The paper presents several new fixed point theorems for some sum operators. Without any compactness or continuity assumptions, we establish sufficient conditions for some operators to have unique fixed points and describe sequences converging to the fixed points. The main results are obtained by the cone theory and monotone iterative technique. Besides, as applications, these new fixed point theorems are used to study the existence and uniqueness of solutions for a class of nonlinear fractional differential equations.

In this paper, we consider existence and regularity of positive solutions of a degenerate fourth order elliptic problem. Firstly, a new Caffarelli-Kohn-Nirenberg type inequality for the fourth order case is established. Then, by the use of the corresponding embedding, we obtain the existence of positive solutions of a degenerate fourth order elliptic problem. Finally, the regularity of the positive solutions is also studied.

In this paper we are interested in the existence of solutions for the following Kirchhoff type elliptic systems \begin{equation*}\begin{cases}\displaystyle -M\Bigg(\sum^m_{j=1}\|u_j\|^2\Bigg)\Delta u_i=f_i(x,u_1,\ldots,u_m) &\mbox{in } \Omega,\\ u_1=\ldots=u_m=0 &\mbox{on } \partial\Omega, \end{cases}\end{equation*}where $\Omega$ is a bounded domain in $\mathbb{R}^2$, $M$ is a Kirchhoff type function, $\|u_i\|^2:=\int_\Omega|\nabla u_i|^2{d}x$, $f_i$ behaves like $\exp(\beta s^2)$ when $|s|\rightarrow \infty$ for some $\beta> 0$, $i=1,\ldots,m$. By variational methods with the Trudinger-Moser inequality, we obtain the existence of solutions for the above systems.

In this paper we are interested in a class of semiclassical Schrödinger-Poisson system with Hartree-type nonlinearity. Firstly, we prove the existence of groundstate for autonomous system by using the subcritical approximation and the Pohozaev constraint method. Secondly, we prove the existence of semiclassical state solutions and multiplicity for system with critical frequency by using the genus. Finally, we study multiplicity and concentration behavior for solutions of system with general potential by using the Lusternik-Schnirelman theory.

We investigate the existence and properties of ground state solutions for a class of nonlinear Choquard equations. Proofs are mainly based on the variational method.

We establish a fixed point theorem for a class of mappings called generic 2-generalized hybrid mappings in the setting of a real Hilbert space. Two examples of that class of mappings are presented herein. The mappings are not quasi-nonexpansive even though they have fixed points. One of these maps is even not continuous. The fixed point theorem proved in this article improves many previous works in the literature.

With the advancement of science and technology, many physical and engineering models require more sophisticated mathematical functional spaces to be studied and well understood. For example, in fluid dynamics, electrorheological fluids (smart fluids) have the property that the viscosity changes (often drastically) when exposed to an electrical field. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools to study such problems as well as other models like the image processing. In this work, we consider the following nonlinear wave equation with variable exponents:\[ u_{tt}-\Delta u-\Delta u_{t}+|u_{t}|^{m(\cdot)-2}u_{t}=0,\quad \text{in }\Omega \times (0,T),\]where $\Omega $ is a bounded domain and $T> 0$, and show that weak solutions decay exponentially or polynomially depending on the range of the variable exponent $m$. We also give two numerical examples to illustrate our theoretical results.

We give an upper bound for the topological entropy of maps on inverse limit spaces in terms of their set-valued components. In a special case of a diagonal map on the inverse limit space $\underleftarrow{\lim}(I,f)$, where every diagonal component is the same map $g\colon I\to I$ which strongly commutes with $f$ (i.e. $f^{-1}\circ g=g\circ f^{-1}$), we show that the entropy equals $\max\{{\rm Ent}(f),{\rm Ent}(g)\}$. As a side product, we develop some techniques for computing topological entropy of set-valued maps.

The Reeb graph of a smooth function on a smooth manifold is the graph obtained as the space of all connected components of preimages (level sets) such that the set of all vertices coincides with the set of all the connected components of preimages containing some singular points. Reeb graphs are fundamental and important tools in algebraic topological and differential topological theory of Morse functions and their variants. In the present paper, as a related fundamental and important study, for given graphs, we construct certain smooth functions inducing the graphs as the Reeb graphs. Such results have been demonstrated by Masumoto, Michalak, Saeki, Sharko, among others, and also by the author since 2000s. We construct good smooth functions on suitable $3$-dimensional connected, closed and orientable manifolds.

