Bounded Solutions of Semilinear Time Delay Hyperbolic Differential and Difference Equations

: In this paper, we study the initial value problem for a semilinear delay hyperbolic equation in Hilbert spaces with a self-adjoint positive deﬁnite operator. The mean theorem on the existence and uniqueness of a bounded solution of this differential problem for a semilinear hyperbolic equation with unbounded time delay term is established. In applications, the existence and uniqueness of bounded solutions of four problems for semilinear hyperbolic equations with time delay in unbounded term are obtained. For the approximate solution of this abstract differential problem, the two-step difference scheme of a ﬁrst order of accuracy is presented. The mean theorem on the existence and uniqueness of a uniformly bounded solution of this difference scheme with respect to time stepsize is established. In applications, the existence and uniqueness of a uniformly bounded solutions with respect to time and space stepsizes of difference schemes for four semilinear partial differential equations with time delay in unbounded term are obtained. In general, it is not possible to get the exact solution of semilinear hyperbolic equations with unbounded time delay term. Therefore, numerical results for the solution of difference schemes for one and two dimensional semilinear hyperbolic equation with time delay are presented. Finally, some numerical examples are given to conﬁrm the theoretical analysis.

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.
A large cycle of works on difference schemes for hyperbolic equations (see, e.g., [38][39][40][41][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||A h || → 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][46][47][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      d 2 u(t) dt 2 + Au(t) = f (t, u(t), u t (t − w), u(t − w)), t > 0, u(t) = ϕ(t), −ω ≤ t ≤ 0 (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: i. u(t) is twice continuously differentiable function on the interval [0, ∞), f (t, u(t), u t (t − w), u(t − w)) is continuous and bounded function on [0, ∞) ii. The element u(t) belongs to D(A) for all t ∈ [0, ∞), and the function Au(t) is continuous on the interval [0, ∞). iii. u(t) satisfies the equation and initial conditions in Equation (1).
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.

Main Existence and Uniqueness Theorem of the Differential Problem
Throughout this paper, c(t) and s(t) are operator-functions defined by formulas We will give some auxiliary statements which will be useful in the sequel.
Recall that the norm A H→H of a bounded operator A : H → H is by definition the smallest number C for which estimate

Lemma 1.
For t ≥ 0, the following estimates hold: 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]) 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 in [0, ∞) × H and the application of successive approximations. Note that on (m − 1)w ≤ t ≤ mw, m = 1,2,..., u t (t − w) and u(t − w) are given. Therefore, the recursive formula for the solution of problem (1) is Besides let f : [0, ∞) × H × H × H −→ H be continuous and bounded function, that is in [0, ∞) × H × H × H and Lipschitz condition holds uniformly with respect to t, v and z Here Proof. Let 0 ≤ t ≤ ω. Then, according to Equation (4), we get for all i = 1, 2, .... Therefore, where Applying the triangle inequality and estimates (3) and (5), we get Applying formulas (8) and (9) and the triangle inequality and estimates (3) and (6), we get Using the triangle inequality, we get Applying formulas (8) and (9) and estimates (3), (6), and (7), we get Then Then, we obtain Therefore, are true for any n, n ≥ 1 by mathematical induction. In a similar manner, for any n, we can obtain From that and formulas (10) and (11) it follows that which proves the existence of a bounded solution of problem (1) Let mω ≤ t ≤ (m + 1) ω, m = 1, 2, .... Then, according to Equation (4), we can write Therefore, where Assume that problem (1) has a bounded solution in [(m − 1)ω, mw] × H and Applying estimates (3) and (16), we get Applying formulas (12) and (13) and estimates (3) and (6), we get Using the triangle inequality, we get Applying formulas (9) and (12) and estimates (3), (6), and (7), we get Then Then, we obtain Therefore, are true for any n, n ≥ 1 by mathematical induction. From that and formulas (14) and (15) it follows that 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 Therefore, Applying estimates (3) and (6), we get Applying the integral inequality, we get 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.

