A Fixed Point Approach to the Hyers-Ulam-Rassias Stability Problem of Pexiderized Functional Equation in Modular Spaces

2021 ◽  
Vol 78 (1) ◽  
pp. 59-72
Author(s):  
Parbati Saha ◽  
Pratap Mondal ◽  
Binayak S. Chqudhury
Keyword(s):  
Fixed Point ◽  
Stability Problem ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Modular Space ◽  
Generalized Contraction ◽  
Mathematical Structures ◽  
Fixed Point Approach ◽  
Modular Spaces ◽  
Rassias Stability

Abstract In this paper, we consider pexiderized functional equations for studying their Hyers-Ulam-Rassias stability. This stability has been studied for a variety of mathematical structures. Our framework of discussion is a modular space. We adopt a fixed-point approach to the problem in which we use a generalized contraction mapping principle in modular spaces. The result is illustrated with an example.

A Fixed Point Approach to the Stability of Quadratic Functional Equations in Modular Spaces

2017 ◽  
Vol 6 (1) ◽  
pp. 171-175
Author(s):  
Seong Sik Kim ◽  
Soo Hwan Kim
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Method ◽  
Quadratic Functional ◽  
Fixed Point Approach ◽  
Modular Spaces ◽  
The Stability ◽  
Positive Integers ◽  
Rassias Stability

In this paper, we investigate the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation f(kx + y) + f(kx – y) = 2k2f(x) + 2f(y) for any fixed positive integers k ∈ Ζ+ in modular spaces by using fixed point method.

Nonlinear iterative algorithms for quasi contraction mapping in modular space

2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Robabe Moradi ◽  
Abdolrahman Razani
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Double Sequence ◽  
Iterative Algorithms ◽  
Modular Space ◽  
Matlab Software ◽  
Numerical Examples ◽  
Modular Spaces

AbstractIn this paper, we introduce new nonlinear iterative algorithms. These algorithms are used to study the convergence of generated iterative sequences in modular spaces. Moreover, we introduce a new double sequence iteration and prove that sequences converge strongly to a fixed point of a ρ-quasi contraction mapping in modular spaces. Finally, some illustrative numerical examples (using the Matlab software) are presented.

scholarly journals Stability analysis for a class of implicit fractional differential equations involving Atangana–Baleanu fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Asma ◽  
Sana Shabbir ◽  
Kamal Shah ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Stability Results ◽  
Rassias Stability ◽  
Banach Contraction Mapping ◽  
Implicit Fractional Differential Equations

AbstractSome fundamental conditions and hypotheses are established to ensure the existence, uniqueness, and stability to a class of implicit boundary value problems (BVPs) with Atangana–Baleanu–Caputo type derivative and integral. The required results are established by utilizing the Banach contraction mapping principle and fixed point theorem of Krasnoselskii. In addition, various types of stability results including Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability are formulated for the problem under consideration. Pertinent examples are given to justify the results we obtain.

scholarly journals Discrete fractional order two-point boundary value problem with some relevant physical applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
A. George Maria Selvam ◽  
Jehad Alzabut ◽  
R. Dhineshbabu ◽  
S. Rashid ◽  
M. Rehman
Keyword(s):  
Boundary Value Problem ◽  
Fractional Order ◽  
Contraction Mapping ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Difference Operators ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Fractional Difference ◽  
Rassias Stability

Abstract The results reported in this paper are concerned with the existence and uniqueness of solutions of discrete fractional order two-point boundary value problem. The results are developed by employing the properties of Caputo and Riemann–Liouville fractional difference operators, the contraction mapping principle and the Brouwer fixed point theorem. Furthermore, the conditions for Hyers–Ulam stability and Hyers–Ulam–Rassias stability of the proposed discrete fractional boundary value problem are established. The applicability of the theoretical findings has been demonstrated with relevant practical examples. The analysis of the considered mathematical models is illustrated by figures and presented in tabular forms. The results are compared and the occurrence of overlapping/non-overlapping has been discussed.

