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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.

##### MSC:
 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