By using the least action principle and the saddle point theorem, some existence theorems are obtained for second-order $p$-Laplacian systems with or without impulsive effects under weak sublinear growth conditions, we improve some existing results in the literature.

We first prove two identities which are analogous to Entry 3.3.4 in Ramanujan's lost notebook. The identities in Entry 3.3.4 come out equal to a cubic theta function of Borwein and Borwein [5]. In our case they come out equal to $\frac{(q^4;q^4)^2}{(q^2;q^4)^2} C^2(q)$. We also express $C(q)$ in terms of theta functions $\phi(q)$ and $\psi(q)$. A series expansion of $\log C(q)$ is also given. One of the identities (9) is equivalent to a Theorem in partitions.

We study solutions of the equation \begin{equation*} \begin{aligned} & u_t = \sum^n_{i,j=1} {\partial \over \partial x_j} \left(a^{ij} {\partial \over \partial x_i} u^m \right) + h u^p \quad\quad {\rm on}\quad \Omega \times(0,T) \\ & u = 0 \quad\quad {\rm on} \quad \partial \Omega \times (0,T),\end{aligned} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary, and $a^{ij} = a^{ij}(x)$ is uniformly positive definite and $h = h(x) \gt 0$ on $\Omega$. When $$0 \lt m \lt 1 \lt p \lt m + \frac{2}{n+1} \quad \textrm{or} \quad 1 \lt m \lt p \leq m + \frac{2}{n+1},$$ we will show that if $u$ is a non-nagative solution and blows up at $T$, then $$u(x,t) \leq C |T-t|^{-1/(p-a)}.$$ The proof relies on rescaling arguments and some, old and new, Fujita-type results.

In this paper, we first establish some new types of fixed point theorem which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem and many results in [W.-S. Du, Some new results and generalizations in metric fixed point theory, Nonlinear Anal. 73 (2010), 1439-1446] and references therein. Applying those new results, we also present some existence theorems of coincidence point and others.

Let $H({\mathbb D})$ denote the class of all analytic functions on the open unit disk ${\mathbb D}$ of ${\mathbb C}$. Let $\varphi$ be an analytic self-map of ${\mathbb D}$ and $u \in H({\mathbb D})$. The generalized weighted composition operator is defined by $$ D^n_{\varphi,u} f = uf^{(n)} \circ \varphi, \quad f \in H({\mathbb D}).$$ The boundedness and compactness of generalized weighted composition operators from area Nevanlinna spaces to Bloch-type spaces and little Bloch-type spaces are characterized in this paper.

We study the boundary layer equation $f^{\prime\prime\prime}(\eta) + R((f^{\prime}(\eta))^2 - f(\eta) f^{\prime\prime}(\eta)) = K$, subjects to the boundary conditions $f(0) = f^{\prime\prime}(0) = 0$, $f(1) = 1$ and $f^{\prime}(1) + \varphi f^{\prime\prime}(1) = 0$. The given problem arises from the study of steady laminar flows in channels with two equally porous walls, where $R$ relates to the Reynold's number, and $K$ is an integration constant. We are able to obtain the homogeneity property and classify all types of solutions for the prescribed positive slip coefficient $\varphi$. In particular, the existence of the continuums in the $R-K$ plane has been verified, and this leads to the existence of multiple solutions for large $R$.

By using a recent result of Ricceri, we prove the existence of multiple solutions for perturbed $n$-Laplacian equations with Dirichlet boundary conditions. The Trudinger Moser inequality allows us to deal with perturbations with exponential growth.

We consider $6$-regular plane graphs whose faces have size $1$, $2$ or $3$. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to $53$ vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-Coxeter construction that takes a $6$-regular plane graph $G_0$, two integers $k$ and $l$ and returns two $6$-regular plane graphs. Then in the final section, we consider the notions of zigzags and central circuits for the considered graphs. We introduced the notions of tightness and weak tightness for them and prove an upper bound on the number of zigzags and central circuits of such tight graphs. We also classify the tight and weakly tight graphs with simple zigzags or central circuits.

In this paper, a class of generalized symmetric strong vector quasi-equilibrium problems in real locally convex Hausdorff topological vector spaces is studied. By using the Kakutani-Fan-Glicksberg fixed point theorem, an existence theorem of solutions for the generalized symmetric strong vector quasi-equilibrium problems is obtained. Moreover, the closedness of the solution set and a stability result of solutions for such problem are also derived.

