We study the asymptotic behavior of nonautonomous differential inclusions with delays in Banach spaces by analyzing their pullback attractors. Our aim is to give a recipe expressed by measures of noncompactness to prove the asymptotic compactness of the process generated by our system. This approach is effective for various differential systems regardless of the compactness of the semigroup governed by linear part.

Since the Hadamard Theorem, several metric and topological conditions have emerged in the literature to date, yielding global inverse theorems for functions in different settings. Relevant examples are the mappings between infinite-dimensional Banach-Finsler manifolds, which are the focus of this work. Emphasis is given to the nonlinear Fredholm operators of nonnegative index between Banach spaces. The results are based on good local behavior of $f$ at every $x$, namely, $f$ is a local homeomorphism or $f$ is locally equivalent to a projection. The general structure includes a condition that ensures a global property for the fibres of $f$, ideally expecting to conclude that $f$ is a global diffeomorphism or equivalent to a global projection. A review of these results and some relationships between different criteria are shown. Also, a global version of the Graves Theorem is obtained for a suitable submersion $f$ with image in a Banach space: given $r>0$ and $x_0$ in the domain of $f$ we give a radius $\varrho(r)> 0$, closely related to the hypothesis of the Hadamard Theorem, such that $B_{\varrho}(f(x_0))\subset f(B_r(x_0))$.

In this paper, we apply Morse theory and local linking to study the existence of nontrivial solutions for Kirchhoff type equations involving the nonlocal fractional $p$-Laplacian with homogeneous Dirichlet boundary conditions: \[ \begin{cases} \bigg[M\bigg(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dxdy\bigg)\bigg]^{p-1} (-\Delta)_p^su(x)=f(x,u)&\mbox{in }\Omega,\\ u=0&\mbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $(-\Delta)_p^s$ is the fractional $p$-Laplace operator with $0<s<1<p<\infty$ with $sp<N$, $M \colon \mathbb{R}^{+}_{0}\rightarrow \mathbb{R}^{+}$ is a continuous and positive function not necessarily satisfying the increasing condition and $f$ is a Carathéodory function satisfying some extra assumptions.

We consider the two-species quasi-linear chemotaxis system generalizing the prototype \begin{equation} \begin{cases} u_t=\nabla\cdot(D_1(u)\nabla u)-\chi_1\nabla\cdot(S_1(u)\nabla v), &x\in \Omega,\ t>0,\\ 0=\Delta v- v +w, &x\in \Omega,\ t>0,\\ w_t=\nabla\cdot(D_2(w)\nabla w)-\chi_2\nabla\cdot(S_2(w)\nabla z), & x\in \Omega,\ t>0,\\ {0=\Delta z- z +u}, & x\in \Omega,\ t>0, \end{cases} \tag*{$(0.1)$} \end{equation} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subseteq \mathbb{R}^N$ $(N\geq1)$. Here $D_i(u)=(u+1)^{m_i-1}$, $S_i(u)=u(u+1)^{q_i-1}$ $(i=1,2)$, with parameters $m_i\geq1$, $q_i>0$ and $\chi_1,\chi_2\in \mathbb{R}$. Hence, (0.1) allows the interaction of attraction-repulsion, with attraction-attraction and repulsion-repulsion type. It is proved that (i) in the attraction-repulsion case $\chi_1<0$: if $q_1<m_1+{2}/{N}$ and $q_2<m_2+{2}/{N}-{(N-2)^+}/{N}$, then for any nonnegative smooth initial data, there exists a unique global classical solution which is bounded; (ii) in the doubly repulsive case $\chi_1= \chi_2 < 0$: if $q_1<m_1+{2}/{N}-{(N-2)^+}/{N}$ and $q_2<m_2+{2}/{N}-{(N-2)^+}/{N}$, then for any nonnegative smooth initial data, there exists a unique global classical solution which is bounded; (iii) in the attraction-attraction case $\chi_1= \chi_2 >0$: if $q_1<{2}/{N}+m_1-1$ and $q_2<{2}/{N}+m_2-1$, then for any nonnegative smooth initial data, there exists a unique global classical solution which is bounded. In particular, these results demonstrate that the circular chemotaxis mechanism underlying (0.1) goes along with essentially the same destabilizing features as known for the quasi-linear chemotaxis system in the doubly attractive case. These results generalize the results of Tao and Winkler (Discrete Contin. Dyn. Syst. Ser. B. 20 (9) (2015), 3165-3183) and also enlarge the parameter range $q>{2}/{N}-1$ (see Cieślak and Winkler (Nonlinearity 21 (2008), 1057-1076)).

