On the asymptotic formula for the solution of nonlocal boundary value perturbation problems for hyperbolic equations.

In the present paper we consider the nonlocal boundary value perturbation problem { ε 2 ∂ 2 u (t , x) ∂ t 2 − (a (x) u x (t , x)) x + δ u (t , x) = f (t , x) , 0 < t < T , x ∈ (0 , l) , u (0 , x) = α u (T , x) + φ (x) , x ∈ [ 0 , l ] , u ′ (0 , x) = β u ′ (T , x) + ψ (x) , x ∈ [ 0 , l ] , u (t , 0) = u (t , l) , u x (t , 0) = u x (t , l) , 0 ≤ t ≤ T for hyperbolic equation with an arbitrary ε ∈ (0, ∞) parameter multiplying the derivative term. An asymptotic formula for the solution of this problem with a small ε parameter is presented. [ABSTRACT FROM AUTHOR]