \to H} \|f(y, u\_0(y), u\_\mathcal{Y}(y-w), u(y-w))\|\_H dy \le \delta^{-\frac{1}{2}} \bar{M} \left(t - m\omega\right),$$

$$\|u\_1'(t) - u\_0'(t)\|\_H \le \int\_{m\omega}^t \|c(t-y)\|\_{H \to H} \|f(y, u\_0(y), u\_\mathcal{Y}(y-w), u(y-w))\|\_H dy \le \bar{M} \left(t - m\omega\right).$$

Using the triangle inequality, we get

$$\|\boldsymbol{u}\_{1}(t)\|\_{H} \leq \delta^{-\frac{1}{2}} \left[ \mathcal{M}\_{m-1} + \tilde{\mathcal{M}}\_{m-1} \right] + \delta^{-\frac{1}{2}} \bar{\mathcal{M}}\left(t - m\omega\right),$$

$$\|\boldsymbol{u}\_{1}^{\prime}(t)\|\_{H} \leq \mathcal{M}\_{m-1} + \tilde{\mathcal{M}}\_{m-1} + \bar{\mathcal{M}}\left(t - m\omega\right).$$

Applying formulas (9) and (12) and estimates (3), (6), and (7), we get

$$\|\|u\_2(t) - u\_1(t)\|\|\_{H} \le \|\|A^{-\frac{1}{2}}\|\|\_{H \to H}$$

$$\begin{split} \|\lambda \times \int\_{0}^{t} \|A^{1/2}\mathbf{s}\left(t - y\right)\|\_{H \to H} \|f(y, u\_{1}(y), u\_{y}(y - w), u(y - w)) - f(y, u\_{0}(y), u\_{y}(y - w), u(y - w))\|\_{H} dy \\ \leq \delta^{-\frac{1}{2}} L \int\_{m\omega}^{t} \|u\_{1}(y) - u\_{0}(y)\|\_{H} dy \leq \delta^{-1} L \tilde{M} \int\_{m\omega}^{t} (y - m\omega) \, dy = \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}L\_{1}(t - m\omega))^{2}}{2!}, \\ \|u\_{2}'(t) - u\_{1}'(t)\|\_{H} \end{split}$$

$$\begin{split} &\leq \int\_{m\omega}^{t} \|\boldsymbol{c}\left(t-\boldsymbol{y}\right)\|\_{H\rightarrow H} \|\boldsymbol{f}\left(\boldsymbol{y},\boldsymbol{u}\_{1}(\boldsymbol{y}),\boldsymbol{u}\_{\boldsymbol{y}}(\boldsymbol{y}-\boldsymbol{w}),\boldsymbol{u}(\boldsymbol{y}-\boldsymbol{w})\right) - \boldsymbol{f}\left(\boldsymbol{y},\boldsymbol{u}\_{0}(\boldsymbol{y}),\boldsymbol{u}\_{\boldsymbol{y}}(\boldsymbol{y}-\boldsymbol{w}),\boldsymbol{u}(\boldsymbol{y}-\boldsymbol{w})\right)\|\_{H} d\boldsymbol{y} \\ &\leq L \int\_{m\omega}^{t} \|\boldsymbol{u}\_{1}(\boldsymbol{y}) - \boldsymbol{u}\_{0}(\boldsymbol{y})\|\_{H} d\boldsymbol{y} \leq L\tilde{M}\delta^{-\frac{1}{2}} \int\_{m\omega}^{t} \left(\boldsymbol{y} - m\omega\right) d\boldsymbol{y} = \frac{\bar{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}\left(t - m\omega\right))^{2}}{2!}. \end{split}$$

Then

$$\|\boldsymbol{u}\_{2}(t)\|\_{H}\leq\delta^{-\frac{1}{2}}\left[\mathcal{M}\_{m-1}+\tilde{\mathcal{M}}\_{m-1}\right]+\delta^{-\frac{1}{2}}\bar{\mathcal{M}}\left(t-m\omega\right)+\frac{\bar{\mathcal{M}}}{L}\frac{(\delta^{-\frac{1}{2}}L\left(t-m\omega\right))^{2}}{2!},$$

$$\|\boldsymbol{u}\_{2}'(t)\|\_{H}\leq\mathcal{M}\_{m-1}+\tilde{\mathcal{M}}\_{m-1}+\bar{\mathcal{M}}\left(t-m\omega\right)+\frac{\bar{\mathcal{M}}}{L\delta^{-\frac{1}{2}}}\frac{(L\delta^{-\frac{1}{2}}\left(t-m\omega\right))^{2}}{2!}.$$

