A rigorous proof of an exponentially small estimate for a boundary value arising from an ordinary differential equation

1990 ◽  
Vol 114 (3-4) ◽  
pp. 243-258 ◽  
Author(s):  
J. G. B. Byatt-Smith ◽  
A. M. Davie
Keyword(s):  
Asymptotic Expansion ◽  
Integral Equations ◽  
Complex Plane ◽  
Existence And Uniqueness ◽  
Asymptotic Estimate ◽  
Contraction Mapping ◽  
Contraction Mapping Theorem ◽  
Oscillatory Solutions ◽  
Connection Problem ◽  
Image Position

SynopsisThe equationhas a solution y(t) which is non-oscillating on the interval (0, ∞) and has the asymptotic expansionEach term of this expansion is even in t so that formally is zero to all orders of ɛ. The estimate of has been obtained by Byatt-Smith [3] who corrects (2) in the complex plane near t = i where the series ceases to be valid. This requires asolution of the equationthe equation for the first Painlevé transcedent. Here we prove rigorously that this method gives the correct asymptotic estimatewhereThe proof involves converting (1) and (3) to integral equations. The existence and uniqueness of these integral equations are established by use of the contraction mapping theorem. We also prove that the appropriate solution to (3) provides a uniformly valid approximation to (2) over a suitably defined region of the complex plane.We also consider the connection problem for the oscillatory solutions of (1) which have asymptotic expansionswhere Ã+, Ã− φ+, and φ− are constants. The connection problem is to determine the asymptotic expansion at +∞ of a solution which has a given asymptotic expansion at −∞. In other words, we wish to find (Ã+,φ+) as a function of Ã−and φ−. We prove that there is a unique solution to the connection problem, provided Ã− is small enough, and obtain bounds on the estimate of Ã+

scholarly journals A New Existence and Uniqueness Theorem for Continuous Games

2011 ◽  
Vol 11 (1) ◽  
Author(s):  
Seamus D Hogan
Keyword(s):  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Jacobian Matrix ◽  
Iterative Sequence ◽  
Response Functions ◽  
Unique Equilibrium ◽  
Contraction Mapping Theorem ◽  
Best Response ◽  
General Sufficient Condition ◽  
General Game

This paper derives a general sufficient condition for existence and uniqueness in continuous games using a variant of the contraction mapping theorem applied to mappings from a subset of the real line on to itself. We first prove this contraction mapping variant, and then show how the existence of a unique equilibrium in the general game can be shown by proving the existence of a unique equilibrium in an iterative sequence of games involving such mappings. Finally, we show how a general condition for this to occur is that a matrix derived from the Jacobian matrix of best-response functions has positive leading principal minors, and how this condition generalises some existing uniqueness theorems for particular games. In particular, we show how the same conditions used in those theorems to show uniqueness, also guarantee existence in games with unbounded strategy spaces.

scholarly journals Some axially symmetric potential problems

1972 ◽  
Vol 18 (1) ◽  
pp. 55-76 ◽  
Author(s):  
F. G. Leppington ◽  
H. Levine
Keyword(s):  
Integral Equation ◽  
Asymptotic Expansion ◽  
Integral Equations ◽  
Asymptotic Estimate ◽  
Axially Symmetric ◽  
Primary Interest ◽  
Symmetric Potential ◽  
Approximate Integral ◽  
Potential Problems ◽  
Higher Order Terms

AbstractSome axially symmetric boundary value problems of potential theory are formulated as integral equations of the first kind. In each case the kernel admits an expansion, for small values of a parameter of the problem, that leads to an approximate integral equation whose solution provides a direct asymptotic estimate for the physical quantity of primary interest. A manipulation of the original and modified integral equations provides an efficient formula for calculating higher order terms in the asymptotic expansion.

Evaluation of the Forward-Backward Sweep Load Flow Method using the Contraction Mapping Principle

2016 ◽  
Vol 6 (6) ◽  
pp. 3229
Author(s):  
Diego Issicaba ◽  
Jorge Coelho
Keyword(s):  
Distribution System ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
System Analysis ◽  
Feasible Solution ◽  
Contraction Mapping Principle ◽  
Load Flow ◽  
Contraction Mapping Theorem ◽  
Flow Method ◽  
Theoretical Results

<p>This paper presents an assessment of the forward-backward sweep load flow method to distribution system analysis. The method is formally assessed using fixed-point concepts and the contraction mapping theorem. The existence and uniqueness of the load flow feasible solution is supported by an alternative argument from those obtained in the literature. Also, the closed-form of the convergence rate of the method is deduced and the convergence dependence of loading is assessed. Finally, boundaries for error values per iteration between iterates and feasible solution are obtained. Theoretical results have been tested in several numerical simulations, some of them presented in this paper, thus fostering discussions about applications and future works.</p>

scholarly journals Periodic solutions of Volterra integral equations

1988 ◽  
Vol 11 (4) ◽  
pp. 781-792 ◽  
Author(s):  
M. N. Islam
Keyword(s):  
Periodic Solutions ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Volterra Integral Equations ◽  
System Of Equations ◽  
Convolution Kernel ◽  
Basic Tool ◽  
Resolvent Function

Consider the system of equationsx(t)=f(t)+∫−∞tk(t,s)x(s)ds,           (1)andx(t)=f(t)+∫−∞tk(t,s)g(s,x(s))ds.       (2)Existence of continuous periodic solutions of (1) is shown using the resolvent function of the kernelk. Some important properties of the resolvent function including its uniqueness are obtained in the process. In obtaining periodic solutions of (1) it is necessary that the resolvent ofkis integrable in some sense. For a scalar convolution kernelksome explicit conditions are derived to determine whether or not the resolvent ofkis integrable. Finally, the existence and uniqueness of continuous periodic solutions of (1) and (2) are btained using the contraction mapping principle as the basic tool.