scholarly journals Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses

2021 ◽  
Vol 7 (2) ◽  
pp. 3169-3185
Author(s):  
Kaihong Zhao ◽  
◽  
Shuang Ma
Keyword(s):  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Integral Boundary ◽  
Nonlinear Functional Analysis ◽  
Nonlinear Functional ◽  
Hadamard Fractional Differential Equations ◽  
Analysis Technique ◽  
Fractional Differential ◽  
Rassias Stability ◽  
Hadamard Fractional Integral

<abstract><p>This paper considers a class of nonlinear implicit Hadamard fractional differential equations with impulses. By using Banach's contraction mapping principle, we establish some sufficient criteria to ensure the existence and uniqueness of solution. Furthermore, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of this system are obtained by applying nonlinear functional analysis technique. As applications, an interesting example is provided to illustrate the effectiveness of main results.</p></abstract>

scholarly journals Fixed Point Theorems for a New Class of Mappings in Modular Spaces Endowed with a Graph

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Nour-eddine El Harmouchi ◽  
Karim Chaira ◽  
El Miloudi Marhrani
Keyword(s):  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Fixed Point Theorems ◽  
Iterative Scheme ◽  
Modular Space ◽  
New Class ◽  
Modular Spaces

In this paper, we discuss a class of mappings more general than ρ-nonexpansive mapping defined on a modular space endowed with a graph. In our investigation, we prove the existence of fixed point results of these mappings. Then, we also introduce an iterative scheme for which proves the convergence to a fixed point of such mapping in a modular space with a graph.

scholarly journals On the Generalized Ulam-Hyers-Rassias Stability of Quadratic Mappings in Modular Spaces withoutΔ2-Conditions

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Kittipong Wongkum ◽  
Parin Chaipunya ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
Fixed Point Theory ◽  
Lower Semicontinuous ◽  
Point Theory ◽  
Quadratic Functional ◽  
Modular Spaces ◽  
Quadratic Mappings ◽  
Rassias Stability

We approach the generalized Ulam-Hyers-Rassias (briefly, UHR) stability of quadratic functional equations via the extensive studies of fixed point theory. Our results are obtained in the framework of modular spaces whose modulars are lower semicontinuous (briefly, lsc) but do not satisfy any relatives ofΔ2-conditions.

scholarly journals Periodicity and positivity in nonlinear neutral integro-dynamic equations with variable delay

Mathematica ◽  
2020 ◽  
Vol 62 (85) (2) ◽  
pp. 117-132
Author(s):  
Malik Belaid ◽  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Dynamic Equations ◽  
Variable Delay ◽  
Positive Periodic Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Completely Continuous ◽  
Continuous Map ◽  
Uniqueness Results

Let T be a periodic time scale. We use Krasnoselskii's fixed point theorem for a sum of two operators to show new results on the existence of periodic and positive periodic solutions of a nonlinear neutral integro-dynamic equation with variable delay. We invert this equation to construct a sum of a contraction and a completely continuous map which is suitable for applying Krasnoselskii's theorem. The uniqueness results of this equation are studied by the contraction mapping principle.

scholarly journals Analytical Methods for Transport Equations in Similarity Form

2007 ◽  
Author(s):  
Abhishek Tiwari ◽  
Kaveh A. Tagavi ◽  
J. M. McDonough
Keyword(s):  
Fixed Point ◽  
Analytical Solutions ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Transport Equations ◽  
Fixed Point Iteration ◽  
Novel Approach ◽  
Derivative Term ◽  
Range Of Values ◽  
Limit Of Solution

We present a novel approach for deriving analytical solutions to transport equations expressed in similarity variables. We apply a fixed-point iteration procedure to these transformed equations by formally solving for the highest derivative term and, from this (via requirements for convergence given by the contraction mapping principle), deduce a range of values for the outer limit of solution domain, for which the fixed-point iteration gives a converged solution.

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