In this paper, we show that if $Q(x)$ satisfies some suitable conditions, then the quasilinear elliptic Dirichlet problem $-\Delta_p u + |u|^{p-2} u = Q(x)|u|^{q-2}u$ in an unbounded cylinder domain $\Omega$ has at least two solutions in which one is a positive ground state solution and the other is a nodal solution.

In this paper, we study the existence of solutions for vector saddle points problems in a general setting. Our approach, first, is based on the KKM lemma and a relaxation of the $C-$ lower semicontinuity notion introduced by T.Tanaka by means of an extension to the vector setting of a Brézis-Nirenberg-Stampacchia condition, and arguments from generalized convexity. This leads us to generalize and improve some new existence results on vector saddle points problems. In the second approach, we establish an existence result for vector saddle point problems under a paracompacity assumption.

In this paper, we introduce two iterative algorithms based on the relaxed extragradient-like method for finding a common element of the solution set of a general system of generalized mixed equilibria and the fixed point set of a strictly pseudocontractive mapping in a real Hilbert space. We will prove the weak convergence of the iterative algorithm under some mild conditions, while the strong convergence is obtained under some more restrictive conditions.

In this note, we establish new inequalities of Hadamard type involving several log-convex functions and log-concave functions.

We find the almost product (locally product) structures of general natural lift type on the tangent bundle of a Riemannian manifold. We get the conditions under which the tangent bundle endowed with such a structure and with a general natural lifted metric is a Riemannian almost product (locally product) or an (almost) para-Hermitian manifold. We give a characterization of the general natural (almost) para-Hermitian structures, which are (almost) para-Kählerian on the tangent bundle.

This paper proves that if $G$ is a triangle-free planar graph in which no 4-cycle is adjacent to a 4-cycle or a 5-cycle, then $G$ is on-line 3-choosable.

In an indefinite metric $g.f.f$-manifold, we study half lightlike submanifolds $M$ tangent to the characteristic vector fields. We discuss the existence of totally umbilical half lightlike submanifolds of an indefinite $\cal{S}$-space form.

An $n$-$sun$ is the graph with $2n$ vertices consisting of an $n$-cycle with $n$ pendent edges which form a 1-factor. In this paper we show that the necessary and sufficient conditions for the decomposition of complete tripartite graphs with at least two partite sets having the same size into $3$-suns and give another construction to get a $3$-sun system of order $n$, for $n\equiv 0,1,4,9$ (mod 12). In the construction we metamorphose a Steiner triple system into a $3$-sun system. We then embed a cyclic Steiner triple system of order $n$ into a $3$-sun system of order $2n-1$, for $n\equiv 1$ (mod 6).

Let $A$, $B$ be two regular commutative unital Banach algebras such that $B$ is integral over $A$. In 2003, Dawson and Feinstein showed that the topological stable rank $\operatorname{tsr}(B) = 1$ whenever $\operatorname{tsr}(A) = 1$. In this note, we investigate whether we will have $\operatorname{tsr}(A) = \operatorname{tsr}(B)$ in general. For instance, when $A$ is a commutative unital $C^*$-algebra, we show that $\operatorname{tsr}(A) \leq \operatorname{tsr}(B)$, and the equality holds at least when the integral extension is separable. In general, $A$ and $B$ have the same Bass stable ranks $\operatorname{Bsr}(A) = \operatorname{Bsr}(B)$.

It is well-known that any abelian C*-algebra can be identified, by Gelfand transform, as $C_0(\Omega)$. Here, the spectrum $\Omega$ is locally compact Hausdorff in general, and compact exactly when the algebra is unital. In this survey article, we collect some results about the relation between the metrizability of $\Omega$ and the separability of $C_0(\Omega)$. A brief discussion for the topological structure of the spectrum of a general separable C*-algebra $A$ is also given.

We consider the asymptotic behavior of minimizers of the Ericksen functional with the Dirichlet boundary condition in a general domain. We examine the asymptotic behavior of minimizers of the functional as some of the elastic coefficients tend to the infinity, according to the de Gennes predictions with respect to nematic smectic-A phase transition.

A class of continuous-time quadratic programming problems (CQP) is discussed in this paper. We propose a discrete approximation procedure to find numerical solutions of (CQP). The provided computational procedure can yield feasible numerical solutions and bounds on the error introduced by the numerical approximation. It can also be shown that the searched sequence of approximate solution functions weakly-star converges to an optimal solution of (CQP). Besides, some numerical examples are provided to implement our proposed method and to show the quality of the proposed error bound.

The $(2,1)$-total number $\lambda_2^t(G)$ of a graph $G$ is the width of the smallest range of integers that suffices to label the vertices and edges of $G$ such that no two adjacent vertices or two adjacent edges have the same label and the difference between the label of a vertex and its incident edges is at least $2$. In this paper, we characterize completely the $(2,1)$-total number of the join of two paths and the join of two cycles.

In this paper we are interested in the multiplicity of weak solutions to the following Neumann problem involving the $p(x)$-Laplacian operator $$ \left\{ \begin{array}{ll} -\delta_{p(x)}u + \mid u \mid^{p(x)-2}u = \lambda \alpha(x) f(u) + \beta(x) g(u) \ \ \ & in \ \Omega \\ \frac{\partial u}{\partial v} = 0 \ \ \ & on \ \Omega\end{array} \right. $$ We establish the existence of at least three solutions to this problem by using, as main tool, a recent variational principle due to Ricceri.

