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On nonlocal fractional boundary value problems. (English) Zbl 1230.26003
Summary: We study a new class of non-local boundary value problems of nonlinear differential equations of fractional order. We extend the idea of a three-point non-local boundary condition (x(1)=αx(η),α,0<η<1) to a non-local strip condition of the form: x(1)=η ν τ x(s)ds,0<ν<τ<1. In fact, this strip condition corresponds to a continuous distribution of the values of the unknown function on an arbitrary finite segment of the interval. In the limit ν0,τ1, this strip condition takes the form of a typical integral boundary condition. Some new existence and uniqueness results are obtained for this class of non-local problems by using standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also discussed.
MSC:
26A33Fractional derivatives and integrals (real functions)
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
34A40Differential inequalities (ODE)