scholarly journals SHARPER EXISTENCE AND UNIQUENESS RESULTS FOR SOLUTIONS TO THIRD-ORDER BOUNDARY VALUE PROBLEMS

2020 ◽  
Vol 25 (3) ◽  
pp. 409-420 ◽  
Author(s):  
Saleh S. Almuthaybiri ◽  
Christopher C. Tisdell
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Boundary Value ◽  
Uniqueness Of Solutions ◽  
Third Order ◽  
Contraction Mapping Theorem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Banach’S Fixed Point Theorem

The purpose of this note is to sharpen Smirnov’s recent work on existence and uniqueness of solutions to third-order ordinary differential equations that are subjected to two- and three-point boundary conditions. The advancement is achieved in the following ways. Firstly, we provide sharp and sharpened estimates for integrals regarding various Green’s functions. Secondly, we apply these sharper estimates to problems in conjunction with Banach’s fixed point theorem. Thirdly, we apply Rus’s contraction mapping theorem in a metric space, where two metrics are employed. Our new results improve those of Smirnov by showing that a larger class of boundary value problems admit a unique solution.

The solution of Bessel function dual integral equations by a multiplying-factor method

1963 ◽  
Vol 59 (2) ◽  
pp. 351-362 ◽  
Author(s):  
B. Noble
Keyword(s):  
Analytic Continuation ◽  
Bessel Function ◽  
Integral Equations ◽  
Complex Plane ◽  
Real Variable ◽  
Dual Integral Equations ◽  
Variable Technique ◽  
Mellin Transforms ◽  
Factor Method ◽  
Image Position

In this paper we first of all consider the dual integral equationswhere f(ρ), g(ρ) are given, A(t) is unknown, and α is a given constant. This system, with g(ρ) = 0, was originally considered by Titchmarsh ((13), p. 337), and Busbridge (1), who obtained a solution by the use of Mellin transforms and analytic continuation in the complex plane. The method described in this paper involves the application of certain multiplying factors to the equations. In the present case it is relatively easy to guess the multiplying factors and then the method is essentially a real-variable technique. It is presented in this way in § 2 below.

On a class of Volterra and Fredholm non-linear integral equations

1976 ◽  
Vol 79 (2) ◽  
pp. 329-335 ◽  
Author(s):  
P. J. Bushell
Keyword(s):  
Integral Equations ◽  
Kernel Function ◽  
Existence And Uniqueness ◽  
Volterra Integral Equations ◽  
Fredholm Equations ◽  
Non Linear ◽  
Linear Volterra Integral Equations ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

This paper concerns the existence and uniqueness of non-negative solutions of non-linear Volterra integral equations of the typeandwhere the kernel function k(.,.) is non-negative and sufficiently smooth, and either 0 < p < 1 or – 1 < p < 1. We will consider also the corresponding Fredholm equationsand

scholarly journals Analysis of 2 + 1 diffusive–dispersive PDE arising in river braiding

2016 ◽  
Vol 27 (5) ◽  
pp. 756-780
Author(s):  
SALEH TANVEER ◽  
CHARIS TSIKKOU
Keyword(s):  
Boundary Condition ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Local Existence ◽  
Dispersive Equation ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Local Existence And Uniqueness ◽  
Formula Group ◽  
Image Position

We present local existence and uniqueness results for the following 2 + 1 diffusive–dispersive equation due to P. Hall arising in modelling of river braiding: $$\begin{equation*} u_{yyt} - \gamma u_{xxx} -\alpha u_{yyyy} - \beta u_{yy} + \left ( u^2 \right )_{xyy} = 0 \end{equation*}$$ for (x,y) ∈ [0, 2π] × [0, π], t > 0, with boundary condition uy=0=uyyy at y=0 and y=π and 2π periodicity in x, using a contraction mapping argument in a Bourgain-type space Ts,b. We also show that the energy ∥u(·, ·, t)∥2L2 and cumulative dissipation ∫0t∥uy (·, ·, s)∥L22dt are globally controlled in time t.

scholarly journals ON TIME PERIODIC SOLUTIONS TO THE GENERALIZED BBM-BURGERS EQUATION WITH TIME-DEPENDENT PERIODIC EXTERNAL FORCE

2020 ◽  
Vol 25 (2) ◽  
pp. 184-197
Author(s):  
Yinxia Wang
Keyword(s):  
Periodic Solutions ◽  
External Force ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Space Dimension ◽  
Burgers Equation ◽  
Time Dependent ◽  
Contraction Mapping Theorem ◽  
Time Periodic Solutions ◽  
Time Periodic

In this paper, we consider the generalized BBM-Burgers equation with periodic external force in Rn. Existence and uniqueness of time periodic solutions that have the same period as the external force are established in some suitable function space for the space dimension n≥ 3. Moreover, we also discuss the time asymptotic stability of the time periodic solution. The proof is mainly based on the contraction mapping theorem and continuous argument.

Fixed points in C∗-algebra valued b-metric spaces endowed with a graph

2018 ◽  
Vol 68 (3) ◽  
pp. 639-654 ◽  
Author(s):  
Sushanta Kumar Mohanta
Keyword(s):  
Fixed Points ◽  
Unique Solution ◽  
Operator Equation ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Contraction Mapping Theorem ◽  
C Algebra

Abstract We discuss the existence and uniqueness of fixed points for a self-mapping defined on a C∗-algebra valued b-metric space endowed with a graph. Our results extend and supplement several recent results in the literature. Some examples are provided to illustrate our results. Finally, as an application of G-contraction mapping theorem, existence of unique solution for a type of operator equation is given.

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