In this paper, four types of Levitin-Polyak well-posedness of vector quasi-equilibrium problems with a functional constraint, as well as an abstract set constraint are investigated. Some criteria and characterizations for these types of Levitin-Polyak well-posedness with or without gap functions of vector quasi-equilibrium problems are obtained. The results in this paper generalize and extend some known results in the literature.

The discontinuous Sturm-Liouville problem defined on $[0,1]$ with jump conditions at point $d \in (0,1)$ is considered. The interlacing of the spectra between the discontinuous Sturm-Liouville problem and two Sturm-Liouville problems defined on $[0,d]$ and $[d,1]$ is provided. As the application of this interlacing to inverse problems, we prove that the potential is determined uniquely by the three spectra generated by the discontinuous Sturm-Liouville problem and two Sturm-Liouville problems defined on $[0,d]$ and $[d,1]$.

In this paper, we study the strong Levitin-Polyak well-posedness for a class of generalized quasi-variational inclusion problems. We establish some metric characterizations of the strong Levitin-Polyak well-posedness for the generalized quasi-variational inclusion problem. We also prove that under suitable conditions, the strong Levitin-Polyak well-posedness of the generalized quasi-variational inclusion problem is equivalent to the existence and uniqueness of solutions, and that the strong Levitin-Polyak well-posedness of generalized quasi-variational inclusion problem in the generalized sense is equivalent to the existence of solutions. As applications, we obtain some results concerned with Levitin-Polyak well-posedness for several kinds of equilibrium problems.

In this paper, we survey and study definitions and properties of tropical polynomials, tropical rational functions and in general, tropical mero-morphic functions, emphasizing practical techniques that can really carry out computations. For instance, we introduce maximally represented tropical poly-nomials and tropical polynomials in compact forms to quickly find roots of given tropical polynomials. We also prove the existence and uniqueness of tropical theorems for meromorphic functions with prescribed roots and poles. Moreover, we explain the relations between classical and tropical meromor-phic functions. Different definitions and applications of tropical meromorphic functions are discussed. Finally, we point out the properties of tropical mero-morphic functions are very similar to complex ones and prove some tropical analogues of theorems in complex analysis.

In this paper we prove the boundedness of certain convolution operator in a weighted Lebesgue space with kernel satisfying the generalized Hörmander's condition. The sufficient conditions for the pair of weights ensuring the validity of two-weight inequalities of a strong type and of a weak type for singular integral with kernel satisfying the generalized Hörmander's condition are found.

In this paper we introduce a new monotonicity concept for multivalued operators, respectively, a new convexity concept for real valued functions, which generalize several monotonicity, respectively, convexity notions already known in literature. We present some fundamental properties of the operators having this monotonicity property. We show that if such a monotonicity property holds locally then the same property holds globally on the whole domain of the operator. We also show that these two new concepts are closely related. As an immediate application we furnish some surjectivity results in finite dimensional spaces.

B. Y. Chen proved in [4] an optimal inequality for Lagrangian submanifolds in complex space forms in terms of the Ricci curvature and the squared mean curvature, well-known as the Chen-Ricci inequality. Recently, the Chen-Ricci inequality was improved in [7, 11] for Lagrangian submanifolds in complex space forms. In this article we extend the improved Chen-Ricci inequality to Kaehlerian slant submanifolds in complex space forms. We also investigate the equality case of the inequality.

Let $R$ be a prime ring, $I$ a nonzero ideal of $R$ and $n$ a fixed positive integer. If $R$ admits a generalized derivation $F$ associated with a nonzero derivation $d$ such that $(F(x \circ y))^{n} = x \circ y$ for all $x,y \in I$, then $R$ is commutative. We also examine the case where $R$ is a semiprime ring.

In this paper, we establish equilibrium version of Ekeland's variational principle without assuming any kind of semicontinuity of the bifunction involved in the formulation of the principle. By using such principle, we derive some existence results for a solution of equilibrium problems with or without compactness assumption on the underlying set. A coercivity condition is introduced to obtain a solution of an equilibrium problem for noncompact case. Our results extend and improve several known results in the literature.

In the present paper we shall first introduce the notion of the algebra ${\cal F}(S,T)$ of two topological $*$-semigroups $S$ and $T$ in terms of bounded and weakly continuous $*$-representations of $S$ and $T$ on Hilbert spaces. In the case where both $S$ and $T$ are commutative foundation $*$-semigroups with identities it is shown that ${\cal F}(S,T)$ is identical to the algebra of the Fourier transforms of bimeasures in $BM(S^*,T^*)$, where $S^*$ ($T^*$, respectively) denotes the locally compact Hausdorff space of all bounded and continuous $*$-semicharacters on $S$ ($T$, respectively) endowed with the compact open topology. This result has enabled us to make the bimeasure Banach space $BM(S^*,T^*)$ into a Banach algebra. It is also shown that the Banach algebra ${\cal F}(S,T)$ is amenable and $K\big(\sigma (\overline{{\cal F}(S,T)})\big)$ is a compact topological group, where $\sigma (\overline {{\cal F}(S,T)})$ denotes the spectrum of the commutative Banach algebra $\overline{{\cal F}(S,T)}$ as a closed subalgebra of wap $(S \times T)$, the Banach algebra of weakly almost periodic continuous functions on $S \times T$.