In this paper we use the penalty approach in order to study inequality-constrained minimization problems with locally Lipschitz objective and constraint functions in Banach spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. In this paper we show that the generalized exact penalty property is stable under perturbations of objective functions, constraint functions and the right-hand side of constraints.

In this paper we define and characterize the class $(V_{2}^{\sigma},V_{2}^{\sigma})$ and establish a core theorem, where $V_{2}^{\sigma}$ is the space of $\sigma$-convergent double sequences $x = (x_{jk})$. We further determine a Tauberian condition for core inclusion and core equivalence.

Using a fundamental identity concerning fans called ``Fan Theorem'' we give new proofs of classical edge colouring theorems. We also derive from the same identity a formula for the chromatic index of Class 2 multigraphs (i.e. multigraphs $G$ such that $\chi'(G) \gt \Delta(G)$) and a new generalization of Vizing's Adjacency Lemma to multigraphs, which is more general than the one given by the author in [D. Cariolaro, On fans in multigraphs, J. Graph Theory, 51 (4), 2006, 301--318].

In this paper, some new existence theorems of equilibrium and maximal element for generalized abstract fuzzy economies with uncountable number of agents and qualitative fuzzy games are proved in $G$-convex spaces, respectively. As applications, some existence theorems of equilibria for abstract economies are given in $G$-convex spaces. The results presented in this paper generalize some known results in the literature.

Let $C$ be a nonempty closed convex proper subset of a real uniformly convex and uniformly smooth Banach space $E$, let $S: C \to C$ be a relatively nonexpansive mapping, and let $T: C \to E$ be an asymptotically weakly suppressive operator. Using the notion of generalized projection, iterative methods for approximating common fixed points of the mappings $S$ and $T$ are studied. In terms of the modified Ishikawa iteration and modified Halpern one for relatively nonexpansive mappings, we propose two modified versions of Chidume, Khumalo and Zegeye's iterative algorithms [C.E. Chidume, M. Khumalo and H. Zegeye, Generalized projection and approximation of fixed points of nonself maps, J. Appro. Theory, 120 (2003) 242-252] for finding approximate common fixed points of the mappings $S$ and $T$. Moreover, it is proved that these two iterative algorithms converge strongly to the same common fixed point of the mappings $S$ and $T$.

In this paper, we introduce the BMO space via heat kernels on $\widetilde{M}$, where $\widetilde{M} = M_1 \times \cdots \times M_n$ is the Shilov boundary of the product domain in $\mathbb{C}^{2n}$ defined by Nagel and Stein ([16], see also [17]), each $M_i$ is the boundary of a weakly pseudoconvex domain of finite type in $\mathbb{C}^2$ and the vector fields of $M_i$ are uniformly of finite type ([14]). And we prove that it is the dual space of product Hardy space $H^1(\widetilde{M})$ introduced in [11].

Motivated essentially by several recent investigations which claimed to have generalized, improved and extended such classical inequalities as the well-known Opial's inequality, here we establish some general weighted Opial-type inequalities on time scales. We also provide counterexamples, corrections and modifications of the aforementioned recent claims by Wong et al. [Taiwanese J. Math., 12 (2008), 463-471].

Let $A$ be a sequence of positive integers. An element $\alpha$ of $A$ is called an $s$-near relative prime number ($s$-near relprime in short) if $\alpha$ is coprime to any distinct element in $A$ except exactly $s$ elements of $A$. In this paper, we study the existence of an arithmetic sequence with no 1-near relprimes.

The paper will investigate the local existence, uniqueness of solution to integrodifferential equations with infinite delay. We assume that the linear part is not necessarily densely defined and satisfies a Hille-Yosida condition. Moreover, the continuity of solutions with respect to initial conditions is also studied.

A refinement of the discrete Jensen's inequality for convex functions defined on a convex subset in linear spaces is given. Application for $f$-divergence measures including the Kullback-Leibler and Jeffreys divergences are provided as well.

