Modified Extragradient Methods for a System of Variational Inequalities in Banach Spaces

In this paper, we introduce a new system of general variational inequalities in Banach spaces. We establish the equivalence between this system of variational inequalities and fixed point problems involving the nonexpansive mapping. This alternative equivalent formulation is used to suggest and analyze a modified extragradient method for solving the system of general variational inequalities. Using the demi-closedness principle for nonexpansive mappings, we prove the strong convergence of the proposed iterative method under some suitable conditions.     

Around Jensen’s inequality for strongly convex functions

In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen–Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen’s operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if \(Sp\left( A \right) \subset \left( 1,\infty \right) \), then

$$\begin{aligned} {{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right) ,\quad r\ge 2 \end{aligned}$$

and if \(Sp\left( A \right) \subset \left( 0,1 \right) \), then

$$\begin{aligned} \left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle }^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle }^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right) ,\quad 0<r<1 \end{aligned}$$

for each positive operator A and \(x\in \mathcal {H}\) with \(\left\| x \right\| =1\).

     

Some iterative methods for nonconvex variational inequalities

It is well known that the nonconvex variational inequalities are equivalent to the fixed point problems. We use this equivalent alternative formulation to suggest and analyze a new class of two-step iterative methods for solving the nonconvex variational inequalities. We discuss the convergence of the iterative method under suitable conditions. We also introduce a new class of Wiener – Hopf equations. We establish the equivalence between the nonconvex variational inequalities and the Wiener – Hopf equations. This alternative equivalent formulation is used to suggest some iterative methods. We also consider the convergence analysis of these iterative methods. Our method of proofs is very simple compared to other techniques.     

Two-dimensional fin with convective base condition

When solving a two-dimensional model of an isolated fin, researchers have mainly concentrated on either a constant or a periodic fin base temperature. It is possible to obtain a numerical solution by a convective boundary condition on the fin base. However, in an analytical solution, one cannot calculate an arbitrary constant because of the convective boundary condition of the separation of variables. Therefore a heat balance is applied here to resolve this difficulty. In addition, a modified solution is presented which does not involve any additional mathematics with respect to the classical approach of solving a one-dimensional model. For different values of the Biot number $B_{22}$, a comparison of one- and two-dimensional solutions is given. Relative errors of the heat flow rates predicted by the classical and modified one-dimensional solutions, and the respective exact two-dimensional solution with respect to an, are computed. It is found that, for large values of $B_{22}$ (say 50.0) modified solution, by using a convective condition at the fin base gives significant accuracy improvements in comparison to the classical one-dimensional technique.     

Mechanical Modeling of Particle-Droplet Interaction Motivated by the Study of Self-Cleaning Mechanisms

Motivated by the study of self-cleaning surfaces, the interaction between a spherical solid particle and a water droplet is studied on the microsale level, in terms of the balance forces and the surface properties. To define the forces acting on the solid-liquid interface, the meniscus depression is computed from the Young-Laplace equation, using a non-linear finite element formulation. The equilibrium force is derived in terms of the contact angle, particle size, and the penetration. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

CYCLOTOMIC ELEMENTS IN K_2F,REVISITED

Basing on results of Xu and Qin [10], and Guo [12] on cyclotomic elements in K2F for local fields F, we prove that every element in K2Q is a finite or infinite product of cyclotomic elements. Next, we extend this result to finite extensions of Q satisfying some additional conditions.     

On convexity for energy decay rates of a viscoelastic equation with boundary feedback

In this paper we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary. We establish an explicit and general decay rate result, using some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function.     

ChemInform Abstract: Band Edge Electronic Structure of BiVO4: Elucidating the Role of the Bi s and V d Orbitals.

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