Existence and uniqueness results for three-point nonlinear fractional (arbitrary order) boundary value problem. (English) Zbl 1488.34135

Kumar, Sachin; Vats, Ramesh Kumar; Nashine, Hemant Kumar

Mat. Vesn. 70, No. 4, 314-325 (2018).

Summary: We present here a new type of three-point nonlinear fractional boundary value problem of arbitrary order of the form \[ \begin{aligned} &^cD^qu(t) = f(t,u(t)), \quad t \in [0,1], \\ &u(\eta) = u'(0)= u''(0) = \cdots = u^{n-2}(0) = 0, \quad I^pu(1) = 0, \quad 0 < \eta < 1, \end{aligned} \] where \(n-1 < q \leq n\), \(n \in \mathbb{N}\), \(n \geq 3\) and \(^cD^q\) denotes the Caputo fractional derivative of order \(q\), \(I^p\) is the Riemann-Liouville fractional integral of order \(p\), \(f : [0,1] \times \mathbb{R} \rightarrow \mathbb{R}\) is a continuous function and \(\eta^{n-1} \neq \frac{\Gamma(n)}{(p+n-1)(p+n-2)\dots(p+1)} \). We give new existence and uniqueness results using Banach contraction principle, Krasnoselskii, Schaefer’s fixed point theorem and Leray-Schauder degree theory. To justify the results, we give some illustrative examples.

MSC:

34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34A08	Fractional ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

Caputo derivative; boundary value problem; fixed point theorem

References:

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