**5. Applications of Theorem 8**

First, we consider the initial value problem (17) for the one dimensional semininear hyperbolic differential equation with time delay term and with nonlocal conditions.

The discretization of problem (17) is provided in two steps. To the differential operator *A* generated by problem (17), we assign the difference operator *A<sup>x</sup> <sup>h</sup>* by the formula

$$A\_h^x \varrho\_h(\mathbf{x}) = \{-(a(\mathbf{x})\varrho\_{\overline{\mathbf{x}}})\_{\mathbf{x},r} + \delta \varrho\_r(\mathbf{x})\}\_1^{K-1},\tag{41}$$

acting in the space of grid functions *<sup>ϕ</sup>h*(*x*) = {*ϕr*}*<sup>K</sup>* <sup>0</sup> satisfying the conditions *ϕ*<sup>0</sup> = *ϕK*, *ϕ*<sup>1</sup> − *ϕ*<sup>0</sup> = *<sup>ϕ</sup><sup>K</sup>* <sup>−</sup> *<sup>ϕ</sup>K*−1. It is known that *<sup>A</sup><sup>x</sup> <sup>h</sup>* is a self-adjoint positive definite operator in *L*2*<sup>h</sup>* = *L*2([0, *l*]*h*) [51]. With the help of *A<sup>x</sup> <sup>h</sup>*, we arrive at the initial value problem

$$\begin{cases} \begin{aligned} \frac{d^2 u^h(t, \mathbf{x})}{dt^2} + A^x\_h u^h(t, \mathbf{x}) &= f^h(t, \mathbf{x}, u^h(t, \mathbf{x}), u^h\_t(t - w, \mathbf{x}), u^h(t - w, \mathbf{x})), \\\\ 0 < t < \infty, &\ge \in [0, l]\_h \\\\ u^h(t, \mathbf{x}) &= \mathbf{p}^h(t, \mathbf{x}), \ -\omega \le t \le 0, \ \mathbf{x} \in [0, l]\_h. \end{aligned} \end{cases} \tag{42}$$

In the second step, we replace problem (42) by first order of accuracy difference scheme (28)

$$\begin{cases} \frac{\boldsymbol{u}\_{k+1}^{h}(\boldsymbol{x}) - 2\boldsymbol{u}\_{k}^{h}(\boldsymbol{x}) + \boldsymbol{u}\_{k-1}^{h}(\boldsymbol{x})}{\tau^{2}} + A\_{h}^{x}\boldsymbol{u}\_{k+1}^{h}(\boldsymbol{x}) = f(t\_{k}, \boldsymbol{x}, \boldsymbol{u}\_{k}^{h}(\boldsymbol{x}), \frac{\boldsymbol{u}\_{k-N}^{h}(\boldsymbol{x}) - \boldsymbol{u}\_{k-N-1}^{h}(\boldsymbol{x})}{\tau}, \boldsymbol{u}\_{k-N}^{h}(\boldsymbol{x})), \\\\ \quad t\_{k} = k\tau, \ 1 \le k < \infty, \ N\tau = \omega, \\\\ \quad \left(I + \tau^{2}A\_{h}^{x}\right)\frac{\boldsymbol{u}\_{k+1}^{h}(\boldsymbol{x}) - \boldsymbol{u}\_{k}^{h}(\boldsymbol{x})}{\tau} = \frac{\boldsymbol{u}\_{k}^{h}(\boldsymbol{x}) - \boldsymbol{u}\_{k-1}^{h}(\boldsymbol{x})}{\tau}, k = mN, m = 0, 1, \ldots, \end{cases} \tag{43}$$
 
$$\boldsymbol{u}\_{k}^{h}(\boldsymbol{x}) = \boldsymbol{p}^{h}(t\_{k}, \boldsymbol{x}), \ t\_{k} = k\tau, \ -N \le k \le 0, \ \boldsymbol{x} \in [0, l]\_{l}.$$

**Theorem 9.** *Suppose that assumptions of Theorem 2 hold. Then, there exists a unique solution* \$ *uh k* %<sup>∞</sup> *<sup>k</sup>*=<sup>0</sup> *of difference scheme (43) which is bounded in* [*mω*,(*m* + 1) *w*]*<sup>τ</sup>* × *L*2*h*, *m* = 0, 1, ··· *of uniformly with respect to τ and h.*

**Proof.** Difference scheme (43) can be written in abstract form (28) in a Hilbert space *L*2*<sup>h</sup>* = *L*2([0, *l*]*h*) with self-adjoint positive definite operator *Ah* = *A<sup>x</sup> <sup>h</sup>* by formula (41). Here,

