Homoclinic solutions in periodic partial difference equations

Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.

Multiple homoclinic solutions for the discrete [formula omitted]-Laplacian via critical point theory

Existence and multiplicity results for Kirchhoff-type problem with sublinear nonlinearity

In this paper, we deal with existence and multiplicity of solutions of Kirchhoff-type problems Under more relaxed assumptions on f, we establish some existence criteria to guarantee that the preceding problem has at least one or infinitely many nontrivial solutions by using the genus properties in critical point theory. Copyright © 2013 John Wiley & Sons, Ltd.

By the weak linking theorem and the linking theorem, we study the existence of periodic solutions for the following system of delay differential equations: 1 where , and is a given constant. Two existence theorems of 4r-periodic solutions of (1) are obtained.

We study the existence of homoclinic solutions for the second order Hamil- tonian system ¨ u+ Vu(t, u) = f(t). Let V (t, u) = −K(t, u)+ W(t, u) ∈ C 1 (R × R n , R) be T-periodic in t, where K is a quadratic growth function and W may be asymptotically quadratic or super-quadratic at infinity. One homoclinic solution is obtained as a limit of solutions of a sequence of periodic second order differential equations.

This paper is devoted to the study of which rational difference equations of order k, with non negative coefficients, are periodic. Our main result is that for and for only the well known periodic difference equations and their natural extensions appear.

By properly constructing a functional and by using the critical point theory, we establish the existence of homoclinic solutions for a class of subquadratic second-order Hamiltonian systems. Our result generalizes and improves some existing ones. An example is given to show that our theorem applies, while the existing results are not applicable.

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of 2 n th-order nonlinear discrete system are obtained. The main approach used in our paper is variational technique and the linking theorem. The problem is to solve the existence of periodic solutions with prescribed minimal period of 2 n th-order discrete systems.

In this paper we discuss some problems about critical point theory. In the first part of the paper we study existence and multiplicity results of semilinear second order elliptic equation: $$ \begin{cases} -\Delta u=f(x,u) &\text{for } x\in \Omega, \\ u=0 &\text{for } x\in \partial \Omega, \end{cases} $$ In [Z. Li. Liu and S. J. Li, Contractibility of level sets of functionals associated with some elliptic boundary value problems and applications , NoDEA 10 (2003), 133–170], the authors study the contractibility of level sets of functionals associated with some elliptic boundary value problems. In this paper by using Morse theory and minimax method we give a more precise description of topological construction of level set of critical value of energy functional for mountain pass type critical point. It is well known that nondegenerate critical point is isolated, so if a critical point is not isolated, it must be a degenerate critical point. In the second part we will give an example that all the critical points of functional of a class of oscillating equation with Neumann boundary condition are isolated and the equation has only constant solutions. Moreover, critical groups of each critical point of the functional are trivial. The elliptic sine-Gordon equation originates from the static case of the hyperbolic sine-Gordon equation modelling the Josephson junction in superconductivity, which is of contemporary interest to physicists. The problem is similar to the elliptic sine-Gordon equation so we believe that it derives from profound physical backdrop.

On positive solutions of the nonlocal Yamabe type equation in compact Riemannian manifold

We study the existence and multiplicity of classical solutions for second-order impulsive Sturm-Liouville equation with Neumann boundary conditions. By using the variational method and critical point theory, we give some new criteria to guarantee that the impulsive problem has at least one solution, two solutions, and infinitely many solutions under some different conditions, respectively. Some examples are also given in this paper to illustrate the main results.

Preface. Introduction. 1. Elements of Nonsmooth Analysis. 1. Generalized Gradients of Locally Lipschitz Functionals. 2. Palais Smale Condition and Coerciveness for a Class of Nonsmooth Functionals. 3. Nonsmooth Analysis in the Sense of Degiovanni. 2. Variational Methods. 1. Critical Point Theory for Locally Lipschitz Functionals. 2. Critical Point Theory for Convex Perturbations of Locally Lipschitz Functionals. 3. A Critical Point Theory in Metric Spaces. 3. Variational Methods. 1. Critical Point Theory for Convex Perturbations of Locally Lipschitz Functionals in the Limit Case. 2. Examples. 4. Multivalued Elliptic Problems in Variational Form. 1. Multiplicity for Locally Lipschitz Periodic Functionals. 2. The Multivalued Forced-pendulum Problem. 3. Hemivariational Inequalities Associated to Multivalued Problems with Strong Resonance. 4. A Parallel Nonsmooth Critical Point Theory. Approach to Stationary Schrodinger Type Equations in Constraints. 8. Non-Symmetric Perturbations of Symmetric Eigenvalue Problems. 1. Non-Symmetric Perturbations of Eigenvalue Problems for Periodic Hemivariational Inequalities with Constraints. 2. Perturbations of Double of Eigenvalue Problems for General Hemivariational Inequalities with Constraints. 3. location of Solutions by Minimax Methods of Variational Hemivariational Inequalities. 10. Nonsmooth Evolution Problems. 1. First Order Evolution Variational Inequalities. 2. Second Order Evolution Variational Inequalities. 3. Stability Problems for Evolution Variational Inequalities. 11. Inequality Problems in BV and Geometric Applications. 1. The General Framework. 2. Area Type Functionals. 3. A Result of Clark Type. 4. An Inequality Problem with Superlinear Potential.

The existence of periodic solutions for the third‐order differential equation is studied. We give some conditions for this equation in order to reduce it to a second‐order nonlinear differential equation. We show that the existence of periodic solutions for the second‐order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bifurcation theorem for the second‐order equation and obtain many periodic solutions for it. Also we show that the above equation has many homoclinic solutions if has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001).