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Zbl 1230.26003
On nonlocal fractional boundary value problems.
(English)
[J] Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 18, No. 4, 535-544 (2011). ISSN 1201-3390; ISSN 1918-2538/e

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 $(x(1) = \alpha x(\eta ), \alpha \in \bbfR, 0 < \eta < 1)$ to a non-local strip condition of the form: $x(1) = \eta \int ^\tau _\nu x(s)ds, 0 < \nu < \tau < 1$. 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 $\nu \rightarrow 0, \tau \rightarrow 1$, 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 2000:
*26A33 Fractional derivatives and integrals (real functions)
34A12 Initial value problems for ODE
34A40 Differential inequalities (ODE)

Keywords: fractional differential equations; nonlocal boundary conditions; fixed point theorems; Leray-Schauder degree

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