We develop a degree theory for variational inequalities which contain multivalued (S$_+$)-perturbations of maximal monotone operators. The multivalued operators need not necessarily be convex-valued. The result is simultaneously an extension of a degree theory for variational inequalities (developed by Benedetti, Obukhovskii and Zecca) and of the Skrypnik-Browder degree and extensions thereof.

We consider a parametric semilinear Dirichlet problem driven by the Laplacian plus an indefinite unbounded potential and with a reaction of superdifissive type. Using variational and truncation techniques, we show that there exists a critical parameter value $\lambda_{\ast} > 0$ such that for all $\lambda > \lambda_{\ast}$ the problem has at least two positive solutions, for $\lambda= \lambda_{\ast}$ the problem has at least one positive solution, and no positive solutions exist when $\lambda\in(0,\lambda_{\ast})$. Also, we show that for $\lambda\geq\lambda_{\ast}$ the problem has a smallest positive solution.

Sufficient conditions are established for the existence results of fractional order semilinear Volterra integrodifferential equations in Banach spaces. Results are obtained by using the theory of fractional cosine families and fractional powers of operators.

We study a quasi-static evolution of the thermo-visco-elastic model. We act with external forces on a non-homogeneous material body, which is a subject of our research. Such action may cause deformation of this body and may change its temperature. Mechanical part of the model contains two kinds of deformation: elastic and visco-elastic. The mechanical deformation is coupled with temperature and both of them may influence each other. Since the constitutive function on evolution of the visco-elastic deformation depends on temperature, the visco-elastic properties of material also depend on temperature. We consider the thermodynamically complete model related to a hardening rule with growth condition in generalized Orlicz spaces. We provide the proof of existence of solutions for such class of models.

By using variational methods and critical point theory, in particular, a generalized weak linking theorem, we study a first order nonlinear Hamiltonian system with asymptotically linear nonlinearity at infinity. We obtain the existence of ground state homoclinic orbits for this nonlinear Hamiltonian system. In particular, we obtain a necessary and sufficient condition for the existence of ground state homoclinic orbits. To the best of our knowledge, there is no published result focusing on necessary and sufficient conditions of the existence of ground state homoclinic orbits for this system.

In this paper, we consider the pullback attractors for a non-autonomous semilinear degenerate parabolic equation $u_{t}-{\rm div}(\sigma(x)\nabla u)+ f(u)=g(x,t)$ defined on a bounded domain $\Omega\subset \mathbb{R}^N$ with smooth boundary. We provide that the usual $(L^{2}(\Omega), L^{2}(\Omega))$ pullback $\mathscr{D}_{\lambda}$-attractor indeed can attract the $\mathscr{D}_{\lambda}$-class in $L^{2+\delta}(\Omega)$, where $\delta \in [0, \infty)$ can be arbitrary.

We prove multiplicity of solutions for a class of quasilinear problems in $ \mathbb{R}^{N} $ involving variable exponents and nonlinearities of concave-convex type. The main tools used are variational methods, more precisely, Ekeland's variational principle and Nehari manifolds.

In this paper, we consider the well-posedness of mixed quasi-variational-hemivariational inequalities ((MQVHVI) for short). By introducing a new concept of the $\alpha$-$\eta$-monotone mappings, we establish several metric characterizations and equivalent conditions of well-posedness for the (MQVHVI).

We introduce a class of subsets of Riemannian manifolds called the $L$-retract. Next we consider a topological degree for set-valued upper semicontinuous maps defined on open sets of compact $L$-retracts in Riemannian manifolds. Then, we present a theorem on the existence of equilibria (or zeros) of an upper semicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition defined on a compact $L$-retract in a Riemannian manifold.

We study the existence of standing wave solutions for the following class of elliptic Hamiltonian-type systems: \[ \begin{cases} -\hbar^2\Delta u+ V(x)u = g(v) & \mbox{in } \mathbb{R}^N, \\ -\hbar^2\Delta v+ V(x)v = f(u) & \mbox{in } \mathbb{R}^N, \end{cases} \] with $N\geq 2$, where $\hbar$ is a positive parameter and the nonlinearities $f,g$ are superlinear and can have arbitrary growth at infinity. This system is in variational form and the associated energy functional is strongly indefinite. Moreover, in view of unboundedness of the domain $\mathbb{R}^N$ and the arbitrary growth of nonlinearities we have lack of compactness. We use a dual variational approach in combination with a mountain-pass type procedure to prove the existence of positive solution for $\hbar$ sufficiently small.

