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### Open Access Journals

DCDS-S

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.

DCDS

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.

DCDS

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.

DCDS

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.

DCDS

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.

DCDS

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.

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