# Advance Fixed Point Theorems and Its Application to Ordinary and Fractional Differential Equations

Year : 2024 | Volume :01 | Issue : 01 | Page : 1-15
By

### Soni Pathak

1. Assistant Professor Ajeenkya D.Y. Patil University (SOE), Charholi Budruk, Pune Maharashtra India
2. Professor Chhatrapati Shivaji Maharaj University, Panvel, Navi Mumbai Maharashtra India
3. Assistant Professor Ajeenkya D.Y. Patil University (SOE), Charholi Budruk, Pune Maharashtra India

#### Abstract

In the present research, advanced results on the fixed point theorem are applied to typical boundary value problems. Finding a differential equation’s solution under certain boundary conditions is the goal of typical boundary value topics. The article appears to extend new fixed point results to a new context, likely involving fractional operators with unique kernels, notably the Caputo-type fractional operator. This extension involves applying the fixed point theorem to a broader set of problems. Caputo fractional derivatives are a generalization of ordinary derivatives to non-integer orders, commonly used in fractional calculus. This extends the scope to fractional boundary value problems, where the differential equation involves fractional derivatives, and the conditions are given in terms of fractional order. Integral type boundary conditions suggest that the conditions involve integrals, which could be part of the fractional differential equation or the boundary conditions. We have used inequalities on a triplet (U, d, T) and quatern (U, d, T, θ). We have developed a novel class of mappings and investigated a fixed point criterion for them, using Geraghty contraction as inspiration. In addition, we demonstrated two applications: one with singular kernels for fractional derivatives in a system of nonlinear differential equations and the other with a two-point boundary value problem of a second order ordinary differential equations.

## Keywords: Fractional differential equations, Ordinary differential Equations, Generalized α-p- -contractions, Weakly contractive mapping, Geraghty function, θ-orbital permissible

[This article belongs to Recent Trends in Mathematics(rtm)]

How to cite this article: Shirish Prabhakarrao Kulkarni, Gooty Rohan, Soni Pathak. Advance Fixed Point Theorems and Its Application to Ordinary and Fractional Differential Equations. Recent Trends in Mathematics. 2024; 01(01):1-15.
How to cite this URL: Shirish Prabhakarrao Kulkarni, Gooty Rohan, Soni Pathak. Advance Fixed Point Theorems and Its Application to Ordinary and Fractional Differential Equations. Recent Trends in Mathematics. 2024; 01(01):1-15. Available from: https://journals.stmjournals.com/rtm/article=2024/view=148826

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## Recent Trends in Mathematics

 Volume 01 Issue 01 Received January 24, 2024 Accepted March 1, 2024 Published June 3, 2024