We consider perturbed Hammerstein integral equations of the form \[ y(t)=\gamma_1(t)H_1(\varphi_1(y))+\gamma_2(t)H_2(\varphi_2(y))+\lambda\int_0^1G(t,s)f(s,y(s)) \,ds \] in the case where $H_1$ and $H_2$ are continuous functions, which can be either linear or nonlinear subject to some restrictions, and $\varphi_1$, $\varphi_2$ are linear functionals. We demonstrate that by using a specially constructed order cone one can equip $\varphi_1$ and $\varphi_2$ with coercivity conditions that are useful in proving existence of multiple positive solutions. In addition, we demonstrate that the methodology can be superior to competing methodologies. We provide an application to the modeling of the deflection of an elastic beam.

This paper is concerned with the existence of homoclinic orbits for a class of second order Hamiltonian systems considering a non-periodic potential and a weaker Ambrosetti-Rabinowitz condition. By considering an auxiliary problem, we show the existence of two different approximative sequences of periodic solutions, the first one of mountain pass type and the second one of local minima. We obtain two different homoclinic orbits by passing to the limit in such sequences. As a relevant application, we obtain another homoclinic solution for the Hamiltonian system studied in [5].

We study positive solutions to singular boundary value problems of the form: \begin{equation*} \begin{cases} -u'' = h(t) \dfrac{f(u)}{u^\alpha} &\text{for } t \in (0,1), \\ u(0) = 0, \\ u'(1) + c(u(1)) u(1) = 0, \end{cases} \end{equation*} where $0< \alpha< 1$, $h\colon(0,1]\rightarrow(0,\infty)$ is continuous such that $h(t)\leq {d}/{t^\beta}$ for some $d> 0$ and $\beta\in[0,1-\alpha)$ and $c\colon [0,\infty)\rightarrow [0,\infty)$ is continuous such that $c(s)s$ is nondecreasing. We assume that $f\colon[0,\infty)\rightarrow(0,\infty)$ is continuously differentiable such that $[(f(s)-f(0))/s^\alpha]+\tau s$ is strictly increasing for some $\tau\geq 0$ for $s\in(0,\infty)$. When there exists a pair of sub-supersolutions $(\psi,\phi)$ such that $0\leq \psi\leq\phi$, we first establish a minimal solution $\underline u$ and a maximal solution $\overline u$ in $[\psi,\phi]$. When there exist two pairs of sub-supersolutions $(\psi_1,\phi_1)$ and $(\psi_2,\phi_2)$ where $0\leq \psi_1 \leq \psi_2 \leq \phi_1$, $\psi_1 \leq \phi_2 \leq \phi_1$ with $\psi_2\not \leq \phi_2$, and $\psi_2$, $\phi_2$ are not solutions, we next establish the existence of at least three solutions $u_1$, $u_2$ and $u_3$ satisfying $u_1\in [\psi_1,\phi_2], u_2\in [\psi_2,\phi_1]$ and $u_3\in [\psi_1,\phi_1]\setminus ([\psi_1,\phi_2]\cup [\psi_2,\phi_1])$.

In the paper we describe the classes of unique solvability of the Dirichlet and mixed two point boundary value problems for the second order linear integro-differential equation $$ u''(t)=p_0(t)u(t)+p_1(t)u(\tau_1(t))+\int_{a}^{b}p(t,s)u(\tau(s))\,ds+ q(t). $$ On the basis of the obtained and, in some sense, optimal results for the linear problems, by the a priori boundedness principle we prove the theorems of solvability and unique solvability for the second order nonlinear functional differential equations under the mentioned boundary conditions.