In this paper we investigate the existence of a center-stable manifold for solutions to the semi-linear evolution equation of the form $u(t) = U(t,s) u(s) + \int_s^t U(t,\xi) f(\xi,u(\xi)) d\xi$, $t \ge s \ge 0$, when its linear part, the evolution family $(U(t,s))_{t \ge s \ge 0}$, has an exponential trichotomy on the half-line and the nonlinear forcing term $f$ satisfies the $\varphi$-Lipschitz condition, i.e., $\|f(x)-f(y)\| \le \varphi(t)\|x-y\|$ where $\varphi(t)$ belongs to some class of admissible function spaces on the half-line. Moreover, we consider the existence of unstable manifolds and their attraction property for evolution equations defined on the whole line. Our methods are the Lyapunov-Perron method, the rescaling procedures, and the use of admissible function spaces as in [14, 15].

In this paper, we consider generalized vector variational-like inequalities involving Dini subdifferential. Some relations among these inequalities and vector optimization problems are presented.

In 1981, Alspach conjectured that if $3 \leq m_{i} \leq v$, $v$ is odd and $v(v-1)/2 = m_{1} + m_{2} + \dots + m_{t}$, then the complete graph $K_{v}$ can be decomposed into $t$ cycles of lengths $m_{1}$, $m_{2}$, $\dots$, $m_{t}$ respectively; if $v$ is even, $v(v-2)/2 = m_{1} + m_{2} + \dots + m_{t}$, then the complete graph minus a one-factor $K_{v}-F$ can be decomposed into $t$ cycles of lengths $m_{1}$, $m_{2}$, $\dots$, $m_{t}$ respectively. In this paper, we extend the study to the decomposition of the complete equipartite graph $K_{m(n)}$. For $m_{i} \in \{4,5\}$, we prove that the trivial necessary conditions are also sufficient.

Let $A \in M_n(C)$. The numerical range of $A$ is defined as $W(A) = \{x^*Ax: x \in C^n, |x|=1 \}$. We give a necessary and sufficient condition for $W(A)$ to be an elliptic disk. The criterion is applied to prove the elliptic numerical ranges of certain bordered matrices.

Conditions on $\beta$ are determined so that $1 + \beta zp'(z)$ subordinated to $\sqrt{1+z}$ implies $p$ is subordinated to $\sqrt{1+z}$. Analogous results are also obtained involving the expressions $1 + \beta{zp'(z)}/{p(z)}$ and $1 + \beta{zp'(z)}/{p^2(z)}$. These results are applied to obtain sufficient conditions for normalized analytic functions $f$ to satisfy the condition $|\left({zf'(z)}/{f(z)}\right)^2-1| \lt 1$.

In this paper, with respect to Hausdorff metric, we establish the Lipschitz continuity of an approximate solution mapping for a parametric scalar equilibrium problem. Simultaneously, we give some applications to parametric optimization problems and parametric variational inequalities. Our main results are new and strengthen some results in the recent literature.

A Graded Morita context is constructed for any comodule of a group coring. For any right $G$-$\underline{\mathcal{C}}$-comodule $\underline{M}$ with dual graded ring $R$, we define a graded ring $T=HOM^{G,\underline{\mathcal{C}}}(M,M) = \bigoplus_{g \in G} HOM^{G,\underline{\mathcal{C}}}(M, M)_g$, and a $G$-graded $R$-$T$ bimodule $Q = \bigoplus_{g \in G} Q^g$, where $Q^g$ is a family of right $A$-linear maps $q^g_{\alpha};\,M_{\alpha} \rightarrow R_{g\alpha}$ in $\mathcal{M}_A$. We construct a graded Morita context $M = (T,\,\,\,R,\,\,\,\bigoplus_{\alpha\in G}M_{\alpha},\,\,\,Q, \,\,\,\tau,\,\,\,\mu)$ with connecting homomorphisms $\tau:\,\, _{T}(\bigoplus_{\alpha\in G}M_{\alpha})\otimes_R Q_T\rightarrow T,\,\,m\otimes q\mapsto mq(-)$, $\,\mu:\, _{R}Q\otimes_T (\bigoplus_{\alpha\in G}M_{\alpha})_{R}\rightarrow R,\,\,q\otimes m\mapsto q(m)$, which generalized the Morita context in [3, 5-7, 10, 13]. Meanwhile, we prove the graded Morita-Takeuchi theory as a generalization of Morita-Takeuchi theory which characterize the equivalence of comodule over field.

In this paper, we introduce iterative schemes for finding a common element of the set of common fixed points for a left amenable semigroup of nonexpansive mappings, the set of solutions of the variational inequalities for a family of $\alpha$-inverse-strongly monotone mappings and the set of solutions of a system of equilibrium problems in a Hilbert space. We establish weak and strong convergence theorems for the sequences generated by our proposed schemes. Moreover, we present various applications to the additive semigroup of nonnegative real numbers and families of strictly pseudocontractive mappings.

