The existence of solutions for nonlinear fractional multipoint boundary value problems at resonance

Abstract

A class of nonlinear fractional multipoint boundary value problems at resonance is considered in this article. The existence results are obtained by the method of the coincidence degree theory of Mawhin. An example is given to illustrate the results.

Fractional multipoint boundary value problems at resonance with kernel dimension greater than one

“…We start with some useful definitions and basic lemmas concerned with the fractional calculus. For more details, the reader may refer to Bai and Zhang 17 and Su and Zhang 7 . We denote by C [0,1] the Banach space of all continuous functions on [0,1] with the sup‐norm and L 1 [0,1] stands for the Banach space of measurable functions that are Lebesgue integrable on [0,1] with the norm .…”

“…However, most of these works have dealt with the case dim ker L =1, where L is the differential operator involved in the equation. In contrast, BVPs at resonance with dim ker L >1 have been investigated only by a few authors who obtained existence results based on an assumption that a specific algebraic expression is not equal to zero (see other studies 7,10–16 ). For example in a previous work, 10 the authors referred to the crucial condition as follows: …”

“…Motivated by the works mentioned above, in particular Zhang 13 and Su and Zhang, 7 we will investigate in this work the existence of solutions for problems () and () when suitable growth conditions are imposed on the nonlinear term f . Actually, we have replaced the condition like () by a weaker one.…”

“…For more details, we refer the reader, for example, to previous studies. [1][2][3][4] Many mathematical interesting results on fractional boundary value problems (FBVPs for short) with resonant conditions have been obtained in the recent literature; see, for instance other works [5][6][7][8][9] and references therein. However, most of these works have dealt with the case dim ker L = 1, where L is the differential operator involved in the equation.…”

Existence of Solutions for Two-Point Boundary Value Problem of Fractional Differential Equations at Resonance

“…Recently, more and more authors have paid their attentions to the boundary value problems of fractional differential equations; see [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Moreover, there have been many works related to the existence of solutions for boundary value problems at resonance; see [12][13][14][15][16][17][18][19][20][21]. It is considerable that there are many papers that have dealt with the solutions of multipoint boundary value problems of fractional differential equations at resonance (see, e.g., [12,16]).…”