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Fixed point theorems on perturbed metric space with an application

Dipti Barman, T. Bag

Abstract

Following the definition of perturbed metric space, in this paper, some fixed point theorems are established for $ F $-perturbed mappings in complete perturbed metric spaces and justify the result by counter example. Finally, an application of this theorem for the existence of a solution for the second-order boundary value problem is given.

Fixed point theorems on perturbed metric space with an application

Abstract

Following the definition of perturbed metric space, in this paper, some fixed point theorems are established for -perturbed mappings in complete perturbed metric spaces and justify the result by counter example. Finally, an application of this theorem for the existence of a solution for the second-order boundary value problem is given.

Paper Structure

This paper contains 6 sections, 6 theorems, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Existence of a Solution to the Boundary Value Problem
  5. Numerical Example
  6. Another generalized fixed-point theorem

Key Result

Proposition 2.3

Let $D, P, Q : X \times X \to [0, \infty)$ be three given mappings and $\alpha > 0$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Curve type picture indicates the value of $D(x, y) = |x - y| + x^2y^4$ where as rectangular type picture indicates the value of $d(x, y) = |x - y|$
  • Figure 2: The upper picture indicates $F( D(x, y))$ where, as the lower picture indicates $\tau + F ( D (Tx, Ty))$
  • Figure 3: (a) Iteration $u_n (t )$ vs $t$ (b) $||u_{n +1} (t ) - u_n (t) | |_\infty$ vs $n$

Theorems & Definitions (19)

  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Theorem 2.8
  • Lemma 3.1
  • Definition 3.2
  • ...and 9 more