In the present paper, we study the theory of dissipative impulsive semidynamical systems. We define different types of dissipativity as point, compact, local and bounded. The center of Levinson is defined for compact dissipative impulsive semidynamical systems and its topological properties are investigated. Also, we present results giving necessary and sufficient conditions to obtain dissipativity, and we include some examples to point out that the concepts of the different kinds of dissipativity are not equivalent in general.

In this paper we prove several existence results for a general class of systems of nonlinear hemivariational inequalities by using a fixed point theorem of Lin (Bull. Austral. Math. Soc. 34, (1986), 107-117). Our analysis includes both the cases of bounded and unbounded closed convex subsets in real reflexive Banach spaces. In the last section we apply the abstract results obtained to extend some results concerning nonlinear hemivariational inequalities, to establish existence results of Nash generalized derivative points and to prove the existence of at least one weak solution for an electroelastic contact problem.

In this paper we study a simplified model of thermoviscoplasticity. We prove local in time existence and uniqueness of solution. Moreover, for sufficiently small data, global in time existence is proved.

Metrical fixed point theory is accomplished by a wide class of terms:

$\bullet$ operators (bounded, Lipschitz, contraction, contractive, nonexpansive, noncontractive, expansive, dilatation, isometry, similarity, Picard, weakly Picard, Bessaga, Janos, Caristi, pseudocontractive, accretive, etc.),

$\bullet$ convexity (strict, uniform, hyper, etc.),

$\bullet$ deffect of some properties (measure of noncompactness, measure of nonconvexity, minimal displacement, etc.),

$\bullet$ data dependence (stability, Ulam stability, well-posedness, shadowing property, etc.),

$\bullet$] attractor,

$\bullet$ basin of attraction$\ldots$

The purpose of this paper is to study several properties of these concepts with respect to equivalent metrics.

In this paper, we study nonlinear fractional differential equations with Hadamard derivative and Ulam stability in the weighted space of continuous functions. Firstly, some new nonlinear integral inequalities with Hadamard type singular kernel are established, which can be used in the theory of certain classes of fractional differential equations. Secondly, some sufficient conditions for existence of solutions are given by using fixed point theorems via a prior estimation in the weighted space of the continuous functions. Meanwhile, a sufficient condition for nonexistence of blowing-up solutions is derived. Thirdly, four types of Ulam-Hyers stability definitions for fractional differential equations with Hadamard derivative are introduced and Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability results are presented. Finally, some examples and counterexamples on Ulam-Hyers stability are given.

Dropping the condition of convexity on the domain of a nonexpansive mapping is a difficult and unusual task in metric fixed point theory. Hilbert geometry has been one of the most fruitful at which authors have succeeded to drop such condition. In this work we revisit some of the results in that direction to study their validity in ${\rm CAT}(0)$ spaces (geodesic spaces of global nonpositive curvature in the sense of Gromov). We show that, although the geometry of ${\rm CAT}(0)$ spaces resembles at certain points that one of Hilbert spaces, much more than the ${\rm CAT}(0)$ condition is required in order to obtain counterparts of fixed point results for non-convex sets in Hilbert spaces. We provide significant examples showing this fact and give positive results for spaces of constant negative curvature as well as $R$-trees.

Consider a fixed class of maps $F$ for which there is a degree theory for the coincidence problem $F(x)=\varphi(x)$ with compact $\varphi$. It is proved that under very natural assumptions this degree extends to a degree for function triples which in particular provides a degree for coincidence inclusions $F(x)\in\Phi(x)$.

In the paper we provide generalizations of the classical Vietoris-Begle mapping theorem for not necessarily closed maps with respect to the Alexander-Spanier cohomology on paracompact space.

In this paper, a generalization of Nadler's fixed point theorem is presented. In the sequel, we consider a nonconvex integral inclusion and prove a Filippov type existence theorem by using an appropriate norm on the space of selection of the multifunction and a $H^+$-type contraction for set-valued maps.