Existence of solutions for some boundary value problems of fractional \(p\)-Laplacian equation at resonance. (English) Zbl 1283.34004

Chen, Taiyong; Liu, Wenbin; Liu, Jiaying

Bull. Belg. Math. Soc. - Simon Stevin 20, No. 3, 503-517 (2013).

The paper is concerned with the existence of solutions of the following two boundary value problems for \(p\)-Laplacian differential equations at resonance: \[ D_{0^+}^\beta\phi_p(D_{0^+}^\alpha x(t))=f(t,x(t),D_{0^+}^\alpha x(t)),\quad t\in[0,1], \]
\[ x(0)=0,\quad D_{0^+}^\alpha x(0)=D_{0^+}^\alpha x(1); \] and \[ D_{0^+}^\beta\phi_p(D_{0^+}^\alpha x(t))=f(t,x(t),D_{0^+}^\alpha x(t)),\quad t\in[0,1], \]
\[ x(1)=0,\quad D_{0^+}^\alpha x(0)=D_{0^+}^\alpha x(1). \] The main tool is the coincidence degree theory. An example is given as an application of their results.

Reviewer: Xueyan Liu (Chattanooga)

MSC:

34A08	Fractional ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

fractional differential equation; \(p\)-Laplacian operator; Caputo fractional derivative; boundary value problem; coincidence degree; resonance