A second order semilinear impulsive functional differential inclusion in a separable Hilbert space is considered. Without imposing hypotheses of the compactness on the cosine families of operators, some sufficient conditions of approximate controllability are formulated in the case where the multivalued nonlinearity of the inclusion is a completely continuous map dominated by a function. By the use of resolvents of controllability Gramian operators and developing appropriate computing techniques for the Hausdorff product measures of noncompactness, the results of approximate controllability for position and velocity are derived. An example is also given to illustrate the application of the obtained results.

The paper is dedicated to the theory of concentration-compactness principles for inhomogeneous fractional Sobolev spaces. This subject for the local case has been studied since the publication of the celebrated works due to P.-L. Lions, which laid the broad foundations of the method and outlined a wide scope of its applications. Our study is based on the analysis of the profile decomposition for the weak convergence following the approach of dislocation spaces, introduced by K. Tintarev and K.-H. Fieseler. As an application, we obtain existence of nontrivial and nonnegative solutions and ground states for fractional Schrödinger equations for a wide class of possible singular potentials, not necessarily bounded away from zero. We consider possible oscillatory nonlinearities for both cases, subcritical and critical which are superlinear at the origin, without the classical Ambrosetti and Rabinowitz growth condition. In some of our results we prove existence of solutions by means of compactness of Palais-Smale sequences of the associated functional at the mountain pass level. To this end we study and provide the behavior of the weak profile decomposition convergence under the related functionals. Moreover, we use a Pohozaev type identity in our argument to compare the minimax levels of the energy functional with the ones of the associated limit problem. Motivated by this fact, in our work we also prove that this kind of identities hold for a larger class of potentials and nonlinearities for the fractional framework.

We consider the diffusional transport in an $r$-component solid solution. The model is expressed by the nonlinear system of strongly coupled parabolic differential equations with initial and nonlinear boundary conditions. The techniques involved are the local mass conservation law for fluxes, which are a sum of the diffusional and Darken drift terms, and the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. The theorems on existence, uniqueness and properties of global weak solutions are proved. The main tool used in the proof of the existence result is the Galerkin approximation method. The agreement between the theoretical results, numerical simulations and experimental data is shown.

We present some existence and uniqueness results on impulsive functional differential equations with multiple delay with fractional Brownian motion. Our approach is based on the Perov fixed point theorem and a new version of Schaefer's fixed point in generalized metric and Banach spaces.

The two-dimensional Rayleigh-Bénard problem for a thermomicropolar fluids model is considered. The existence of suitable weak solutions which may not be unique, and the existence of the unique strong solution are proved. The global attractor for the m-semiflow associated with weak solutions and the global attractor for semiflow associated with strong solutions are shown to be equal.

In this work we give a generalization of the Filippov-Ważewski Theorem to the fourth order differential inclusions in a separable complex Banach space $\mathbb{X}$ \begin{equation*} \mathcal{D}y=y^{\prime \prime \prime \prime }-( A^{2}+B^{2}) y^{\prime \prime }+A^{2}B^{2}y\in F( t,y) , \end{equation*} with the initial conditions in $c\in [ 0,T]$ \begin{equation} y( c) =\alpha ,\qquad y^{\prime }( c) =\beta ,\qquad y^{\prime \prime }( c) =\gamma ,\qquad y^{\prime \prime \prime }( c) =\delta ,\tag{*} \end{equation} We assume that the multifunction $F\colon[ 0,T] \times \mathbb{X} \leadsto c( \mathbb{X}) $ is Lipschitz continuous in $y$ with the integrable Lipschitz constant $l( .) $, while $A^{2},B^{2}\in B( \mathbb{X}) $ are the infinitesimal generators of two cosine families of operators. The main result is the following version of Filippov Lemma:

Theorem: Let $y_{0}\in W^{4,1}=W^{4,1}([ 0,T] ,\mathbb{X}) $ be such function with (*) that \begin{equation*} \mathrm{dist}( \mathcal{D}y_{0}( t) ,F( t,y_{0}( t) ) ) \leq p_{0}( t) \quad \text{a.e. in } [ c,d] \subset [ 0,T] , \end{equation*} where $p_{0}\in L^{1}[ 0,T] $. Then there are $\mathcal{\sigma }_{0}$ $($depending on $p_{0})$ and $\varphi $ such that for each $\varepsilon >0$ there exists a solution $y\in W^{4,1}$ of the above problem such that almost everywhere in $t\in [c,d]$ we have $\vert \mathcal{D}y( t) -\mathcal{D}y_{0}( t) \vert \leq \mathcal{\sigma }_{0}( t) $, \begin{alignat*}2 \vert y( t) -y_{0}( t) \vert &\leq(\varphi \ast _{c}\sigma _{0})( t) , &\qquad \vert y^{\prime }( t) -y_{0}^{\prime }( t) \vert \leq ( \varphi ^{\prime }\ast _{c}\sigma _{0}t) ( t ) , \\ \vert y^{\prime \prime }( t) -y_{0}^{\prime \prime }( t) \vert &\leq ( \varphi ^{\prime \prime }\ast _{c}\sigma _{0}) ( t) &\qquad \vert y^{\prime \prime \prime }( t) -y_{0}^{\prime \prime \prime }( t) \vert \leq( \varphi ^{\prime \prime \prime }\ast _{c}\sigma _{0})( t) , \end{alignat*} where $\ast _{c}$ stands for the convolution started at $c$.

Our estimates are constructive and more precise then those in the known versions of Filippov Lemma.

We investigate the following nonlinear biharmonic equations with pure power nonlinearities: \begin{equation*} \begin{cases} \triangle^2u-\triangle u+V(x)u= u^{p-1}u & \text{in } \mathbb{R}^N,\\ u>0 &\text{for } u\in H^2(\mathbb{R}^N), \end{cases} \end{equation*} where $2 < p< 2^*={2N}/({N-4})$. Under some suitable assumptions on $V(x)$, we obtain the existence of ground state solutions. The proof relies on the Pohožaev-Nehari manifold, the monotonic trick and the global compactness lemma, which is possibly different to other papers on this problem. Some recent results are extended.

In this paper, a class of singular impulsive Dirichlet boundary value problems is considered. By using variational method and critical point theory, different parameter ranges are obtained to guarantee existence and multiplicity of positive classical solutions of the problem when nonlinearity exhibits different growths.

In this paper, we are concerned with the following elliptic equations: \begin{equation*}\label{e:JG} \begin{cases} -\Delta_{\mathbb{M}}u = \lambda f &\text{in } z:= (r,x,t) \in \mathbb{M}_0,\\ u= 0 &\text{on } \partial\mathbb{M}. \end{cases} \end{equation*} Here, $\lambda >0$ and $M=[0,1)\times X\times[0,1)$ as a local model of stretched corner-manifolds, that is, the manifolds with corner singularities with dimension $N=n+2\geq 3$. Here $X$ is a closed compact submanifold of dimension $n$ embedded in the unit sphere of $\mathbb{R}^{n+1}$. We study the existence of nontrivial weak solutions for the semilinear corner degenerate elliptic equations without the Ambrosetti and Rabinowitz condition via the mountain pass theorem and fountain theorem.

In the paper [Probab. Theory Relat. Fields, 100 (1994), 417-428] Xue-Mei Li has shown that the moment stability of an SDE is closely connected with the topology of the underlying manifold. In particular, she gave sufficient condition on SDE on a manifold $M$ under which the fundamental group $\pi_1 M=0$. We prove that under similar analytical conditions the manifold $M$ is contractible, that is all homotopy groups $\pi_n M$, $n\geq1$, vanish.

Let $f$ be a map from a one-relator model two-complex $K_{\mathcal{P}}$ into the real projective plane. The composition $\varrho\circ f_{\#}$ of the homomorphism $f_{\#}$ induced by $f$ on fundamental groups with the action $\varrho$ of $\pi_1(\mathbb{R}\mathrm{P}^2)$ over $\pi_2(\mathbb{R}\mathrm{P}^2)$ provides a local integer coefficient system $f_{\#}^{\varrho}$ over $K_{\mathcal{P}}$. We prove that if the twisted integer cohomology group $H^2(K_{\mathcal{P}};_{f_{\#}^{\varrho}}\mathbb Z)=0$, then $f$ is homotopic to a non-surjective map. As an intermediary step for the proof, we show that if $H^2(K_{\mathcal{P}};_{\beta}\mathbb Z)=0$ for some local integer coefficient system $\beta$ over $K_{\mathcal{P}}$, then $K_{\mathcal{P}}$ is aspherical.