Let

$$\|\|u\_n(t) - u\_{n-1}(t)\|\|\_{H} \le \frac{\bar{M}}{L} \frac{(\delta^{-\frac{1}{2}}L\left(t - m\omega\right))^n}{n!}, \quad \|\|u\_n'(t) - u\_{n-1}'(t)\|\|\_{H} \le \frac{\bar{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}\left(t - m\omega\right))^n}{n!}.$$

and

$$\|\|u\_n(t)\|\|\_{H} \le \delta^{-\frac{1}{2}} \left[M\_{m-1} + \tilde{M}\_{m-1}\right] + \delta^{-\frac{1}{2}}\tilde{M}(t - m\omega) + \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}L(t - m\omega))^2}{2!} + \dots + \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}L(t - m\omega))^n}{n!},$$

$$\|u\_n'(t)\|\_{H} \le M\_{m-1} + \tilde{M}\_{m-1} + \tilde{M}(t - m\omega) + \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}(t - m\omega))^2}{2!} + \dots + \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}(t - m\omega))^n}{n!}.$$

$$\text{Then, we obtain }$$

$$\|u\_{n+1}(t) - u\_n(t)\|\_{H} \le \|A^{-\frac{1}{2}}\|\_{H \to H}$$

$$\begin{split} \lambda \times \int\_{m\omega}^{t} \|A^{1/2}s(t-y)\|\_{H\to H} \|f(y,u\_{n}(y),u\_{\mathcal{Y}}(y-w),u(y-w)) - f(y,u\_{n-1}(y),u\_{\mathcal{Y}}(y-w),u(y-w))\|\_{H} dy \\ \leq \delta^{-\frac{1}{2}}L \int\_{m\omega}^{t} \|u\_{n}(y) - u\_{n-1}(y)\|\_{H} dy \leq \delta^{-\frac{1}{2}}L \int\_{m\omega}^{t} \frac{\tilde{M}\left(\delta^{-\frac{1}{2}}L\left(y-m\omega\right)\right)^{n}}{n!} dy = \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}L\left(t-m\omega\right))^{n+1}}{(n+1)!}, \\ \|u\_{n+1}'(t) - u\_{n}'(t)\|\_{H} \end{split}$$

$$\begin{split} &\leq \int\_{m\omega}^{t} \|\boldsymbol{c}\left(t-\boldsymbol{y}\right)\|\_{H\rightarrow H} \|\boldsymbol{f}\left(\boldsymbol{y},\boldsymbol{u}\_{n}(\boldsymbol{y}),\boldsymbol{u}\_{\boldsymbol{y}}(\boldsymbol{y}-\boldsymbol{w}),\boldsymbol{u}(\boldsymbol{y}-\boldsymbol{w})\right) - \boldsymbol{f}\left(\boldsymbol{y},\boldsymbol{u}\_{n-1}(\boldsymbol{y}),\boldsymbol{u}\_{\boldsymbol{y}}(\boldsymbol{y}-\boldsymbol{w}),\boldsymbol{u}(\boldsymbol{y}-\boldsymbol{w})\right)\|\_{H} d\boldsymbol{y} \\ &\leq \int\_{m\omega}^{t} \boldsymbol{L} \|\boldsymbol{u}\_{n}(\boldsymbol{y}) - \boldsymbol{u}\_{n-1}(\boldsymbol{y})\|\_{H} ds \leq \int\_{m\omega}^{t} \boldsymbol{L} \frac{\bar{M}}{\bar{\boldsymbol{L}}} \frac{(\boldsymbol{L}\delta^{-\frac{1}{2}} \left(\boldsymbol{y} - m\omega\right))^{n}}{n!} d\boldsymbol{y} = \frac{\bar{M}}{\bar{\boldsymbol{L}}\delta^{-\frac{1}{2}}} \frac{(\boldsymbol{L}\delta^{-\frac{1}{2}} \left(t - m\omega\right))^{n+1}}{(n+1)!}. \end{split}$$