*f*(*tk*, *x*, *u<sup>h</sup> <sup>k</sup>* (*x*), *<sup>u</sup><sup>h</sup> <sup>k</sup>*−*N*(*x*)−*u<sup>h</sup> <sup>k</sup>*−*N*−1(*x*) *<sup>τ</sup>* , *<sup>u</sup><sup>h</sup> <sup>k</sup>*−*<sup>N</sup>* (*x*)) and *<sup>u</sup><sup>h</sup> <sup>k</sup>* = *<sup>u</sup><sup>h</sup> <sup>k</sup>* (*x*) are abstract mesh functions defined on [0, *l*]*<sup>h</sup>* with the values in *H* = *L*2*h*. Therefore, the proof of Theorem 9 is based on Theorem 8 and symmetry properties of the difference operator *A<sup>x</sup> h*.

Second, we study the initial nonlocal boundary value problem (22) for one dimensional semilinear delay hyperbolic equations type with involution. The discretization of problem (22) is provided in two steps. To the differential operator *A* generated by problem (22), we assign the difference operator *Ax <sup>h</sup>* by the formula

$$A\_h^x \boldsymbol{q}^h(\mathbf{x}) = \left\{-(a(\mathbf{x})\boldsymbol{q}\_{\overline{\mathbf{x}}}(\mathbf{x}))\_{\mathbf{x},r} - \beta(a(-\mathbf{x})\boldsymbol{q}\_{\overline{\mathbf{x}}}(-\mathbf{x}))\_{\mathbf{x},r} + \delta \boldsymbol{q}^r(\mathbf{x})\right\}\_{-M+1'}^{M-1} \tag{44}$$

acting in the space of grid functions *<sup>ϕ</sup>h*(*x*) = {*ϕr*}*<sup>M</sup>* <sup>−</sup>*<sup>M</sup>* satisfying the conditions *<sup>ϕ</sup>*−*<sup>M</sup>* <sup>=</sup> *<sup>ϕ</sup><sup>M</sup>* <sup>=</sup> 0. It is known that *A<sup>x</sup> <sup>h</sup>* is a self-adjoint positive definite operator in *L*2*<sup>h</sup>* = *L*2([−*l*, *l*]*h*) [52]. With the help of *Ax <sup>h</sup>*, we arrive at the initial value problem

$$\begin{cases} \frac{d^2 u^h(t, \mathbf{x})}{dt^2} + A\_h^x u^h(t, \mathbf{x}) = f^h(t, \mathbf{x}, u^h(t, \mathbf{x}), u\_t^h(t - w, \mathbf{x}), u^h(t - w, \mathbf{x})), \\\\ 0 < t < \infty, \ \mathbf{x} \in [-l\_\prime, l]\_{h\prime} \\\\ u^h(t, \mathbf{x}) = \boldsymbol{q}^h(t, \mathbf{x}), \ -\boldsymbol{\omega} \le t \le 0, \ \mathbf{x} \in [-l\_\prime, l]\_h. \end{cases} \tag{45}$$

In the second step, we replace problem (45) by first order of accuracy difference scheme (28)

$$\begin{cases} \frac{\boldsymbol{u}\_{k+1}^{h}(\boldsymbol{x}) - 2\boldsymbol{u}\_{k}^{h}(\boldsymbol{x}) + \boldsymbol{u}\_{k-1}^{h}(\boldsymbol{x})}{\tau^{2}} + A\_{h}^{x}\boldsymbol{u}\_{k+1}^{h}(\boldsymbol{x}) = \boldsymbol{f}(t\_{k}, \boldsymbol{x}, \boldsymbol{u}\_{k}^{h}(\boldsymbol{x}), \frac{\boldsymbol{u}\_{k-N}^{h}(\boldsymbol{x}) - \boldsymbol{u}\_{k-N-1}^{h}(\boldsymbol{x})}{\tau}, \boldsymbol{u}\_{k-N}^{h}(\boldsymbol{x})), \\\\ \boldsymbol{t}\_{k} = k\tau, \ 1 \le k < \infty, \ N\tau = \omega, \\\\ \left(\boldsymbol{I} + \tau^{2}A\_{h}^{x}\right) \frac{\boldsymbol{u}\_{k+1}^{h}(\boldsymbol{x}) - \boldsymbol{u}\_{k}^{h}(\boldsymbol{x})}{\tau} = \frac{\boldsymbol{u}\_{k}^{h}(\boldsymbol{x}) - \boldsymbol{u}\_{k-1}^{h}(\boldsymbol{x})}{\tau}, \ k = mN, \ m = 0, 1, \ldots, \end{cases} \tag{46}$$
 
$$\boldsymbol{u}\_{k}^{h}(\boldsymbol{x}) = \boldsymbol{q}^{h}(t\_{k}, \boldsymbol{x}), \ t\_{k} = k\tau, \ -N \le k \le 0, \ \boldsymbol{x} \in [-l, l]\_{h}.$$

