scholarly journals On uniformly contractive systems and quadratic equations in Banach space

1995 ◽  
Vol 52 (3) ◽  
pp. 441-455 ◽  
Author(s):  
David K. Ruch
Keyword(s):  
Banach Space ◽  
Systems Theory ◽  
Pointwise Convergence ◽  
Contraction Mapping ◽  
Continuation Method ◽  
Approximate Solutions ◽  
Contraction Mapping Principle ◽  
Uniqueness Result ◽  
Convergence Criteria ◽  
Quadratic Equations

The solution of quadratic equations using the contraction mapping principle is considered. A uniqueness result extending that given by Argyros is proved. Uniformly contractive systems theory is used to find approximate solutions and convergence criteria are given. In particular, only pointwise convergence of approximating operators is required to guarantee convergence of the approximate solutions. A theorem and algorithm for a continuation method are presented, and illustrated on Chandrasekhar's equation.

scholarly journals Quadratic equations and applications to Chandrasekhar's and related equations

1985 ◽  
Vol 32 (2) ◽  
pp. 275-292 ◽  
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Integral Equation ◽  
Contraction Mapping ◽  
New Technique ◽  
Contraction Mapping Theorem ◽  
Quadratic Equations ◽  
A New Technique

A new technique, using the contraction mapping theorem, for solving quadratic equations in Banach space is introduced. The results are then applied to solve Chandrasekhar's integral equation and related equations without the usual positivity assumptions.

Analysis of fractional integro-differential equations with nonlocal Erdélyi-Kober type integral boundary conditions

2020 ◽  
Vol 23 (5) ◽  
pp. 1401-1415
Author(s):  
Palanisamy Duraisamy ◽  
Thangaraj Nandha Gopal ◽  
Muthaiah Subramanian
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Contraction Mapping ◽  
Fixed Point Theorems ◽  
Integral Boundary Conditions ◽  
Contraction Mapping Principle ◽  
Uniqueness Result ◽  
Uniqueness Of Solutions ◽  
Integral Boundary ◽  
Type Integral

Abstract In this article, we study the existence and uniqueness of solutions for nonlinear fractional integro-differential equations with nonlocal Erdélyi-Kober type integral boundary conditions. The existence results are based on Krasnoselskii’s and Schaefer’s fixed point theorems, whereas the uniqueness result is based on the contraction mapping principle. Examples illustrating the applicability of our main results are also constructed.

scholarly journals On the ultradifferentiable normalization

2021 ◽  
Author(s):  
Hao Wu ◽  
Xingdong Xu ◽  
Dongfeng Zhang
Keyword(s):  
Banach Space ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Weighted Norms

AbstractWe show the theory of the formal ultradifferentiable normalization. The tools utilized here are KAM methods and Contraction Mapping Principle in the Banach space fixed with weighted norms.

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.

Semi-local convergence of a Newton-like method for solving equations with a singular derivative

2018 ◽  
Vol 27 (1) ◽  
pp. 01-08
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
GEORGE SANTHOSH ◽  
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Real Line ◽  
Local Convergence ◽  
Approximate Solutions ◽  
Convergence Domain ◽  
Solving Equations ◽  
Solutions Of Equations ◽  
Local Convergence Analysis ◽  
Special Case

We present a semi-local convergence analysis for a Newton-like method to approximate solutions of equations when the derivative is not necessarily non-singular in a Banach space setting. In the special case when the equation is defined on the real line the convergence domain is improved for this method when compared to earlier results. Numerical results where earlier results cannot apply but the new results can apply to solve nonlinear equations are also presented in this study.

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 On singular systems of nonlinear equations involving 3n-Caputo derivatives

2020 ◽  
Vol 23 (2) ◽  
pp. 179-192
Author(s):  
Amele Taïeb
Keyword(s):  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Singular Systems ◽  
Nonlinear Differential Equations ◽  
Contraction Mapping Principle ◽  
Systems Of Nonlinear Equations ◽  
Fractional Systems ◽  
Caputo Derivatives ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results

We study singular fractional systems of nonlinear differential equations involving 3n-Caputo derivatives. We investigate existence and uniqueness results using the contraction mapping principle. We also discuss the existence of at least one solution by means of Schauder fixed point theorem. Moreover, we define and discuss the Ulam–Hyers stability and the generalized Ulam–Hyers stability of solutions for such systems. To illustrate the main results, we present some examples.

scholarly journals Exponential Stability of Periodic Solution to Wilson-Cowan Networks with Time-Varying Delays on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Jinxiang Cai ◽  
Zhenkun Huang ◽  
Honghua Bin
Keyword(s):  
Periodic Solution ◽  
Stability Analysis ◽  
Exponential Stability ◽  
Time Scales ◽  
Continuous Time ◽  
Lyapunov Functional ◽  
Contraction Mapping ◽  
Sufficient Conditions ◽  
Contraction Mapping Principle ◽  
Time Varying

We present stability analysis of delayed Wilson-Cowan networks on time scales. By applying the theory of calculus on time scales, the contraction mapping principle, and Lyapunov functional, new sufficient conditions are obtained to ensure the existence and exponential stability of periodic solution to the considered system. The obtained results are general and can be applied to discrete-time or continuous-time Wilson-Cowan networks.

scholarly journals Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Yu-Zhu Wang
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Hyperbolic Equation ◽  
Contraction Mapping ◽  
Dimensional Space ◽  
Contraction Mapping Principle ◽  
Nonlinear Hyperbolic Equation ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped nonlinear hyperbolic equation inn-dimensional space. Under small condition on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained by the contraction mapping principle.

scholarly journals POINTWISE CONVERGENCE AND SEMIGROUPS ACTING ON VECTOR-VALUED FUNCTIONS

2011 ◽  
Vol 84 (1) ◽  
pp. 44-48 ◽  
Author(s):  
MICHAEL G. COWLING ◽  
MICHAEL LEINERT
Keyword(s):  
Banach Space ◽  
Pointwise Convergence ◽  
Function Spaces ◽  
Semigroup Of Operators ◽  
Almost Everywhere ◽  
Vector Valued Function ◽  
Vector Valued Functions ◽  
Complex Valued ◽  
Vector Valued

AbstractA submarkovian C0 semigroup (Tt)t∈ℝ+ acting on the scale of complex-valued functions Lp(X,ℂ) extends to a semigroup of operators on the scale of vector-valued function spaces Lp(X,E), when E is a Banach space. It is known that, if f∈Lp(X,ℂ), where 1<p<∞, then Ttf→f pointwise almost everywhere. We show that the same holds when f∈Lp(X,E) .

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