Positive Solutions for a Fourth-Order Riemann–Stieltjes Integral Boundary Value Problem

Abstract

In this paper, we use the fixed point index to study the existence of positive solutions for the fourth-order Riemann–Stieltjes integral boundary value problem−x4t=ft,xt,x′t,x″t,x″′t, t∈0,1x0=x′0=x″′1=0,x″0=αx″t, wheref:0,1×ℝ+×ℝ+×ℝ+×ℝ+⟶ℝ+is a continuous function andαx″denotes a linear function. Two existence theorems are obtained with some appropriate inequality conditions on the nonlinearityf, which involve the spectral radius of related linear operators. These conditions allowft,z1,z2,z3,z4to have superlinear or sublinear growth inzi,  i=1,2,3,4.