We present a combinatorial approach to the symmetric motion planning in polyhedra using finite spaces. For a finite space $P$ and a positive integer $k$, we introduce two types of combinatorial invariants, $\mathrm{CC}^{S}_k(P)$ and $\mathrm{CC}^{\Sigma}_k(P)$, that are closely related to the design of symmetric robotic motions in the $k$-iterated barycentric subdivision of the associated simplicial complex $\mathcal{K}(P)$. For the geometric realization $\mathcal{B}(P)=|\mathcal{K}(P)|$, we show that the first $\mathrm{CC}^{S}_k(P)$ converges to Farber-Grant's symmetric topological complexity $\mathrm{TC}^{S}(\mathcal{B}(P))$ and the second $\mathrm{CC}^{\Sigma}_k(P)$ converges to Basabe-González-Rudyak-Tamaki's symmetrized topological complexity $\mathrm{TC}^{\Sigma}(\mathcal{B}(P))$ as $k$ becomes larger.

This paper is concerned with the existence of positive bound state solutions for Schrödinger-Poisson systems with critical nonlocal term: \begin{equation} \begin{cases} -\Delta u=\phi|u|^{3}u+\lambda Q(x)|u|^{q-2}u &\text{in } \mathbb{R}^3, \\ -\Delta \phi=|u|^5 & \text{in } \mathbb{R}^3. \end{cases} \tag{$\mathcal{P}$} \end{equation} Under certain assumptions on $Q$ and $\lambda$, we prove that $(\mathcal{P})$ has multiple positive bound state solutions by decomposition the Nehari manifold and fine estimates.

We study the behaviour of solutions of limit periodic difference systems over (infinite) fields with absolute values. The considered systems are described by the coefficient matrices that belong to commutative groups whose boundedness is not required. In particular, we are interested in special systems with solutions which vanish at infinity or which are not asymptotically almost periodic. We obtain a transparent condition on the matrix groups which ensures that the special systems form a dense subset in the space of all considered systems, i.e. that, in any neighbourhood of any considered limit periodic system, there exists asystem which have non-asymptotically almost periodic or vanishing solutions. The presented results improve and extend known ones.

In this paper, without assuming operators to be continuous or compact, by employing monotone iterative technique on ordered Banach space, we at first establish new fixed point theorems for some kinds of nonlinear mixed monotone operators in set $P_{h,e}$. Then, we study a new form of fractional two point boundary value problem depending on a certain constant and give the existence and uniqueness of solutions. We also show that the unique solution exists in set $P_{h,e}$ or $P_{h}$ and can be uniformly approximated by constructing two iterative sequences for any initial values. At the end, a concrete example is given to illustrate our abstract results. The conclusions of this article enrich the fixed point theorems and provide new methods to deal with nonlinear differential equations.

The study of topologically nontrivial field configurations is an important topic in many branches of physics and applied sciences. In this paper we are interested to the existence of such structures, the so-called skyrmions, in the context of liquid crystals. More precisely, we consider a two-dimensional nematic or cholesteric liquid crystal. In the nematic case we use a Bogomol'nyi type decomposition in order to get a topological lower bound on the configurations with a given degree for the full Oseen-Frank energy functional, and so we can find a global minimum of degree $\pm 1$ for the energy. Then we consider the cholesteric case in presence of an electric field under the one constant approximation assumption, and, by using the concentration-compactness method, we prove the existence of a minimum again on the configurations of degree $\pm 1$, for sufficiently large electric fields.

In the paper the existence, uniqueness and the multiplicity of solutions for a quasilinear elliptic problems driven by the $\Phi$-Laplacian operator is established. Here we consider the non-reflexive case taking into account the Orlicz and Orlicz-Sobolev framework. The non-reflexive case occurs when the $N$-function $\widetilde{\Phi}$ does not verify the $\Delta_{2}$-condition. In order to prove our main results we employ variational methods, regularity results and truncation arguments.

In this work, we develop a fixed point index theory for the sum of $k$-set contractions and expansive mappings with constant $h> 1$ when $0\le k< h-1$ as well as in the limit case $k=h-1$. After computing this new index, several fixed point theorems and recent results are derived, including Krasnosel'skii type theorems. Two examples of application illustrate the theoretical results.

