**Title:** Cyclic and Noncyclic Geraghty Type Condensing Operators and Optimal Solutions of Nonlocal Integro-Differential Equations

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-20-00061; **Volume 3 / Issue 3 / Year 2021 (Special Issue)**, Pages 344-357

**Document Type:** Research Paper

**Author(s):** Pradip Ramesh Patle ^{a} , Dhananjay Gopal ^{b} , Rabha Waell Ibrahim ^{c}

^{a}Department of Mathematics, Amity School of Engineering and Technology Amity University, Madhya Pradesh Gwalior 474020, India

^{b}Department of Mathematics, Guru ghasidas Vishvavidyalaya, Koni- Bilaspur-495009, India

^{c}IEEE: 94086547, Kuala Lumpur, 59200, Malaysia

Received: 7 January 2021, Accepted: 6 April 2021, Published: 25 April 2021.

**Corresponding Author:** Rabha Waell Ibrahim (Email address: rabhaibrahim@yahoo.com)

**Full Text:** PDF

**Abstract**

The present work considers a family of cyclic (non-cyclic) relatively Geraghty type condensing functions and primarily aims to study the existence of (coupled) points and pairs of best proximity in Banach spaces. The occurrence of optimal solutions for a system of non-local integro-differential equations is demonstrated as an application. As numerical illustrations, we present the optimal solution of integro-differential systems (type cell growth), which can be optimized by using fractal entropy (the measurement of complexity). The fractal power is playing an important role to state the stability and maximization of solutions.

**Keywords:** oupled point (pair) of best proximity, Cyclic (non-cyclic) relatively Geraghty condensing operator, Optimal solution, Non-local integro-differential equation, Fractal, Entropy, Fractional calculus

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