scholarly journals Ulam–Hyers stabilities of a differential equation and a weakly singular Volterra integral equation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ozgur Ege ◽  
Souad Ayadi ◽  
Choonkil Park
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Fixed Point ◽  
Volterra Integral Equation ◽  
Continuous Functions ◽  
Banach Fixed Point Theorem ◽  
Space Of Continuous Functions ◽  
Weakly Singular ◽  
Banach Contraction ◽  
The Banach Contraction Principle

AbstractIn this work we study the Ulam–Hyers stability of a differential equation. Its proof is based on the Banach fixed point theorem in some space of continuous functions equipped with the norm $\|\cdot \|_{\infty }$ ∥ ⋅ ∥ ∞ . Moreover, we get some results on the Ulam–Hyers stability of a weakly singular Volterra integral equation using the Banach contraction principle in the space of continuous functions $C([a,b])$ C ( [ a , b ] ) .

A Topological Banach Fixed Point Theorem for Compact Hausdorff Spaces

1994 ◽  
Vol 37 (4) ◽  
pp. 552-555 ◽  
Author(s):  
Juris Steprans ◽  
Stephen Watson ◽  
Winfried Just
Keyword(s):  
Fixed Point ◽  
Hausdorff Space ◽  
Unique Fixed Point ◽  
Banach Fixed Point Theorem ◽  
Hausdorff Spaces ◽  
Compact Metric Space ◽  
Banach Contraction ◽  
Compact Metrizable Space ◽  
Admissible Metric ◽  
The Banach Contraction Principle

AbstractWe propose an analogue of the Banach contraction principle for connected compact Hausdorff spaces. We define a J-contraction of a connected compact Hausdorff space. We show that every contraction of a compact metric space is a J-contraction and that any J-contraction of a compact metrizable space is a contraction for some admissible metric. We show that every J-contraction has a unique fixed point and that the orbit of each point converges to this fixed point.

scholarly journals Nonlinear Volterra Integral Equation of the Second Kind and Biorthogonal Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
M. I. Berenguer ◽  
D. Gámez ◽  
A. I. Garralda-Guillem ◽  
M. C. Serrano Pérez
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Volterra Integral Equation ◽  
Biorthogonal System ◽  
Banach Fixed Point Theorem ◽  
Nonlinear Volterra Integral Equation ◽  
Numerical Resolution

We obtain an approximation of the solution of the nonlinear Volterra integral equation of the second kind, by means of a new method for its numerical resolution. The main tools used to establish it are the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.

scholarly journals Existence results for fractional neutral integro-differential systems with nonlocal condition through resolvent operators

2019 ◽  
Vol 27 (1) ◽  
pp. 107-124
Author(s):  
D. Mallika ◽  
D. Baleanu ◽  
S. Suganya ◽  
M. Mallika Arjunan
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Existence Results ◽  
Resolvent Operators ◽  
Differential Systems ◽  
Banach Contraction ◽  
Krasnoselskii Fixed Point Theorem ◽  
The Banach Contraction Principle

Abstract The manuscript is primarily concerned with the new existence results for fractional neutral integro-differential equation (FNIDE) with nonlocal conditions (NLCs) in Banach spaces. Based on the Banach contraction principle and Krasnoselskii fixed point theorem (FPT) joined with resolvent operators, we develop the main results. Ultimately, an representation is also offered to demonstrate the accomplished theorem.

scholarly journals Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Infinite Delay

2021 ◽  
Vol 5 (2) ◽  
pp. 52
Author(s):  
Kaihong Zhao ◽  
Yue Ma
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Normed Space ◽  
Existence Of Solutions ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Banach Contraction ◽  
The Banach Contraction Principle

The existence of solutions for a class of nonlinear neutral Hadamard-type fractional integro-differential equations with infinite delay is researched in this paper. By constructing an appropriate normed space and utilizing the Banach contraction principle, Krasnoselskii’s fixed point theorem, we obtain some sufficient conditions for the existence of solutions. Finally, we provide an example to illustrate the validity of our main results.

scholarly journals PRECOMPACTNESS OF THE SET OF TRAJECTORIES OF THE CONTROLLABLE SYSTEM DESCRIBED BY A NONLINEAR VOLTERRA INTEGRAL EQUATION

2012 ◽  
Vol 17 (5) ◽  
pp. 686-695 ◽  
Author(s):  
Anar Huseyin ◽  
Nesir Huseyin
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Admissible Control ◽  
Continuous Functions ◽  
Closed Ball ◽  
Controllable System ◽  
Space Of Continuous Functions ◽  
Nonlinear Volterra Integral Equation ◽  
Control Functions

In this paper the controllable system whose behaviour is described by a nonlinear Volterra integral equation, is studied. The set of admissible control functions is the closed ball of the space L p (p > 1) with radius µ 0 and centered at the origin. It is shown that the set of trajectories of the system is a bounded and precompact subset of the space of continuous functions.

scholarly journals On Existence and Uniqueness of an Integrable Solution for a Fractional Volterra Integral Equation on 𝑹+

2020 ◽  
pp. 122-125
Author(s):  
Faez N. Ghaffoori
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Uniqueness Theorem ◽  
Point Theorem ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Banach Fixed Point Theorem ◽  
Existence And Uniqueness Theorem ◽  
Integrable Solution

In this paper, by using the Banach fixed point theorem, we prove the existence and uniqueness theorem of a fractional Volterra integral equation in the space of Lebesgue integrable 𝐿1(𝑅+) on unbounded interval [0,∞).

A generalization of Reich’s fixed point theorem for multi-valued mappings

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3295-3305 ◽  
Author(s):  
Antonella Nastasi ◽  
Pasquale Vetro
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Contractive Condition ◽  
Complete Metric Spaces ◽  
New Class ◽  
Banach Contraction ◽  
The Banach Contraction Principle

Motivated by a problem concerning multi-valued mappings posed by Reich [S. Reich, Some fixed point problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 57 (1974) 194-198] and a paper of Jleli and Samet [M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl. 2014:38 (2014) 1-8], we consider a new class of multi-valued mappings that satisfy a ?-contractive condition in complete metric spaces and prove some fixed point theorems. These results generalize Reich?s and Mizoguchi-Takahashi?s fixed point theorems. Some examples are given to show the usability of the obtained results.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

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