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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 ๐‘‘ 2 ๐‘ข ( ๐‘ก ) / ๐‘‘ ๐‘ก 2 + ๐ด ๐‘ข ( ๐‘ก ) = ๐‘“ ( ๐‘ก ) ( 0 โ‰ค ๐‘ก โ‰ค 1 ) , ๐‘– ( ๐‘‘ ๐‘ข ( ๐‘ก ) / ๐‘‘ ๐‘ก ) + ๐ด ๐‘ข ( ๐‘ก ) = ๐‘” ( ๐‘ก ) ( โˆ’ 1 โ‰ค ๐‘ก โ‰ค 0 ) , ๐‘ข ( 0 + ) = ๐‘ข ( 0 โˆ’ ) , ๐‘ข ๐‘ก ( 0 + ) = ๐‘ข ๐‘ก ( 0 โˆ’ ) , ๐ด ๐‘ข ( โˆ’ 1 ) = ๐›ผ ๐‘ข ( ๐œ‡ ) + ๐œ‘ , 0 < ๐œ‡ โ‰ค 1 , for hyperbolic Schrödinger equations in a Hilbert space ๐ป with the self-adjoint positive definite operator ๐ด 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.