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

Weerawat Sudsutad; Chatthai Thaiprayoon; Sotiris K. Ntouyas; Weerawat Sudsutad; Chatthai Thaiprayoon; Sotiris K. Ntouyas

doi:10.3934/math.2021244

AIMS Mathematics

2021, Volume 6, Issue 4: 4119-4141. doi: 10.3934/math.2021244

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

1.
Department of Applied Statistics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok 10800, Thailand
2.
Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand
3.
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
4.
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received: 19 December 2020 Accepted: 01 February 2021 Published: 05 February 2021
MSC : 26A33, 34A08, 34A12, 34B15

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.
- existence,
- uniqueness,
- $ \psi $-Hilfer fractional derivative,
- nonlocal boundary condition
Citation: Weerawat Sudsutad, Chatthai Thaiprayoon, Sotiris K. Ntouyas. Existence and stability results for $ \psi $-Hilfer fractional integro-differential equation with mixed nonlocal boundary conditions[J]. AIMS Mathematics, 2021, 6(4): 4119-4141. doi: 10.3934/math.2021244

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Abstract

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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