The paper concerns calculus of certain tangent sets in the multifunction setting. To this end the multifunction linear openness is defined in linear topological spaces. The relationship with some metric notions is investigated and some metric openness results are derived.

In this paper, we study the parametric generalized quasi vector equilibrium problem (PGQVEP). We investigate existence of solution for PGQVEP and continuities of the solution mappings of PGQVEP. In particular, resulta concerning the lower semicontinuity of the solution mapping of PGQVEP are presented.

The Kubo–Yokoi and Donsker delta functions are well known generalized functions in infinite dimensional distribution theory. In this paper we develop the delta function for an affine subspace and show that it is a generalization of the Kubo–Yokoi and Donsker delta functions. The Wiener– Itˆo expansion of the delta function for an affine subspace is also given.

In this paper we study the famous Hayman's conjecture for transcendental meromorphic functions in a $p$-adic field by using methods of $p$-adic analysis and particularly the $p$-adic Nevanlinna theory. In $\mathbb{C}$, W. K. Hayman's stated that if $f$ is a transcendental meromorphic function, then $f' + af^m$ has infinitely many zeros that are not zeros of $f$ for each integer $m \geq 3$ and $a \in \mathbb{C} \setminus \{0\}$, which was proved in [2], [6], [8] and [11]. Here we examine the problem in an algebraically closed complete ultrametric field $\mathbb{K}$ of characteristic zero. Considering the function $f' + Tf^m$ with $T \in \mathbb{K}(x)$, we show that Hayman's statement holds for each $m \geq 5$ and $m = 1$. Further, if the residue characteristic of $\mathbb{K}$ is zero, then the statement holds for each positive integer $m$ different from $2$. We also examine the problem inside an ``open'' disc.

We prove an overshooting property of a multistep Newton method for polynomials with all real zeros, a special case of which is a classical result for the double-step Newton method. This result states, in essence, that a double Newton step from a point to the left of the smallest zero of a polynomial with all real zeros never overshoots the first critical point of the polynomial. Our result here states, in essence, that a Newton $(k+1)$-step from a point to the left of the smallest zero never overshoots the $k$th critical point of the polynomial, thereby generalizing the double-step result. Analogous results hold when starting from a point to the right of the largest zero. We also derive a version of the aforementioned classical result that, unlike that result, takes into account the multiplicities of the first or last two zeros.

In this paper we consider the following boundary value problems for $p$-Laplacian functional dynamic equations on time scales $$\begin{array}{rl} & \left[ \Phi _p(u^{\bigtriangleup }(t))\right]^{\bigtriangledown }+a(t)f(u(t),u(\mu (t)))=0,t\in \left( 0,T\right) _{\mathbf{T}},\\ u_0(t)& =\varphi (t), \, \, t\in \left[ -r,0\right] _{\mathbf{T}},\, \, u(0)-B_0(u^{\bigtriangleup }(\eta ))=0,\, \, u^{\bigtriangleup }(T)=0, \mbox{ or}\\ u_0(t)& =\varphi (t),\, \, t\in \left[ -r,0\right] _{\mathbf{T}},\, \, u^{\bigtriangleup }(0)=0,u(T)+B_1(u^{\bigtriangleup }(\eta ))=0. \end{array} $$ Some existence criteria of at least three positive solutions are established by using the well-known Leggett-Williams fixed-point theorem. An example is also given to illustrate the main results.

To solve the boundary integral equations (BIE) of mixed boundary conditions, we propose the mechanical quadrature methods (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote $h_{m}$ as the mesh width of a curved edge $\Gamma_{m}$ ($m=1,...,d)$ of polygons. Then the multivariate asymptotic expansions of solution errors are found to be $O(h^{3}),$ where $h=\max_{1\leq m\leq d}h_{m}.$ Hence, by using the splitting extrapolation methods (SEM), the high convergence rates as $O(h^{5})$ can be achieved. Moreover, numerical examples are provided to support our theoretical analysis.

In this work, we establish a generalized KKM theorem in a hyperconvex metric space, and then we use this theorem to get a fixed point theorem, the matching theorem, the coincidence theorem, minimax inequality theorems and the variational inequality theorems.

This paper is concerned with the problem of nonlinear best simultaneous approximations in conditional complete lattice Banach spaces with a strong unit. Characterization results of the best simultaneous approximation from simultaneous suns and suns are established. A counterexample, to which the characterization theorem for convex sets due to Mohebi ( Numer. Funct. Anal. Optim., 25 (2004), 685-705) fails, is provided and a corrected version of the theorem is presented.

Let $(\Omega, \Sigma, \mu)$ be a $\sigma-$finite measure space, $1\le p \lt \infty$, $X$ be a Banach space $X$ and ${\cal B} :X\times Y \to Z$ be a bounded bilinear map. We say that an $X$-valued function $f$ is $p-$integrable with respect to ${\cal B}$ whenever $\sup\{\int_\Omega\|{\cal B}(f(w),y)\|^pd\mu: \|y\|=1\}$ is finite. We identify the spaces of functions integrable with respect to the bilinear maps arising from H\"older's and Young's inequalities. We apply the theory to give conditions on $X$-valued kernels for the boundedness of integral operators $T_{{\cal B}}(f) (w)=\int_{\Omega'}{{\cal B}}(k(w,w'),$ $f(w'))d\mu'(w')$ from ${\mathrm L}^p(Y)$ into ${\mathrm L}^p(Z)$, extending the results known in the operator-valued case, corresponding to ${\cal B}:{\mathrm L}(X,Y)\times X\to Y$ given by ${\cal B}(T,x)=Tx$.