$$\|\|u\_{n+1}(t) - u\_n(t)\|\|\_{H} \le \frac{\bar{M}}{L} \frac{(\delta^{-\frac{1}{2}}L\left(t - m\omega\right))^{n+1}}{(n+1)!}, \ \|\|u'\_{n+1}(t) - u'\_n(t)\|\|\_{H} \le \frac{\bar{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}(t - m\omega))^{n+1}}{(n+1)!}$$

and

$$\|\|u\_{n+1}(t)\|\|\_{H} \leq \delta^{-\frac{1}{2}} \left[M\_{m-1} + \bar{M}\_{m-1}\right] + \delta^{-\frac{1}{2}}\bar{M}\left(t - m\omega\right) + \frac{\bar{M}}{L} \frac{(\delta^{-\frac{1}{2}}L(t - m\omega))^{2}}{2!} \cdot \dots + \frac{\bar{M}}{L} \frac{(\delta^{-\frac{1}{2}}L(t - m\omega))^{n+1}}{(n+1)!},$$

$$\|u'\_{n+1}(t)\|\|\_{H} \leq M\_{m-1} + \bar{M}\_{m-1} + \bar{M}\left(t - m\omega\right) + \frac{\bar{M}}{L^{s-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}(t - m\omega))^{2}}{2!} + \dots + \frac{\bar{M}}{L^{s-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}(t - m\omega))^{n+1}}{(n+1)!},$$

are true for any *n*, *n* ≥ 1 by mathematical induction. From that and formulas (14) and (15) it follows that

$$\|u(t)\|\_{H} \le \|u\_0(t)\|\_{H}$$

$$+ \sum\_{i=0}^{\infty} \|u\_{i+1}(t) - u\_i(t)\|\_{H} \le \delta^{-\frac{1}{2}} \left[M + \tilde{M}\right] + \sum\_{i=0}^{\infty} \frac{\tilde{M}}{L} \frac{(\delta^{-\frac{1}{2}}L\left(t - m\omega\right))^{i+1}}{(i+1)!}$$

$$\le \delta^{-\frac{1}{2}} \left[M\_{m-1} + \tilde{M}\_{m-1}\right] + \frac{\tilde{M}}{L} e^{\delta^{-\frac{1}{2}}L(t - m\omega)}, \ m\omega \le t \le (m+1)w,$$

$$\|u'(t)\|\_{H} \le \|u'\_0(t)\|\_{H} + \sum\_{i=0}^{\infty} \|u'\_{i+1}(t) - u'\_i(t)\|\_{H}$$

$$\le \delta^{-\frac{1}{2}} \left[M + \tilde{M}\right] + \sum\_{i=0}^{\infty} \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} \frac{(L\delta^{-\frac{1}{2}}(t - m\omega))^{i+1}}{(i+1)!}$$

$$\le M\_{m-1} + \tilde{M}\_{m-1} + \frac{\tilde{M}}{L\delta^{-\frac{1}{2}}} e^{L\delta^{-\frac{1}{2}}(t - m\omega)}, \ m\omega \le t \le (m+1)w$$

which proves the existence of a bounded solution of problem (1) in [*mω*,(*m* + 1) *w*] × *H*.

Now we will prove uniqueness of the bounded solution of the problem. Suppose that there is a bounded solution *v*(*t*) of problem (1) and *v*(*t*) = *u*(*t*). Denoting *z*(*t*) = *v*(*t*) − *u*(*t*) and using Equation (1), we get

$$\begin{cases} \frac{d^2z(t)}{dt^2} + Az(t) = f(t, v(t), v\_l(t-w), v(t-w)) - f(t, u(t), u\_l(t-w), u(t-w)), & t > 0, \\ z(t) = 0, \ -w \le t \le 0 \end{cases}$$

for *z*(*t*).

Let 0 ≤ *t* ≤ *w*. Since *v*(*t* − *w*) = *u*(*t* − *w*) = *ϕ*(*t* − *w*), we can write

$$\begin{cases} \frac{d^2z(t)}{dt^2} + Az(t) = f(t, v(t), q\_t(t-w), q(t-w)) - f(t, u(t), q\_t(t-w), q(t-w)), & t > 0, \\\ z(t) = 0, \ -w \le t \le 0. \end{cases}$$

Therefore,

$$z(t) = \int\_0^t s\left(t - y\right) \left[ f(y, v(y), \varphi\_y(y - w), \varphi(y - w)) - f(y, u(y), \varphi\_y(y - w), \varphi(y - w)) \right] dy.$$

