scholarly journals Existence of Homoclinic Orbits for a Class of Asymptoticallyp-Linear Difference Systems withp-Laplacian

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Qiongfen Zhang ◽  
X. H. Tang
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Homoclinic Orbits ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Difference System ◽  
Difference Systems ◽  
Linear Difference Systems

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptoticallyp-linear difference system withp-LaplacianΔ(|Δu(n-1)|p-2Δu(n-1))+∇[-K(n,u(n))+W(n,u(n))]=0, wherep∈(1,+∞),n∈ℤ,u∈ℝN,K,W:ℤ×ℝN→ℝare not periodic inn, and W is asymptoticallyp-linear at infinity.

scholarly journals Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Damped Vibration Problems with Impulsive Effects

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qiongfen Zhang
Keyword(s):  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Impulsive Effects ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity ◽  
Vibration Problems ◽  
Damped Vibration Problems

This paper is concerned with the existence and multiplicity of fast homoclinic solutions for a class of damped vibration problems with impulsive effects. Some new results are obtained under more relaxed conditions by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory. The results obtained in this paper generalize and improve some existing works in the literature.

scholarly journals On Neumann hemivariational inequalities

2002 ◽  
Vol 7 (2) ◽  
pp. 103-112 ◽  
Author(s):  
Halidias Nikolaos
Keyword(s):  
Critical Point ◽  
Nontrivial Solution ◽  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Mountain Pass ◽  
Hemivariational Inequality ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz

We derive a nontrivial solution for a Neumann noncoercive hemivariational inequality using the critical point theory for locally Lipschitz functionals. We use the Mountain-Pass theorem due to Chang (1981).

scholarly journals Positive Solutions for a Nonhomogeneous Kirchhoff Equation with the Asymptotical Nonlinearity inR3

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Ling Ding ◽  
Lin Li ◽  
Jin-Ling Zhang
Keyword(s):  
Variational Principle ◽  
Critical Point ◽  
Positive Solutions ◽  
Critical Point Theory ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Kirchhoff Equation ◽  
Mountain Pass ◽  
Asymptotically Linear ◽  
Ekeland's Variational Principle

We study the following nonhomogeneous Kirchhoff equation:-(a+b∫R3‍|∇u|2dx)Δu+u=k(x)f(u)+h(x),  x∈R3,  u∈H1(R3),  u>0,  x∈R3, wherefis asymptotically linear with respect totat infinity. Under appropriate assumptions onk,f, andh, existence of two positive solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.

Existence of the Solutions for a Class of Nonlinear Difference Systems Boundary Value Problem

2013 ◽  
Vol 281 ◽  
pp. 312-318
Author(s):  
Fang Su ◽  
Xue Wen Qin
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Critical Point ◽  
Critical Point Theory ◽  
Boundary Value ◽  
Point Theory ◽  
Difference Systems

In this paper, by using critical point theory, we obtain a new result on the existence of the solutions for a class of difference systems boundary value problems. Results obtained extend or improve existing ones.

MULTIPLE HOMOCLINICS IN A HAMILTONIAN SYSTEM WITH ASYMPTOTICALLY OR SUPER LINEAR TERMS

2006 ◽  
Vol 08 (04) ◽  
pp. 453-480 ◽  
Author(s):  
YANHENG DING
Keyword(s):  
Hamiltonian System ◽  
Critical Point ◽  
Critical Point Theory ◽  
Variational Formulation ◽  
Homoclinic Orbits ◽  
Point Theory ◽  
Asymptotically Linear ◽  
Recent Information ◽  
Existence And Multiplicity ◽  
The Matrix

This paper is concerned with homoclinic orbits in the Hamiltonian system [Formula: see text] where H is periodic in t with Hz(t, z) = L(t)z + Rz(t, z), Rz(t, z) = o(|z|) as z → 0. We find a condition on the matrix valued function L to describe the spectrum of operator [Formula: see text] so that a proper variational formulation is presented. Supposing Rz is asymptotically linear as |z| → ∞ and symmetric in z, we obtain infinitely many homoclinic orbits. We also treat the case where Rz is super linear as |z| → ∞ with assumptions different from those studied previously in relative work, and prove existence and multiplicity of homoclinic orbits. Our arguments are based on some recent information on strongly indefinite functionals in critical point theory.

