Abstract
The nonlinear alternative of the Leray Schauder type and the Banach contraction principle are used to investigate the existence of solutions for second-order differential equations with integral boundary conditions. The compactness of solutions set is also investigated.
Highlights
This paper is concerned with the existence of solutions for the second-order boundary value problem
Boundary value problems with integral boundary conditions have been studied by a number of authors, for example 10–14
A map f : 0, 1 × R → R is said to be L1-Caratheodory if i t → f t, u is measurable for each u ∈ R, ii u → f t, u is continuous for almost each t ∈ 0, 1, iii for every r > 0 there exists hr ∈ L1 0, 1, R such that f t, u ≤ hr t for a.e. t ∈ 0, 1 and all |u| ≤ r
Summary
−y t f t, y t , a.e. t ∈ 0, 1 , 1.1 y 0 0, y 1 g s y s ds, where f : 0, 1 × R → R is a given function and g : 0, 1 → R is an integrable function. Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers 1–9 and the references therein. Boundary value problems with integral boundary conditions have been studied by a number of authors, for example 10–14. Our approach here is based on the Banach contraction principle and the Leray-Schauder alternative 15
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