Applications
First, we consider the initial value problem for a semilinear hyperbolic equation with time delay and with nonlocal conditions 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).
and Lipschitz condition is satisfied uniformly with respect to t, z, w

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) The proof of Theorem 2 is based on the abstract Theorem 1, on the self-adjointness and positivity in L 2 [0, l] of a differential operator A defined by the formula [51] and on estimates Second, we consider the initial value problem for a semilinear hyperbolic equation with time delay and with involution where a(x) and ϕ(t, x) are given sufficiently smooth functions, δ > 0 is the sufficiently large number.
Theorem 3. Suppose the following hypotheses: and Lipschitz condition is satisfied uniformly with respect to t, z, w Assume that all compatibility conditions are satisfied. Then there exists a unique solution to problem (22) The proof of Theorem 3 is based on the abstract Theorem 1, on the self-adjointness and positivity in L 2 [−l, l] of a differential operator A defined by the formula and on the estimate Third, let Ω ⊂ R m be a bounded open domain with smooth boundary S, Ω = Ω ∪ S. In [0, ∞) × Ω we consider the initial boundary value problem for a multidimensional semilinear delay differential equation of hyperbolic type where a r (x) and ϕ(t, x) are given sufficiently smooth functions and δ > 0 is the sufficiently large number and a r (x) > 0.
Theorem 4. Suppose the following hypotheses: and Lipschitz condition is satisfied uniformly with respect to t, z, w Assume that all compatibility conditions are satisfied. Then there exists a unique solution to problem (23) 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 with domain [53] and on the estimate 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 Fourth, in [0, ∞) × Ω we consider the initial boundary value problem for a multidimensional semilinear delay hyperbolic equation where a r (x) and ϕ(t, x) are given sufficiently smooth functions and δ > 0 is the sufficiently large number and a r (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 with domain 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

The Main Theorem on Existence and Uniqueness of a Uniformly Bounded Solution of the Difference Scheme
In the present section for the approximate solution of Equation (1) we will study the first order of accuracy difference scheme The approach of proof of the theorem on the existence and uniqueness of a bounded solution of difference scheme (28) uniformly with respect to τ is based on reducing this difference scheme to an equivalent nonlinear equations. Equivalent nonlinear equations for the difference scheme (28) is in C τ (H) and the use of successive approximations. Here and in future R = (I + τiA The recursive formula for the solution of difference scheme (28) is From Equations (29) and (30) it follows where Let us give the lemma that will be needed below.

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 x h by the formula acting in the space of grid functions ϕ h (x) = {ϕ r } K 0 satisfying the conditions [51]. With the help of A x h , we arrive at the initial value problem In the second step, we replace problem (42) by first order of accuracy difference scheme (28) Theorem 9. Suppose that assumptions of Theorem 2 hold. Then, there exists a unique solution u h k ∞ k=0 of difference scheme (43) which is bounded in [mω, (m + 1) w] τ × L 2h , 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 2h = L 2 ([0, l] h ) with self-adjoint positive definite operator A h = A x h by formula (41). Here, , u h k−N (x)) and u h k = u h k (x) are abstract mesh functions defined on [0, l] h with the values in H = L 2h . Therefore, the proof of Theorem 9 is based on Theorem 8 and symmetry properties of the difference operator A x 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 A x h by the formula acting in the space of grid functions [52]. With the help of A x h , we arrive at the initial value problem In the second step, we replace problem (45) by first order of accuracy difference scheme (28) Theorem 10. Suppose that assumptions of Theorem 3 hold. Then, there exists a unique solution u h k ∞ k=0 of difference scheme (46) which is bounded in [mω, (m + 1) w] τ × L 2h , 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 2h = L 2 ([−l, l] h ) with self-adjoint positive definite operator A h = A x h by formula (44). Here, 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 Ω h = {x = x r = (h 1 j 1 , · · · , h m j m ) , j = (j 1 , · · · , j m ) , 0 ≤ j r ≤ N r , We introduce the Banach space L 2h = L 2 (Ω h ) of the grid functions ϕ h (x) = {ϕ(h 1 r 1 , ..., h m r m )} defined on Ω h , equipped with the norm to the differential operator A generated by problem (23), we assign the difference operator A x h by the formula acting in the space of grid functions u h (x), satisfying the conditions u h (x) = 0(∀ x ∈ S h ). It is known that A x h is a self-adjoint positive definite operator in L 2h . With the help of A x h , we arrive at the initial value problem In the second step, we replace problem (48) by first order of accuracy difference scheme (28) Theorem 11. Suppose that assumptions of Theorem 4 hold. Then, there exists a unique solution u h k ∞ k=0 of difference scheme (49) which is bounded in [mω, (m + 1) w] τ × L 2h , 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 2h = L 2 (Ω h ) with self-adjoint positive definite operator A h = A x h by formula (47). Here, ) and u h k = u h k (x) are abstract mesh functions defined on Ω h with the values in H = L 2h . Therefore, the proof of Theorems 11 is based on the abstract Theorem 8 and symmetry properties of the difference operator A x h defined by formula (47) and the following theorem on coercivity inequality for the solution of the elliptic problem in L 2h [53].

