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.

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 A study of fractional integro-differential equations via Hilfer-Hadamard fractional derivative

2019 ◽  
Vol 27 (1) ◽  
pp. 71-84
Author(s):  
D. Vivek ◽  
K. Kanagarajan ◽  
E. M. Elsayed
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Existence Of Solution ◽  
Ulam Stability ◽  
Hadamard Fractional Derivative ◽  
Stability Results

Abstract In this paper, we investigate the existence of solution of integro-differential equations (IDEs) with Hilfer-Hadamard fractional derivative. The main results are obtained by using Schaefer’s fixed point theorem. Some Ulam stability results are presented.

scholarly journals An extension of Darbo’s fixed point theorem for a class of system of nonlinear integral equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Amar Deep ◽  
Deepmala ◽  
Jamal Rezaei Roshan ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Equations ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Existence Results ◽  
Nonlinear Integral Equations ◽  
Existence Of A Solution ◽  
Darbo’S Fixed Point Theorem ◽  
Novi Sad

Abstract We introduce an extension of Darbo’s fixed point theorem via a measure of noncompactness in a Banach space. By using our extension we study the existence of a solution for a system of nonlinear integral equations, which is an extended result of (Aghajani and Haghighi in Novi Sad J. Math. 44(1):59–73, 2014). We give an example to show the specified existence results.

Simulation functions and Meir–Keeler condensing operators with application to integral equations

2021 ◽  
Author(s):  
Moosa Gabeleh ◽  
Mehdi Asadi ◽  
Pradip Ramesh Patle
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Integral Equations ◽  
Point Theorem ◽  
Darbo’S Fixed Point Theorem ◽  
Simulation Functions ◽  
Condensing Operators

We propose a new concept of condensing operators by using a notion of measure of non-compactness in the setting of Banach spaces and establish a new generalization of Darbo’s fixed point theorem. We also show the applicability of our results to integral equations. A concrete example will be presented to support the application part.

scholarly journals Generalizations of Darbo’s fixed point theorem for new condensing operators with application to a functional integral equation

2019 ◽  
Vol 52 (1) ◽  
pp. 166-182 ◽  
Author(s):  
Habib ur Rehman ◽  
Dhananjay Gopal ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Coupled Fixed Point ◽  
Functional Integral ◽  
Nonlinear Functional ◽  
Darbo’S Fixed Point Theorem ◽  
Condensing Operators ◽  
Functional Integral Equations

AbstractIn this paper, we provide some generalizations of the Darbo’s fixed point theorem associated with the measure of noncompactness and present some results on the existence of the coupled fixed point theorems for a special class of operators in a Banach space. To acquire this result, we defineα-ψandβ-ψcondensing operators and using them we propose new fixed point results. Our results generalize and extend some comparable results from the literature. Additionally, as an application, we apply the obtained fixed point theorems to study the nonlinear functional integral equations.

scholarly journals Uniqueness and Stability Results on Non-local Stochastic Random Impulsive Integro-Differential Equations

2021 ◽  
Vol 2 (3) ◽  
pp. 9-20
Author(s):  
VARSHINI S ◽  
BANUPRIYA K ◽  
RAMKUMAR K ◽  
RAVIKUMAR K
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Sufficient Conditions ◽  
Banach Fixed Point Theorem ◽  
Local Conditions ◽  
Non Local ◽  
Inequality Techniques ◽  
Stability Results

The paper is concerned with stochastic random impulsive integro-differential equations with non-local conditions. The sufficient conditions guarantees uniqueness of mild solution derived using Banach fixed point theorem. Stability of the solution is derived by incorporating Banach fixed point theorem with certain inequality techniques.

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 About the Existence Results of Fractional Neutral Integrodifferential Inclusions with State-Dependent Delay in Fréchet Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Selvaraj Suganya ◽  
Dumitru Baleanu ◽  
Siva Selvarasu ◽  
Mani Mallika Arjunan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Semigroup Theory ◽  
Existence Results ◽  
Fréchet Spaces ◽  
State Dependent ◽  
Nonlinear Alternative ◽  
State Dependent Delay ◽  
Multivalued Contractions

A recent nonlinear alternative for multivalued contractions in Fréchet spaces thanks to Frigon fixed point theorem consolidated with semigroup theory is utilized to examine the existence results for fractional neutral integrodifferential inclusions (FNIDI) with state-dependent delay (SDD). An example is described to represent the hypothesis.

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