The Existence of Solutions to Integral Boundary Value Problems of Fractional Differential Equations at Resonance

Abstract

This paper deals with the integral boundary value problems of fractional differential equations at resonance. By Mawhin’s coincidence degree theory, we present some new results on the existence of solutions for a class of differential equations of fractional order with integral boundary conditions at resonance. An example is also included to illustrate the main results.

Resonant Integral Boundary Value Problems for Caputo Fractional Differential Equations

“…Similar arguments for related issues are treated in [12]. In [7], by mean of coincidence degree theory, the existence of solution of integral BVP for nonlinear differential equation is proved:…”

Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance

“…We use the classical Banach space Y = C[0, 1] with the norm u ∞ = max t∈[0,1] |u(t)| and the Banach space [22,23]). After further discussion for problems (1.1), we define two operators L and N as follows:…”

“…where C D α 1-and D become a popular research field. At present, many researchers study the existence of solutions of fractional differential equations such as the Riemann-Liouville fractional derivative problem at nonresonance [6][7][8][9][10][11][12][13][14][15][16], the Riemann-Liouville fractional derivative problem at resonance [17][18][19][20][21][22][23], the Caputo fractional boundary value problem [6,24,25], the Hadamard fractional boundary value problem [26][27][28], conformable fractional boundary value problems [29][30][31][32], impulsive problems [33][34][35], boundary value problems [8,[36][37][38][39][40][41][42][43], and variational structure problems [44,45].…”

“…For example, Tang et al [24] investigated the existence of solutions for the four-point boundary value problems of fractional differential equations where D α 0+ denotes the Caputo fractional derivative with 1 < α ≤ 2. Zou and He [23] investigated the integral boundary value problem for resonant frac-…”

Positive Solutions for a Hadamard Fractional p-Laplacian Three-Point Boundary Value Problem

“…where D α 0+ denotes the Riemann-Liouville fractional derivative. Positive solutions [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] and nontrivial solutions [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52] were also studied for fractional-order equations. For example, the authors in [16] used the Guo-Krasnoselskii's fixed-point theorem and the Leggett-Williams fixed-point theorem to study the existence and multiplicity of positive solutions for the fractional boundary-value problem…”