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### Open Access Journals

DCDS-S

We show that the trajectories of a conserved phase-field model with
memory are compact in the space of continuous functions and, for an
exponential relaxation kernel, we establish the convergence of
solutions to a single stationary state as time goes to infinity. In
the latter case, we also estimate the rate of decay to equilibrium.

DCDS

We establish the existence of solutions to an anti-periodic non-monotone
boundary value problem. Our approach relies on a combination of monotonicity and
compactness methods.

DCDS

In this paper we first conduct a study of the spectrum of the negative
$p$-Laplacian with Neumann boundary conditions. More precisely we investigate
the first nonzero eigenvalue. We produce alternative variational
characterizations, we examine its dependence on $p\in( 1,\infty)
$ and on the weight function $m\in L^{\infty}(Z) _{+}$ and we
prove that the isolation of the principal eigenvalue $\lambda_{0}=0,$ is
uniform for all $p$ in a bounded closed interval. All these results are then
used to prove an index formula (jumping theorem) for the $d_{( S)_{+}}-$degree map at the first nonzero eigenvalue. Finally the index formula
is used to prove a multiplicity result for problems with a multivalued
crossing nonlinearity.

DCDS

A nonlinear Volterra inclusion associated to a family of time-dependent $m$-accretive operators,
perturbed by a multifunction, is considered in a Banach space. Existence results
are established
for both nonconvex and convex valued perturbations.
The class of extremal solutions is also
investigated.

CPAA

We study a nonlinear Dirichlet problem driven by the sum of a $p-$Laplacian ($p>2$) and a Laplacian and which at $\pm\infty$ is resonant with respect to the spectrum of $\left(-\triangle_{p}, W_{0}^{1, p}\left(\Omega\right)\right) $ and at zero is resonant with respect to the spectrum of $\left(-\triangle, H_{0}^{1}\left(\Omega\right) \right) $ (double resonance). We prove two multiplicity theorems providing three and four nontrivial solutions respectivelly, all with sign information. Our approach uses critical point theory together with truncation and comparison techniques and Morse theory.

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