The decomposition set-valued mapping in a Banach space $E$ with cones $K_i$, $i = 1, \ldots, n$ describes all decompositions of a given element on addends, such that addend $i$ belongs to the $i$-th cone. We examine the decomposition mapping and its dual.

We study conditions that provide the additivity of the decomposition mapping. For this purpose we introduce and study the Riesz interpolation property and lattice properties of spaces with respect to several preorders. The notion of 2-vector lattice is introduced and studied. Theorems that establish the relationship between the Riesz interpolation property and lattice properties of the dual spaces are given.

This paper deals with the classical distance function to closed sets and its extension to the case of set-valued mappings. It has been well recognized that the distance functions play a crucial role in many aspects of variational analysis, optimization, and their applications. One of the most remarkable properties of even the classical distance function is its intrinsic nonsmoothness, which requires the usage of generalized differential constructions for its study and applications. In this paper we present new results in theser directions using mostly the generalized differential constructions introduced earlier by the first author, as well as their recent modifications. We pay the main attention to studying subgradients of the distance functions in out-of-set points, which is essentially more involved in comparison with the in-set case. Most of the results obtained are new in both finite-dimensional and infinite-dimensional settings; some of them of provide essential improvements of known results even for convex sets.

In this note we show how the Hamiltonian Cycle problem can be reduced to solving a system of polynomial equations related to the adjacency matrix of a graph. This system of equations can be solved using the method of Gröbner bases, but we also show how a symbolic determinant related to the adjacency matrix can be used to directly decide whether a graph has a Hamiltonian cycle.

Let $C$ be a nonempty closed convex subset of a real Hilbert space and let $\{T_n\}$ be a family of mappings of $C$ into itself such that the set of all common fixed points of $\{T_n\}$ is nonempty. We consider a sequence $\{x_n\}$ generated by the hybrid method in mathematical programming and give the conditions of $\{T_n\}$ under which $\{x_n\}$ converges strongly to a common fixed point of $\{T_n\}$.

The weighted quasi-variational inequalities over product of sets (for short, WQVIP) and system of weighted quasi-variational inequalities (for short, SWQVI) are introduced. It is shown that these two problems are equivalent. A relationship between SWQVI and system of vector quasi-variational inequalities is given. The concept of normalized solutions of WQVIP and SWQVI is introduced. A relationship between solution (respectively, normalized solution) of SWQVI and solution of weighted constrained Nash equilibrium problem (respectively, normalized weight Nash equilibrium) is also given. The scalar quasi-equilibrium problem (for short, QEP), which includes WQVIP as a particular case, is also considered. By introducing the concept of densely pseudomonotonicity of bifunctions, the existence of a solution of QEP is established. As a consequence, existence results for solutions of WQVIP and constrained Nash equilibrium problems for vector valued functions are derived.

In this paper, we prove the following. Let $E$ be a strictly convex Banach space. Let $\{ T(t) : t \geq 0 \}$ be a one-parameter strongly continuous semigroup of nonexpansive mappings on a subset $C$ of $E$. Then \[ \bigcap_{t \geq 0} F(T(t)) = F\left( \frac{1}{2} T(1) + \frac{1}{2} T(\sqrt{2}) \right) \] holds, where $F(T(t))$ is the set of fixed points of $T(t)$ for each $t \geq 0$.

In this paper, a vector equilibrium problem (VEP) with multifunctions is considered. Using the asymptotic cone of the solution set of (VEP), we give conditions under which the solution set is nonempty and compact, and then extend it to a random vector equilibrium problem with multifunctions.

In this note we find some relations between $k−$free numbers. We obtain some asymptotes of the error term in the summation of $k−$free integers $R_k(x)$. Further, we determine some constants which happen in the summation of $k−$free integers.

