Nonlinear Langevin equations and inclusions involving mixed fractional order derivatives and variable coefficient with fractional nonlocal-terminal conditions

Bashir Ahmad; Ahmed Alsaedi; Sotiris K. Ntouyas; Bashir Ahmad; Ahmed Alsaedi; Sotiris K. Ntouyas

doi:10.3934/math.2019.3.626

AIMS Mathematics

2019, Volume 4, Issue 3: 626-647. doi: 10.3934/math.2019.3.626

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Nonlinear Langevin equations and inclusions involving mixed fractional order derivatives and variable coefficient with fractional nonlocal-terminal conditions

1.
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
2.
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received: 30 March 2019 Accepted: 29 May 2019 Published: 10 June 2019
MSC : 26A33, 34A08, 34B15, 34A60

In this paper, we discuss the existence and uniqueness of solutions for a new kind of Langevin equation involving Riemann-Liouville as well as Caputo fractional derivatives, and variable coefficient, supplemented with nonlocal-terminal fractional integro-differential conditions. The proposed study is based on modern tools of functional analysis. We also extend our discussion to the associated inclusions problem. For the applicability of the obtained results, several examples are constructed. Some interesting observations are also presented.
- fractional derivatives,
- fractional integral,
- Langevin equations,
- nonlocal-terminal value problems,
- existence,
- uniqueness,
- fixed point theorems
Citation: Bashir Ahmad, Ahmed Alsaedi, Sotiris K. Ntouyas. Nonlinear Langevin equations and inclusions involving mixed fractional order derivatives and variable coefficient with fractional nonlocal-terminal conditions[J]. AIMS Mathematics, 2019, 4(3): 626-647. doi: 10.3934/math.2019.3.626

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Abstract

In this paper, we discuss the existence and uniqueness of solutions for a new kind of Langevin equation involving Riemann-Liouville as well as Caputo fractional derivatives, and variable coefficient, supplemented with nonlocal-terminal fractional integro-differential conditions. The proposed study is based on modern tools of functional analysis. We also extend our discussion to the associated inclusions problem. For the applicability of the obtained results, several examples are constructed. Some interesting observations are also presented.

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