We investigate selection principles which are motivated by Arhangel'ski${\rm\check i}$'s $\alpha_i$-properties, $i=1,2,3,4$, and their relations with classical selection principles. It will be shown that they are closely related to the selection principle ${\sf S}_1$ and often are equivalent to it.

n this paper, we introduce the concept of a ($\alpha$-) quasi-Armendariz module, principally quasi-Baer module and syudy its some properties. In particular, we show: (1) For an $\alpha$-quasi-Armendariz module $M_R$, $M_R$ is a principally quasi-Baer module if and only if $M[x;\alpha]_{R[x;\alpha]}$ is a principally quasi-Baer module. (2) A necessary and sufficient condition for a trivial extensions to be quasi-Armendariz is obtained. Consequently, new families of quasi-Armendariz rings are presented.

In this paper, we extend Moudafi’s result in a Hilbert space to that in a Banach spaces. Then, we introduce implicit and explicit sequences for an infinite family of nonexpansive mappings in Banach spaces and prove strong convergence theorems for finding a common fixed point of the family of mappings.

In this paper we study the following initial value problem for the nonlinear equation, \[ \left\{ \begin{array}[c]{l} u^{\prime\prime}(t)={u(t)}^{p}(c_{1}+c_{2}u^{\prime}(t)^{q}),~~~p,~q\geq 1,\ c_{1}\geq0,c_{2}\geq0,\\ u(0)=u_{0},~u^{\prime}(0)=u_{1}. \end{array} \right. \] We are interested in the properties of solutions of the above problem. We have found blow-up phenomena and obtained some results on blow-up rates, blow-up constants and life-spans.

Sufficient and necessary conditions are established for the second-order difference equations \[ \Delta (r_{n - 1} \Delta x_{n - 1} ) + p_n x^\gamma _n = 0,n = 1,2, \ldots \] where $\gamma $ is the quotient of odd positive integers. Our results extend the well known oscillation theorem which was proved in [1,JMAA,91:9-29,1983], and answer an open problem in [2] when $r_n=1,\gamma=1$.

In this paper, we first discuss the exact controllability of a linear parabolic system governed by bilinear control as a coefficient like $uy$ and then give a result of the locally exact null-controllability of a semilinear parabolic system. Our main method is based on the Implicit Function Theorem.

In this paper, we introduce a type of parametric vector equilibrium problem on topological vector spaces. This problem includes parametric vector variational inequality problem and ε-vector equilibrium problem and so on, as special cases. Existence and some continuity properties for the of solution mapping of the parametric vector equilibrium problem are established.

This paper deals with relationship between singular measure and corresponding $p(x)$-capacity. It we have proved that the value of capacity of an arbitrary capacitor is completely determined by an absolutely continuous component of the measure, and the contribution of a singular part of the measure appears to be zero.

Existence, uniqueness and regularity of weak solutions of semilinear second order Volterra integro-differential equations in Hilbert space are established by the variational method. As an application we give a well-posedness result for the semilinear viscoelastic equations with long memory.

In this paper, we establish some inequalities of Hermite-Hadamard-Fejér type for convex functions and convex functions on the co-ordinates defined in a rectangle from the plane.

We establish necessary and sufficient conditions for efficiency of multiobjective fractional programming problems involving r-invex functions. Using the optimality conditions, we investigate the parametric type dual, Wolfe type dual and Mond-Weir type dual for multiobjective fractional programming problems concerning r-invexity. Some duality theorems are also proved for such problem in the framework of r-invexity.

The space $BMOA$ can be characterized by the boundedness of Garsia norm. In this paper like as the $BMOA$ function we characterize the holomorphic weighted Lipschitz function by the boundedness of the Grasia-type norm on the unit ball in $\mathbb C^n$.

In this paper, we introduce a generalized implicit hybrid projectionproximal point algorithm for finding zeros of a maximal monotone operator in a Hilbert space setting. The global convergence of the method for the weak topology under appropriate assumptions on the algorithm parameters is established.

An approximation theorem is given for abstract quasilinear evolution equations in the sense of Hadamard. A stability condition is proposed under which a sequence of approximate solutions converges to the solution. The result obtained in this paper is a generalization of an approximation theorem of regularized semigroups and is applied to an approximation problem for a degenerate Kirchhoff equation.

Banach spaces. Let $\phi$ be a bijection from $B(X)$ onto $B(Y)$ satisfying ${\phi}([A,B])=[\phi(A),\phi(B)]$ for all $A, B\in B(X)$. Then $\phi=\psi+\tau$, where $\psi$ is a ring isomorphism or a negative of a ring anti-isomorphism from $B(X)$ onto $B(Y)$, and $\tau$ is a map from $B(X)$ into $\Bbb CI$ satisfying $\tau([A,B])=0$ for all $A, B\in B(X)$.

In this paper we prove new fixed point theorems for multi valued mappings with an implicit relations on complete uniform space.

Jank and Terglane gave a unicity condition of three meromorphic functions of class $\mathcal{A}$. We generalize this unicity condition for arbitrary $q$ meromorphic functions, and prove that the condition is sharp in the cases $q=3$ and $4$. Moreover, we provide a conjecture concerning this aspect.

In the present paper the direct theorem on approximation of classes of harmonic functions with singularities on closed quasiconformal curves by the harmonic rational function is obtained.

The $q$-Konhauser polynomials, which were introduced and investigated in several recent works, are $q$-biorthogonal with respect to the weight function $x^{\alpha }\;e_{q}(-x)$ over the semi-infinite interval $(0,\infty )$. In the present paper, we derive various families of multilinear and multilateral generating functions for these $q$-Konhauser polynomials. We also briefly consider several special cases and consequences of the results presented in this paper.