Applying estimates (3) and (6), we get

$$||z(t)||\_H \le ||A^{-\frac{1}{2}}||\_{H \to H}$$

$$\begin{aligned} \|\lambda \times \int\_0^t \|A^{1/2}s(t-y)\| \|f(y,v(y),\rho\_\mathcal{Y}(y-w),\rho(y-w)) - f(y,u(y),\rho\_\mathcal{Y}(y-w),\rho(y-w))\|\_H dy \\ \leq L\delta^{-\frac{1}{2}} \int\_0^t \|v(y) - u(y)\|\_H ds \leq L\delta^{-\frac{1}{2}} \int\_0^t \|z(y)\|\_H dy. \end{aligned}$$

Applying the integral inequality, we get

$$\|\|z(t)\|\|\_{H} \le 0.$$

From that it follows that *z*(*t*) = 0 which proves the uniqueness of a bounded solution of problem (1) in [0, *w*] × *H*. Using the same method and mathematical induction, we can establish the uniqueness of a bounded solution of problem (1) in [0, ∞) × *H*. Theorem 1 is proved.

#### **3. Applications**

First, we consider the initial value problem for a semilinear hyperbolic equation with time delay and with nonlocal conditions

$$\begin{cases} \begin{aligned} u\_{tt}(t, \mathbf{x}) - (a(\mathbf{x}) u\_x(t, \mathbf{x}))\_x + \delta u(t, \mathbf{x}) &= f(t, \mathbf{x}, u(t, \mathbf{x}), u\_t(t - \mathbf{w}, \mathbf{x}), u(t - \mathbf{w}, \mathbf{x})), \\ &0 < t < \infty, \ \mathbf{x} \in (0, l), \\\\ u(t, \mathbf{x}) = \boldsymbol{\varrho}(t, \mathbf{x}), \ -\boldsymbol{\omega} &\leq t \leq 0, \ \mathbf{x} \in [0, l], \\\\ u(t, 0) = u(t, l), u\_x(t, 0) &= u\_x(t, l), \ 0 \leq t < \infty \end{aligned} \end{cases} \tag{17}$$

where *a*(*x*) and *ϕ*(*t*, *x*) are given sufficiently smooth functions, *δ* > 0 is the sufficiently large number. Suppose that *a*(*x*) ≥ *a* > 0 and *a*(*l*) = *a*(0).

**Theorem 2.** *Suppose the following hypotheses:*

*1. For any t*, −*w* ≤ *t* ≤ 0

$$\|\|\boldsymbol{\varrho}(t\_{\prime}.)\|\_{\mathcal{W}^{2}\_{2}[0,l]} \leq M\_{\prime} \quad \|\|\boldsymbol{\varrho}^{\prime}(t\_{\prime}.)\|\_{\mathcal{W}^{1}\_{2}[0,l]} \leq \tilde{M}.\tag{18}$$

*2. The function f* : [0, ∞) × (0, *l*) × *L*<sup>2</sup> [0, *l*] × *L*<sup>2</sup> [0, *l*] × *L*<sup>2</sup> [0, *l*] → *L*<sup>2</sup> [0, *l*] *be continuous and bounded, that is*

$$\|f(t, \cdot, u, v, w)\|\_{L\_2[0, l]} \le \overline{M} \tag{19}$$

*and Lipschitz condition is satisfied uniformly with respect to t*, *z*, *w*

$$\|\|f(t, \cdot, u, z, w) - f(t, \cdot, v, z, w)\|\|\_{L\_2[0, l]} \le L \|u - v\|\_{L\_2[0, l]}.$$

*3. Here and in future, M*, *M*:, *M*, Ł *are positive constants. Assume that all compatibility conditions are satisfied. Then there exists a unique solution to problem (17) which is bounded in* [0, ∞) × *L*<sup>2</sup> [0, *l*]*.*

The proof of Theorem 2 is based on the abstract Theorem 1, on the self-adjointness and positivity in *L*<sup>2</sup> [0, *l*] of a differential operator *A* defined by the formula

$$Au = -\frac{d}{dx}\left(a(x)\frac{du}{dx}\right) + \delta u\tag{20}$$

with domain *D*(*A*) = *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*<sup>2</sup> <sup>2</sup> [0, *l*] : *u* (0) = *u* (*l*), *u* (0) = *u* (*l*) [51] and on estimates

$$\|\|\boldsymbol{\varepsilon}\{\boldsymbol{t}\}\|\|\_{L\_2[0,l]\to L\_2[0,l]} \le 1,\\ \|\left(\boldsymbol{A}\right)^{\frac{1}{2}}\boldsymbol{s}\{\boldsymbol{t}\}\|\|\_{L\_2[0,l]\to L\_2[0,l]} \le 1, \ t \ge 0. \tag{21}$$

