Existence and stability results for $ \psi $-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions

In this paper, we discuss the existence, uniqueness and stability of boundary value problems for $ \psi $-Hilfer fractional integro-differential equations with mixed nonlocal (multi-point, fractional derivative multi-order and fractional integral multi-order) boundary conditions. The uniqueness result is proved via Banach's contraction mapping principle and the existence results are established by using the Krasnosel'ski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$'s fixed point theorem and the Larey-Schauder nonlinear alternative. Further, by using the techniques of nonlinear functional analysis, we study four different types of Ulam's stability, i.e. , Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. Some examples are also constructed to demonstrate the application of main results.

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