Existence and uniqueness of positive solutions for second-order nonlinear integral boundary value problem

Abstract:

In this paper,by using the method of generalized Krasnoselskii's contractive fixed point theorem in bimetric spaces we study the existence and uniqueness of the positive solutions for the following second-order nonlinear integral boundary value problem\$$u''+a(t)f(t,u(t),u'(t))=0,~t\in(0,1),$$ $$u(0)=0,~\alpha\int_{0}^{\eta}u(s)ds=u(1),$$ where$~0<\eta<1,~0<\alpha<\dfrac{2}{\eta^{2}},~a\in C([0,1],[0,\infty)),~t_{0}\in[\eta,1],~a(t_{0})>0,~$and$~f:[0,1]\times[0,\infty)\times R\rightarrow[0,\infty)~$is continuous.