scholarly journals Infinitely Many Homoclinic Solutions for Nonperiodic Fourth Order Differential Equations with General Potentials

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Liu Yang
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Critical Point ◽  
Fourth Order ◽  
Critical Point Theory ◽  
Point Theory ◽  
Homoclinic Solutions

We investigate a class of nonperiodic fourth order differential equations with general potentials. By using variational methods and genus properties in critical point theory, we obtain that such equations possess infinitely homoclinic solutions.

Existence of Solutions to Boundary Value Problems for a Class of Nonlinear Difference Systems

2018 ◽  
Vol 19 (5) ◽  
pp. 531-537
Author(s):  
Tao Zhou ◽  
Xia Liu ◽  
Haiping Shi
Keyword(s):  
Boundary Value Problems ◽  
Critical Point ◽  
Critical Point Theory ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Point Theory ◽  
Mountain Pass Lemma ◽  
Variational Technique ◽  
Difference Systems ◽  
Existence Criteria

AbstractThis paper is devoted to investigate a question of the existence of solutions to boundary value problems for a class of nonlinear difference systems. The proof is based on the notable mountain pass lemma in combination with variational technique. By using the critical point theory, some new existence criteria are obtained.

scholarly journals Existence and Multiplicity of Fast Homoclinic Solutions for a Class of Nonlinear Second-Order Nonautonomous Systems in a Weighted Sobolev Space

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Qiongfen Zhang ◽  
Yuan Li
Keyword(s):  
Continuous Function ◽  
Mountain Pass Theorem ◽  
Weighted Sobolev Space ◽  
Point Theory ◽  
Second Order ◽  
Homoclinic Solutions ◽  
Mountain Pass ◽  
Nonautonomous Systems ◽  
Symmetric Mountain Pass Theorem ◽  
Existence And Multiplicity

This paper is concerned with the following nonlinear second-order nonautonomous problem:ü(t)+q(t)u̇(t)-∇K(t,u(t))+∇W(t,u(t))=0, wheret∈R,u∈RN, andK,W∈C1(R×RN,R)are not periodic intandq:R→Ris a continuous function andQ(t)=∫0t‍q(s)dswithlim|t|→+∞⁡Q(t)=+∞. The existence and multiplicity of fast homoclinic solutions are established by using Mountain Pass Theorem and Symmetric Mountain Pass Theorem in critical point theory.

scholarly journals On the Struwe—Jeanjean—Toland monotonicity trick

2012 ◽  
Vol 142 (1) ◽  
pp. 155-169 ◽  
Author(s):  
Marco Squassina
Keyword(s):  
Critical Point ◽  
Banach Spaces ◽  
Critical Point Theory ◽  
Point Theory ◽  
Energy Functional ◽  
Real Parameter ◽  
Mountain Pass ◽  
Abstract Version ◽  
Symmetry Assumption ◽  
Continuous Functionals

The abstract version of Struwe's monotonicity trick developed by Jeanjean and Toland for functionals depending on a real parameter is strengthened in the sense that it provides, for almost every value of the parameter, the existence of a bounded almost symmetric Palais–Smale sequence at the mountain-pass level whenever a mild symmetry assumption is set on the energy functional. In addition, the whole theory is extended to the case of continuous functionals on Banach spaces, in the framework of non-smooth critical point theory.

scholarly journals Existence of Solutions for Higher Order Bvp with Parameters via Critical Point Theory

2015 ◽  
Vol 48 (1) ◽  
Author(s):  
Mariusz Jurkiewicz ◽  
Bogdan Przeradzki
Keyword(s):  
Critical Point ◽  
Critical Point Theory ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Linear Part ◽  
Point Theory ◽  
Higher Order ◽  
Mountain Pass ◽  
Mountain Pass Lemma

AbstractThis paper is concerned with the existence of at least one solution of the nonlinear 2k-th order BVP. We use the Mountain Pass Lemma to get an existence result for the problem, whose linear part depends on several parameters.

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