Abstract
In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving ϕ-Laplacian operator D0+αϕx,D0+βux+fx,ux=0,x∈0,1,D0+σu0=D0+βu0=0,u1=∫01 gtutdt+∑n=1n=+∞ αnuηn, where ϕ:0,1×R→R is a continuous function and D0+p is the Riemann-Liouville derivative for p∈α,β,σ. By using some properties of fixed point index, we obtain the existence results and give an example at last.
Highlights
Our aim in this article is to study the existence of a nontrivial positive solution to the following integral and infinite point boundary-value problem involving a two-dimensional φ-Laplacian operator
Noting that the generalized φ-Laplacian operator can turn into the well-known pðtÞ-Laplacian operator when we replace φ by φpðtÞðxÞ = jxjpðtÞ−2x, so our results extend and enrich some existing papers
By using the homotopy deformation property of the fixed point index, our paper aims at investigating the existence of at least one positive solution for bvp 1
Summary
Ð0, 1Þ, ð1Þ n=1 where Dp0+ is the Riemann-Liouville derivative for p ∈ fα, β, σg, 0 < α ≤ 1 ≤ β ≤ 2 and. Boundary value problems with fractional-order differential equations involving pðtÞ-Laplacian are of great importance and are an interesting class of problems. Such kind of BVPs in Banach space has been studied by many authors, see, for example [10,11,12,13] and the references therein. By using the homotopy deformation property of the fixed point index, our paper aims at investigating the existence of at least one positive solution for bvp 1. We give our main results and their proofs and we end by giving as an example, a problem involving a sum of many pðtÞ-Laplacian operators
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