This paper constitutes a short survey of existence methods for variational inequalities of parabolic type. After discussing several illustrative examples in detail, we discuss some of the common methods for proving existence of solutions to such problems: the translation method, Rothe’s method, and the penalty method. As these methods rely on existence results for elliptic variational inequalities, we also provide a summary of basic results and techniques for static problems.

We study the iterative subgradient methods for nonsmooth convex constrained optimization problems in a uniformly convex and uniformly smooth Banach space, followed by metric and generalized projections onto the feasible sets. The normalized stepsizes $\alpha_n$ are chosen {\em apriori}, satisfying the conditions $\sum_{n=0}^\infty\alpha_n=\infty$, $\alpha_n \to 0.$ We prove that the every sequence generated in this way is weakly convergent to a minimizer in the average if the problem has solutions. In addition, we show that the perturbed $\epsilon_n$-subgradient method is stable when $\epsilon_n \to 0.$ More general case of variational inequalities with monotone (possibly) nonpotential operators is also considered.

We study the following semilinear elliptic problem: $$ \left\{\begin{array}{l} \displaystyle\sum_{i,j=1}^N a_{ij}(x, u)\displaystyle\frac{\partial^2 u}{\partial x_i\partial x_j} +\displaystyle\sum_{i=1}^N b_i(x, u)\displaystyle\frac{\partial u}{\partial x_i}+c(x, u)u=f(x)\quad\mbox{ in }B, \\ u=0 \qquad\mbox{ on }{\partial B},\end{array} \right. $$ where $B$ is a ball in ${\Bbb R}^N$, $N\geq 3$, $a_{ij}=a_{ij}(x,r)\in C^{0,1} (\bar{B}\times{\Bbb R})$, $a_{ij}$, $\partial a_{ij}/\partial x_i$, $\partial a_{ij}/\partial r$, $b_i$, $c\in L^\infty(B\times{\Bbb R})$, with $i, j=1, 2, \cdots, N$ and $c\leq 0$, and $f\in L^p(B)$. For each $p$, $p\geq N$, there exists a strong solution $u\in W^{2,p}(B)\cap W_0^{1,p}(B)$ provided the oscillations of $a_{ij}$ with respect to $r$ are sufficiently small. Moreover, for $N/2

In Chen and Chiang [2] and Chen, Thompson and Hung [3], the symmetric trimmed mean has been shown, for various linear models, to have the efficiency of having asymptotic covariance matrices close to the Cr\'amer-Rao lower bounds for some heavy tail error distributions. In this paper, we investigate some further theoretical results for this symmetric trimmed mean for the linear regression model. From the nonparametric point of view, we develop a robust version of the Gauss-Markov theorem for the problem of estimating regression parameter vector $\beta$ and parametric vector function $C\beta$ where the best estimators are this trimmed mean and $C$ multiplied by it, respectively. In addition, we show that these best estimators are the best Mallows-type bounded influence linear symmetric trimmed means. Finally, from the parametric aspect, we show that the symmetric trimmed mean is Rao$'$s first order efficient for a heavy tail error distribution.

In this paper, as a continuation of the authors’work, the existence of solutions of IVVI for noncompact valued multifunctions is provided under generalized pseudomonotonicity assumption. This generalizes Konnov and Yao’s results [6, Theorems 3.1 and 4.1]. Also another IVVI without the generalized pseudomonotonicity assumption is proposed and the existence of solutions of the IVVI is proved by using a selection theorem. This IVVI can be regarded as an extension of the VVI studied in Lee et al. [10].

It is proved that six-dimensional $G1$-submanifolds of the octave algebra are $W_1\oplus W_3$ - manifolds.

In this article, we study translation surfaces in the 3-dimensional Minkowski space whose Gauss map $G$ satisfies the condition $\Delta G=AG,A\in {\rm Mat}(3,\Bbb R)$, where $\Delta$ denotes the Laplacian of the surface with respect to the induced metric and Mat(3, ${\Bbb R}$) the set of $3\times 3$ real matrices, and also obtain the complete classification theorem for those.

Let $X$ be a compact Hausdorff space and $C(X)$ the space of continuous functions defined on $X$. There are three versions of the Banach-Stone theorem. They assert that the Banach space geometry, the ring structure, and the lattice structure of $C(X)$ determine the topological structure of $X$, respectively. In particular, the lattice version states that every disjointness preserving linear bijection $T$ from $C(X)$ onto $C(Y)$ is a weighted composition operator $Tf=h\cdot f\circ\varphi$ which provides a homeomorphism $\varphi$ from $Y$ onto $X$. In this note, we manage to use basically algebraic arguments to give this lattice version a short new proof. In this way, all three versions of the Banach-Stone theorem are unified in an algebraic framework such that different isomorphisms preserve different ideal structures of $C(X)$.

In [5] the following problem was posed. Is there a solution of the following difference equation $$ x_{n+1}=\displaystyle\frac{\beta x_{n-1}}{\beta+x_n},\quad x_{-1},x_0\gt 0,\ \beta \gt 0, \quad n=0,1,2,... $$ such that $x_n\to 0$ as $n\to\infty.$ We prove a result which, as a special case, solves the above problem.

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset such that every vertex in the graph is dominated by at least k vertices in this set. The present paper studies the k-tuple domination problem in graphs from an algorithmic point of view. In particular, we give a linear-time algorithm for the 2-tuple domination problem in trees by employing a labeling method.

We study the null boundary controllability for a one-dimensional fourth order parabolic equation. We show that if the initial data is continuous then the fourth order parabolic equation is controllable.

In this paper, we show that the ¸ - multiplier convergent of series depends completely upon the AK - property of sequence space ¸. From this conclusion we obtain a lot of new important theorems.