In this paper we prove the existence of solutions for hemivariational inequalities with delays and then investigate optimal control problems for some cost functions.

In this note, a modified version of a conjectured Nesbitt type inequality is given and a generalization of Nesbitt’s inequality involving three parameters is established.

In this paper, we first consider classes of nonlinear mappings containing the class of firmly nonexpansive mappings which can be deduced from an equilibrium problem in a Hilbert space. Further, we deal with fixed point theorems and ergodic theorems for these nonlinear mappings.

Recently, Han and Lu [HL] developed a discrete Littlewood-Paley-Stein analysis and multi-parameter Hardy space theory associated with isotropic flag singular integral operators initially studied by Muller-Ricci-Stein [MRS] and Nagel-Ricci-Stein [NRS]. The purpose of this paper is to carry out the multi-parameter Hardy space theory associated with non-isotropic flag singular integrals. The boundedness of such flag singular integral operator $T$ from $H^p_F$ to $H^p_F$ and from $H^p_F$ to $L^p$ are established in this paper. Discrete Calderón's identity and Min-Max principle derived here are the main tools used to establish the non-isotropic multi-parameter Hardy space theory.

Almost eleven decades ago, Barnes introduced and made a systematic investigation on the multiple Gamma functions $\Gamma_n$. In about the middle of 1980s, these multiple Gamma functions were revived in the study of the determinants of Laplacians on the $n$-dimensional unit sphere ${\bf S}^n$ by using the multiple Hurwitz zeta functions $\zeta_n(s,a)$. In this paper, we first aim at presenting a generalized Hurwitz formula for $\zeta_n(s,a)$ together with its various special cases. Secondly, we give analytic continuations of multiple Hurwitz-Euler eta function $\eta_n(s,a)$ in two different ways. As a by-product of our second investigation, a relationship between $\eta_n(-\ell,a)$ $(\ell \in \mathbb{N}_0)$ and the generalized Euler polynomials $E_\ell^{(n)}(n-a)$ is also presented.

Surfaces in 4D Riemannian space forms have been investigated extensively. In contrast, only few results are known for surfaces in 4D neutral indefinite space forms $R^4_2(c)$. Thus, in this paper we study space-like surfaces in $R^4_2(c)$ satisfying certain simple geometric properties. In particular, we classify space-like surfaces in $\mathbb E^4_2$ with constant mean and Gauss curvatures and null normal curvature. We also classify Wintgen ideal surfaces in $R^4_2(c)$ whose Gauss and normal curvatures satisfy $K^D = 2K$.

Let $\mathcal M$ be a class of (mono)morphisms in a category $\mathcal A$. To study mathematical notions, such as injectivity, tensor products, flatness, one needs to have some categorical and algebraic information about the pair $(\mathcal{A},\mathcal{M})$. In this paper we take $\mathcal A$ to be the category Act-S of acts over a semigroup $S$, and ${\mathcal M}_d$ to be the class of sequentially dense monomorphisms (of interest to computer scientists, too) and study the categorical properties, such as limits and colimits, of the pair $(\mathcal{A},\mathcal{M})$. Injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, and the authors who used it to obtain information about injectivity relative to monomorphisms.

In this paper, we give an alternative proof that the Kurzweil-Henstock integral using partition of unity is equivalent to the Lebesgue integral in the $n$-dimensional Euclidean space. We also define and discuss the Kurzweil-Henstock integral on manifolds.

In this paper, we establish some weighted Ostrowski integral inequality for mappings of bounded variation.

We introduced the concept of an asymptotically $\kappa$-strict pseudocontractive type mapping in the intermediate sense which is not necessarily Lipschitzian. We proved that the modified Mann iteration process: $x_{n+1} = (1-\alpha_n) x_n + \alpha_n T^n x_n$, $\forall n \geq 1$ where $\{\alpha_n\}$ is a sequence in $(0,1)$ with $\delta \leq \alpha_n \leq 1 - \kappa - \delta$ for $\delta \in (0,1)$ converges weakly to a fixed point of an asymptotically $\kappa$-strict pseudocontractive type mapping $T$ in the intermediate sense. Furthermore, a CQ method which generates a strongly convergent sequence for this class of mappings is proposed and strong convergence result for this CQ method is established.

