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

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.

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
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Full Text: Euclid