**Theorem 10.** *Suppose that assumptions of Theorem 3 hold. Then, there exists a unique solution* \$ *uh k* %<sup>∞</sup> *k*=0 *of difference scheme (46) which is bounded in* [*mω*,(*m* + 1) *w*]*<sup>τ</sup>* × *L*2*h*, *m* = 0, 1, ··· *uniformly with respect to τ and h.*

**Proof.** Difference scheme (46) can be written in abstract form (28) in a Hilbert space *L*2*<sup>h</sup>* = *L*2([−*l*, *l*]*h*) with self-adjoint positive definite operator *Ah* = *A<sup>x</sup> <sup>h</sup>* by formula (44). Here,

*f*(*tk*, *x*, *u<sup>h</sup> <sup>k</sup>* (*x*), *<sup>u</sup><sup>h</sup> <sup>k</sup>*−*N*(*x*)−*u<sup>h</sup> <sup>k</sup>*−*N*−1(*x*) *<sup>τ</sup>* , *<sup>u</sup><sup>h</sup> <sup>k</sup>*−*<sup>N</sup>* (*x*)) and *<sup>u</sup><sup>h</sup> <sup>k</sup>* = *<sup>u</sup><sup>h</sup> <sup>k</sup>* (*x*) are abstract mesh functions defined on [−*l*, *l*]*<sup>h</sup>* with the values in *H* = *L*2*h*. Therefore, the proof of Theorem 10 is based on Theorem 8 and symmetry properties of the difference operator *A<sup>x</sup> h*.

Third, we study the initial boundary value problem (23) for multidimensional semilinear delay hyperbolic equations.

The discretization of problem (23) is provided in two steps. In the first step, here and in future, we define the grid space

$$\begin{aligned} \overline{\Omega}\_{\text{fl}} &= \{ \mathbf{x} = \mathbf{x}\_{\text{r}} = (h\_1 j\_1, \dots, h\_m j\_m), \ j = (j\_1, \dots, j\_m), \ 0 \le j\_{\text{r}} \le N\_{\text{r}}, \\\mathcal{N}\_{\text{l}} h\_{\text{l}} &= 1, \ \mathbf{r} = 1, \ \mathbf{\cdot} &= \mathbf{1}, \ \mathbf{\cdot} &= \overline{\Omega}\_{\text{l}} \cap \Omega, \ \mathcal{S}\_{\text{l}} = \overline{\Omega}\_{\text{l}} \cap \mathcal{S}. \end{aligned}$$

We introduce the Banach space *L*2*<sup>h</sup>* = *L*2(Ω*h*) of the grid functions *ϕh*(*x*) = {*ϕ*(*h*1*r*1, ..., *hmrm*)} defined on Ω*h*, equipped with the norm

$$||q\_h||\_{L\_{2h}} = \left(\sum\_{\mathbf{x}\in\overline{\mathbf{T}}\_h} |q\_h(\mathbf{x})|^2 \, h\_1 \, \cdots \, h\_m\right)^{1/2}$$

to the differential operator *A* generated by problem (23), we assign the difference operator *A<sup>x</sup> <sup>h</sup>* by the formula

$$A\_h^x u\_h(\mathbf{x}) = -\sum\_{r=1}^m \left( a\_r(\mathbf{x}) u\_{\overline{x}\_r, h} \right)\_{\mathbf{x}\_r, \mathbf{j}\_r} \tag{47}$$

acting in the space of grid functions *uh*(*x*), satisfying the conditions *uh*(*x*) = 0(∀ *x* ∈ *Sh*). It is known that *A<sup>x</sup> <sup>h</sup>* is a self-adjoint positive definite operator in *<sup>L</sup>*2*h*. With the help of *<sup>A</sup><sup>x</sup> <sup>h</sup>*, we arrive at the initial value problem

$$\begin{cases} \begin{aligned} \frac{d^2 u\_h(t, \mathbf{x})}{dt^2} + A\_h^x u\_h(t, \mathbf{x}) &= f^h(t, \mathbf{x}, u^h(t, \mathbf{x}), u\_h^h(t - w, \mathbf{x}), u^h(t - w, \mathbf{x})), \\\\ 0 < t < \infty, &\ge \Omega\_h, \\\\ u\_h(t, \mathbf{x}) &= \varphi\_h(t, \mathbf{x}), \ -\omega \le t \le 0, \ \mathbf{x} \in \overline{\Omega}\_h. \end{aligned} \end{cases} \tag{48}$$