In this paper, we study the growth of all solutions of a linear differential equation. From this we obtain some uniqueness theorems of a nonconstant entire function and its linear differential polynomials having the same fixed points. The results in this paper also improve some known results. Two example are provided to show that the results in this paper are best possible.

In this paper, we establish some inequalities which improve some Ostrowski type inequalities. Applications for Euler’s Beta mapping and special means are also given.

We shall link the well-established concept of joint spectrum with the interesting and practical periodic eigenvalue problems (PEVPs). A Bauer-Fike perturbation theory, incorporating a Clifford algebra technique, for joint spectrum is applied to PEVPs, producing new perturbation results.

By using the lattice VOA $V_{\sqrt{2}A_n}$, we construct a class of vertex operator algebras $\{M^{(n)}|\, n=2,3,4, \dots\}$ as coset subalgebras. We show that the VOA $M=M^{(n)}$ is generated by its weight $2$ subspace and the symmetric group $S_{n+1}$, which is isomorphic to the Weyl group $W(A_n)$ of the root system of type $A_n$, acts faithfully on $M$. Moreover, some irreducible modules of $M$ are constructed using the coset construction.

IIn this paper, we deal with a class of nonexpansive mappings with the property $D(\overline{co} F_{\frac 1n} (T),F(T))\to 0$, as $n\to \infty$, where $D$ is the Hausdorff metric. We show that nonexpansive mappings with compact domains enjoy this property and give some examples of this kind of mappings with noncompact domains in $l^\infty$. Then we prove a nonlinear ergodic theorem, and a convergence theorem of mann's type for this kind of mappings.

Introducing a higher order modulus of smoothness based on $q$-integers, in this paper first we obtain Jackson-type estimates in approximation by Jackson-type generalizations of the $q$-Picard and $q$-Gauss-Weierstrass singular integrals and give their global smoothness preservation property with respect to the uniform norm. Then, we study approximation and geometric properties of the complex variants for these $q$ -singular integrals attached to analytic functions in compact disks. Finally, we prove approximation properties of these $q$-singular integrals attached to vector-valued functions.

In this paper, we introduce a class $F_{\lambda ,p}^{n}(\alpha ,\beta ,\gamma )$ of meromorphic multivalent functions in $U^{*}=\{z:0\lt |z|\lt 1\}$ by using a differential operator $D_{\lambda ,p}^{n}f(z)$. We obtain coefficient estimates, distortion theorem, radius of convexity, closure theorems and integral transforms for the class $F_{\lambda ,p}^{n}(\alpha ,\beta ,\gamma )$. Several results involving the Hadamard products of functions belonging to the class $F_{\lambda ,p}^{n}(\alpha ,\beta ,\gamma )$ are also derived.

In this study we show that $I_{\Omega ,\alpha ,v}$ and $M_{\Omega ,\alpha ,v},$ the fractional integral and maximal operators the generalized shift operator generated by Bessel differential operator respectively, are bounded operators from $L_{1,v}\left( \left\vert x\right\vert ^{\frac{\beta (n+2\left\vert v\right\vert -\alpha )}{n+2\left\vert v\right\vert }}\right. ,$ $\left.\mathbb{R}_{n}^{+}\right) $ to $L_{\frac{n+2\left\vert v\right\vert }{n+2\left\vert v\right\vert -\alpha },\infty }\left( \left\vert x\right\vert ^{\beta }, \mathbb{R}_{n}^{+}\right) $ where $0\lt \alpha 0,...,v_{n}\gt 0,\left\vert v\right\vert =v_{1}+...+v_{n}$and $-1\lt \beta \lt 0.$.

In this paper we introduce a hybrid relaxed-extragradient method for finding a common element of the set of common fixed points of $N$ nonexpansive mappings and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The hybrid relaxed-extragradient method is based on two well-known methods: hybrid and extragradient. We derive a strong convergence theorem for three sequences generated by this method. Based on this theorem, we also construct an iterative process for finding a common fixed point of $N+1$ mappings, such that one of these mappings is taken from the more general class of Lipschitz pseudocontractive mappings and the rest $N$ mappings are nonexpansive.

In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form \begin{eqnarray*} 0\in G(x,y)+Q(x,y), \end{eqnarray*} where both mappings $G$ and $Q$ are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal solutions requires quite restrictive assumptions. Our techniques are mainly based on modern tools of variational analysis and generalized differentiation revolving around the fundamental extremal principle in variational analysis and its analytic counterpart known as the subdifferential variational principle.

In this paper we study some problems in $C^*$-algebra using elementary algebraic method. Our intention is to demonstrate that in many occasions algebraic properties and techniques may be the natural choice.

The purpose of this paper is to establish some new results on the unique solvability of solution of the first mixed problem for strongly hyperbolic systems in infinite cylinders with nonsmooth base.

In this survey we present some recent results in Euclidean and Heisenberg spaces for Pompeiu problem with sets of higher codimension. We prove a theorem which demonstrates a higher codimension set, together with a full complement of rotations, will possess the Pompeiu property. We also consider some aspects of the Morera side of the problem.