The Leray-Schauder degree is defined for mappings of the form $I-C$, where $C$ is a compact mapping from the closure of an open bounded subset of a Banach space $X$ into $X$. Since the fifties, a lot of work has been devoted in extending this theory to the same type of mappings on some nonlinear spaces, and in extending the class of mappings in the frame of Banach spaces or manifolds. New applications of Leray-Schauder theory and its extensions have also been given, specially in bifurcation theory, nonlinear boundary value problems and equations in ordered spaces. The paper surveys those developments.

Starting from the famous Schauder fixed-point theorem, we present some Lefschetz-like and Nielsen-like generalizations for certain admissible (multivalued) self-maps on metric ANR-spaces. These fixed-point principles are applied for obtaining the existence and multiplicity results for boundary value problems.

Let $\Omega$ be an arbitrary smooth bounded domain in $\mathbb R^2$ and $\varepsilon> 0$ be arbitrary. Write $(x,y)$ for a generic point of $\mathbb R^2$. Squeeze $\Omega$ by the factor $\varepsilon$ in the $y$-direction to obtain the squeezed domain $\Omega_\varepsilon=\{(x,\varepsilon y)\mid (x,y)\in\Omega\}$. Consider the following reaction-diffusion equation on $\Omega_\varepsilon$: \begin{equation} u_t=\Delta u+f(u), t > 0,\ (x,y)\in\Omega_\varepsilon\\ \partial _{\nu_\varepsilon} u=0, \qquad t > 0,\ (x,y)\in\partial\Omega_\varepsilon. \tag{$E_\varepsilon$} \end{equation} Here, $\nu_\varepsilon$ is the exterior normal vector field on $\partial \Omega_\varepsilon$ and $f\colon \mathbb R\to \mathbb R$ is a nonlinearity satisfying some growth and dissipativeness conditions ensuring that (E$_\varepsilon$) generates a semiflow $\pi_\varepsilon$ on $H^1(\Omega_\varepsilon)$ with a global attractor $\mathcal A_\varepsilon$. In this paper we report on some recent results concerning the asymptotic behavior of the equations (E$_\varepsilon$) as $\varepsilon \to 0$.

In this paper we review some old and new results on cohomology of the maximal invariant set inside of an isolating block $B$. In particular, we prove the following one: If $u\cup c v$ is nonzero for some $u\in\overline{H}^*(B)$ and $v\in \overline H^*(B,B^-)$ then the restriction of $u$ to $\overline H^*(S)$ is nontrivial.

We present a topological approach to the problem of the existence of unstable periodic solutions for 2-dimensional, time-periodic ordinary differential equations. This approach makes use of the braid invariant, which is one of the topological invariants for periodic solutions exploiting a concept in the low-dimensional topology. Using the braid invariant, an equivalence relation on the set of periodic solutions is defined. We prove that any equivalence class consisting of at least two solutions must contain an unstable one, except one particular equivalence class. Also, it is shown that more than half of the equivalence classes contain unstable solutions.

A class of elliptic operators in ${\mathbb R}^n$ is considered. It is proved that the operators are Fredholm and proper. The topological degree is constructed. Existence of solutions for a reaction-diffusion system is studied.

We consider the statement of an initial boundary value problem for a generalized Oldroyd model describing both laminar and turbulent flows of a nonlinear visco-elastic fluid. The operator interpretation of a posed problem is presented. The properties of operators forming the corresponding equation are investigated. We introduce approximating operator equations and prove their solvability. On that base the existence theorem for the operator equation equivalent to the stated initial boundary value problem is proved.

The paper is devoted to studying a topological structure of sets of fixed points (or equilibria) of some classes of single-valued or multivalued maps. In our opinion, this paper contains all main results connected with this problem. We deal with Browder-Gupta type results and the inverse systems approach.

A generalization of the theorem of Zhong on the product of spheres to multivalued maps is given. We prove also a stronger result of Bourgin-Yang type.

We consider closed symplectically aspherical manifolds, i.e. closed symplectic manifolds $(M,\omega)$ satisfying the condition $[\omega]|_{\pi_2M}=0$. Rudyak and Oprea [On the Lustrnik–Schnirelmann category of symplectic manifolds and the Arnold conjecture, Math. Z. 230 (1999), 673–678] remarked that such manifolds have nice and controllable homotopy properties. Now it is clear that these properties are mostly determined by the fact that the strict category weight of $[\omega]$ equals 2. We apply the theory of strict category weight to the problem of estimating the number of closed orbits of charged particles in symplectic magnetic fields. In case of symplectically aspherical manifolds our theory enables us to improve some known estimations.

The existence of radial solutions for some semilinear elliptic equations in an exterior domain is established under sublinearity or sign assumption imposed on the nonlinearity.

In the paper we study the existence of bounded solutions for differential equations of the form: $x''-Ax= f(t,x)$, where $A\in L(H)$, $f\colon {\mathbb R}\times H \to H$ ($H$ - a Hilbert space) is a continuous mapping. Using a perturbation of the equation, the Leray-Schauder topological degree and fixed point theory, we overcome the difficulty that the linear problem is non-Fredholm in any resonable Banach space.

In the case of the abstract autonomous semilinear equation in a Banach space we provide conditions which ensure that the approximate cycles given by a semidiscretization method converge to the exact cycle.