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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 687321, 12 pages
http://dx.doi.org/10.1155/2012/687321
Research Article

A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrödinger Equations

Department of Mathematics, Duzce University, Konuralp, 81620 Duzce, Turkey

Received 12 February 2012; Accepted 8 April 2012

Academic Editor: Allaberen Ashyralyev

Copyright © 2012 Yildirim Ozdemir and Mehmet Kucukunal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The nonlocal boundary value problem , , , , , , for hyperbolic Schrödinger equations in a Hilbert space with the self-adjoint positive definite operator is considered. The stability estimates for the solution of this problem are established. In applications, the stability estimates for solutions of the mixed-type boundary value problems for hyperbolic Schrödinger equations are obtained.

1. Introduction

Methods of solutions of nonlocal boundary value problems for partial differential equations and partial differential equations of mixed type have been studied extensively by many researches (see, e.g., [112] and the references given therein).

In the present paper, the nonlocal boundary value problem for differential equations of hyperbolic Schrödinger type in a Hilbert space with self-adjoint positive definite operator is considered.

It is known that various nonlocal boundary value problems for the hyperbolic Schrödinger equations can be reduced to problem (1.1).

A function is called a solution of the problem (1.1) if the following conditions are satisfied.(i) is twice continuously differentiable on the interval (0,1] and continuously differentiable on the segment . The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii)The element belongs to for all , and the function is continuous on the segment .(iii) satisfies the equations and nonlocal boundary condition (1.1).

In the present paper, the stability estimates for the solution of the problem (1.1) for the hyperbolic Schrödinger equation are established. In applications, the stability estimates for the solutions of the mixed-type boundary value problems for hyperbolic Schrödinger equations are obtained.

Finally note that hyperbolic Schrödinger equations play important role in physics and engineering (see, e.g., [1316] and the references given therein).

Furthermore, the investigation of the numerical solution of initial value problems and Schrödinger equations is the subject of extensive research activity during the last decade (indicatively [1725] and the references given therein).

2. The Main Theorem

Let be a Hilbert space, and let be a positive definite self-adjoint operator with , where . Throughout this paper, is a strongly continuous cosine operator function defined by

Then, from the definition of the sine operator function

it follows that

For the theory of cosine operator function, we refer to Fattorini [26] and Piskarev and Shaw [27].

We begin with two lemmas that will be needed as follows.

Lemma 2.1. The following estimates hold:

Lemma 2.2. Let Then, the operator has an inverse and the estimate holds, where does not depend on and .

Proof. Actually, the proof of estimate (2.8) is based on the following estimate: Using the definitions of cosine and sine operator functions, , (positivity), and (self-adjointness property), we obtain Since we have that Hence, Lemma 2.2 is proved.

Now, we will obtain the formula for solution of problem (1.1). It is known that for smooth data of initial value problems

there are unique solutions of problems (2.13), and following formulas hold: Using (2.14), (2.15), and (1.1), we can write

Now, using the nonlocal boundary condition

we obtain the operator equation:

Since the operator

has an inverse

for the solution of the operator equation (2.18), we have the formula

Thus, for the solution of the nonlocal boundary value problem (1.1) we obtain (2.15), (2.16), and (2.21).

Theorem 2.3. Suppose that , and . Let be continuously differentiable on and let be twice continuously differentiable on functions. Then, there is a unique solution of the problem (1.1) and the following stability inequalities hold, where is independent of , and .

Note that there are three inequalities in Theorem 2.3 on the stability of solution, stability of first derivative of solution and stability of second derivative of solution. That means the solution of problem (1.1) and its first and second derivatives are continuously dependent on and .

Proof. First, estimate (2.22) will be obtained. Using formula (2.21) and integration by parts, we obtain Using estimates (2.4), and (2.8), we get Applying to the formula (2.25) and using estimates (2.4) and (2.8), we can write Using formulas (2.15) and (2.16) and integration by parts, we obtain Using estimates (2.4) we get Then, from estimates (2.26), (2.27), and (2.29) it follows (2.22).
Second, (2.23) will be obtained. Applying to the formula (2.25) and using estimates (2.4),and (2.8), we obtain Applying to the formula (2.25) and using estimates (2.4), (2.8), we get Applying to the formulas (2.28), and using estimates (2.4) we can write Combining estimates (2.30), (2.31),and (2.32), we get estimate (2.23).
Third, estimate (2.24) will be obtained. Using formula (2.25) and integration by parts, we obtain Applying to formula (2.33) and using estimates (2.4) and (2.8), we get Applying to formula (2.33) and using estimates (2.4), and (2.8) we can write Using formulas (2.28), and integration by parts, we obtain Applying to the formulas (2.36), and using estimates (2.4), we get From (2.34) and (2.35) and estimates (2.37) it follows (2.24). This completes the proof of Theorem 2.3.

Remark 2.4. We can obtain the same stability results for the solution of the following multipoint nonlocal boundary value problem: for differential equations of mixed type in a Hilbert space with self-adjoint positive definite operator .

3. Applications

Initially, the mixed problem for the hyperbolic Schrödinger equation is considered, where . The problem (3.1) has a unique smooth solution for smooth , , , and functions.

We introduce the Hilbert space of all the square integrable functions defined on and Hilbert spaces and equipped with norms

respectively. This allows us to reduce the mixed problem (3.1) to the nonlocal boundary value problem (1.1) in Hilbert space with a self-adjoint positive definite operator defined by problem (3.1).

Theorem 3.1. The solutions of the nonlocal boundary value problem (3.1) satisfy the following stability estimates: where does not depend on not only and but also .

The proof of Theorem 3.1 is based on the abstract Theorem 2.3 and symmetry properties of the space operator defined by problem (3.1).

Next, we consider the mixed nonlocal boundary value problem for the multidimensional hyperbolic Schrödinger equation: where is the unit open cube in the -dimensional Euclidean space :

with boundary and . Here, , and are given smooth functions in and .

We introduce the Hilbert space of all square integrable functions defined on , equipped with the norm

and Hilbert spaces and defined on , equipped with norms

respectively. The problem (3.4) has a unique smooth solution for smooth , and functions. This allows us to reduce the mixed problem (3.4) to the nonlocal boundary value problem (1.1) in Hilbert space with a self-adjoint positive definite operator defined by problem (3.4).

Theorem 3.2. The following stability inequalities for solutions of the nonlocal boundary value problem (3.4) hold. Here, is independent of , and .

The proof of Theorem 3.2 is based on the abstract Theorem 2.3, symmetry properties of the space operator defined by problem (3.4), and the following theorem on the coercivity inequality for the solution of the elliptic differential problem in in Sobolevskii [28].

Theorem 3.3. For the solutions of the elliptic differential problem the following coercivity inequality holds:

Acknowledgment

The authors would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey) for his helpful suggestions to the improvement of this paper.

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