Existence Of Positive Solutions For Singular And Nonsingular Fredholm Integral Equations

This thesis is composed of two chapters. In the first chapter, we introduce the background of problems which will be investigatedand the main results of this paper. In Chapter 2, the followingFredholm integral equation u(t) =∫₀¹ k(t, s)f(s, u(s))ds is considered. The second chapter is the main text,we first establish the two existence results for positive solutions of singular and nonsingularFredholm integral equations and prove them. We then apply the main results to singular fourth-order boundary value problems and singular 2nth Lidstone boundary value problems.The purpose of this paper is to establish the new existence results of positive solutions of singular and nonsingular Fredholm integral equations by applying the fixed point theorem in cones and the fixed point index theorem in cones under some conditions concerning the firsteigenvalue of the relevant linear operator. The main results show that the problem has a positive solution at least provided growth rates of nonlinear term are appropriate.To our knowledge, the results of uniform nonresonance at the first eigenvalue of singular Fredholm integral equations have not been seen to now. Our main results improve and generalize the some known results. The generality of the results obtained is illustrated through applications to several well-known boundary value problems.