We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, which allows the phase space to be not separable. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and $N\setminus E$ need not to be a neighbourhood of the compact invariant set. Moreover, we define H-shape cohomology groups and the H-shape cohomology index that is used to develop the Morse equations. Particularly, we apply H-shape index theory to an abstract retarded nonautonomous parabolic equation to illustrate these advantages for the new shape index theory, in which we obtain a special existence property of bounded full solutions.

We study the existence of nonzero nonnegative or strictly positive solutions of second order Neumann boundary value problems with nonlinearities which are allowed to take negative values via a recently established fixed point theorem for $r$-nowhere normal-outward maps in Banach spaces. As applications, we obtain results on the existence of strictly positive solutions for some models of population inhabiting one dimensional heterogeneous environments with perfect barriers, where the local rate of change in the population density changes sign.

Conley index theory is a powerful topological tool for obtaining information about invariant sets in dynamical systems. A key feature of Conley theory is that the index is robust under perturbation; given a continuous family of flows $\{\varphi_\lambda\}$, the index remains constant over a range of parameter values, avoiding many of the complications associated with bifurcations. This theory is well-developed for flows and homeomorphisms, and has even been extended to certain classes of semiflows. However, in recent years mathematicians and scientists have become interested in differential inclusions. Here the theory has also been studied for inclusions which satisfy certain bounding properties. In this paper we extend some of these results-in particular, the stability of isolating neighbourhoods under perturbation-to inclusions which do not satisfy these bounding properties. We do so by utilizing a novel approach to the solution set of differential inclusions which results in an object called a multiflow. This perspective allows us to relax the assumptions of the earlier work and also to develop tools needed to extend the continuation of Conley's attractor-repeller decomposition to differential inclusions, a result which is addressed in subsequent work. Our interest in these results is in the study of piecewise-continuous differential equations-which are typically reframed as a certain type of differential inclusion called Filippov systems-and how these discontinuous equations relate to families of smooth systems which limit to them. Therefore this paper also discusses in some detail how the generalization of Conley index theory applies to Filippov systems.

The Conley index theory is a powerful topological tool for describing the basic structure of dynamical systems. One important feature of this theory is the attractor-repeller decomposition of isolated invariant sets. In this decomposition, all points in the invariant set belong to the attractor, its associated dual repeller, or a connecting region. In this connecting region, points tend towards the attractor in forwards time and the repeller in backwards time. This decomposition is also, in a certain topological sense, stable under perturbation. Conley theory is well-developed for flows and homeomorphisms, and has also been extended to some more abstract settings such as semiflows and relations. In this paper we aim to extend the attractor-repeller decomposition, including its stability under perturbation, to continuous time set-valued dynamical systems. The most common of these systems are differential inclusions such as Filippov systems. Of particular importance for this generalization is the perturbation of Filippov systems to nearby smooth systems.

Motivated by [12] and [11], we use a geometric approach to define the Maslov index for heteroclinic orbits of non-Hamiltonian systems on a two-dimensional phase space, and we proceed by explaining the Maslov index is equal to the sum of the nullity of a family of Fredholm operators. As an application, we illustrate the role of our results in the Nagumo equation.

The goal of this paper is to consider fractional differential hemivariational inequalities (FDHVIs, for short) in the framework of Banach spaces. Our first aim is to investigate the existence of mild solutions to FDHVIs by means of a fixed point technique avoiding the hypothesis of compactness on the semigroup. The second step of the paper is to study the existence of decay mild solutions to FDHVIs via giving asymptotic behavior of Mittag-Leffler function.

The aim of this paper is to discuss Penot's problem on a generalization of Caristi's fixed point theorem. We settle this problem in the negative and we present some new theorems on the existence of fixed points of set-valued mappings in ordered metric spaces and reflexive Banach spaces.

We study multiplicity of solutions to an asymptotically linear Dirichlet problem associated with a planar system of second order ordinary differential equations. The existence of two sign-preserving component-wise solutions is guaranteed when the Morse indexes of the linearizations at zero and at infinity do not coincide, and one of the asymptotic problems has zero-index. The proof is developed in the framework of topological and shooting methods and it is based on a detailed analysis and characterization of the phase angles in a two-dimensional setting.

In this paper, stability and Hopf bifurcation in a diffusive predator-prey system are discussed considering prey species' group behavior. The interaction term is of Holling type II with the prey density X under the square root. The local behavior is first discussed for the corresponding homogeneous system. Then, the diffusive system's linear stability is discussed around a homogeneous equilibrium state followed by bifurcations in the infinite-dimensional system. By the linear stability analysis, we see that a Hopf bifurcation occurs in the homogeneous system. By choosing a proper bifurcation parameter, we prove that a Hopf bifurcation occurs in the nonhomogeneous system. Furthermore, the direction of the Hopf bifurcation is obtained.

