Existence solutions for nonlocal fractional differential equation with nonlinear boundary conditions

Document Type: Regular Paper

Authors

1 Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran

2 Department of Mathematics, Payame Noor University, Iran

Abstract

 
In this paper, by employing the Guo-Krasnoselskii fixed point theorem in a cone, we study the existence of positive solutions to the following nonlocal fractional boundary value problems
{ 𝐷0+ 𝛼 𝑒(𝑑) = 𝑓(𝑑, 𝑒(𝑑)), 𝑑 ∈ (0,1), 𝑒𝑑 + 𝑒′(0) = 12 [𝐻1(πœ‘π‘’) + ∫ 𝐻2 𝐸(𝑠 𝑒𝑠)𝑑𝑠 , 𝑒1) + 𝑒′(1) = 0, c
where c𝐷0+ 𝛼is the standard Caputo derivative of order 𝛼 1 < 𝛼< 2, 𝐸⊆ (0,1) is some measurable set. We provide conditions on 𝑓 𝐻1, 𝐻2 and πœ‘such that the problem exhibits at least one positive solution.

Keywords