Multiplicity of Positive Solutions for Fractional Differential Equation withp-Laplacian Boundary Value Problems

Abstract

We investigate the existence of multiple positive solutions of fractional differential equations withp-Laplacian operatorDa+β(ϕp(Da+αu(t)))=f(t,u(t)),  a<t<b,uja=0,  j=0,1,2,…,n-2,u(α1)(b)=ξu(α1)(η),ϕp(Da+αu(a))=0=Da+β1(ϕp(Da+αu(b))), whereβ∈(1,2],α∈(n-1,n],  n≥3,ξ∈(0,∞),η∈(a,b),β1∈(0,1],α1∈{1,2,…,α-2}is a fixed integer, andϕp(s)=|s|p-2s,  p>1,  ϕp-1=ϕq,  (1/p)+(1/q)=1, by applying Leggett–Williams fixed point theorems and fixed point index theory.