Abstract
The nonlocal boundary value problem ð ‘‘ 2 ð ‘¢ ( ð ‘i ) / ð ‘‘ ð ‘i 2 + ð ´ ð ‘¢ ( ð ‘i ) = ð ‘“ ( ð ‘i ) ( 0 ≤ ð ‘i ≤ 1 ) , ð ‘– ( ð ‘‘ ð ‘¢ ( ð ‘i ) / ð ‘‘ ð ‘i ) + ð ´ ð ‘¢ ( ð ‘i ) = ð ‘” ( ð ‘i ) ( − 1 ≤ ð ‘i ≤ 0 ) , ð ‘¢ ( 0 + ) = ð ‘¢ ( 0 − ) , ð ‘¢ ð ‘i ( 0 + ) = ð ‘¢ ð ‘i ( 0 − ) , ð ´ ð ‘¢ ( − 1 ) = ð ›¼ ð ‘¢ ( 𠜇 ) + 𠜑 , 0 < 𠜇 ≤ 1 , for hyperbolic Schrodinger equations in a Hilbert space ð » with the self-adjoint positive definite operator ð ´ 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 Schrodinger equations are obtained.
Highlights
A Note on Nonlocal Boundary Value Problems for Hyperbolic Schrodinger EquationsThe nonlocal boundary value problem d2u t /dt[2] Au t f t 0 ≤ t ≤ 1 , i du t /dt Au t g t −1 ≤ t ≤ 0 , u 0 u 0− , ut 0 ut 0− , Au −1 αu μ φ, 0 < μ ≤ 1, for hyperbolic Schrodinger equations in a Hilbert space H with the self-adjoint positive definite operator A is considered
From 2.34 and 2.35 and estimates 2.37 it follows 2.24. This completes the proof of Theorem 2.3
We introduce the Hilbert space L2 0, 1 of all the square integrable functions defined on 0, 1 and Hilbert spaces W21 0, 1 and W22 0, 1 equipped with norms φ W21 0,1 φ W22 0,1 φ x 2dx φ x 2dx φx x 2dx
Summary
The nonlocal boundary value problem d2u t /dt[2] Au t f t 0 ≤ t ≤ 1 , i du t /dt Au t g t −1 ≤ t ≤ 0 , u 0 u 0− , ut 0 ut 0− , Au −1 αu μ φ, 0 < μ ≤ 1, for hyperbolic Schrodinger equations in a Hilbert space H with the self-adjoint positive definite operator A is considered. The stability estimates for the solution of this problem are established. The stability estimates for solutions of the mixed-type boundary value problems for hyperbolic Schrodinger equations are obtained
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