We deal with the problem $$ \begin{cases} -\Delta u = \lambda u^{q(x)} & \text{if } x\in \Omega,\\ u = 0 & \text{if } x\in \p\Omega, \end{cases} $$ where $\Omega\subset \mathbb R^N$ is a bounded smooth domain, $\lambda> 0$ is a parameter and the exponent $q(x)$ is a continuous positive function that takes values both greater than and less than one in $\overline{\Omega}$. It is therefore a kind of concave-convex problem where the presence of the interphase $q=1$ in $\overline{\Omega}$ poses some new difficulties to be tackled. The results proved in this work are the existence of $\lambda^* > 0$ such that no positive solutions are possible for $\lambda > \lambda^*$, the existence and structural properties of a branch of minimal solutions, $u_\lambda$, $0 < \lambda < \lambda^*$, and, finally, the existence for all $ \lambda \in (0,\lambda^*)$ of a second positive solution.

We introduce two tools, dynamical thickening and flow selectors, to overcome the infamous discontinuity of the gradient flow endpoint map near non-degenerate critical points. More precisely, we interpret the stable fibrations of certain Conley pairs $(N,L)$, established in [J. Weber, Stable foliations and semi- ow Morse homology, arXiv 1408.3842], as a dynamical thickening of the stable manifold. As a first application and to illustrate efficiency of the concept we reprove a fundamental theorem of classical Morse theory, Milnor's homotopical cell attachment theorem [J. Milnor, Morse theory, Ann. Math. Stud. No. 51. Princeton University Press, Princeton, 1963]. Dynamical thickening leads to a conceptually simple and short proof.

We prove that for every integer sequence $I$ satisfying Dold relations there exists a map $f \colon \mathbb{R}^d\to \mathbb{R}^d$, $d \ge 2$, such that ${\rm Per}(f) ={\rm Fix}(f) =\{o\}$, where $o$ denotes the origin, and $(i(f^n, o))_n = I$.

For an orthogonal $\Gamma$-representation $V$ ($\Gamma$ is a finite group) and for an even $\Gamma$-invariant $C^2$-functional $f\colon V\to \mathbb R$ satisfying the condition $0< \theta \nabla f(x)\bullet x \leq \nabla^2 f(x)x\bullet x$ (for $\theta> 1$ and $x\in V\setminus \{0\}$), we consider the odd Newtonian system $\ddot x(t)=-\nabla f(x(t))$ and establish the existence of multiple periodic solutions with a minimal period $p$ (for any given $p > 0$). As an example, we prove the existence of arbitrarily many periodic solutions with minimal period $p$ for a specific $D_n$-symmetric Newtonian system.

In this paper, we give two equivalent definitions of tail pressure involving open covers and establish a variational principle which exhibits the relationship between tail pressure and measure-theoretic tail entropy.

We examine a logistic equation with local and non-local reaction terms both for time dependent and steady-state problems. Mainly, we use bifurcation and monotonicity methods to prove the existence of positive solutions for the steady-state equation and sub-supersolution method for the long time behavior for the time dependent problem. The results depend strongly on the size and sign of the parameters on the local and non-local terms.

Bifurcations of the class $A_{\mu}$, in Arnold's classification, in non-autonomous $p$-periodic difference equations generated by parameter depending families with $p$ maps, are studied. It is proved that the conditions of degeneracy, non-degeneracy and unfolding are invariant relatively to cyclic order of compositions for any natural number $\mu$. The main tool for the proofs is the local topological conjugacy. Invariance results are essential for proper definition of bifurcations of the class $A_{\mu}$ and associated lower codimension bifurcations, using all possible cyclic compositions of fiber families of maps of the $p$-periodic difference equation. Finally, we present two examples of the class $A_{3}$ or swallowtail bifurcation occurring in period two difference equations for which bifurcation conditions are invariant.

We provide necessary conditions on a given map, between two compact differential manifolds, for its critical set to be finite. As consequences of these conditions we also provide several examples of pairs of compact differential manifolds such that every map between them has infinite critical set.

We study the existence of harmonic and subharmonic solutions for the suplinear Duffing equation with delay. Our proofs are based on the twisting theorem due to W.Y. Ding.

The aim of this paper is to give an explicit motion planning algorithm for configuration spaces in the higher dimensional case.

We find all Heegaard diagrams with the property "alternating" or "weakly alternating"' on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two $3$-manifolds, each admits an automorphism whose non-wandering set consists of two Williams solenoids, one attractor and one repeller. These manifolds contain half of Prism manifolds, Poincaré's homology $3$-sphere and many other Seifert manifolds, all integer Dehn surgeries on the figure eight knot, also many connected sums. The result shows that many kinds of $3$-manifolds admit a kind of "translation" with certain stability.