Existence of Solutions for Fractional Multi-Point Boundary Value Problems on an Infinite Interval at Resonance

Abstract

This paper aims to investigate a class of fractional multi-point boundary value problems at resonance on an infinite interval. New existence results are obtained for the given problem using Mawhin’s coincidence degree theory. Moreover, two examples are given to illustrate the main results.

Resonant fractional order differential equation with two-dimensional kernel on the half-line

“…and the mapping L domL∩ KerP : domL ∩ KerP → ImL is invertible. We denote the inverse of L domL∩KerP by K p : ImL → domL ∩ KerP and the generalized inverse of L is denoted by K P,Q : H → domL ∩ KerP where [17,26]).…”

“…For other studies in which the dim Ker(L) = 2 on finite interval (0, 1) see [25,26]. Recently, Djebali and Aoun [4] studied the following class of fractional multipoint boundary value problem at resonance with dim Ker(L) = 1 on (0, +∞),…”

Existence Results for an m-Point Mixed Fractional-Order Problem at Resonance on the Half-Line

“…When the corresponding homogeneous equation of a fractional boundary value problem (FBVP) has a trivial solution then the FBVP is a non-resonance problem and its solution can be obtained using fixed point theorems, see [4][5][6][7] and the references cited therein. When the homogeneous equation of a FBVP has a non-trivial solution then the problem is a resonance problem and the solution can be obtained using topological degree methods [8][9][10][11][12][13][14][15].…”

“…In [16], the authors consider a higher-order fractional boundary value problem involving mixed fractional derivatives: Guezane Lakoud et al [17] obtained existence results for a fractional boundary value problem at resonance on the half-line: Zhang and Liu [15] considered the following FBVP…”