In this paper, which is a sequel to [1], we extend the spectral convergence result from [5] to a larger class of singularly perturbed families of scalar linear differential operators. This also extends the Conley index continuation principles from [1].

We prove, by topological methods, new results on the existence of nonzero positive weak solutions for a class of multi-parameter second order elliptic systems subject to functional boundary conditions. The setting is fairly general and covers the case of multi-point, integral and nonlinear boundary conditions. We also present a non-existence result. We provide some examples to illustrate the applicability of our theoretical results.

We prove that, in every infinite dimensional Hilbert space, there exists $t_0>-1$ such that the smallest Lipscthiz constant of retractions from the unit ball onto its boundary is the same as the smallest Lipschitz constant of retractions from the unit ball onto its $t$-spherical cup for all $t\in[-1,t_0]$.

In this paper, we study the existence of periodic solutions of perturbed planar Hamiltonian systems of the form $$ \begin{cases} x'=f(y)+p_1(t,x,y), \\ y'=-g(x)+p_2(t,x,y). \end{cases} $$ We prove a continuation lemma for a given planar system and further use it to prove that this system has at least one $T$-periodic solution provided that $g$ has some sub-quadratic potentials.

In this paper we give uniform stable spatial bounds for the resolvent operator families of the abstract fractional Cauchy problem on $\mathbb{R}_+$. Such bounds allow to prove existence and uniqueness of $\mu$-pseudo almost automorphic $\epsilon$-mild regular solutions to the nonlinear fractional Cauchy problem in the whole real line. Finally, we apply our main results to the fractional heat equation with critical nonlinearities.

We consider a generalization of the Allen-Cahn type equation in divergence form $-{\rm div}(\nabla G(\nabla u(x,y)))+F_u(x,y,u(x,y))=0$. This is more general than the usual Laplace operator. We prove the existence and regularity of heteroclinic solutions under standard ellipticity and $m$-growth conditions.

Inspired by the recent articles of T. Ohmoto and M. Shiota [9], [10] on $C^1$-triangulations of semialgebraic sets, we prove here by using different methods the following theorem: Let $R$ be a real closed field and let an expansion of $R$ to an o-minimal structure be given. Then for any closed bounded definable subset $A$ of $ R^n$ and a finite family $B_1,\dots,B_r$ of definable subsets of $A$ there exists a definable triangulation $h\colon |\mathcal K|\rightarrow A$ of $A$ compatible with $B_1,\dots,B_r$ such that $\mathcal K$ is a simplicial complex in $R^n$, $h$ is a $C^1$-embedding of each (open) simplex $\Delta\in \mathcal K$ and $h$ extends to a definable $C^1$-mapping defined on a definable open neighborhood of $|\mathcal K|$ in $R^n$. This improves Ohmoto-Shiota's theorem in three ways; firstly, $h$ is a $C^1$-embedding on each simplex; secondly, the simplicial complex $\mathcal K$ is in the same space as $A$ and thirdly, our proof is performed for any o-minimal structure. The possibility to have $h$ with the first of these properties was stated by Ohmoto and Shiota as an open problem (see [9]).

We investigate the problem of the realization of a given graph as the Reeb graph $\mathcal{R}(f)$ of a smooth function $f\colon M\rightarrow \mathbb{R}$ with finitely many critical points, where $M$ is a closed manifold. We show that for any $n\geq2$ and any graph $\Gamma$ admitting the so-called good orientation there exist an $n$-manifold $M$ and a Morse function $f\colon M\rightarrow \mathbb{R} $ such that its Reeb graph $\mathcal{R}(f)$ is isomorphic to $\Gamma$, extending previous results of Sharko and Masumoto-Saeki. We prove that Reeb graphs of simple Morse functions maximize the number of cycles. Furthermore, we provide a complete characterization of graphs which can arise as Reeb graphs of surfaces.