Let $T$ be a linear power bounded operator on a Banach space $X$ and let $S_{\lambda} = (1−\lambda)I + \lambda T$ be the averaged map of $T$, where $\lambda \in (0,1)$. It is shown that $S_{\lambda}$ is asymptotically regular on $X$; that is, $\lim_{n \to \infty} \| S_{\lambda}^{n}x - S_{\lambda}^{n+1}x \| = 0$ for every $x \in X$. Hence the sequence $\{S_{\lambda}^{n}x\}$ converges strongly provided it has a weak cluster point.

$\varepsilon$-Entropy theoretic solutions to the homeomorphism problem of the entire funcition spaces equipped with the compact open topology and the homeomorphism problem of the analytic function spaces equipped with the norm topology are given.

We give quantitative estimates of the rate of convergence of equiuniform approximation processes of integral operators in Banach spaces in terms of the modulus of continuity of functions to be approximated and higher order absolute moments of approximate kernels. Furthermore, applications are presented for various equi-uniform summation processes, interpolation type operators, convolution type operators, and several concrete examples of approximating operators are also provided.

In this paper, we consider nonsmooth multiobjective fractional programming problems involving locally Lipschitz functions. We introduce the property of generalized invexity for fractional function. We present necessary optimality conditions, sufficient optimality conditions and duality relations for nonsmooth multiobjective fractional programming problems, which is for a weakly efficient solution under suitable generalized invexity assumptions.

We obtain general coincidence theorems and related results for multimaps in very large classes defined on $\omega$-connected spaces. Our typical consequence is as follows: Let $X$ be a compact $\omega$-connected topological space, and $F : X \multimap X$ a multimap with nonempty values and open fibers such that, for each open subset $O \subset X$, $\bigcap_{x \in O} Fx$ is empty or $\omega$-connected. Then $F$ has a fixed point.

The purpose of this paper is to investigate the convergence of a proximal method for solving pseudomonotone type variational-like inequalities. The main result is given for the finite-dimensional case. However convergence can still be obtained in an infinite-dimensional Hilbert space under a strong pseudomonotone type assumption or a pseudo-Dunn type one on the operator involved.

This paper deals with the convergence of the algorithm built on the auxiliary problem principle for solving a class of pseudomonotone type mixed variational-like inequalities. The convergence criteria of this method are presented under a joint pseudo-Dunn property or a joint strong pseudomonotonicity assumption on the operators involved in the mixed variational-like inequality problem. Moreover a posteriori error estimation for approximate solutions is also given.

This paper is primarily concerned with approximation at 0 of an $(a,k)$-regularized resolvent family $R(\cdot)$ for Volterra integral equation. We shall consider convergence rates of some kind of local means $Q_m(t)$, $t \geq 0$, $m \geq 0$, of $R(t)/k(t)$. Some approximation theorems and local ergodic theorems with rates will be deduced from general approximation theorems for regularized approximation processes.

Let $q \gt 1$ and $E$ be a real $q$-uniformly smooth Banach space, $K$ be a nonempty closed convex subset of $E$ and $T : K \to K$ be a Lipschitz continuous mapping. Let $\{u_n\}$ and $\{v_n\}$ be bounded sequences in $K$ and $\{\alpha_n\}$ and $\{\beta_n\}$ be real sequences in $[0,1]$ satisfying some restrictions. Let $\{x_n\}$ be the sequence generated from an arbitrary $x_1 \in K$ by the Ishikawa iteration process with errors: $y_n = (1 − \beta n) x_n + \beta_n Tx_n + v_n$, $x_{n+1} = (1 − \alpha_n) x_n + \alpha_n Ty_n + u_n$, $n \geq 1$. Sufficient and necessary conditions for the strong convergence $\{x_n\}$ to a fixed point of $T$ is established.

In this paper, we study the convergence theorems of the Ishikawa iterative sequences with mixed errors for the accretive Lipschitzian mappings in Banach spaces.

This paper is concerned with a nonlinear scalarization for vectorvalued functions. We consider applying the scalarization to existence of cone saddle-points. Some properties of the scalarization about cone-continuity and cone-convexity are described and then as an application, an existence theorem for vector saddle-points is treated.