*Mathematics* **2019**, *7*, 1163

Second, we consider the initial value problem for a semilinear hyperbolic equation with time delay and with involution

$$\begin{cases} \begin{aligned} &u\_{tt}(t, \mathbf{x}) - (a(\mathbf{x})u\_x(t, \mathbf{x}))\_x + \delta u(t, \mathbf{x}) - \beta \left(a(-\mathbf{x})u\_x\left(t, -\mathbf{x}\right)\right)\_x \\ &= f(t, \mathbf{x}, u(t, \mathbf{x}), u\_t(t - \mathbf{w}, \mathbf{x}), u(t - \mathbf{w}, \mathbf{x})), \ 0 < t < \infty, \ \mathbf{x} \in (-l, l), \\\\ &u(t, \mathbf{x}) = \boldsymbol{\varrho}(t, \mathbf{x}), \ -\boldsymbol{\omega} \le t \le \mathbf{0}, \ \mathbf{x} \in [-l, l], \\\\ &u(t, -l) = u(t, l) = 0, \ 0 \le t < \infty, \end{aligned} \end{cases} \tag{22}$$

where *a*(*x*) and *ϕ*(*t*, *x*) are given sufficiently smooth functions, *δ* > 0 is the sufficiently large number. Suppose that *a* ≥ *a* (*x*) = *a* (−*x*) ≥ *δ* > 0, *δ* − *a* |*β*| ≥ 0.

**Theorem 3.** *Suppose the following hypotheses:*

*1. For any t*, −*w* ≤ *t* ≤ 0

$$\|\|\boldsymbol{\varrho}(t\_{\prime}.)\|\|\_{\mathcal{W}^{2}\_{2}[-l,l]} \leq \mathcal{M}\_{\prime} \; \left\|\left|\boldsymbol{\varrho}^{\prime}(t\_{\prime}.)\right\|\right\|\_{\mathcal{W}^{1}\_{2}[-l,l]} \leq \widetilde{\mathcal{M}}.$$

*2. The function f* : [0, ∞) × (−*l*, *l*) × *L*<sup>2</sup> [−*l*, *l*] × *L*<sup>2</sup> [−*l*, *l*] × *L*<sup>2</sup> [−*l*, *l*] → *L*<sup>2</sup> [−*l*, *l*] *be continuous and bounded, that is*

$$||f(t, \cdot, u, v, w)||\_{L\_2[-l, l]} \le \overline{M}$$

*and Lipschitz condition is satisfied uniformly with respect to t*, *z*, *w*

$$||f(t, \cdot, u, z, w) - f(t, v, z, w)||\_{L\_2[-l, l]} \le L ||u - v||\_{L\_2[-l, l]}.$$

*Assume that all compatibility conditions are satisfied. Then there exists a unique solution to problem (22) which is bounded in* [0, ∞) × *L*<sup>2</sup> [−*l*, *l*]*.*

The proof of Theorem 3 is based on the abstract Theorem 1, on the self-adjointness and positivity in *L*<sup>2</sup> [−*l*, *l*] of a differential operator *A* defined by the formula

$$Av(\mathbf{x}) = -\left(a(\mathbf{x})v\_{\mathbf{x}}(\mathbf{x})\_{\mathbf{x}} - \beta \left(a(-\mathbf{x})v\_{\mathbf{x}}(-\mathbf{x})\right)\_{\mathbf{x}} + \delta v\left(\mathbf{x}\right)\right)$$

with the domain *D*(*A*) = *<sup>u</sup>* <sup>∈</sup> *<sup>W</sup>*<sup>2</sup> <sup>2</sup> [−*l*, *l*] : *u* (−*l*) = *u* (*l*) = 0 [52] and on the estimate

$$||\mathfrak{c}\{\mathfrak{t}\}||\_{L\_2[-l,l]\to L\_2[-l,l]} \le \mathbf{1}, \; ||\mathfrak{t}\{\mathbf{A}\}||\_{L\_2[-l,l]\to L\_2[-l,l]} \le \mathbf{1}, \; \mathfrak{t} \ge 0.$$