Infinite state Markov chains with block-structured transition matrix find extensive applications in many areas, including telecommunications, and queueing systems, for example. In this paper, we use the censoring technique to study a MAP/G/1 queue with set-up time and multiple vacations, whose transition matrix can be transformed to a block-Toeplitz or block-repeating structured. Hence, we are able to relate the boundary conditions of the system to a Markov chain of M/G/1 type. This leads to a solution of the boundary equations, which is crucial for solving the system of differential equations. We also provide expressions for the distribution of stationary queue length, virtual waiting time and the busy period, respectively.

This paper deals with the existence and multiplicity of solutions for a class of semilinear elliptic problems of the form \begin{equation*} \begin{cases} -\Delta u = \displaystyle \frac{\mu}{|x|^2}u + f(x,u) & \text{ in } & \Omega, \\ u = 0 & \text{ on } & \partial \Omega, \end{cases} \end{equation*} where $\Omega = \Omega_1 \times \Omega_2 \subset \mathbb{R}^N$ ($N \geqq 5$) is a bounded domain having cylindrical symmetry, $\Omega_1 \subset \mathbb{R}^m$ is a bounded regular domain and $\Omega_2$ is a $k-$dimensional ball of radius $R$, centered in the origin and $m+k = N$, and $m \geqq 2$, $k \geqq 3$, $0 \leqq \mu \lt \mu^\star = \left(\frac{N-2}{2}\right)^2$. The proofs rely essentially on the critical point theory tools combined with the Hardy inequality.

In this paper the authors give a characterization of the compactness for the commutator $[b,\mu_\Omega]$ in the Morrey spaces $L^{p,\,\lambda}(\mathbb R^n)$, where $\mu_\Omega$ denotes the Marcinkiewicz integral. More precisely, the authors prove that if $b\in \mathrm{VMO}(\mathbb{R}^n)$, the $\mathrm{BMO}(\mathbb{R}^n)$-closure of $C_c^\infty(\mathbb{R}^n)$, then the commutators $[b,\mu_\Omega]$ is a compact operator in the Morrey spaces $L^{p,\,\lambda}(\mathbb{R}^n)$ for $1 \lt p \lt \infty$ and $0 \lt \lambda \lt n$. Conversely, if $b \in \mathrm{BMO}(\mathbb{R}^n)$ and $[b, \mu_{\Omega}]$ is a compact operator in $L^{p,\,\lambda}(\mathbb{R}^n)$ for some $p \in (1, \infty)$ and $\lambda \in (0,n)$, then $b \in \mathrm{VMO}(\mathbb{R}^n)$. In the above results, the kernel function $\Omega$ of the operator $\mu_{\Omega}$ is assumed to satisfy a very weak condition on $S^{n-1}$.

In this paper a generalized format for a constrained extremum problem is considered. Subsequently, the paper investigates and deepens some aspects concerning the linear separation between two sets in the Euclidean space, that are a convex cone and a generic set. A condition equivalent to their linear separation is given. Moreover, a condition equivalent to regular linear separation is proposed; this condition includes also the nonconvex case and it is finalized to the application to the generalized constrained extremum problems.

A new system of generalized mixed implicity equilibrium problems is introduced and studied in real Banach spaces. The notion of the Yosida approximation introduced by Moudafi in Hilbert spaces is first generalized to reflexive Banach spaces. By using the notion of the Yosida approximation, a system of generalized equation problems is considered and its equivalence with the system of generalized mixed implicity equilibrium problems is also proved. By using the system of generalized equation problems, a new iterative algorithm for solving the system of generalized mixed implicity equilibrium problems is suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithm is proved under suitable conditions. These results are new and unify and generalize some recent results in this field.

A set is called a unique range set (counting multiplicities) for a particular family of functions if the inverse image of the set counting multiplicities uniquely determines the function in the family. So far, almost all constructions of unique range sets for meromorphic functions are zero sets of polynomials which satisfy an injectivity condition introduced by Fujimoto. A polynomial $P(z)$ satisfies the injectivity condition if $P$ is injective on the zeros of its derivative. In this paper, we will construct examples of unique range sets for meromorphic functions without assuming an injectivity condition.