The purpose of this article is to deal with the following biharmonic critical Choquard equation\begin{align*}\begin{cases}\Delta^{2}u = \lambda f(x) |u|^{q-2}u+ g(x)\bigg(\displaystyle \int_{\Omega}\frac{g(y)|u(y)|^{2_\alpha^*}}{|x-y|^{\alpha}}dy\bigg)|u|^{2_\alpha^*-2}u & \text{in } \Omega,\\u,\ \nabla u = 0 & \text{on } \partial\Omega,\end{cases}\end{align*}where $\Omega$ is a bounded domain in $\mathbb R^N$ with smooth boundary $\partial \Omega$, $N\geq 5$, $1< q < 2$, $0< \alpha < N$, $2_\alpha^*=({2N-\alpha})/({N-4})$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and $\lambda > 0$ is a parameter. The functions $f, g\colon \overline {\Omega}\rightarrow \mathbb R$ are continuous sign-changing weight functions. Using the Nehari manifold and fibering map analysis, we prove the existence of two nontrivial solutions of the problem with respect to parameter $\lambda$.

The present paper deals with the notions of past attractors and repellers for nonautonomous dynamical systems. This uses the topological method of extending functions in order to describe the nonautonomous attractors by means of the prolongational limit sets in the extended phase space. Essentially, for a given nonautonomous dynamical system $(\theta ,\varphi ) $ with base set $P=\mathbb{T}$, where $\mathbb{T}$ is the time $\mathbb{Z}$ or $\mathbb{R}$, and with base flow $\theta $ as the addition, the limit sets $\omega ^{-}( 0) $ and $\omega^{+}( 0) $ in the Stone-Čech compactification $\beta \mathbb{T}$ determine respectively the past and the future of the conduction system.

This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation:\begin{equation*}(-\Delta)^{s_1}_{p}u+(-\Delta)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\beta} \quad \text{in } \mathbb R^N,\end{equation*}where $1< q< p$, $0< s_2\leq s_1< 1$, $ps_1=N$, $\beta\in[0,N)$, and $V,K\colon \mathbb R^N\to\mathbb R$, $f\colon \mathbb R\to \mathbb R$ are continuous functions satisfying some natural hypotheses. We are concerned both with the case when $f$ has a subcritical growth and with the critical framework with respect to the exponential nonlinearity. By combining a Moser-Trudinger type inequality for fractional Sobolev spaces with Schwarz symmetrization techniques and related variational and topological methods, we prove the existence of nonnegative solutions.

We introduce four, a priori different, concepts of topological pressure for possibly discontinuous semiflows acting on a compact metric space and observe that they all agree with the classical one when restricted to the continuous setting. Moreover, for a class of impulsive semiflows, which appear to be examples of discontinuous systems, we prove a variational principle. As a consequence, we conclude that for this class of systems the four notions of pressure coincide and, moreover, they also coincide with a concept of the topological pressure introduced in [3].

In the paper we study the Liouville-type theorems for generalized Hénon-Lane-Emden elliptic system in $\mathbb{R}^N$. By the methods of spherical averages, Rellich-Pohozaev type identities, Sobolev inequalities on $S^{N-1}$, feedback and measure arguments, and scale invariance of the solutions, we show that if the pair of exponents is subcritical, then this system has no positive solutions for $N=2$ and no bounded positive solutions for $N=3$.

We study the topological regularity of solutions to the Cauchy problem of a (spatial) third-order partial differential equation with a multi-valued perturbation and an impulsive effect. In the framework of the functional space, the principal part of the differential operator corresponds to an Airy operator generating a noncompact $C_0$-group of unitary operators. Our attention is concerned with the $R_\delta$-structure of the solution set for the Cauchy problem. Geometric aspects of the corresponding solution map are also considered. In our main results, no any compactness condition on the impulsive functions is needed. Moreover, we give illustrating examples for the nonlinearity and impulsive functions.

We study existence of positive solutions of the following heterogeneous diffusive logistic equation with a harvesting term,\begin{equation*}-\Delta u =\lambda a(x) u -b(x) u^2 - c h(x), \quad\text{in } \mathbb{R}^N,\qquad \lim_{|x|\rightarrow\infty}u(x)=0,\end{equation*}where $\lambda$ and $c$ are positive constant, $h(x)$, $b(x)$ are nonnegative and there exists a bounded region $\Omega_0$ such that $\overline{\Omega}_0 = \{ x : b(x)=0 \}$. Under the strong growth rate assumption, that is, when $\lambda \geq \lambda_1(\Omega_0)$, the first eigenvalue of weighted eigenvalue problem $-\Delta v=\mu a(x)v$ in $\Omega_0$ with Dirichlet boundary condition, we will show that if $h \equiv 0$ in $\mathbb{R}^N\setminus\overline{\Omega}_0$ then our equation has a unique positive solution for all $c$ large, provided that $\lambda$ is in a right neighborhood of $\lambda_1 (\Omega_0)$. In addition we prove a new result on the positive solution set of this equation in the weak growth rate case complimenting existing results in the literature.