Existence solutions for nonlocal fractional differential equation with nonlinear boundary conditions

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 𝑢(𝑡) = 𝑓(𝑡, 𝑢(𝑡)), 𝑡 ∈ (0,1), 𝑢𝑡 + 𝑢′(0) = 12 [𝐻1(𝜑𝑢) + ∫ 𝐻2 𝐸(𝑠 𝑢𝑠)𝑑𝑠 , 𝑢1) + 𝑢′(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 𝜑such that the problem exhibits at least one positive solution.