Theorem 12. For the solutions of the elliptic difference problem
the coercivity inequality n ∑ r=1 u h x r x r L 2h is satisfied, where M 1 does not depend on h and ω h .
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 x h by the formula acting in the space of grid functions u h (x), satisfying the conditions D h u h (x) = 0 (∀ x ∈ S h ). Here D h is the approximation of operator ∂ ∂ − → p . With the help of A x h , 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).
Proof. Difference scheme (49) can be written in abstract form (28) in a Hilbert space L 2h = L 2 (Ω h ) with self-adjoint positive definite operator A h = A x h by formula (50). Here, ) and u h k = u h k (x) are abstract mesh functions defined on Ω h with the values in H = L 2h . Therefore, the proof of Theorems 13 is based on the abstract Theorem 8 and symmetry properties of the difference operator A x h defined by formula (50) and the following theorem on coercivity inequality for the solution of the elliptic problem in L 2h [53].

Theorem 14.
For the solutions of the elliptic difference problem is satisfied, where M 2 does not depend on h and ω h .

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.

One Dimensional Case
For the numerical experiment, we consider the mixed problem for the semilinear delay one dimensional hyperbolic differential equation with nonlocal boundary conditions. The exact solution of problem (51) is u (t, x) = e −t sin x. We will consider the following iterative difference scheme of first order of approximation in t for the numerical solution of problem (51) for the semilinear delay hyperbolic equation. Here and in future j denotes the iteration index and an initial guess 0 u k n , k ≥ 1, 0 ≤ n ≤ M is to be made. For solving difference scheme (52), the numerical steps are given below. For 0 ≤ k < N, 0 ≤ n ≤ M the algorithm is as follows : 1. j = 1. 2. j−1 u k n is known. 3. j u k n is calculated. 4. If the max absolute error between j−1 u k n and j u k n is greater than the given tolerance value, take j = j + 1 and go to step 2. Otherwise, terminate the iteration process and take j u k n as the result of the given problem.

in matrix form
Here R, A, B, and C are (M + 1) × (M + 1) matrices given below: Finally, here ϕ( j−1 u k , u k−N ) and j u s , s = k, k ± 1 are (M + 1) × 1 column vectors as So, we have the initial value problem for the second order difference equation with respect to k with matrix coefficients. From Equations (53) and (54) it follows that Here, ψ u k , u k−1 is (M + 1) × 1 column vector defined by formula (54).
In computations the initial guess is chosen as 0 u k = {sin x n } M n=0 and when the maximum errors between two consecutive results of iterative difference scheme (52) become less than 10 −8 , the iterative process is terminated. We present numerical experimental results for different values of N and M and u k n represent the numerical solutions of difference scheme (52) at (t k , x n ) . The table of numerical results is constructed for N = M = 30, 60, 120 in t ∈ [0, 1] , t ∈ [1, 2] , t ∈ [2, 3], respectively and the errors are computed by the following formula As can be seen from tables, these numerical experiments support the theoretical statements. The number of iterations and maximum errors are decreasing with the increase of grid points.
In Table 1, as we increase values of M and N each time starting from M = N = 30 by a factor of 2 the errors in the first order of accuracy difference scheme decrease approximately by a factor of 1/2. The errors presented in Table 1 indicate the first order of accuracy of the difference scheme.

Two-Dimensional Case
For the numerical experiment, we consider the mixed boundary value problem − cos e −t sin x sin y u (t − 1, x, y) , 0 < t < ∞, 0 < x, y < π, u (t, x, y) = e −t sin x sin y, 0 ≤ x, y ≤ π, −1 ≤ t ≤ 0, u (t, 0, y) = u (t, π, y) = 0, 0 ≤ y ≤ π, t ≥ 0, for the semilinear two dimensional delay hyperbolic equation. The exact solution of problem (56) is u (t, x) = e −t sin x sin y. We will consider the following iterative difference scheme of first order of approximation in t for the numerical solution of the initial-boundary value problem (56) for the semilinear delay hyperbolic equation. Here and in the future j denotes the iteration index and an initial guess 0 u k n,i , k ≥ 1, 0 ≤ n, i ≤ M is to be made. For solving difference scheme (57), the numerical steps are given below. For 0 ≤ k < N, 0 ≤ n, i ≤ M the algorithm is as follows : 1. j = 1. 2. j−1 u k n,i is known. 3. j u k n,i is calculated. 4. If the max absolute error between j−1 u k n,i and j u k n,i is greater than the given tolerance value, take j = j + 1 and go to step 2. Otherwise, terminate the iteration process and take j u k n,i as the result of the given problem.

