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A nonlocal boundary value problem for a nonlinear fractional differential equation with two indices. (English) Zbl 1275.34005
Summary: This paper is devoted to a study of a nonlocal boundary value problem for a nonlinear differential equation depending on two fractional orders \(\alpha,\beta\in(1,2]\). The problem is inverted and an equivalent integral equation is constructed; as applications, the contraction mapping principle and a Krasnosel’skii fixed point theorem are applied to obtain sufficient conditions for the existence of solutions. An example illustrates the results. In the case that \(\alpha=\beta=2\), results for fourth order ordinary differential equations are obtained.

34A08 Fractional ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations