We collect, improve, and generalize very recent results due to Mongkolkeha et al. (2014) in three directions: firstly, we study g-best proximity points; secondly, we employ more general test functions than can be found in that paper, which lets us prove best proximity results using different kinds of control functions; thirdly, we introduce and handle a weak version of the P-property. Our results can also be applied to the study of coincidence points between two mappings as a particular case. As a consequence, the contractive condition we introduce is more general than was used in the mentioned paper.

We study the concept of $\phi$-module amenability of Banach algebras, which are Banach modules over another Banach algebra with compatible actions. Also, we compare the notions of $\phi$-amenability and $\phi$-module amenability of Banach algebras. As a consequence, we show that, if $S$ is an inverse semigroup with finite set $E$ of idempotents and $l^1\left(S\right)$ is a commutative Banach $l^1\left(E\right)$-module, then $l^1{\left(S\right)}^{**}$ is ${\phi }^{**}$-module amenable if and only if $S$ is finite, when $\phi \in {\text{H}\text{o}\text{m}}_{l^1\left(E\right)}\left(l^1\left(S\right)\right)$ is an epimorphism. Indeed, we have generalized a well-known result due to Ghahramani et al. (1996).

We introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such that X becomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spaces $L^{\varphi }(\mu )$ and $E^{\varphi }(\mu )$ over nonatomic measure spaces in terms of the function $\varphi$. We will prove that the Fréchet differentiability of the farthest point map and the conditions $\varphi \in {\mathrm{\Delta }}_2$ and $\varphi >0$ in reflexive Musielak-Orlicz function spaces are equivalent.

In this paper, we consider topology and shape optimization problem related to the nonstationary Navier-Stokes system. The minimization of dissipated energy in the fluid flow domain is discussed. The proposed approach is based on a sensitivity analysis of a design function with respect to the insertion of a small obstacle in the fluid flow domain. Some numerical results show the efficiency and accurate of the proposed approach.

For two kinds of nonlinear constrained optimization problems, we propose two simple penalty functions, respectively, by augmenting the dimension of the primal problem with a variable that controls the weight of the penalty terms. Both of the penalty functions enjoy improved smoothness. Under mild conditions, it can be proved that our penalty functions are both exact in the sense that local minimizers of the associated penalty problem are precisely the local minimizers of the original constrained problem.

The purpose of this paper is to elicit some interesting extensions of generalized almost contraction mappings to the case of non-self-mappings with $\alpha$-proximal admissible and prove best proximity point theorems for this classes. Moreover, we also give some examples and applications to support our main results.

In the very recent paper of Akbar and Gabeleh (2013), by using the notion of $P$-property, it was proved that some late results about the existence and uniqueness of best proximity points can be obtained from the versions of associated existing results in the fixed point theory. Along the same line, in this paper, we prove that these results can be obtained under a weaker condition, namely, weak $P$-property.

In Internet traffic modeling, many authors presented models based on particular fractal shot noise representations. The inconvenience of these approaches is the multitude of assumptions and the lack of tools to check them. In this paper we propose a unified model based on a general Poisson shot noise representation for the cumulative input process (CIP). We present a procedure of approximation of this process; then we give a procedure for controlling the bandwidth of Internet providers. The approximation and control go via limit theorems for functionals of the CIP, namely, the supremum process, the right inverse, and the storage mapping.

Recently, sufficient descent property plays an important role in the global convergence analysis of some iterative methods. In this paper, we propose a new iterative method for solving unconstrained optimization problems. This method provides a sufficient descent direction for objective function. Moreover, the global convergence of the proposed method is established under some appropriate conditions. We also report some numerical results and compare the performance of the proposed method with some existing methods. Numerical results indicate that the presented method is efficient.

Recently, Basha (2013) addressed a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. He assumed that if $A$ and $B$ are nonvoid subsets of a partially ordered set that is equipped with a metric and $S$ is a non-self-mapping from $A$ to $B$, then the mapping $S$ has an optimal approximate solution, called a best proximity point of the mapping $S$, to the operator equation $Sx=x$, when $S$ is a continuous, proximally monotone, ordered proximal contraction. In this note, we are going to obtain his results by omitting ordering, proximal monotonicity, and ordered proximal contraction on $S$.

Let A and B be two nonempty subsets of a Banach space X. A mapping T : $A\cup B\to A\cup B$ is said to be cyclic relatively nonexpansive if T(A) $\subseteq B$ and T(B) $\subseteq A$ and $∥Tx-Ty∥\le ∥x-y∥$ for all ($x\text{,}y$) $\in A\times B$. In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach space X. It is shown that if (A, B) is a nonempty, weakly compact, and convex pair and (A, B) has seminormal structure, then a cyclic relatively nonexpansive mapping T : $A\cup B\to A\cup B$ has a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.

When the domain is a polygon of ${ℝ}^{\mathrm{2}}$, the solution of a partial differential equation is written as a sum of a regular part and a linear combination of singular functions. The purpose of this paper is to present explicitly the singular functions of Stokes problem. We prove the Kondratiev method in the case of the crack. We finish by giving some regularity results.