We write Equation (57) in matrix form
Here R, A, B, and C are (M + 1) × (M + 1) × (M + 1) given matrices and ϕ( j−1 u k , u k−N ) and j u s , s = k, k ± 1 are given (M + 1) × (M + 1) × 1 column vectors. Therefore, we will use the same algorithm as the one dimensional case.
So, we have the initial value problem for the second order difference equation with respect to k with matrix coefficients. From Equations (53) and (54) it follows that Here, ψ u k , u k−1 is the given (M + 1) × (M + 1) × 1 column vector.
In computations the initial guess is chosen as 0 u k = {sin x n sin x i } M n,i=0 and when the maximum errors between two consecutive results of iterative difference scheme (57) become less than 10 −6 , the iterative process is terminated. We present numerical results for different values of N and M and u k n,i represent the numerical solutions of this difference scheme at (t k , x n , x i ) . The table is  As can be seen from table, these numerical experiments support the theoretical statements. The number of iterations and maximum errors are decreasing with the increase of grid points.
In Table 2, as we increase values of M and N each time starting from M = N = 30 by a factor of 2 the errors in the first order of accuracy difference scheme decrease approximately by a factor of 1/2. The errors presented in tables indicate the the time convergence order is one. This result fits with the theoretical results perfectly.  1. In the present paper, the main theorem on the existence and uniqueness of a bounded solution of the initial value problem for a semilinear hyperbolic equation with time delay in a Hilbert space with the self adjoint positive definite operator 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 the abstract problem is presented. The theorem on the existence and uniqueness of an uniformly bounded solution of this difference scheme with respect to τ is established. In applications, the existence and uniqueness of a uniformly bounded solutions with respect to time and space stepsizes of difference schemes for four semilinear partial differential equations with time delay are obtained. Numerical results for the solution of difference schemes for one and two dimensional semilinear delay hyperbolic equation are presented.
2. We are interested in studying uniformly boundedness of solutions of high order of accuracy difference schemes uniformly with respect to time stepsize of approximate solutions of this initial-value problem, in which bounded solutions were established without any assumptions in respect to the grid steps τ and h. We have not been able to establish such type of results for the solution of the very well-known second order difference scheme , k = mN, m = 1, ..., Note that absolute stable two-step difference schemes of the high order of approximation for hyperbolic partial differential equations were presented and investigated in papers [11,54]. Applying methods of the present paper and papers [11,54] we can establish the similar stability and convergence results of this paper for the solution of the absolute stable two-step difference schemes of high order of approximation for semilinear delay hyperbolic equations.
3. Investigate the uniform to-step difference schemes and asymptotic formulas for the solution of initial value perturbation problem      ε 2 u (t) + Au(t) = f (t, u(t), u t (t − w), u(t − w)), t > 0, u(t) = ϕ(t), −w ≤ t ≤ 0 for a semilinear delay hyperbolic equation in a Hilbert space H with the self adjoint positive definite operator A and with ε ∈ (0, ∞) parameter multiplying the highest order derivative term.
In [31], the uniform difference schemes and asymptotic formulas for the solution of initial value perturbation problem for a linear hyperbolic equation in a Hilbert space with the self adjoint positive definite operator and with ε ∈ (0, ∞) parameter multiplying the highest order derivative term were presented and investigated.
4. Investigate the initial value problem u tt (t)dt + Au(t)dt = f (t, u(t), u t (t − w), u(t − ω))dw t , w t = √ tξ, ξ ∈ N(0, 1), t > 0, u(t) = 0, −ω ≤ t ≤ 0 for a semilinear stochastic hyperbolic equation with time delay in a Hilbert space H with the self adjoint positive definite operator A. Here, w t is a standard Wiener process given on the probability space (Q, F, P). Note that absolute stable difference schemes for stochastic linear hyperbolic equations in Hilbert spaces were presented and investigated in [30].
Finally, in paper [55], a Lie algebra approach is applied to solve an SIS model where infection rate and recovery rate are time-varying. The method presented here has been used widely in chemical and physical sciences but not in epidemic applications due to insufficient symmetries.