Eigenvalue Criteria for Existence of Positive Solutions to Fractional Boundary Value Problem

Wen Lian ,Zhanbing Bai

doi:10.1155/2020/8196918

Abstract

The existence and multiplicity of positive solutions for the nonlinear fractional differential equation boundary value problem (BVP) DC0+αyx+fx,yx=0, 0<x<1, y0=y′1=y″0=0 is established, where 2<α≤3, CD0+α is the Caputo fractional derivative, and f:0,1×0,∞⟶0,∞ is a continuous function. The conclusion relies on the fixed-point index theory and the Leray-Schauder degree theory. The growth conditions of the nonlinearity with respect to the first eigenvalue of the related linear operator is given to guarantee the existence and multiplicity.

Highlights

We concentrate on the existence and multiplicity of positive solutions for the following problem: CDα0+yðxÞ + f ðx, yðxÞÞ = 0, 0 < x < 1, ð1Þ
Yð0Þ = y′ð1Þ = y′′ð0Þ = 0, Journal of Function Spaces is given by ð1
Let K be a cone in a Banach space X, and Ω be a bounded open set in K

Summary

Introduction

We concentrate on the existence and multiplicity of positive solutions for the following problem: CDα0+yðxÞ + f ðx, yðxÞÞ = 0, 0 < x < 1, ð1Þ yð0Þ = y′ð1Þ = y′′ð0Þ = 0, ð2Þ where 2 < α ≤ 3, CDα0+ is the Caputo fractional derivative, and f : 1⁄20, 1 × 1⁄20,+∞Þ ⟶ 1⁄20,+∞Þ is a continuous function. Bai and Qiu [22, 23] have investigated the existence and multiplicity of positive solutions of (1) and (2) by using the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem in a cone, but they did not consider its eigenvalue criteria.

Background

Existence and Multiplicity

Example