scholarly journals A fixed point theorem and the Hyers–Ulam stability in non-Archimedean spaces

2013 ◽  
Vol 400 (1) ◽  
pp. 68-75 ◽  
Author(s):  
Janusz Brzdȩk ◽  
Krzysztof Ciepliński
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Ulam Stability

ON APPROXIMATELY ADDITIVE MAPPINGS IN 2-BANACH SPACES

2019 ◽  
Vol 101 (2) ◽  
pp. 299-310 ◽  
Author(s):  
JANUSZ BRZDĘK ◽  
EL-SAYED EL-HADY
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Stability Result ◽  
Main Tool ◽  
Ulam Stability ◽  
Cauchy Equation ◽  
Stability Issues ◽  
Additive Mappings

We show how some Ulam stability issues can be approached for functions taking values in 2-Banach spaces. We use the example of the well-known Cauchy equation $f(x+y)=f(x)+f(y)$, but we believe that this method can be applied for many other equations. In particular we provide an extension of an earlier stability result that has been motivated by a problem of Th. M. Rassias. The main tool is a recent fixed point theorem in some spaces of functions with values in 2-Banach spaces.

scholarly journals Existence, uniqueness, approximation of solutions and Ealpha-Ulam stability results for a class of nonlinear fractional differential equations involving psi-Caputo derivative with initial conditions

2021 ◽  
Vol 25 (1) ◽  
pp. 1-30
Author(s):  
Choukri Derbazi ◽  
Zidane Baitiche ◽  
Mouffak Benchohra ◽  
Gaston N'guérékata
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Ulam Stability ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
Stability Results

The main purpose of this paper is to study the existence, uniqueness, Ea-Ulam stability results, and other properties of solutions for certain classes of nonlinear fractional differential equations involving the ps-Caputo derivative with initial conditions. Modern tools of functional analysis are applied to obtain the main results. More precisely using Weissinger's fixed point theorem and Schaefer's fixed point theorem the existence and uniqueness results of solutions are proven in the bounded domain. While the well known Banach fixed point theorem coupled with Bielecki type norm are used with the end goal to establish sufficient conditions for existence and uniqueness results on unbounded domains. Meanwhile, the monotone iterative technique combined with the method of upper and lower solutions is used to prove the existence and uniqueness of extremal solutions. Furthermore, by means of new generalizations of Gronwall's inequality, different kinds of Ea-Ulam stability of the proposed problem are studied. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility and correctness of the main results.

scholarly journals Existence, Uniqueness, and Mittag–Leffler–Ulam Stability Results for Cauchy Problem Involving ψ -Caputo Derivative in Banach and Fréchet Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Choukri Derbazi ◽  
Zidane Baitiche ◽  
Mouffak Benchohra ◽  
G. N’Guérékata
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Ulam Stability ◽  
Fréchet Spaces ◽  
Existence Of A Solution ◽  
Gronwall’S Inequality ◽  
Condensing Operators ◽  
Stability Results

Our aim in this paper is to investigate the existence, uniqueness, and Mittag–Leffler–Ulam stability results for a Cauchy problem involving ψ -Caputo fractional derivative with positive constant coefficient in Banach and Fréchet Spaces. The techniques used are a variety of tools for functional analysis. More specifically, we apply Weissinger’s fixed point theorem and Banach contraction principle with respect to the Chebyshev and Bielecki norms to obtain the uniqueness of solution on bounded and unbounded domains in a Banach space. However, a new fixed point theorem with respect to Meir–Keeler condensing operators combined with the technique of Hausdorff measure of noncompactness is used to investigate the existence of a solution in Banach spaces. After that, by means of new generalizations of Grönwall’s inequality, the Mittag–Leffler–Ulam stability of the proposed problem is studied on a compact interval. Meanwhile, an extension of the well-known Darbo’s fixed point theorem in Fréchet spaces associated with the concept of measures of noncompactness is applied to obtain the existence results for the problem at hand. Finally, as applications of the theoretical results, some examples are given to illustrate the feasibility of the main theorems.