Let $R$ be a ring, $M$ a right $R$-module and $(S,\leq)$ a strictly totally ordered monoid. It is shown that $[[M^{S,\leq}]]$, the module of generalized power series with coefficients in $M$ and exponents in $S$, is a p.q.Baer right $[[R^{S,\leq}]]$-module if and only if the right annihilator of any $S$-indexed family of cyclic submodules of $M$ in $R$ is generated by an idempotent of $R$. Furthermore, we will show that for a ring $R$ with all left semicentral idempotents are central, the ring $[[R^{S,\leq}]]$ consisting of generalized power series over $R$ is a right p.q.Baer ring if and only if $R$ is a right p.q.Baer ring and any $S$-indexed family of central idempotents of $R$ has a generalized join in $I(R)$, where $I(R)$ is the set of all idempotents of $R$.

In this paper we study the existence, multiplicity and nonexistence of positive solutions for $p$-Laplacian problems with $L^1$-indefinite weight. As an application, we give some existence and multiplicity results for Emden-Fowler type $p$-Laplacian radial problems defined on an exterior domain depending on the boundary value which plays the role of a parameter.

The notion of sequential purity for acts over the monoid $\mathbb{N}^\infty$, called projection algebras, was introduced and studied by Mahmoudi and Ebrahimi. This paper is devoted to the study of this notion and its relation to injectivity of $S$-acts for a semigroup $S$. We prove that in general injectivity implies absolute sequential purity and they are equivalent for acts over some classes of semigroups.

Suppose that $C$ is a nonempty subset of a real Banach space $X$. In this article, we construct two types of iterative schemes with errors for a finite family $\{T_i\}_{i=1}^k$ of nonself generalized asymptotically quasi-nonexpansive mappings of $C$ into $X$. Furthermore, not only a necessary and sufficient condition for $\{x_n\}$ generated by each of those iterations to converge to a common fixed point of $\{T_i\}_{i=1}^k$ is obtained, but also the weak and strong convergence theorems of $\{x_n\}$ in uniformly convex Banach spaces are established as well.

In this paper we prove some results related to a certain vector field satisfying the second equation of Obata [8] on vector fields.

Our purpose in this paper is first to discuss nonlinear operators and nonlinear projections in Banach spaces which are related to the resolvents of $m$-accretive operators and maximal monotone operators. Some of these operators in Banach spaces are new. Next, we discuss some properties for such nonlinear operators and nonlinear projections in Banach spaces. Further, using these properties, we prove strong convergence theorems by hybrid methods for nonlinear operators with equilibrium problems in Banach spaces.

We prove that every surjective isometry between unit spheres of $L^\infty(\Sigma,\Omega, \mu)$ and a Banach space $F$ can be linearly and isometrically extended to the whole space, which means that if the unit sphere of a Banach space $F$ is isometric to the unit sphere of $L^\infty(\Sigma,\Omega, \mu)$, then $F$ is linearly isometric to $L^\infty(\Sigma,\Omega, \mu)$.

Waveform relaxation methods are decoupling or splitting methods for large scale ordinary differential equations. In this paper, we apply the meth- ods directly to semi-linear parabolic partial functional differential equations. Taking into consideration of the complicated forms of these parabolic equa- tions, we propose a kind of embedded waveform relaxation methods, which are in fact two-level waveform relaxation methods and which can also be ap- plied to some other systems. We provide explicit iterative expressions of the embedded methods and exhibit the superlinear rate of convergence on finite time intervals. We also apply the two-level idea to the functional differential equations derived from semi-discretization of the original system. The win- dowing technique is employed for the situation of long time intervals. Finally, two numerical experiments are performed to confirm our theory.

This paper deal with the approximate controllability for the nonlinear functional differential control problem governed by the variational inequality. Sufficient conditions for the approximate controllability of the system are discussed under the bounded condition on the controller operator independent of the the time interval. We also prove the regularity and norm estimations for solutions of the given problems.

This paper presents a sufficient condition for two vector fields $X$ and $Y$ to have the squares noncommutative, i.e. $[X^2, Y^2] \not= 0$. We prove that if the vector fields $X$, $Y$ span a nilpotent distribution with nilpotence class 2, then the squares of the vector fields do not commute.

The main object of this paper is to give sufficient conditions for certain families of integral operators, which are defined here by means of the normalized form of the generalized Bessel functions, to be univalent in the open unit disk. In particular cases, we find the corresponding simpler conditions for integral operators involving the Bessel function, the modified Bessel function and the spherical Bessel function.

In this note, we show that the homogeneous Triebel-Lizorkin-type spaces $\dot{F}^{\alpha,\tau}_{p,q}$ with four parameters defined in [7, 8] is essentially same as the generalized Carleson measure space $CMO^{\alpha,q}_r$ introduced in [6] with equivalent norms.