In this paper, we deal with the following Schrödinger-Bopp-Podolsky system:\begin{equation}\begin{cases}-\Delta u+u+\phi u=f(u),\\-\Delta\phi +\varepsilon^{2}\Delta^{2}\phi =4\pi u^{2},\end{cases}\tag{${\rm P}_{\varepsilon}$}\quad \hbox{in }\mathbb{R}^{3},\end{equation}where $\varepsilon> 0$ and $f$ is a continuous, superlinear and subcritical nonlinearity. By using a perturbation approach and the method of invariant sets of descending flow incorporated with minimax arguments, we prove the existence and multiplicity of sign-changing solutions of system $({\rm P}_\varepsilon)$. Moreover, the asymptotic behavior of sign-changing solutions is also established. Our results mainly extend the results in Liu, Wang and Zhang ([21], Ann. Mat. Pura Appl. 2016).

In this paper we establish the existence and uniqueness result of right continuous bounded variation solution for a perturbed differential inclusion governed by time-dependent maximal monotone operators.

The aim of this paper is to investigate the existence of weak bounded solutions of the gradient-type quasilinear elliptic system\begin{equation}\tag{P}\begin{cases}- {\rm div} ( a_i(x, u_i, \nabla u_i) )+ A_{i, t} (x, u_i, \nabla u_i) = G_i(x, \mathbf{u}) \quad \hbox{in $\Omega$ for $i\in\{1,\dots,m\}$,}\\ {\mathbf{u}} = 0\quad \hbox{on $\partial\Omega$,}\end{cases}\end{equation}with $m\geq 2$ and $\mathbf{u}=(u_1,\dots, u_{m})$, where $\Omega\subset{\mathbb R}^N$ is an open bounded domain and some functions $A_i\colon \Omega\times{\mathbb R}\times{\mathbb R}^N\rightarrow{\mathbb R}$,$i\in\{1,\dots,m\}$, and $G\colon \Omega\times{\mathbb R}^m\rightarrow{\mathbb R}$ exist such that $a_i(x,t,\xi) = \nabla_{\xi} A_i(x,t,\xi)$, $A_{i, t} (x,t,\xi) = \frac{\partial A_i}{\partial t} (x,t,\xi)$, and $G_{i}(x,\mathbf{u}) = \frac{\partial G}{\partial u_i}(x,\mathbf{u})$. We prove that, under suitable hypotheses, the functional ${\mathcal J}$ related to problem (P) is $\mathcal{C}^1$ on a "good" Banach space $X$ and satisfies the weak Cerami-Palais-Smale condition. Then, generalized versions of the Mountain Pass Theorems allow us to prove the existence of at least one critical point and, if additionally ${\mathcal J}$ is even, of infinitely many critical points.

A subdiffusion problem in which the diffusion term is related to a stable stochastic process is introduced. Linear models of these systems have been studied in a general way, but non-linear models require a more specific analysis. The model presented in this work corresponds to a non-linear evolution equation with fractional time derivative and a pseudo-differential operator acting on the spatial variable. This type of equations has a couple of fundamental solutions, whose estimates for the $L_p$-norm are found to obtain three main results concerning mild and global solutions. The existence and uniqueness of a mild solution is based on the conditions required in some parameters, one of which is the order of stability of the stochastic process. The existence and uniqueness of a global solution is found for the case of small initial conditions and another for non-negative initial conditions. The relationship between the Fourier analysis and Markov processes, together with the theory of fixed points in Banach spaces, is particularly exploited. In addition, the present work includes the asymptotic behavior of global solutions as a linear combination of the fundamental solutions with $L_p$-decay.

We study $n$-player games where players form non-additive beliefs about opponent's decisions and answer with pure strategies. The concept of an equilibrium under uncertainty was introduced by J. Dow and S. Werlang (1994) for two players and was extended to $n$-player games by J. Eichberger and D. Kelsey (2000). The authors consider payoff functions expressed by Choquet integral. The concept of an equilibrium under uncertainty with payoff functions expressed by the Sugeno integral were considered by T. Radul (2018). We consider a generalization of this result with payoff functions expressed by fuzzy integral generated by arbitrary continuous $t$-norm.

Let $S$ be a right reversible semitopological semigroup, and let ${\rm LUC}(S)$ be the space of left uniformly continuous functions on $S$. Suppose that ${\rm LUC}(S)$ has a left invariant mean. Let $K$ be a weakly compact convex subset of a Banach space not necessarily with normal structure. We show that there always exists a common fixed point for any jointly weakly continuous and super asymptotically nonexpansive action of $S$ on $K$. Several variances involving the weak* compactness, the RNP, the distality of $K$ and/or the left reversibility of $S$ are also provided.

In this paper, we are concerned with the existence of Pohožaev-type ground state solutions for the Choquard equation with a singular potential and a critical exponent. By virtue of a generalized version of the Lions-type theorem and the Pohožaev manifold, we obtain the existence of a Pohožaev-type ground state solution for the above problem. Some recent results from the literature are improved and extended.