Third, let <sup>Ω</sup> <sup>⊂</sup> *<sup>R</sup><sup>m</sup>* be a bounded open domain with smooth boundary *<sup>S</sup>*, <sup>Ω</sup> <sup>=</sup> <sup>Ω</sup> <sup>∪</sup> *<sup>S</sup>*. In [0, <sup>∞</sup>) <sup>×</sup> <sup>Ω</sup> we consider the initial boundary value problem for a multidimensional semilinear delay differential equation of hyperbolic type

$$\begin{cases} \frac{\partial^2 u(t, \mathbf{x})}{\partial t^2} - \sum\_{r=1}^m (a\_r(\mathbf{x}) u\_{\mathbf{x}}) \mathbf{x}\_r + \delta u(t, \mathbf{x}) \\\\ \quad = f(t, \mathbf{x}, u(t, \mathbf{x}), u\_t(t - w, \mathbf{x}), u(t - w, \mathbf{x})), \ 0 < t < \infty, \ \mathbf{x} = (\mathbf{x}\_1, \dots, \mathbf{x}\_m) \in \Omega, \\\\ u(t, \mathbf{x}) = \boldsymbol{\varrho}(t, \mathbf{x}), \ -\boldsymbol{\omega} \le t \le 0, \ \mathbf{x} \in \overline{\Omega}, \\\\ u(t, \mathbf{x}) = 0, \ \mathbf{x} \in \mathcal{S}, \ 0 \le t < \infty, \end{cases} \tag{23}$$

where *ar*(*x*) and *ϕ*(*t*, *x*) are given sufficiently smooth functions and *δ* > 0 is the sufficiently large number and *ar*(*x*) > 0.

**Theorem 4.** *Suppose the following hypotheses:*

*1. For any t*, −*w* ≤ *t* ≤ 0*,*

$$\|\|\boldsymbol{\varrho}(t\_{\prime}.)\|\|\_{\mathcal{W}^{2}\_{2}(\square)} \leq \mathcal{M}\_{\prime} \quad \|\|\boldsymbol{\varrho}^{\prime}(t\_{\prime}.)\|\|\_{\mathcal{W}^{1}\_{2}(\square)} \leq \widetilde{\mathcal{M}}.$$

*2. The function f* : [0, ∞) × *Q* × *L*2(Ω) × *L*2(Ω) × *L*2(Ω) → *L*2(Ω) *be continuous and bounded, that is*

$$\|f(t, u, v, w)\|\_{L\_2(\square)} \le M$$

*and Lipschitz condition is satisfied uniformly with respect to t*, *z*, *w*

$$\|\|f(t,u,z,w) - f(t,v,z,w)\|\|\_{L\_2(\overline{\Omega})} \le L \left\|u - v\right\|\_{L\_2(\overline{\Omega})}.$$

*Assume that all compatibility conditions are satisfied. Then there exists a unique solution to problem (23) which is bounded in* [0, ∞) × *L*2(Ω).

The proof of Theorem 4 is based on the abstract Theorem 1, on the self-adjointness and positivity in *L*2(Ω) of a differential operator *A* defined by the formula

$$Au(\mathbf{x}) = -\sum\_{r=1}^{m} (a\_r(\mathbf{x})u\_{\mathbf{x}\_r})\_{\mathbf{x}\_r} + \delta u(\mathbf{x})\tag{24}$$

with domain [53]

$$D(A) = \left\{ u(\mathbf{x}) : u(\mathbf{x}), \ u\_{\mathbf{x}\_r}(\mathbf{x}), \ (a\_r(\mathbf{x})u\_{\mathbf{x}\_r})\_{\mathbf{x}\_r} \in L\_2(\overline{\Omega}), \ 1 \le r \le m, \ u(\mathbf{x}) = 0, \ \mathbf{x} \in S \right\}$$

and on the estimate

$$\|\boldsymbol{c}\{\boldsymbol{t}\}\|\_{L\_{2}(\overline{\mathbf{T}})\to L\_{2}(\overline{\mathbf{T}})} \leq 1,\ \|\boldsymbol{(A})^{\frac{1}{2}}\boldsymbol{s}\{\boldsymbol{t}\}\|\_{L\_{2}(\overline{\mathbf{T}})\to L\_{2}(\overline{\mathbf{T}})} \leq 1,\ \boldsymbol{t}\geq \mathbf{0}\tag{25}$$

and on the following theorem on the coercivity inequality for the solution of the elliptic differential problem in *L*2(Ω).

