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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 $\left(x\left(1\right)=\alpha x\left(\eta \right),\alpha \in ℝ,0<\eta <1\right)$ to a non-local strip condition of the form: $x\left(1\right)=\eta {\int }_{\nu }^{\tau }x\left(s\right)ds,0<\nu <\tau <1$. 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 $\nu \to 0,\tau \to 1$, 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:
 26A33 Fractional derivatives and integrals (real functions) 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 34A40 Differential inequalities (ODE)