TITLE

# A General Version of the Retract Method for Discrete Equations

AUTHOR(S)
Diblík, Josef; Růžičková, Irena; Růžičková, Miroslava
PUB. DATE
February 2007
SOURCE
Acta Mathematica Sinica;Feb2007, Vol. 23 Issue 2, p341
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In this paper we study a problem concerning the compulsory behavior of the solutions of systems of discrete equations u( k + 1) = F( k, u( k)), k âˆˆ N( a) = { a, a + 1, a + 2, . . . }, a âˆˆ â„•, â„• = {0, 1, . . . } and F : N( a) Ã— â„ n â†’ â„ n . A general principle for the existence of at least one solution with graph staying for every k âˆˆ N( a) in a previously prescribed domain is formulated. Such solutions are defined by means of the corresponding initial data and their existence is proved by means of retract type approach. For the development of this approach a notion of egress type points lying on the defined boundary of a given domain and with respect to the system considered is utilized. Unlike previous investigations, the boundary can contain points which are not points of egress type, too. Examples are inserted to illustrate the obtained result.
ACCESSION #
24487045

## Related Articles

• Inverse problems for differential systems on graphs with regular singularities. Yurko, V. // Mathematical Notes;Sep2014, Vol. 96 Issue 3/4, p617

The article presents a study on inverse spectral problem for differential equations of arbitrary order on compact startype graphs when differential equations have regular singularities at boundary vertices. It mentions that Boundary value problems on graphs often appear in natural sciences and...

• Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaupâ€“Kupershmidt Equations. Shuang Ping Tao; Shang Bin Cui // Acta Mathematica Sinica;Aug2005, Vol. 21 Issue 4, p881

This paper is devoted to studying the initial value problems of the nonlinear Kaupâ€“Kupershmidt equations  \frac{{\partial u}} {{\partial t}} + a_{1} \frac{{u\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}}...

• Solvability and Trajectory-Final Controllability of Pseudohyperbolic Systems. Nomirovskii, D. A. // Ukrainian Mathematical Journal;Mar2005, Vol. 57 Issue 3, p440

We consider the problem of solvability and optimization for a pseudohyperbolic operator of the general form. We prove theorems on existence and uniqueness for various right-hand sides of the equation. The results obtained are applied to the problem of trajectory-final controllability.

• Gap phenomenon in the homogenization of parabolic optimal control problems. D'APICE, C.; DE MAIO, U.; KOGUT, P. I. // IMA Journal of Mathematical Control & Information;Dec2008, Vol. 25 Issue 4, p461

In this paper, we study the asymptotic behaviour of a parabolic optimal control problem in a domain Î©Îµ âŠ‚ , whose boundary âˆ‚Î©Îµ contains a highly oscillating part. We consider this problem with two different classes of Dirichlet boundary controls, and, as a result, we...

• Application of the Boundary Function Method to Finding Slow Periodic Solutions of Singularly Perturbed Bang-Bang Systems. Fridman, L. M. // Differential Equations;Feb2003, Vol. 39 Issue 2, p256

Shows that slow periodic solutions of singularly perturbed bang-bang systems (SPBBS) have interior boundary layers appearing in the transition from a neighborhood of one leaf of the integral manifold into a neighborhood of another. Application of the boundary function method; Existence of...

• Strong Solvability of Boundary Value Contact Problems. Giuffr�, Sofia // Applied Mathematics & Optimization;May/Jun2005, Vol. 51 Issue 3, p361

Strong solvability in Sobolev spaces is proved for a unilateral contact boundary value problem for a class of nonlinear discontinuous operators. The operator is assumed to be of Caratheodory type and to satisfy a suitable ellipticity condition. Only measurability with respect to the independent...

• CONCENTRATION PHENOMENA FOR FOURTH-ORDER ELLIPTIC EQUATIONS WITH CRITICAL EXPONENT. Hammami, Mokhless // Electronic Journal of Differential Equations;2004, Vol. 2004, p1

We consider the nonlinear equation Î”Â²u = un+4/n-4 - Îµu with u > 0 in Î© and u = Î”u = 0 on âˆ‚Î©. Where Î© is a smooth bounded domain in Rn, n â‰¥ 9, and Îµ is a small positive parameter. We study the existence of solutions which concentrate around one or two points...

• On nonlocal calculation for inhomogeneous indefinite Neumann boundary value problems. Il�yasov, Yavdat; Runst, Thomas // Calculus of Variations & Partial Differential Equations;Jan2005, Vol. 22 Issue 1, p101

This paper concerns with a family of inhomogeneous Neumann boundary value problems having indefinite nonlinearities which depend on a real parameter. We discuss the existence and the multiplicity of positive solutions with respect to. Developing the fibering method further, we can introduce a...

• Global Existence and Blowâ€“up of Solutions for Parabolic Systems Involving Crossâ€“Diffusions and Nonlinear Boundary Conditions. Xiu Hui Yang; Fu Cai Li; Chun Hong Xie // Acta Mathematica Sinica;Aug2005, Vol. 21 Issue 4, p923

In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions: Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper...

Share

## Read the Article

Courtesy of THE LIBRARY OF VIRGINIA

Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library