**Theorem 5.** *For the solutions of the elliptic differential problem*

$$\begin{cases} A^x u(\mathbf{x}) = \omega(\mathbf{x}), & \mathbf{x} \in \Omega, \\ u(\mathbf{x}) = 0, & \mathbf{x} \in \mathcal{S}, \end{cases}$$

*the coercivity inequality [53]*

$$\sum\_{r=1}^{m} \left|| \boldsymbol{u}\_{\mathcal{X}\_r \mathcal{X}\_r} \right||\_{L\_2(\overline{\Omega})} \leq M\_1 ||\boldsymbol{\omega}||\_{L\_2(\overline{\Omega})}.$$

*is satisfied. Here M*<sup>1</sup> *is independent of ω*(*x*).

*Mathematics* **2019**, *7*, 1163

Fourth, in [0, ∞) × Ω we consider the initial boundary value problem for a multidimensional semilinear delay hyperbolic equation

$$\begin{cases} \begin{aligned} \frac{\partial^2 u(t, \mathbf{x})}{\partial t^2} - \sum\_{r=1}^m (a\_r(\mathbf{x}) u\_{\mathbf{x}\_r}) \mathbf{x}\_r + \delta u(t, \mathbf{x}) \\\\ \mathbf{x} = f(t, \mathbf{x}, u(t, \mathbf{x}), u\_t(t - w, \mathbf{x}), u(t - w, \mathbf{x})), \ \mathbf{x} = (\mathbf{x}\_1, \dots, \mathbf{x}\_m) \in \Omega, \\\\ u(t, \mathbf{x}) = \varrho(t, \mathbf{x}), \ -\omega \le t \le 0, \ \mathbf{x} \in \overline{\Omega}, \\\\ \frac{\partial u}{\partial \overline{\mathcal{V}}}(t, \mathbf{x}) = 0, \ \mathbf{x} \in \mathcal{S}, \ 0 \le t < \infty, \end{aligned} \end{cases} \tag{26}$$

where *ar*(*x*) and *ϕ*(*t*, *x*) are given sufficiently smooth functions and *δ* > 0 is the sufficiently large number and *ar*(*x*) > 0. Here, −→*p* is the normal vector to Ω.

**Theorem 6.** *Suppose that assumptions of Theorem 4 hold. Assume that all compatibility conditions are satisfied. Then same stability estimates for the solution of (26) hold.*

The proof of Theorem 6 is based on the abstract Theorem 3, on the self-adjointness and positivity of a differential operator *A* defined by the formula

$$Au(\mathbf{x}) = -\sum\_{r=1}^{m} (a\_r(\mathbf{x})u\_{\mathbf{x}\_r})\_{\mathbf{x}\_r} + \delta u(\mathbf{x}) \tag{27}$$

with domain

$$D(A) = \left\{ u(\mathbf{x}) : u(\mathbf{x}), \ u\_{\mathbf{x}\_r}(\mathbf{x}), \ (a\_r(\mathbf{x})u\_{\mathbf{x}\_r})\_{\mathbf{x}\_r} \in L\_2(\overline{\Omega}), \ 1 \le r \le m\_r \ \frac{\partial}{\partial \overline{p'}} u(\mathbf{x}) = 0, \ \mathbf{x} \in S \right\}$$

in *L*2(Ω) and on the following theorem on the coercivity inequality for the solution of the elliptic differential problem in *L*2(Ω).

**Theorem 7.** *For the solutions of the elliptic differential problem*

$$\begin{cases} A^x u(x) = \omega(x), & x \in \Omega, \\ \frac{\partial u(x)}{\partial \overline{p'}} = 0, & x \in S, \end{cases}$$

*the coercivity inequality [53]*

$$\sum\_{r=1}^{m} ||u\_{\mathbf{x}\_r \mathbf{x}\_r}||\_{L\_2(\mathbf{T})} \le M\_1(\delta) ||\omega||\_{L\_2(\mathbf{T})}$$

*is satisfied. Here M*1(*δ*) *is independent of ω*(*x*).