In the second step, we replace problem (48) by first order of accuracy difference scheme (28)

$$\begin{cases} \frac{\boldsymbol{u}\_{k}^{k+1}(\boldsymbol{x}) - 2\boldsymbol{u}\_{h}^{k}(\boldsymbol{x}) + \boldsymbol{u}\_{h}^{k-1}(\boldsymbol{x})}{\tau^{2}} + A\_{h}^{x}\boldsymbol{u}\_{h}^{k+1}(\boldsymbol{x}) = f(t\_{k}, \boldsymbol{x}, \boldsymbol{u}\_{h}^{k}(\boldsymbol{x}), \frac{\boldsymbol{u}\_{k-N}^{k}(\boldsymbol{x}) - \boldsymbol{u}\_{k-N-1}^{k}(\boldsymbol{x})}{\tau}, \boldsymbol{u}\_{k-N}^{k}(\boldsymbol{x})), \\\\ \quad t\_{k} = k\tau, \ 1 \le k < \infty, \ N\tau = \omega, \ \boldsymbol{x} \in \Omega\_{h}. \\\\ \quad \left(I + \tau^{2}A\_{h}^{x}\right)\frac{\boldsymbol{u}\_{h}^{k+1}(\boldsymbol{x}) - \boldsymbol{u}\_{h}^{k}(\boldsymbol{x})}{\tau} = \frac{\boldsymbol{u}\_{h}^{k}(\boldsymbol{x}) - \boldsymbol{u}\_{h}^{k-1}(\boldsymbol{x})}{\tau}, \ k = mN, \ m = 0, 1, \ldots, \end{cases} \tag{49}$$

**Theorem 11.** *Suppose that assumptions of Theorem 4 hold. Then, there exists a unique solution* \$ *uh k* %<sup>∞</sup> *k*=0 *of difference scheme (49) which is bounded in* [*mω*,(*m* + 1) *w*]*<sup>τ</sup>* × *L*2*h*, *m* = 0, 1, ··· *uniformly with respect to τ and h.*

**Proof.** Difference scheme (49) can be written in abstract form (28) in a Hilbert space *L*2*<sup>h</sup>* = *L*2(Ω*h*) with self-adjoint positive definite operator *Ah* = *A<sup>x</sup> <sup>h</sup>* by formula (47). Here, *f*(*tk*, *x*, *u<sup>h</sup> <sup>k</sup>* (*x*), *<sup>u</sup><sup>h</sup> <sup>k</sup>*−*N*(*x*)−*u<sup>h</sup> <sup>k</sup>*−*N*−1(*x*) *<sup>τ</sup>* , *<sup>u</sup><sup>h</sup> <sup>k</sup>*−*<sup>N</sup>* (*x*)) and *<sup>u</sup><sup>h</sup> <sup>k</sup>* = *<sup>u</sup><sup>h</sup> <sup>k</sup>* (*x*) are abstract mesh functions defined on Ω*<sup>h</sup>* with the values in *H* = *L*2*h*. Therefore, the proof of Theorems 11 is based on the abstract Theorem 8 and symmetry properties of the difference operator *A<sup>x</sup> <sup>h</sup>* defined by formula (47) and the following theorem on coercivity inequality for the solution of the elliptic problem in *L*2*<sup>h</sup>* [53].

**Theorem 12.** *For the solutions of the elliptic difference problem*

$$\begin{cases} A\_{\boldsymbol{h}}^{\boldsymbol{x}} \boldsymbol{u}^{\boldsymbol{h}}(\boldsymbol{x}) = \boldsymbol{\omega}^{\boldsymbol{h}}(\boldsymbol{x}), \; \boldsymbol{x} \in \Omega\_{\boldsymbol{h}\prime} \\\\ \boldsymbol{u}^{\boldsymbol{h}}(\boldsymbol{x}) = 0, \; \boldsymbol{x} \in \mathcal{S}\_{\boldsymbol{h}\prime} \end{cases}$$

*the coercivity inequality*

$$\sum\_{r=1}^n \left\| \left| \boldsymbol{\omega}^{\mathrm{h}}\_{\,^{\mathrm{x}\_r \,\mathrm{yr}}} \right| \right\|\_{L\_{2h}} \le M\_1 \left\| \left| \boldsymbol{\omega}^{\mathrm{h}} \right| \right\|\_{L\_{2h}}$$

*is satisfied, where M*<sup>1</sup> *does not depend on h and ωh*.