We prove that if a Banach space $X$ has the weak fixed point property and $Y$ satisfies the condition $M(Y)> 1$, then the direct sum $X\oplus Y$ with a uniformly convex norm has the weak fixed point property.

In this paper we are mainly concerned with the linearization of the flow with prescribed frequency for analytic perturbation of constant vector fields on a torus under weaker non-degeneracy condition and non-resonant condition. As is well known the perturbation of constant vector fields may induce a shift of frequency, when Kolmogorov's non-degeneracy condition is violated. By introducing external parameters and using the polynomial structure to truncate, we prove that if the frequency mapping has the nonzero Brouwer's topological degree at some non-resonant frequency, then the conjugated vector fields will have a linear flow with this frequency.

This paper is concerned with two classes of singularly perturbed Schrödinger-Poisson systems of the form \begin{equation*} \begin{cases} -\varepsilon^2\triangle u+u+ \phi u=f(u), & x\in {\mathbb{R}}^{3},\\ -\triangle \phi=u^2, & x\in \mathbb{R}^3, \end{cases} \end{equation*} and \begin{equation*} \begin{cases} -\varepsilon^2\triangle u+V(x)u+ \phi u=g(x,u)+K(x)u^5, & x\in {\mathbb{R}}^{3},\\ -\triangle \phi=u^2,& x\in \mathbb{R}^3, \end{cases} \end{equation*} where $\varepsilon > 0$ is a small parameter. We prove that: (1) the first system admits a concentrating bounded state for small $\varepsilon$, where $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies Berestycki-Lions assumptions which are almost necessary; (2) there exists a constant $\varepsilon_0> 0$ determined by $V,K$ and $g$ such that for any $\varepsilon\in (0,\varepsilon_0]$ the second system has a nontrivial solution, where $V,K\in \mathcal{C}(\mathbb{R}^3,\mathbb{R})$, $V(x)\ge 0$, $K(x)> 0$, $g\in \mathcal{C}(\mathbb{R}^3\times \mathbb{R},\mathbb{R})$ is an indefinite function. Our results improve and complement the previous ones in the literature.

In this work we study weighted $p$-Laplacian equations in a bounded domain with a variable and generally non-smooth diffusion coefficient having at most a finite number of zeroes. The main attention is focused on the case that the diffusion coefficient $a(x)$ in such equations satisfies the inequality $\liminf_{x\to z}|x-z|^{-p}a(x)> 0$ for every $ z\in \overline\Omega$. We show the existence of weak solutions and global attractors in $L^2(\Omega)$, $L^q(\Omega)$ $(q\geq 2)$ and $D_0^{1,p}(\Omega)$, respectively.

For $a\in [0,+\infty)$, the function space $E_{\geq a}$ ($E_{> a}$; $E_{\leq a}$; $E_{< a}$) of all continuous maps from $[0,1]$ to itself whose topological entropies are larger than or equal to $a$ (larger than $a$; smaller than or equal to $a$; smaller than $a$) with the supremum metric is investigated. It is shown that the spaces $E_{\geq a}$ and $E_{> a}$ are homeomorphic to the Hilbert space $l_2$ and the spaces $E_{\leq a}$ and $E_{< a}$ are contractible. Moreover, the subspaces of $E_{\leq a}$ and $E_{< a}$ consisting of all piecewise monotone maps are homotopy dense in them, respectively.

This work is devoted to investigations of some interesting aspects of a multi-component Reaction-Diffusion system of the form \[ \partial_t z=\mathbf{D}\Delta_x z + M(x)z + W(x)|z|^{p-2}\beta z, \quad z\colon \mathbb R\times\mathbb R^N\to\mathbb R^{2K}, \ N\geq2 \] where $M$, $W$ are external potential functions, $\mathbf{D}$ and $\beta$ are matrices of diffusion coefficients and coupling constants respectively. When the diffusion rate is small, we show that the geometric shapes of the external potential functions will influence the multiplicity of solutions to the system. It is also of interest to know that, for $z=(u,v)$, we shall deal with standard diffusion coefficients $D_u> 0$ and the incongruent diffusion coefficients $D_v< 0$ which has generally been overlooked in the study of Reaction-Diffusion systems.

