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

Abstract

0+ u(t) = f (t, u(t), D β+1 0+ u(t), D β 0+ u(t)), 0 < t < 1, u(0) = u (0) = 0, u(1) = 1 0 u(t) dA(t), where C D α 1-is the left Caputo fractional derivative of order α ∈ (1, 2], and D β 0+ is the right Riemann-Liouville fractional derivative of order β ∈ (0, 1]. The coincidence degree theory is the main theoretical basis to prove the existence of solutions of such problems.

Nonlinear differential equations involving mixed fractional derivatives with functional boundary data

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] However, among several types of fractional differential equations found in the literature, it is imperative to mention that the Caputo and Riemann-Liouville derivatives are studied separately in many cases. Moreover, the recent papers on the study of boundary value problems at resonance having mixed type fractional-order derivatives is not satisfactory, and the topic has not been extensively studied so far (see other studies [18][19][20][21][22][23][24][25][26][27][28][29][30][31] ). The initial attempts in recent years are as follows.…”

Existence of the Unique Nontrivial Solution for Mixed Fractional Differential Equations

“…In [23], Song and Cui concerned the existence of solutions of nonlinear mixed fractional differential equation with the integral boundary value problem under resonance:…”

On resonant mixed Caputo fractional differential equations

“…In [23], the existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance was studied, and a very recent study [24] in-troduced a new method to convert the boundary value problems for impulsive fractional differential equations to integral equations.…”