scholarly journals Extremal solutions and strong relaxation for nonlinear periodic evolution inclusions

2000 ◽  
Vol 43 (3) ◽  
pp. 569-586 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Nikolaos Yannakakis
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Operator ◽  
Periodic Problem ◽  
Distributed Parameter ◽  
Pseudomonotone Operators ◽  
Evolution Inclusions ◽  
Evolution Triple ◽  
Relaxation Theorem ◽  
Strong Relaxation ◽  
Surjectivity Result

AbstractWe study the existence of extremal periodic solutions for nonlinear evolution inclusions defined on an evolution triple of spaces and with the nonlinear operator establish A being time-dependent and pseudomonotone. Using techniques of multivalued analysis and a surjectivity result for L-generalized pseudomonotone operators, we prove the existence of extremal periodic solutions. Subsequently, by assuming that A(t, ·) is monotone, we prove a strong relaxation theorem for the periodic problem. Two examples of nonlinear distributed parameter systems illustrate the applicability of our results.

scholarly journals Existence of anti-periodic solutions for a class of first order evolution inclusions

Filomat ◽  
2014 ◽  
Vol 28 (6) ◽  
pp. 1167-1180 ◽  
Author(s):  
Xiaoyou Liu ◽  
Yiliang Liu
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Evolution ◽  
Maximal Monotone ◽  
Pseudomonotone Operators ◽  
Evolution Inclusions ◽  
Derivative Operator ◽  
Evolution Triple ◽  
First Order ◽  
Nonlinear Parabolic ◽  
Surjectivity Result

The existence of anti-periodic solutions for a class of first order nonlinear evolution inclusions defined in the framework of an evolution triple of spaces is considered. We study the problems under both convexity and nonconvexity conditions on the multivalued right-hand side. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions, the surjectivity result for L-pseudomonotone operators and continuous extreme selection results from multivalued analysis. An example of a nonlinear parabolic problem is given to illustrate our results.

PERIODIC SOLUTIONS FOR EVOLUTION INCLUSIONS WITH TIME-DEPENDENT SUBDIFFERENTIALS

2002 ◽  
Vol 35 (3) ◽  
Author(s):  
Nikolaos Matzakos ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Periodic Solutions ◽  
Time Dependent ◽  
Evolution Inclusions

scholarly journals Existence and Uniqueness of Solutions for the p(x)-Laplacian Equation with Convection Term

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1768
Author(s):  
Bin-Sheng Wang ◽  
Gang-Ling Hou ◽  
Bin Ge
Keyword(s):  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Convection Term ◽  
Pseudomonotone Operators ◽  
Uniqueness Of Solutions ◽  
Laplacian Equation ◽  
Uniqueness Of The Solution ◽  
Gradient Based ◽  
Surjectivity Result ◽  
Quasilinear Elliptic

In this paper, we consider the existence and uniqueness of solutions for a quasilinear elliptic equation with a variable exponent and a reaction term depending on the gradient. Based on the surjectivity result for pseudomonotone operators, we prove the existence of at least one weak solution of such a problem. Furthermore, we obtain the uniqueness of the solution for the above problem under some considerations. Our results generalize and improve the existing results.

Anti-periodic solutions for nonlinear evolution inclusions

2018 ◽  
Vol 18 (2) ◽  
pp. 1025-1047
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Periodic Solutions ◽  
Nonlinear Evolution ◽  
Evolution Inclusions

scholarly journals Nonconvex evolution inclusions generated by time-dependent subdifferential operators

1999 ◽  
Vol 12 (3) ◽  
pp. 233-252
Author(s):  
Kate Arseni-Benou ◽  
Nikolaos Halidias ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Parabolic Problems ◽  
Nonlinear Feedback ◽  
Compact Type ◽  
Evolution Inclusions ◽  
Infinite Dimensional ◽  
Nonlinear Parabolic ◽  
Feedback Control Systems ◽  
Relaxation Theorem ◽  
Subdifferential Operators ◽  
Path Connected

We consider nonlinear nonconvex evolution inclusions driven by time-varying subdifferentials ∂ϕ(t,x) without assuming that ϕ(t,.) is of compact type. We show the existence of extremal solutions and then we prove a strong relaxation theorem. Moreover, we show that under a Lipschitz condition on the orientor field, the solution set of the nonconvex problem is path-connected in C(T,H). These results are applied to nonlinear feedback control systems to derive nonlinear infinite dimensional versions of the “bang-bang principle.” The abstract results are illustrated by two examples of nonlinear parabolic problems and an example of a differential variational inequality.

Optimal bounds for bifurcation values of a superlinear periodic problem

2005 ◽  
Vol 135 (1) ◽  
pp. 119-132 ◽  
Author(s):  
Rafael Ortega ◽  
Meirong Zhang
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Periodic Problem ◽  
Optimal Bounds ◽  
Sobolev Constants ◽  
Dynamic Bifurcation

In this paper we will show that the optimal bounds for certain static and dynamic bifurcation values of periodic solutions of some superlinear differential equations can be expressed explicitly using Sobolev constants.

scholarly journals A Periodic Problem of a Semilinear Pseudoparabolic Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
Yang Cao ◽  
Jingxue Yin ◽  
Chunhua Jin
Keyword(s):  
Periodic Solutions ◽  
Periodic Problem ◽  
Complete Classification ◽  
Pseudoparabolic Equation ◽  
Periodic Problems ◽  
Existence And Nonexistence

A class of periodic problems of pseudoparabolic type equations with nonlinear periodic sources are investigated. A rather complete classification of the exponentpis given, in terms of the existence and nonexistence of nontrivial and nonnegative periodic solutions.

scholarly journals Periodic solutions for semi-linear evolution inclusions

2007 ◽  
Vol 331 (2) ◽  
pp. 1246-1262 ◽  
Author(s):  
Xiaoping Xue ◽  
Jinfeng Yu
Keyword(s):  
Periodic Solutions ◽  
Evolution Inclusions ◽  
Linear Evolution

scholarly journals Anti-periodic problem for semilinear differential inclusions involving Hille-Yosida operators

2021 ◽  
pp. 1-31
Author(s):  
Nguyen Thi Van Anh ◽  
Tran Dinh Ke ◽  
Do Lan
Keyword(s):  
Periodic Solutions ◽  
Differential Inclusions ◽  
Mild Solutions ◽  
Periodic Problem ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Multivalued Maps ◽  
Integrated Semigroups ◽  
Long Time ◽  
Theoretical Results

In this paper we are interested in the anti-periodic problem governed by a class of semilinear differential inclusions with linear parts generating integrated semigroups. By adopting the Lyapunov-Perron method and the fixed point argument for multivalued maps, we prove the existence of anti-periodic solutions. Furthermore, we study the long-time behavior of mild solutions in connection with anti-periodic solutions. Consequently, as the nonlinearity is of single-valued, we obtain the exponential stability of anti-periodic solutions. An application of theoretical results to a class of partial differential equations will be given.

Sign in / Sign up

Export Citation Format

Share Document