Existence of Odd Number of Positive Solutions to a Class of Third-order Integral Boundary Value Problem
In order to develop and perfect the basic theory for nonlocal problems of ordinary differential equations,we establish the existence of odd number of positive solutions to a class of third-order integral boundary value problems by using Guo-Krasnoselskill fixed point theorem in this paper.Firstly,the form of solution is obtained by investigating the corresponding Green's function of lin-ear integral boundary value problem,and the properties of Green's function and the nonnegativity,monotonicity and other properties of solutions are discussed at the same time.Secondly,the existence of solutions to the third-order integral boundary value problem is transformed into a fixed point problem to an operator defined on a cone,and the complete continuity of the operator is examined.Next,with the help of Guo-Krasnoselskill fixed point theorem,it is proven that the operator has an odd number of fixed points when the nonlinear term satisfies specific growth conditions,and thus the existence of odd number of positive solutions for the third-order integral boundary value problem is obtained.Finally,a concrete example is given to illustrate the rationality of our results.Based on the facts above,Guo-Krasnoselskill fixed point theorem,that is often used to establish the existence of one positive solution and two positive solutions at least,is applied to investigate the existenceof infinite(odd)positive solutions in this paper,which extends and perfects the research results of positive solutions to third-order boundary value problems,enriches the research contents of boundary value problems for ordinary differential equations,and provides theoretical basis for wide applications of nonlocal problems of ordi-nary differential equations in applied mathematics and physics.
integral boundary value problemfixed point theorempositive solution
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