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Solutions of the Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space

1
Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
2
Department of Mathematics, Texas A&M University Kingsville, Kingsville 78363, TX, USA
3
Florida Institute of Technology, Melbourne 32901, FL, USA
4
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5
Department of Mathematics, Basic Sciences and Humanities, GMR Institute of Technology, Rajam 532 127, Andhra Pradesh, India
*
Authors to whom correspondence should be addressed.
Symmetry 2019, 11(5), 686; https://doi.org/10.3390/sym11050686
Received: 19 April 2019 / Revised: 14 May 2019 / Accepted: 16 May 2019 / Published: 18 May 2019
PDF [437 KB, uploaded 20 May 2019]

Abstract

The present paper aims to define three new notions: Θ e -contraction, a Hardy–Rogers-type Θ -contraction, and an interpolative Θ -contraction in the framework of extended b-metric space. Further, some fixed point results via these new notions and the study endeavors toward a feasible solution would be suggested for nonlinear Volterra–Fredholm integral equations of certain types, as well as a solution to a nonlinear fractional differential equation of the Caputo type by using the obtained results. It also considers a numerical example to indicate the effectiveness of this new technique.
Keywords: extended b-metric space; Θe-contraction; HR-Θ-contraction; nonlinear Volterra–Fredholm integral equations; nonlinear fractional differential equation of the Caputo type extended b-metric space; Θe-contraction; HR-Θ-contraction; nonlinear Volterra–Fredholm integral equations; nonlinear fractional differential equation of the Caputo type
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Abdeljawad, T.; Agarwal, R.P.; Karapınar, E.; Kumari, P.S. Solutions of the Nonlinear Integral Equation and Fractional Differential Equation Using the Technique of a Fixed Point with a Numerical Experiment in Extended b-Metric Space. Symmetry 2019, 11, 686.

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