On the stability for the fuzzy initial value problem

2020 ◽  
Vol 39 (5) ◽  
pp. 7747-7755
Author(s):  
Ngo Van Hoa ◽  
Tofigh Allahviranloo ◽  
Ho Vu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Initial Value Problem ◽  
Ulam Stability ◽  
Initial Value ◽  
The Stability ◽  
Theoretical Results ◽  
Rassias Stability

In this paper, we present the Hyers–Ulam stability and Hyers–Ulam-Rassias stability (HU-stability and HUR-stability for short) for fuzzy initial value problem (FIVP) by using fixed point theorem. We improve and extend some known results on the stability for FDEs by dropping some assumptions. Some examples illustrate the theoretical results.

scholarly journals Orthogonal stability of the generalized quadratic functional equations in the sense of Rätz

2019 ◽  
Vol 52 (1) ◽  
pp. 523-530
Author(s):  
Laddawan Aiemsomboon ◽  
Wutiphol Sintunavarat
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Algebra ◽  
Point Theorem ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Banach Module ◽  
Unital Banach Algebra

AbstractLet (X, ⊥) be an orthogonality module in the sense of Rätz over a unital Banach algebra A and Y be a real Banach module over A. In this paper, we apply the alternative fixed point theorem for proving the Hyers-Ulam stability of the orthogonally generalized k-quadratic functional equation of the formaf(kx + y) + af(kx - y) = f(ax + ay) + f(ax - ay) + \left( {2{k^2} - 2} \right)f(ax)for some |k| > 1, for all a ɛ A1 := {u ɛ A||u|| = 1} and for all x, y ɛ X with x⊥y, where f maps from X to Y.

scholarly journals The Ulam Stability of Fractional Differential Equation with the Caputo-Fabrizio Derivative

2022 ◽  
Vol 2022 ◽  
pp. 1-9
Author(s):  
Shuyi Wang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Differential Equation ◽  
Point Theorem ◽  
Ulam Stability ◽  
Integral Boundary ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Uniqueness Results ◽  
Fractional Differential

The aim of this paper is to establish the Ulam stability of the Caputo-Fabrizio fractional differential equation with integral boundary condition. We also present the existence and uniqueness results of the solution for the Caputo-Fabrizio fractional differential equation by Krasnoselskii’s fixed point theorem and Banach fixed point theorem. Some examples are provided to illustrate our theorems.

Ulam stability of some functional inclusions for multi-valued mappings

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5489-5495 ◽  
Author(s):  
Janusz Brzdęk ◽  
Magdalena Piszczek
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Positive Integer ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Main Tool ◽  
Ulam Stability ◽  
Nonlinear Operators ◽  
The Family ◽  
Fixed Positive Integer

We show that some multifunctions F : K ? n(Y), satisfying functional inclusions of the form ? (x,F(?1(x)),..., F(?n(x)))? F(x)G(x), admit near-selections f : K ? Y, fulfilling the functional equation ? (x,f (?1(x)),..,, f(?n(x)))= f(x), where functions G : K ? n(Y), ?: K x Yn ? Y and ?1,..., ?n ? KK are given, n is a fixed positive integer, K is a nonempty set, (Y,?) is a group and n(Y) denotes the family of all nonempty subsets of Y. Our results have been motivated by the notion of Ulam stability and some earlier outcomes. The main tool in the proofs is a very recent fixed point theorem for nonlinear operators, acting on some spaces of multifunctions.

scholarly journals Hyers–Ulam Stability and Existence of Solutions for Differential Equations with Caputo–Fabrizio Fractional Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 333 ◽  
Author(s):  
Kui Liu ◽  
Michal Fečkan ◽  
D. O’Regan ◽  
JinRong Wang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Stability Result ◽  
Banach Fixed Point Theorem ◽  
Ulam Stability ◽  
Uniqueness Of Solutions ◽  
Transform Method ◽  
Fractional Differential

In this paper, the Hyers–Ulam stability of linear Caputo–Fabrizio fractional differential equation is established using the Laplace transform method. We also derive a generalized Hyers–Ulam stability result via the Gronwall inequality. In addition, we establish existence and uniqueness of solutions for nonlinear Caputo–Fabrizio fractional differential equations using the generalized Banach fixed point theorem and Schaefer’s fixed point theorem. Finally, two examples are given to illustrate our main results.

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