Fourth, we study the initial boundary value problem (26) for multidimensional semilinear delay hyperbolic equations. The discretization of problem (23) is provided in two steps. To the differential operator *A* generated by problem (26), we assign the difference operator *A<sup>x</sup> <sup>h</sup>* by the formula

$$A\_h^x u^h(\mathbf{x}) = -\sum\_{r=1}^m \left( a\_r(\mathbf{x}) u\_{\overline{\mathbf{x}}\_r}^h \right)\_{\mathbf{x}\_r, \mathbf{j}\_r} + \delta u^h(\mathbf{x}) \tag{50}$$

acting in the space of grid functions *<sup>u</sup>h*(*x*), satisfying the conditions *<sup>D</sup>huh*(*x*) = <sup>0</sup> (<sup>∀</sup> *<sup>x</sup>* <sup>∈</sup> *Sh*). Here *<sup>D</sup><sup>h</sup>* is the approximation of operator *<sup>∂</sup> <sup>∂</sup>*−→*<sup>p</sup>* . With the help of *<sup>A</sup><sup>x</sup> <sup>h</sup>*, we arrive at the initial value problem (48). In the second step, we replace problem (48) by first order of accuracy difference scheme (28), we get Equation (49).

**Theorem 13.** *Suppose that assumptions of Theorem 6 hold. Then, there exists a unique solution* \$ *uh k* %<sup>∞</sup> *k*=0 *of difference scheme (49) which is bounded in* [*mω*,(*m* + 1) *w*]*<sup>τ</sup>* × *L*2*h*, *m* = 0, 1, ··· *uniformly with respect to τ and h.*

**Proof.** Difference scheme (49) can be written in abstract form (28) in a Hilbert space *L*2*<sup>h</sup>* = *L*2(Ω*h*) with self-adjoint positive definite operator *Ah* = *A<sup>x</sup> <sup>h</sup>* by formula (50). Here, *f*(*tk*, *x*, *u<sup>h</sup> <sup>k</sup>* (*x*), *<sup>u</sup><sup>h</sup> <sup>k</sup>*−*N*(*x*)−*u<sup>h</sup> <sup>k</sup>*−*N*−1(*x*) *<sup>τ</sup>* , *<sup>u</sup><sup>h</sup> <sup>k</sup>*−*<sup>N</sup>* (*x*)) and *<sup>u</sup><sup>h</sup> <sup>k</sup>* = *<sup>u</sup><sup>h</sup> <sup>k</sup>* (*x*) are abstract mesh functions defined on Ω*<sup>h</sup>* with the values in *H* = *L*2*h*. Therefore, the proof of Theorems 13 is based on the abstract Theorem 8 and symmetry properties of the difference operator *A<sup>x</sup> <sup>h</sup>* defined by formula (50) and the following theorem on coercivity inequality for the solution of the elliptic problem in *L*2*<sup>h</sup>* [53].

**Theorem 14.** *For the solutions of the elliptic difference problem*

$$\begin{cases} A\_h^x u^h(\mathbf{x}) = \omega^h(\mathbf{x}), \; \mathbf{x} \in \Omega\_{h'} \\\\ D^h u^h(\mathbf{x}) = 0, \; \mathbf{x} \in S\_{h'} \end{cases}$$

*the coercivity inequality*

$$\sum\_{r=1}^n \left\| \left| \boldsymbol{\omega}^{\mathrm{li}}\_{\boldsymbol{x}\_r, \boldsymbol{x} \overline{\boldsymbol{\sigma}}} \right| \right\|\_{L\_{2h}} \le M\_2 ||\boldsymbol{\omega}^{\mathrm{li}}||\_{L\_{2h}}$$

*is satisfied, where M*<sup>2</sup> *does not depend on h and ωh*.

## **6. Numerical Experiments**

In general, it is not able to get the exact solution of semilinear hyperbolic problems. Therefore, numerical results for the solution of difference schemes for one and two dimensional semilinear hyperbolic equations with time delay are presented. These results fit with the theoretical results perfectly.
