On Approximations to Solutions of Nonlinear Integral Equations of the Urysohn Type

1973 ◽  
Vol 16 (1) ◽  
pp. 137-141
Author(s):  
K. A. Zischka
Keyword(s):  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Nonlinear Integral Equations ◽  
Priori Estimates ◽  
The Given ◽  
Uniqueness Of A Solution

This note will derive a priori estimates of the errors due to replacing the given integral operator A by a similar operator A* of the same type when successive approximations are applied to the integral equation φ=Aφ.The existence and uniqueness of solutions to this equation follow easily by applying a well known fixed point theorem in a Banach space to the above mapping [1, 2]. Moreover, sufficient conditions for the existence and uniqueness of a solution to Urysohn's equation are stated explicitly in a note by the author [3].

ASYMPTOTIC BEHAVIOR OF STOCHASTIC DISSIPATIVE QUANTUM ZAKHAROV EQUATIONS

2013 ◽  
Vol 13 (02) ◽  
pp. 1250016 ◽  
Author(s):  
YANFENG GUO ◽  
BOLING GUO ◽  
DONGLONG LI
Keyword(s):  
Asymptotic Behavior ◽  
White Noise ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Galerkin Approximation ◽  
Uniqueness Of Solutions ◽  
Asymptotic Behaviors ◽  
Priori Estimates

The stochastic dissipative quantum Zakharov equations with white noise are studied. The existence and uniqueness of solutions are obtained by using the standard Galerkin approximation method on the basis of the time uniform a priori estimates in various spaces. Moreover, the asymptotic behaviors of the solutions for the stochastic dissipative quantum Zakharov equations with white noise are also investigated.

scholarly journals Global Existence for Stochastic Strongly Dissipative Zakharov Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Xueqin Wang ◽  
Yadong Shang ◽  
Chunlin Lei
Keyword(s):  
Global Existence ◽  
White Noise ◽  
Approximation Method ◽  
Stochastic Partial Differential Equations ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Galerkin Approximation ◽  
Uniqueness Of Solutions ◽  
Priori Estimates

The stochastic strongly dissipative Zakharov equations with white noise are studied. On the basis of the time uniform a priori estimates, we prove the existence and uniqueness of solutions in energy spaces E1 and E2, by using the standard Galerkin approximation method of stochastic partial differential equations.

scholarly journals Attractors for a nonautonomous reaction-diffusion equation with delay

2018 ◽  
Vol 3 (1) ◽  
pp. 127-150 ◽  
Author(s):  
Hafidha Harraga ◽  
Mustapha Yebdri
Keyword(s):  
Diffusion Equation ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Absorbing Set ◽  
Uniqueness Of Solutions ◽  
Priori Estimates ◽  
Equation With Delay

AbstractIn this paper, we discuss the existence and uniqueness of solutions for a non-autonomous reaction-diffusion equation with delay, after we prove the existence of a pullback 𝒟-asymptotically compact process. By a priori estimates, we show that it has a pullback 𝒟-absorbing set that allow us to prove the existence of a pullback 𝒟-attractor for the associated process to the problem.

scholarly journals Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks

2005 ◽  
Vol 2005 (3) ◽  
pp. 281-297 ◽  
Author(s):  
Hong Xiang ◽  
Ke-Ming Yan ◽  
Bai-Yan Wang
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Global Stability ◽  
A Priori Estimates ◽  
A Priori ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
High Order ◽  
Priori Estimates

By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high-order Hopfield-type neural networks. We obtain some easily verifiable sufficient conditions to ensure that there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution.

scholarly journals The Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition

2020 ◽  
Vol 11 (4) ◽  
pp. 1991-2022
Author(s):  
Annamaria Barbagallo ◽  
Vincenzo Esposito
Keyword(s):  
Sobolev Spaces ◽  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Existence And Uniqueness Results ◽  
Mixed Problems ◽  
Uniqueness Results ◽  
Hyperbolic Operators ◽  
Priori Estimates

Abstract The mixed Cauchy–Neumann and Cauchy–Robin problems for a class of hyperbolic operators with double characteristics in presence of transition is investigated. Some a priori estimates in Sobolev spaces with negative indexes are proved. Subsequently, existence and uniqueness results for the mixed problems are obtained.

scholarly journals ON THE SYMMETRY AND UNIQUENESS OF SOLUTIONS OF THE GINZBURG–LANDAU EQUATIONS FOR SMALL DOMAINS

2001 ◽  
Vol 03 (01) ◽  
pp. 1-14 ◽  
Author(s):  
A. AFTALION ◽  
E. N. DANCER
Keyword(s):  
Order Parameter ◽  
A Priori Estimates ◽  
A Priori ◽  
Small Radius ◽  
Uniqueness Of Solutions ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
Priori Estimates ◽  
Ginzburg Landau Equations ◽  
Dimensional Domain

In this paper, we study the Ginzburg–Landau equations for a two dimensional domain which has small size. We prove that if the domain is small, then the solution has no zero, that is no vortex. More precisely, we show that the order parameter Ψ is almost constant. Additionnally, we obtain that if the domain is a disc of small radius, then any non normal solution is symmetric and unique. Then, in the case of a slab, that is a one dimensional domain, we use the same method to derive that solutions are symmetric. The proofs use a priori estimates and the Poincaré inequality.

A difference scheme for equations of ocean dynamics on unstructured grids

2014 ◽  
Vol 29 (3) ◽  
Author(s):  
Alexey V. Drutsa
Keyword(s):  
Difference Scheme ◽  
Existence And Uniqueness ◽  
Large Scale ◽  
A Priori Estimates ◽  
Unstructured Grids ◽  
A Priori ◽  
Ocean Dynamics ◽  
Priori Estimates ◽  
Uniqueness Of The Solution ◽  
Grid Operators

AbstractA difference scheme on unstructured grids is constructed for the system of equations of large scale ocean dynamics. The properties of the grid problem and grid operators are studied, in particular, a series of a priori estimates and the theorem on existence and uniqueness of the solution are proved.

scholarly journals A Result on the Existence and Uniqueness of Stationary Solutions for a Bioconvective Flow Model

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Aníbal Coronel ◽  
Luis Friz ◽  
Ian Hess ◽  
Alex Tello
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Flow Model ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Stationary Solutions ◽  
Local Concentration ◽  
Priori Estimates ◽  
Bioconvective Flow

In this note, we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem. The bioconvective model is a boundary value problem for a system of four equations: the nonlinear Stokes equation, the incompressibility equation, and two transport equations. The unknowns of the model are the velocity of the fluid, the pressure of the fluid, the local concentration of microorganisms, and the oxygen concentration. We derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions.

scholarly journals On nonlocal minimal graphs

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Matteo Cozzi ◽  
Luca Lombardini
Keyword(s):  
Existence And Uniqueness ◽  
A Priori Estimates ◽  
A Priori ◽  
Analytic Approach ◽  
Comparison Principles ◽  
Minimal Graphs ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Priori Estimates ◽  
Rearrangement Inequalities

AbstractWe develop a functional analytic approach for the study of nonlocal minimal graphs. Through this, we establish existence and uniqueness results, a priori estimates, comparison principles, rearrangement inequalities, and the equivalence of several notions of minimizers and solutions.

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