Second order three-point boundary value problems in abstract spaces

Abstract

In this paper we investigate the existence of solutions of the nonhomogeneous three-point boundary value problem $$\left\{ {\begin{array}{*{20}c} {x''(t) + a(t)x'(t) + b(t)x(t) + u(t)f(t,x(t)) = 0, a.e. on [0,1],} \\ {x(0) = 0, x(1) - \zeta x(\eta ) = c.} \\ \end{array} } \right.$$ . The coefficient functions a and b are continuous real-valued functions on [0, 1], η and ζ are some positive constants. Denote by E a Banach space and assume, that u belongs to an Orlicz space i.e., u(·) ∈ LM([0, 1],ℝ), where M is an N-function and c ∈ E.