Introducing the Bolzano property, we present a topological version of the Poincaré-Miranda theorem. One simple, and one algorithmic proof that $n$-cube-like complexes have this property are given. Moreover, we investigate under what conditions the inverse limit preserves the Bolzano property. Finally, we give a characterization of the Bolzano property for locally connected spaces.

This paper is concerned with the existence of solutions for a class of intermediate local-nonlocal boundary value problems of the following type: \begin{equation} -{\rm div} \bigg[a\bigg(\mathop{\,\rlap{-}\!\!\int}\nolimits_{\Omega (x,r)}u(y)\,dy\bigg)\nabla u\bigg] = f(x,u,\nabla u ) \quad \mbox{in } \Omega, \ u\in H_{0}^{1}(\Omega ), \tag{${\rm IP}$} \end{equation} where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$, $a\colon\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, $f\colon \Omega \times \mathbb{R} \times \mathbb{R}^{N}$ is a given function, $r>0$ is a fixed number, $\Omega (x,r)=\Omega \cap B(x,r)$, where $B(x,r)=\{ y\in \mathbb{R}^{N}: |y-x|<r\}$. Here $|\,\cdot\, |$ is the Euclidian norm, $$ \mathop{\,\rlap{-}\!\!\int}\nolimits_{\Omega (x,r)}u(y)\,dy=\frac{1}{{\rm meas}\hspace{.06em}(\Omega (x,r))}\int_{\Omega (x,r)}u(y)\,dy $$ and ${\rm meas}\hspace{.06em}(X)$ denotes the Lebesgue measure of a measurable set $X\subset \mathbb{R}^{N}$.

We consider a Robin boundary value problem driven by the Laplacian plus an indefinite and unbounded potential. We assume that the reaction term of the equation is resonant with respect to the principal eigenvalue and the resonance is strong. Using primarily variational tools we prove two multiplicity theorems producing respectively two and three nontrivial smooth solutions.

In this paper we prove a kind of weighted Trudinger-Moser inequality which is employed to establish sufficient conditions for the existence of solutions to a large class of quasilinear elliptic differential equations with critical exponential growth. The class of operators considered includes, as particular cases, the Laplace, $p$-Laplace and $k$-Hessian operators when acting on radially symmetric functions.

The dissipative wave equation with sup-cubic nonlinearity and lower regular forcing term which belongs to $H^{-1}({\mathbb{R}^3})$ in the whole space $\mathbb{R}^3$ is considered. Well-posedness of a translational regular solution is achieved by establishing extra space-time translational regularity of an energy solution. Furthermore, a global attractor in the naturally defined energy space $H^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3)$ is built.

We study a one-dimensional $p$-Laplacian resonant problem with $p$-sublinear terms and depending on a positive parameter. By using quadrature methods we provide the exact number of positive solutions with respect to $\mu\in\mathopen{]}0,+\infty\mathclose[$. Specifically, we prove the existence of a critical value $\mu_1>0$ such that the problem under examination admits: no positive solutions and a continuum of nonnegative solutions compactly supported in $[0,1]$ for $\mu\in\mathopen{]}0,\mu_1\mathclose[$; a unique positive solution of compacton-type for $\mu=\mu_1$; a unique positive solution satisfying Hopf's boundary condition for $\mu\in\mathopen{]}\mu_1,+\infty\mathclose[$.

In this article, we generalize the Poincaré recurrence theorem to impulsive dynamical systems in $\mathbb R^n$. For a measure preserving system, we present some sufficient conditions to establish an impulsive system that is also measure preserving. Then, two recurrence theorems are proved. Finally, we use two examples to illustrate our results.

We develop some relative index theories for abstract operator equations. As applications, we prove a new Galerkin approximation formula and a new saddle point reduction formula for the $P$-index. We apply these new formulas to the minimal periodic problem for $P$-symmetric periodic solutions of nonlinear Hamiltonian systems.

The weakly convergent sequence coefficient ${\rm WCS}(X)$ is estimated by the James type constant $J_{X,t}(\tau)$, Jordan-von Neumann type constant $C_{t}(X)$ and the Domínguez Benavides coefficient $R(1,X)$, which enable us to obtain some sufficient conditions for normal structure. The results obtained in this paper are more general than other previously known sufficient conditions for normal structure.

