Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential
Jean Mawhin
Discrete & Continuous Dynamical Systems - S 2013, 6(4): 1065-1076 doi: 10.3934/dcdss.2013.6.1065
T-periodic solutions of systems of difference equations of the form\begin{eqnarray*}\Delta \phi[\Delta q(n-1)] = \nabla_q F[n,q(n)] + h(n) \quad (n \in \mathbb{Z})\end{eqnarray*}where $\phi = \nabla \Phi$, with $\Phi$ strictly convex, is a homeomorphism of $\mathbb{R}^N$ onto the ball $B_a \subset \mathbb{R}^N$, or a homeomorphism of the ball $B_{a} \subset \mathbb{R}^N$ onto $\mathbb{R}^N$, are considered when $F(n,u)$ is periodic in the $u_j$. The approach is variational.
keywords: Difference equations variational methods $\phi$-laplacian periodic solutions curvature oscillator relativistic oscillator.
The index at infinity of some twice degenerate compact vector fields
A.M. Krasnosel'skii Jean Mawhin
Discrete & Continuous Dynamical Systems - A 1995, 1(2): 207-216 doi: 10.3934/dcds.1995.1.207
The index at infinity of some compact vector fields associated with Nemytski operators is computed in situations where the linear part is degenerate and the nonlinear part does not satisfy the Landesman-Lazer conditions. Applications are given to the existence and multiplicity of solutions of nonlinear equations depending upon a parameter.
keywords: The index Nemytski operators multiplicity of solutions. Landesman-Lazer conditions
The index at infinity for some vector fields with oscillating nonlinearities
Alexander Krasnosel'skii Jean Mawhin
Discrete & Continuous Dynamical Systems - A 2000, 6(1): 165-174 doi: 10.3934/dcds.2000.6.165
This paper is devoted to the computation of the index at infinity for some asymptotically linear completely continuous vector fields $x-T(x)$, when the principal linear part $x-Ax$ is degenerate ($1$ is an eigenvalue of $A$), and the sublinear part is not asymptotically homogeneous (in particular do not satisfy Landesman-Lazer conditions). In this work we consider only the case of a one-dimensional degeneration of the linear part, i.e.s $1$ is a simple eigenvalue of $A$. For this case we formulate an abstract theorem and give some general examples for vector fields of Hammerstein type and for a two point boundary value problem.
keywords: oscillating nonlinearities. Vector fields
Guiding-like functions for periodic or bounded solutions of ordinary differential equations
Jean Mawhin James R. Ward Jr
Discrete & Continuous Dynamical Systems - A 2002, 8(1): 39-54 doi: 10.3934/dcds.2002.8.39
We state and prove some extensions of the fundamental theorem of the method of guiding functions for periodic and for bounded solutions of ordinary differential systems. Those results unify and generalize previous results of Krasnosel'skii, Perov, Mawhin, Walter and Gossez.
keywords: coincidence degree. Guiding functions periodic solutions bounded solutions
Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities
Jean Mawhin
Discrete & Continuous Dynamical Systems - A 2012, 32(11): 4015-4026 doi: 10.3934/dcds.2012.32.4015
Using a Lusternik-Schnirelman type multiplicity result for some indefinite functionals due to Szulkin, the existence of at least $n+1$ geometrically distinct T-periodic solutions is proved for the relativistic-type Lagrangian system $$(\phi(q'))' + \nabla_qF(t,q) = h(t),$$ where $\phi$ is an homeomorphism of the open ball $B_a \subset \mathbb{R}n$ onto $\mathbb{R}n$ such that $\phi(0) = 0$ and $\phi = \nabla \Phi$, $F$ is $T_j$-periodic in each variable $q_j$ and $h \in L^s(0,T;\mathbb{R}n)$ $(s > 1)$ has mean value zero. Application is given to the coupled pendulum equations $$\left(\frac{q'_j}{\sqrt{1 - \|q\|^2}}\right)' + A_j \sin q_j = h_j(t) \quad (j = 1,\ldots,n).$$ Similar results are obtained for the radial solutions of the homogeneous Neumann problem on an annulus in $\mathbb{R}n$ centered at $0$ associated to systems of the form $$\nabla \cdot \left(\frac{\nabla w_i}{\sqrt{1 - \sum_{j=1}^n \|\nabla w_j\|^2}}\right) + \partial_{w_j} G(\|x\|,w) = h_i(\|x\|), \quad (i = 1,\ldots,n),$$ involving the extrinsic mean curvature operator in a Minkovski space.
keywords: periodic solutions $\phi$-Laplacian relative category. Neumann problems Relativistic pendulum system critical point theory extrinsic mean curvature Hamiltonian systems
Radial solutions for Neumann problems with $\phi$-Laplacians and pendulum-like nonlinearities
Cristian Bereanu Petru Jebelean Jean Mawhin
Discrete & Continuous Dynamical Systems - A 2010, 28(2): 637-648 doi: 10.3934/dcds.2010.28.637
In this paper we study the existence and multiplicity of radial solutions for Neumann problems in a ball and in an annular domain, associated to pendulum-like perturbations of mean curvature operators in Euclidean and Minkowski spaces and of the $p$-Laplacian operator. Our approach relies on the Leray-Schauder degree and the upper and lower solutions method.
keywords: mean curvature and $p$-Laplacian operators Neumann problem Leray-Schauder degree upper and lower solutions. pendulum-like nonlinearities

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