In this article we are concerned with the following Choquard equation \begin{alignat*}2 -\Delta u &= \lambda|u|^{q-2}u +\bigg(\int_{\Omega}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\bigg)|u|^{2^*_{\mu}-2}u,\quad u> 0, &\quad& \text{in }\Omega,\\ u& = 0 &\quad& \text{on } \partial \Omega , \end{alignat*} where $\Omega$ is an open bounded set with continuous boundary in $\mathbb{R}^N$ $( N\geq 3)$, $2^*_{\mu}=({2N-\mu})/({N-2})$ and $q \in [2,2^*)$ where $2^*={2N}/({N-2})$. Using Lusternik-Schnirelman theory, we associate the number of positive solutions of the above problem with the topology of $\Omega$. Indeed, we prove that if $\lambda < \lambda_1$, then problem has $\text{cat}_{\Omega}(\Omega)$ positive solutions whenever $q \in [2,2^*)$ and $N> 3 $ or $4< q< 6 $ and $N=3$.

In this article, we establish the existence of solutions to the fractional $p$-Kirchhoff type equations with a generalized Choquard nonlinearity without assuming the Ambrosetti-Rabinowitz condition.

In this paper we prove the existence of a sign-changing solutions for the equation $$ -\Delta u - \frac{1}{2} ( x \cdot \nabla u) = f(u), \quad x \in \mathbb{R}^2, $$ where $f$ has exponential critical growth in the sense of the Trudinger-Moser inequality. In the proof we apply variational methods.

In this note we are going to present new short proofs concerning either the existence or the non-existence of some limits appearing in the theory of almost periodic functions. Our proofs are completely different from those presented in the papers [1] and [3].

In this paper we consider nonlinear eigenvalue problems for a one-dimensional Dirac equation with spectral parameter in the boundary condition. We investigate local and global bifurcations of nontrivial solutions to these problems. The existence of unbounded continua of nontrivial solutions bifurcating from points and intervals of the line of trivial solutions is shown.

In this paper we address questions on the existence and multiplicity of a class of geometrically motivated mappings with certain symmetries that serve as solutions to the nonlinear system in variation: $$ {\rm ELS} [(u, \mathscr{P}), \Omega] = \begin{cases} [\nabla u]^t {\rm div} [F_\xi\nabla u ] -F_s [\nabla u]^t u = \nabla \mathscr{P} &\text{in } \Omega, \\ \det \nabla u = 1 &\text{in } \Omega, \\ u \equiv x &\text{on } \partial \Omega. \end{cases} $$ Here $\Omega \subset \mathbb R^n$ is a bounded domain, $F=F(r, s, \xi)$ is a sufficiently smooth Lagrangian, $F_s=F_s(|x|, |u|^2, |\nabla u|^2)$ and $F_\xi=F_\xi(|x|, |u|^2, |\nabla u|^2)$ with $F_s$ and $F_\xi$ denoting the derivatives of $F$ with respect to the second and third variables respectively while $\mathscr{P}$ is an a priori unknown hydrostatic pressure resulting from the incompressibility constraint $\det \nabla u =1$. Among other things, by considering twist mappings $u$ with an SO$(n)$-valued twist path, we prove the existence of multiple and topologically distinct solutions to ELS for $n \ge 2$ even versus the only (non) twisting solution $u \equiv x$ for $n \ge 3$ odd. An extremality analysis for twist paths and those of Lie exponential types and a suitable formulation of a differential operator action on twists relating to ELS are the key ingredients in the proof.