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Existence results for perturbed boundary value problem with fractional order

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In this work, we deal with the following class of fractional differential equations with fractional derivative boundary conditions:

$$\begin{aligned} {\left\{ \begin{array}{ll} -D^{\alpha } u(t)+a(t) u(t)=w(t)f(t,u(t)), \ \ \ \ \ t \in (0,1),\\ u^{(j)}(0)=0, \ \ 0 \le j \le n-2, \ \ [D^{\beta } u(t)]_{t=1}=0, \end{array}\right. } \end{aligned}$$

where \(n\ge 3\), \(n-1< \alpha < n\), \(1 \le \beta \le n-2\), \(D^{\alpha }\) and \(D^{\beta }\) are the standard Riemann-Liouville fractional derivatives and a is a continuous function on [0, 1]. The associated Green’s function is derived in term of a series of functions by the perturbed approach. Sharp estimates on it are established. We give sufficient conditions for existence results by the means of Schauder’s fixed point theorem. Some examples are given to illustrate our results.

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Correspondence to Om Kalthoum Wanassi.

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Wanassi, O.K., Toumi, F. Existence results for perturbed boundary value problem with fractional order. Ricerche mat (2021).

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