### Regularity of minimizers for nonconvex vectorial integrals with $p-q$ growth via relaxation methods.

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We use the topological degree theory for condensing multimaps to present an existence result for impulsive semilinear functional differential inclusions in Banach spaces. Moreover, under some additional assumptions we prove the compactness of the solution set.

We study the relation between the solutions set to a perturbed semilinear differential inclusion with nonconvex and non-Lipschitz right-hand side in a Banach space and the solutions set to the relaxed problem corresponding to the original one. We find the conditions under which the set of solutions for the relaxed problem coincides with the intersection of closures (in the space of continuous functions) of sets of δ-solutions to the original problem.

We study a controllability problem for a system governed by a semilinear functional differential inclusion in a Banach space in the presence of impulse effects and delay. Assuming a regularity of the multivalued non-linearity in terms of the Hausdorff measure of noncompactness we do not require the compactness of the evolution operator generated by the linear part of inclusion. We find existence results for mild solutions of this problem under various growth conditions on the nonlinear part and...

The paper deals with the multivalued boundary value problem ${x}^{\text{'}}\in A(t,x)x+F(t,x)$ for a.a. $t\in [a,b]$, $Mx\left(a\right)+Nx\left(b\right)=0$, in a separable, reflexive Banach space $E$. The nonlinearity $F$ is weakly upper semicontinuous in $x$. We prove the existence of global solutions in the Sobolev space ${W}^{1,p}([a,b],E)$ with $1<p<\infty $ endowed with the weak topology. We consider the case of multiple solutions of the associated homogeneous linearized problem. An example completes the discussion.

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