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

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
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### References:

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