Fixed point theorem combined with variational methods for a class of nonlinear impulsive fractional problems with derivative dependence

Adnan Khaliq; Mujeeb ur Rehman; Adnan Khaliq; Mujeeb ur Rehman

doi:10.3934/math.2021118

AIMS Mathematics

2021, Volume 6, Issue 2: 1943-1953. doi: 10.3934/math.2021118

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Research article

Fixed point theorem combined with variational methods for a class of nonlinear impulsive fractional problems with derivative dependence

Adnan Khaliq ^,,
Mujeeb ur Rehman

School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan

Received: 14 October 2020 Accepted: 22 September 2020 Published: 04 December 2020
MSC : 34A08, 34B37, 34G20

In this article, we deal with a class of nonlinear impulsive problems of fractional-order in which nonlinearity is due to the fractional-order derivative term. The investigation involved a fixed point theorem with a combination of variational approach and critical point theory to establish sufficient conditions for the existence of at least one solution. First, a damped problem is discussed by using the critical point theory and variational approach, then the solutions of the damped problem and the main problem are connected with the assistance of a fixed point theorem. Towards the end, to illustrate our outcomes, two examples are given.
- fixed point theorem,
- variational methods,
- fractional impulsive boundary value problem,
- critical points
Citation: Adnan Khaliq, Mujeeb ur Rehman. Fixed point theorem combined with variational methods for a class of nonlinear impulsive fractional problems with derivative dependence[J]. AIMS Mathematics, 2021, 6(2): 1943-1953. doi: 10.3934/math.2021118

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Abstract

In this article, we deal with a class of nonlinear impulsive problems of fractional-order in which nonlinearity is due to the fractional-order derivative term. The investigation involved a fixed point theorem with a combination of variational approach and critical point theory to establish sufficient conditions for the existence of at least one solution. First, a damped problem is discussed by using the critical point theory and variational approach, then the solutions of the damped problem and the main problem are connected with the assistance of a fixed point theorem. Towards the end, to illustrate our outcomes, two examples are given.

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