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Nonlocal boundary value hyperbolic problems involving integral conditions
Boundary Value Problems volume 2014, Article number: 205 (2014)
Abstract
Stability estimates for the solution of the nonlocal boundary value problem with two integral conditions for hyperbolic equations in a Hilbert space H are established. In applications, stability estimates for the solution of the nonlocal boundary value problems for hyperbolic equations are obtained.
MSC: 35L10.
1 Introduction
It is well known that nonlocal boundary value problems with integral conditions are widely used for thermoelasticity, chemical engineering, heat conduction, and plasma physics [1]–[4]. Some problems arising in dynamics of ground waters are defined as hyperbolic equations with nonlocal conditions [5] and [6]. The authors of [7] investigate nonclassical problems for multidimensional hyperbolic equation with integral boundary conditions and the uniqueness of classical solution. In [8] a linear secondorder hyperbolic equation with forcing and integral constraints on the solution is converted to a nonlocal hyperbolic problem. Using the Riesz representation theorem and the Schauder fixed point theorem, existence and uniqueness of a generalized solution are proved. The solutions of hyperbolic equations with nonlocal integral conditions were investigated in [9]–[16]. The method of operators as a tool for investigation of the solution to hyperbolic equations in Hilbert and Banach spaces has been used extensively in [17]–[29].
In [30] the nonlocal boundary value problem
was investigated. Stability estimates for the solution of the problem were established. First order of accuracy difference schemes for the approximate solutions of the problem were presented. Stability estimates for the solution of these difference schemes were established. Theoretical statements were supported by numerical examples.
In the present paper, we consider the nonlocal boundary value problem with integral conditions
in a Hilbert space H with a selfadjoint positive definite operator A. We are interested in studying the stability of solutions of problem (1) under the assumption
As in [30], a function u(t) is called a solution of problem (1) if the following conditions are satisfied:

(i)
u(t) is twice continuously differentiable on the interval (0,1) and continuously differentiable on the segment [0,1].

(ii)
The element u(t) belongs to D(A) for all t\in [0,1], and the function Au(t) is continuous on the segment [0,1].

