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.
Highlights
We use the fixed point index to study the existence of positive solutions for the fourth-order Riemann–Stieltjes integral boundary value problem
We investigate the existence of positive solutions for the following fourth-order Riemann–Stieltjes integral boundary value problem:
When f ∈ C([0, 1] × R+ × R−, R+) and in [2], the authors studied the existence of positive solutions for the fourth-order m-point boundary value problem:
Summary
We use the fixed point index to study the existence of positive solutions for the fourth-order Riemann–Stieltjes integral boundary value problem Two existence theorems are obtained with some appropriate inequality conditions on the nonlinearity f, which involve the spectral radius of related linear operators. We investigate the existence of positive solutions for the following fourth-order Riemann–Stieltjes integral boundary value problem:
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