The nonlocal boundary value problem for differential equation \[\left\{ \begin{array}{l} \frac{d^{2}u(t)}{dt^{2}} + Au(t) = f(t) \quad \ (0 \leq t \leq 1), \\ u(0) = \sum\limits_{r=1}^{n} \alpha _{r} u(\lambda_{r}) + \varphi, u_{t}(0) = \sum\limits_{r=1}^{n} \beta _{r} u_{t}(\lambda_{r}) + \psi, \\ 0 \lt \lambda_{1} \leq \lambda_{2} \leq ... \leq \lambda_{n} \leq 1 \end{array} \right. \] in a Hilbert space $H$ with the self-adjoint positive definite operator $A$ is considered. The stability estimates for the solution of the problem under the assumption \[ \sum_{k=1}^{n} \left\vert \alpha_{k} + \beta_{k} \right\vert + \sum_{k=1}^{n} \left\vert \alpha_{k} \right\vert \sum_{\substack{m=1 \\ m \neq k}}^{n} \left\vert \beta_{m} \right\vert \lt |1 + \sum_{k=1}^{n} \alpha_{k} \beta_{k}| \] are established. The first order of accuracy difference schemes for the approximate solutions of the problem are presented. The stability estimates for the solution of these difference schemes under the assumption \[ \sum_{k=1}^{n} \left\vert \alpha_{k} \right\vert + \sum_{k=1}^{n} \left\vert \beta_{k} \right\vert + \sum_{k=1}^{n} \left\vert \alpha_{k} \right\vert \sum_{k=1}^{n} \left\vert \beta_{k} \right\vert \lt 1 \] are established. In practice, the nonlocal boundary value problems for one dimensional hyperbolic equation with nonlocal boundary conditions in space variable and multidimensional hyperbolic equation with Dirichlet condition in space variables are considered. The stability estimates for the solutions of difference schemes for the nonlocal boundary value hyperbolic problems are obtained.

The boundness of Volterra type operators on $Q_K$ space is investigated in this paper. A new generalized Carleson measure and Logarithmic $Q_K$ spaces has been introduced and studied. In addition, we give a new characterization of $Q_K$ space and $Q_{K,0}$ space.

In this paper we discuss the relationship between the Kurzweil-Henstock-Pettis and Kurzweil-Henstock integrals in Banach spaces. We prove that in Schur spaces the Kurzweil-Henstock-Pettis and Kurzweil-Henstock integrability for measurable functions satisfying the condition $(C)$ are equivalent. In particular, in Schur spaces the Kurzweil-Henstock-Dunford, Kurzweil-Henstock-Pettis and Kurzweil-Henstock integrability for measurable functions satisfying the condition $(C)$ are equivalent.

A three-dimensional homogeneous Lorentzian manifold is either symmetric or locally isometric to a Lie group equipped with a left-invariant Lorentzian metric [4]. We completely classify surfaces with parallel second fundamental form in all non-symmetric homogeneous Lorentzian three-manifolds. Interesting differences arise with respect to the Riemannian case studied in [11, 12].

Suppose that $M$ is a closed subspace of a Banach space $X$ such that $M^{\bot}$ is complemented in the dual space $X^{*}$, where $M^{\perp} = \{x^{*} \in X^{*}: x^{*}(m) = 0$ for all $m \in M\}$. Godefroy and Saphar [4] study the three-space problem for the approximation properties on $(X,M)$. In this paper, we extend some of their results and solve the three-space problem for the weak bounded approximation property on $(X,M)$, which was introduced in Lima and Oja [10].

In this short note we formulate and prove a Banach space version of the Banach-Stone theorem, obtained recently by Ercan and Onal ([4]) for the case of lattice-valued continuous functions.

Existence theorem for a variational inequality problem with local intersection property has been obtained in topological space by relaxing the property of open inverse values from the result of Vetrivel and Nanda [7].

A linear $k$-forest is a graph whose components are paths with lengths at most $k$. The minimum number of linear $k$-forests needed to decompose a graph $G$ is the linear $k$-arboricity of $G$ and denoted by $la_{k}(G)$. In this paper, we settle the cases left in determining the linear $2$-arboricity of the complete graph $K_n$. Mainly, we prove that $la_{2} (K_{12t+10}) = la_{2}(K_{12t+11}) = 9t+8$ for any $t \geq 0$.

Reducing both the number of players and the number of choices, we define a new reduced game for a multi-choice cooperative game with respect to a solution of the game and an action vector. Then, we characterize the multi-choice Shapley value by applying a partially consistent property of the multi-choice Shapley value.