(iii)
u(t) satisfies the equation and nonlocal boundary conditions (1).
2 The main theorem
Let H be a Hilbert space, A be a positive definite selfadjoint operator with A\ge \delta I, where \delta >{\delta}_{0}>0. Throughout this paper, \{c(t),t\ge 0\} is a strongly continuous cosine operatorfunction defined by
Then, from the definition of sine operatorfunction s(t),
it follows that
For the theory of cosine operatorfunction we refer to [23] and [24].
Lemma 2.1
The following estimates hold:
Lemma 2.2
Suppose that assumption (2) holds. Then the operator T,
has the inverse
and the following estimate is satisfied:
Proof
Applying the triangle inequality and estimates (3), we obtain
Estimate (4) follows from this estimate. Lemma 2.2 is proved. □
Now, we will obtain the formula for the solution of problem (1) for \phi \in D(A) and \psi \in D({A}^{1/2}). It is clear that [23] the initial value problem
has a unique solution,
where the function f(t) is not only continuous but also continuously differentiable on [0,1], {u}_{0}\in D(A) and {u}_{0}^{\prime}\in D({A}^{1/2}).
Using (6) and the nonlocal boundary condition
we get
Then
Differentiating both sides of (6), we obtain
Using this formula and the integral condition
we get
Thus,
Now, we have a system of equations (7) and (8) for the solution of {u}_{0} and {u}_{0}^{\prime}. Solving it, we get
and
Hence, for the solution of the nonlocal boundary value problem (1) we have (6), (9), and (10).
Theorem 2.1
Suppose that\phi \in D(A), \psi \in D({A}^{1/2}), andf(t)is a continuously differentiable on[0,1]; assumption (2) holds. Then there is a unique solution of problem (1) and the following stability inequalities:
are valid, where M does not depend onf(t), t\in [0,1], φ, and ψ.
Proof
We take the estimates
from [26] for the solution of problem (5). The proof of Theorem 2.1 is based on estimates (14), (15), (16), and the estimates for the norms of {u}_{0}, {A}^{1/2}{u}_{0}^{\prime}, {A}^{1/2}{u}_{0}, {u}_{0}^{\prime}, A{u}_{0}, {A}^{1/2}{u}_{0}^{\prime}.
First of all, let us find an estimate for {\parallel u(0)\parallel}_{H}. By using (9) and estimates (3), (4), we obtain
Applying {A}^{1/2} to (10), we get
Using estimates (3), (4), we obtain
Thus, estimates (14), (17), and (18) yield estimate (11).
Second, applying operator {A}^{1/2} to (9), we get
Using estimates (3) and (4), we obtain
Using (10), and estimates (3), (4), we get
Then estimate (12) follows from estimates (15), (19), and (20).
Third, applying A to (9) and using Abel’s formula, we have
and using estimates (3), (4), we get
In the same manner, applying {A}^{1/2} to (10) and using Abel’s formula, and estimates (3), (4), we obtain
Thus, estimate (13) follows from estimates (16) and (21), and (22). □
3 Applications
Now, we consider the applications of Theorem 2.1. First, a nonlocal boundary value problem for a hyperbolic equation
under assumption (2) is considered. Problem (23) has a unique smooth solution u(t,x) for (2), smooth functions a(x)\ge a>0 (x\in (0,1)), a(0)=a(1), \phi (x), \psi (x) (x\in [0,1]) and f(t,x) (t,x\in [0,1]), σ a positive constant and under some conditions. This allows us to reduce problem (23) to nonlocal boundary value problem (1) in the Hilbert space H={L}_{2}[0,1] with a selfadjoint positive definite operator {A}^{x} defined by (23).
Theorem 3.1
For the solution of problem (23), we have the following stability inequalities:
where M does not depend on\phi (x), \psi (x), andf(t,x).
The proof of Theorem 3.1 is based on Theorem 2.1 and the symmetry properties of the space operator generated by problem (23).
Proof
Problem (23) can be written in the abstract form
in the Hilbert space {L}_{2}[0,1] of all square integrable functions defined on [0,1] with a selfadjoint positive definite operator A={A}^{x} defined by the formula
with the domain
Here, f(t)=f(t,x) and u(t)=u(t,x) are known and unknown abstract functions defined on [0,1] with the values in H={L}_{2}[0,1]. Therefore, estimates (24) and (25) follow from estimates (11), (12), and (13). Thus, Theorem 3.1 is proved. □
Second, let Ω be the unit open cube in the mdimensional Euclidean space {\mathbb{R}}^{m}:\{x=({x}_{1},\dots ,{x}_{m}):0<{x}_{j}<1,1\le j\le m\} with boundary S, \overline{\Omega}=\Omega \cup S. In [0,1]\times \Omega, let us consider a boundary value problem for the multidimensional hyperbolic equation
under assumption (2). Here, {a}_{r}(x) (x\in \Omega), \phi (x), \psi (x) (x\in \overline{\Omega}) and f(t,x), t\in (0,1), x\in \Omega are given smooth functions and {a}_{r}(x)\ge a>0.
Let us introduce the Hilbert space {L}_{2}(\overline{\Omega}) of all square integrable functions defined on \overline{\Omega}, equipped with the norm
Theorem 3.2
For the solution of problem (27), the following stability inequalities hold:
where M does not depend on\phi (x), \psi (x), andf(t,x) (t\in (0,1), x\in \Omega).
Proof
Problem (27) can be written in the abstract form (26) in the Hilbert space {L}_{2}(\overline{\Omega}) with a selfadjoint positive definite operator A={A}^{x} defined by the formula
with domain
Here, f(t)=f(t,x) and u(t)=u(t,x) are known and unknown abstract functions defined on \overline{\Omega} with the values in H={L}_{2}(\overline{\Omega}). So, estimates (28) and (29) follow from estimates (11), (12), (13), and the following theorem. □
Theorem 3.3
[29]
For the solution of the elliptic differential problem
the following coercivity inequality holds:
where M is independent of ω.
4 Conclusion
This work is devoted to the study of the stability of the nonlocal boundary value problem with integral conditions for hyperbolic equations. For the solution of nonlocal boundary problem (1) in a Hilbert space H with a selfadjoint positive definite operator A, Theorem 2.1 is established. Two applications of Theorem 2.1 are given. Of course, stable twostep difference schemes for approximate solution of problem (1) can be presented. The methods given above permit us to establish the stability of these difference schemes. Applying [31], we can give a numerical support of the theoretical results.
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This work is supported by the Scientific Research Fund of Fatih University (Project No: P50041203B).
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Ashyralyev, A., Aggez, N. Nonlocal boundary value hyperbolic problems involving integral conditions. Bound Value Probl 2014, 205 (2014). https://doi.org/10.1186/s1366101402054
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DOI: https://doi.org/10.1186/s1366101402054