We provide conditions which ensure that the solution set of the Cauchy problem for a singularly perturbed system of differential inclusions in infinite dimensional Banach spaces is upper semicontinuous with respect to the parameter $\varepsilon\ge0$ of the perturbation. The main tools are represented by suitable introduced measures of noncompactness and the topological degree theory in locally convex spaces.

Total and local topological indices are constructed for various types of continuous maps of infinite-dimensional manifolds and ANR's from a broad class. In particular the construction covers locally compact maps with compact sets of fixed points (e.g. maps having a certain finite iteration compact or having compact attractor or being asymptotically compact etc.); condensing maps ($k$-set contraction) with respect to Kuratowski's or Hausdorff's measure of non-compactness on Finsler manifolds; maps, continuous with respect to the topology of weak convergence, etc.

The characteristic point is that all conditions are formulated in internal terms and the index is in fact internal while the construction is produced through transition to the enveloping space. Examples of applications are given.

We consider a conconvex parametrized integral inclusion and we prove that the solution set is a retract of Banach space.

We construct a homotopy invariant appropriate for studying the existence of coincidence points of Fredholm operators of nonnegative index and multivalued admissible maps. Cohomotopy methods are used as a more suitable tool than homological ones. Both finite and infinite dimensional cases are investigated.

In the paper some sufficient conditions are established in order that a continuous map have a fixed point. The results are related to those obtained by R. D. Nussbaum in [Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 171 (1972), 349–375], L. Górniewicz and D. Rozpłoch-Nowakowska in [On the Schauder Fixed Point Theorem, Topology in Nonlinear Analysis, Banach Center Publications, vol. 35, Polish Academy of Sciences, Warszawa, 1996], S. Szufla in [On the application of measure of noncompactness to existence theorems, Rend. Sem. Math. Univ. Padova 75 (1986), 1–14] and D. Bugajewski in [Weak solutions of integral equations with weakly singular kernel in Banach spaces, Comm. Math. 34 (1994), 49–58].

The differential game of pursuit-evasion over a fixed time segment is considered. The problem of construction of the stable absorption operator of control system is investigated. The attainability sets is appointed with the help of the stable absorption operator. The partition of the conjugate space on the finite regions of convexity of Hamiltonian is used for constructing stable absorption operator.

This paper deals with Mayer's problem for control systems with state constraints and, possibly, discontinuous terminal cost. The main result of this paper consists in the characterization of the value function as the unique solution to an Hamilton-Jacobi equation. The above characterization extends results already obtained in the case of regular cost functions and under some controlability assumptions on the boundary of the set of constraints.

Existence results are given for the implicit evolution inclusions $(Bx(t))'+A(t,x(t))\ni f(t)$ and $(Bx(t))'+A(t,x(t))-G(t,x(t))\ni f(t)$ with $B$ a bounded linear operator, $A(t,\cdot)$ a bounded, coercive and pseudo-monotone set-valued mapping and $G$ a set-valued mapping of non-monotone type. Continuity of the solution set of first inclusion with respect to $f$ is also obtained which is used to solve the second inclusion.

Let $\sigma(F)$ denote the point spectrum of a multivalued operator $F \colon H \rightrightarrows H $ defined over a real Hilbert space. The aim of this note is to explore the continuity properties of the spectral mapping $F \rightrightarrows \sigma(F)$.

We present a higher-order extension of the well-known theorem of Ben-Tal and Zowe on second-order necessary optimality conditions in topological vector spaces. We also examine the connection between this extension and the results of Furukawa and Yoshinaga which are stated in terms of higher-order variational sets and Neustadt derivatives.

For a Banach space $E$, a given pair $(\overline p, \overline x)\in[0,a]\times E$, and control system governed by a semilinear functional differential includion of the form $$ x'(t)\in Ax(t) +F(t, x(t), Tx) $$ the existence of a mild trajectory of $x(t)$ satisfying the condition $x(\overline p)=\overline x$ is considered. Using topological methods we develop an unified approach to the cases when a multivalued nonlinearity $F$ is Carathéodory upper semicontinuous or almost lower semicontinuous and an abstract extension operator $T$ allows to deal with variable and infinite delay. For the Carathéodory case, the compactness of the solutions set and, as a corollary, an optimization result are obtained.

We study the space $\mathcal M_m$ of all $m$-accretive operators on a Banach space $X$ endowed with an appropriate complete metrizable uniformity and the space $\overline{\mathcal M}{}^*_m$ which is the closure in $\mathcal M_m$ of all those operators which have a zero. We show that for a generic operator in $\mathcal M_m$ all infinite products of its resolvents become eventually close to each other and that a generic operator in $\overline{\mathcal M}{}_m^*$ has a unique zero and all the infinite products of its resolvents converge uniformly on bounded subsets of $X$ to this zero.

Let $N$ be a norming set in a Banach space $X$. In this paper we prove the lower semicontinuity with respect to the topology $\sigma(X,N)$ of the Kobayashi distance in a bounded, relatively compact in $\sigma(X,N)$, convex and open subset of a Banach space. We apply this result to the Denjoy-Wolff type theorem.

We simplify a proof of Bessaga's theorem given in the monograph of Deimling. Moreover, our argument let us also obtain the following result.

Let $F$ be a selfmap of an arbitrary set $\Omega$ and $\alpha\in (0,1)$. Then $F$ is an $\alpha$-similarity with respect to some complete metric $d$ for $\Omega$ (that is, $d(Fx,Fy)=\alpha d(x,y)$ for all $x,y\in\Omega$) if and only if $F$ is injective and $F$ has a unique fixed point.

Finally we present that the converse to the Contraction Principle for bounded spaces is independent of the Axiom of Choice.

We give an example of $Z_2$-space $X$ with a property that the identity map ${\rm id}_X:X\to X$ as well as its restriction to the fixed point set of the group action ${\rm id}^{Z_2}:X^{Z_2}\to X^{Z_2}$ are deformable to fixed point free maps whereas there is no fixed point free map in the equivariant homotopy class of the identity $[{\rm id}_X]_{Z_2}$.

In this paper, we first establish a coincidence theorem under the noncompact settings. Then we derive some fixed point theorems for a family of functions. We apply our fixed point theorem to study nonempty intersection problems for sets with convex sections and obtain a social equilibrium existence theorem. We also introduce a concept of a quasi-variational inequalities and prove an existence result for a solution to such a system.