We study a partial order in product spaces and then present some new properties of sets via the partial order. Based on these properties and monotone iterative technique, we establish some new fixed point theorems in product spaces. As an application, we utilize the main fixed point theorem to study a nonlinear system of fractional differential equations. We get the existence-uniqueness of positive solutions for this system, which complements the existing results of positive solutions for this nonlinear problem in the literature.

We are concerned with the qualitative analysis of solutions of a class of fractional boundary value problems with Dirichlet boundary conditions. By combining a direct variational approach with the theory of the fractional derivative spaces, we establish the existence of infinitely many distinct positive solutions whose $E^\alpha$-norms and $L^\infty$-norms tend to zero (to infinity, respectively) whenever the nonlinearity oscillates at zero (at infinity, respectively).

In this paper, we consider the following boundary value problem: $$ \begin{cases} ((-u'(t))^n)'=nt^{n-1}f(u(t)) & \text{for }0< t <1, \\ u'(0)=0,\quad u(1)=0, \end{cases} $$ where $n >1$. Using the fixed point theory on a cone and approximation technique, we obtain the existence of positive solutions in which $f$ may be singular at $u=0$ or $f$ may be sign-changing.

We study applicability conditions of the Nehari manifold method to the equation of the form $D_u T(u)-\lambda D_u F(u)=0$ in a Banach space $W$, where $\lambda$ is a real parameter. Our study is based on the development of the Rayleigh quotient theory for nonlinear problems. It turns out that the extreme values of parameter $\lambda$ which define intervals of applicability of the Nehari manifold method can be found through the critical values of the corresponding nonlinear generalized Rayleigh quotient. In the main part of this paper, we provide general results on this relationship. Theoretical results are illustrated by considering several examples of nonlinear boundary value problems. Furthermore, we demonstrate that the introduced tool of nonlinear generalized Rayleigh quotient can also be applied to prove new results on the existence of multiple solutions for nonlinear elliptic equations.

For the multivalued Volterra integral equation defined in a Banach space, the set of solutions is proved to be $R_\delta$, without auxiliary conditions imposed in Theorem 6 [J. Math. Anal. Appl. 403 (2013), 643-666]. It is shown that the solution set map, corresponding to this Volterra integral equation, possesses a continuous singlevalued selection; and the image of a~convex set under the solution set map is acyclic. The solution set to the Volterra integral inclusion in a separable Banach space and the preimage of this set through the Volterra integral operator are shown to be absolute retracts.

We proceed with the investigation of the problem \begin{equation} -\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \quad \mbox{in } \Omega, \qquad \frac{\partial u}{\partial n} = 0\quad \mbox{on } \partial \Omega, \tag{${\rm P}_\lambda$} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb R^N$ ($N \geq 2$), $a,b \in C^\alpha(\overline{\Omega})$ with $0<\alpha<1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of an unbounded subcontinuum of nontrivial nonnegative solutions of $({\rm P}_\lambda)$. Our approach is based on a priori bounds, a regularisation procedure, and Whyburn's topological method.

We study multiplicity results for positive solutions for a fractional elliptic system involving both concave-convex and critical growth terms. With the help of Nehari manifold and Ljusternik-Schnirelmann category, we investigate how the coefficient $h$ of the critical nonlinearity affects the number of positive solutions to this problem and get a relationship between the number of positive solutions and the topology of the global maximum set of $h$.

We deal with the following class of problems: \begin{equation*} \begin{cases} -\Delta u=\lambda u+|x|^{\alpha}g(u_+)+ f(x)&\mbox{in } B_1,\\ u =0&\mbox{on }\partial B_1, \end{cases} \end{equation*} where $B_1$ is the unit ball in $\mathbb R^2$, $g$ is a $C^1$-function in $[0,+\infty)$ which is assumed to be in the subcritical or critical growth range of Trudinger-Moser type and $f\in L^{\mu}(B_1)$ for some $\mu>2$. Under suitable hypotheses on the constant $\lambda$, we prove existence of at least two solutions to this problem using variational methods. In case of $f$ radially symmetric, the two solutions are radially symmetric as well.

We deal with the existence of solutions to a critical elliptic system involving the fractional Laplacian operator. We consider the primitive of the nonlinearity interacting with the spectrum of the operator. The one side resonant case is also considered. Variational methods are used to obtain the existence, and our result improves earlier results of the authors.