×

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 \((x(1) = \alpha x(\eta ), \alpha \in \mathbb{R}, 0 < \eta < 1)\) 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:

26A33 Fractional derivatives and integrals
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A40 Differential inequalities involving functions of a single real variable
PDFBibTeX XMLCite
Full Text: Link