Existence and Positivity of Solutions for a Second-Order Boundary Value Problem with Integral Condition

This work is devoted to the study of uniqueness and existence of positive solutions for a second-order boundary value problem with integral condition. The arguments are based on Banach contraction principle, Leray Schauder nonlinear alternative, and Guo-Krasnosel’skii fixed point theorem in cone. Two examples are also given to illustrate the main results. 1. Introduction Boundary value problem with integral boundary conditions is a mathematical model for of various phenomena of physics, ecology, biology, chemistry, and so forth. Integral conditions come up when values of the function on the boundary are connected to values inside the domain or when direct measurements on the boundary are not possible. The presence of an integral term in the boundary condition leads to great difficulties. Our aim, in this work, is the study of existence, uniqueness, and positivity of solution for the following second-order boundary value problem: with boundary conditions of type where is a given function. Using the nonlinear alternative of Leray Schauder, we establish the existence of nontrivial solution of the BVP (1.1)-(1.2), under the condition where , to prove the uniqueness of solution, we apply Banach contraction principle, by using Guo-Krasnosel'skii fixed point theorem in cone we study the existence of positive solution. As applications, some examples to illustrate our results are given. Various types of boundary value problems with integral boundary conditions were studied by many authors using different methods see [1–9]. In [2] Benchohra et al. have studied (1.1) with the integral condition , the authors assumed that the function depends only on and and the condition (1.3) holds for , so our work is new and more general than [2]. Similar boundary value problems for third-order differential equations with one of the following conditions , , , or , , , were investigated by Zhao et al. in [6], they established the existence and nonexistence and the multiplicity of positive solutions in ordered Banach spaces basing on fixed point theory in cone. For more knowledge about the nonlocal boundary value problem, we refer to the references [10–17]. This paper is organized as follows. In Section 2, we give some notations, recall some concepts and preparation results. In the third Section, we give two main results, the first result based on Banach contraction principle and the second based nonlinear alternative of Leray-Schauder type. In Section 4, we treat the positivity of solutions with the help of Guo-Krasnosel'skii fixed point theorem in cone. Some examples are given