This paper is concerned with the existence of random attractors for the stochastic FitzHugh-Nagumo systems with delay on infinite lattice. Under suitable dissipative conditions, It shows that such a system has a random attractor which is a random compact invariant set.

The purpose of this paper is to investigate the problem of finding a common element of the solution set of a general system of variational inequalities, the solution set of a variational inequality problem and the fixed point set of a strict pseudocontraction in a real Hilbert space. Based on the well-known viscosity approximation method, extragradient method and Mann's iteration method, we propose and analyze a relaxed projection-viscosity approximation method for computing a common element. Under very mild assumptions, we obtain a strong convergence theorem for three sequences generated by the proposed method. Our proposed method is quite general and flexible and develops some iterative methods considered in the earlier and recent literature.

The paper is concerned with the problem of finding a common solution of a variational inequality problem governed by Lipschitz continuous monotone mappings and of a fixed point problem of nonexpansive mappings. To solve this problem, we introduce two new iterative algorithms which are based on Tseng's extragradient method. Moreover we prove the weak and strong convergence of these new algorithms to a solution of the above-stated problem.

We consider a class of nonlinear systems for which a positive solution exists and is unique. Such systems appear quite naturally in several applications concerning difference equations. Moreover, certain extremum problems can be reduced to solving these systems. In order to solve such problems we develop a quasi-Newton algorithm which is very efficient just because the existence and uniqueness of the solution are guaranteed. Several numerical examples illustrate the general results.

Let $C$ be a closed convex subset of a real Hilbert space $H$. Let $A$ be an inverse-strongly monotone mapping of $C$ into $H$ and let $B$ be a maximal monotone operator on $H$ such that the domain of $B$ is included in $C$. We introduce two iteration schemes of finding a point of $(A+B)^{-1}0$, where $(A+B)^{-1}0$ is the set of zero points of $A+B$. Then, we prove two strong convergence theorems of Halpern's type in a Hilbert space. Using these results, we get new and well-known strong convergence theorems in a Hilbert space.

We study hypersurfaces either in the De Sitter space $\mathbb{S}_1^{n+1} \subset \mathbb{R}_1^{n+2}$ or in the anti De Sitter space $\mathbb{H}_1^{n+1} \subset \mathbb{R}_2^{n+2}$ whose position vector $\psi$ satisfies the condition $L_k \psi = A \psi + b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface, for a fixed $k = 0,\dots,n-1$, $A$ is an $(n+2) \times (n+2)$ constant matrix and $b$ is a constant vector in the corresponding pseudo-Euclidean space. For every $k$, we prove that when $A$ is self-adjoint and $b = 0$, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, open pieces of standard pseudo-Riemannian products in $\mathbb{S}_1^{n+1}$ ($\mathbb{S}_1^m(r) \times \mathbb{S}^{n-m}(\sqrt{1-r^2})$, $\mathbb{H}^m(-r) \times \mathbb{S}^{n-m}$ $(\sqrt{1+r^2})$, $\mathbb{S}_1^m(\sqrt{1-r^2}) \times \mathbb{S}^{n-m}(r)$, $\mathbb{H}^m(-\sqrt{r^2-1}) \times \mathbb{S}^{n-m}(r)$), open pieces of standard pseudo-Riemannian products in $\mathbb{H}_1^{n+1}$ ($\mathbb{H}_1^m(-r) \times \mathbb{S}^{n-m}(\sqrt{r^2-1})$, $\mathbb{H}^m(-\sqrt{1+r^2}) \times \mathbb{S}_1^{n-m}(r)$, $\mathbb{S}_1^m(\sqrt{r^2-1}) \times \mathbb{H}^{n-m}(-r)$, $\mathbb{H}^m(-\sqrt{1-r^2}) \times \mathbb{H}^{n-m}(-r)$) and open pieces of a quadratic hypersurface $\{x \in \mathbb{M}_{c}^{n+1} \;|\; = d\}$, where $R$ is a self-adjoint constant matrix whose minimal polynomial is $t^2+at+b$, $a^2-4b \leq 0$, and $\mathbb{M}_{c}^{n+1}$ stands for $\mathbb{S}_1^{n+1} \subset \mathbb{R}_1^{n+2}$ or $\mathbb{H}_1^{n+1} \subset \mathbb{R}_2^{n+2}$. When the $k$-th mean curvature is constant and $b$ is a non-zero constant vector, we show that the hypersurface is totally umbilical, and then we also obtain a classification result (see Theorem 2).