We provide an $L^{2}$-bound for pseudoholomorphic curves in symplectisation of contact manifolds which is the first step for compactness theorem in symplectic field theory introduced by Eliashberg, Givental and Hofer.

In this paper, by using a nonempty intersection lemma due to the authors, we obtain two coincidence theorems involved $\mathfrak{RC}$-maps in abstract convex spaces, which are actually equivalent. We then derive some maximal element theorems for set-valued maps in abstract convex spaces. As an application, we study the existence of solutions for a system of generalized equilibrium problems in abstract convex spaces. We also give some examples to illustrate our results.

We prove in this paper that a piecewise $C^2$ function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}$ is convex if and only if for every $(x,y) \in {\rm gph} \partial\varphi$, the limiting second-order subdifferential mapping $\partial^2 \varphi(x,y): \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ has the so-called positive semi-definiteness (PSD) - in analogy with the notion of positive semi-definiteness of symmetric real matrices. As a by-product, characterization for strong convexity of $\varphi$ is established.

Some properties of remotely almost periodic functions are studied. The existence and uniqueness of remotely almost periodic solution to Parabolic Inverse Problems for a type of boundary value problem are established. Stability of the solution is discussed.

This paper gives alternative theorems to investigate the existence of pseudo almost automorphic solutions for semilinear evolution equations. Moreover, we consider a parabolic equation to illustrate the theorem.

This study is the continuation of our earlier works [3, 4]. Here, we mainly investigate the global approximation behavior of modified Szász-Mirakjan operators presented in the papers mentioned above.

Recently Sun et al [X.-Y. Sun, J.-L. Wu, Y.-W. Wu, J.-F. Hou, Total colorings of planar graphs without adjacent triangles, Discrete Math 309:202-206 (2009)] proved that planar graphs with maximum degree six and with no adjacent triangles are total $8$-colorable. This results implies that if every vertex of a planar graph of maximum degree six is missing either a $3$-cycle or a $4$-cycle, then the graph is total $8$-colorable. In this paper we strengthen that condition by showing that if every vertex of a planar graph of maximum degree six is missing some $k_v$-cycle for $k_v \in \{3,4,5,6,7,8\}$, then the graph is total $8$-colorable.

We determine precise conditions for the boundedness of Bergman projections from Lebesgue classes onto the spaces in the title, which are members of the same one-parameter family of spaces. The projections provide integral representations for the functions in the spaces. We obtain many properties of the spaces as straightforward corollaries of the projections, integral representations, and isometries among the spaces. We solve the Gleason problem and an extremal problem for point evaluations in each space. We establish maximality of these spaces among those that exhibit Mobius-type invariances and possess decent functionals. We find new Hermitian non-Kahlerian metrics that characterize half of these spaces by Lipschitz-type inequalities.

Let $G$ be a connected graph and $\mathcal{L}$ be its normalized Laplacian matrix. Let $\lambda_1$ be the second smallest eigenvalue of $\mathcal{L}$. In this paper we studied the effect on the second smallest normalized Laplacian eigenvalue by grafting some pendant paths.

Let $\alpha \in (0,\infty)$, $q \in [1,\infty]$, $s$ be a nonnegative integer, $\omega \in A_1(\mathbb{R}^n)$ (the class of Muckenhoupt's weights). In this paper, the authors introduce the weighted Morrey-Campanato space $L(\alpha,\,q,\,s,\,\omega;\,\mathbb{R}^n)$ and obtain its equivalence on different $q \in [1,\infty]$ and integers $s \ge \lfloor n\alpha\rfloor$ (the integer part of $n\alpha$). The authors then introduce the weighted Lipschitz space $\wedge(\alpha,\,\omega;\,\mathbb{R}^n)$ and prove that $\wedge(\alpha,\,\omega;\,\mathbb{R}^n) = L(\alpha,\,q,\,s,\,\omega;\,\mathbb{R}^n)$ when $\alpha \in (0,\infty)$, $s \ge \lfloor n \alpha \rfloor$ and $q \in [1,\infty]$. Using this, the authors further establish a new characterization of $L(\alpha,\,q,\,s,\,\omega;\,\mathbb{R}^n)$ by using the convolution $\varphi_{t_{B}} \ast f$ to replace the minimizing polynomial $P_{B}^{s}f$ on any ball $B$ of a function $f$ in its norm when $\alpha \in (0,\infty)$, $s \ge \lfloor n \alpha \rfloor$, $\omega \in A_1(\mathbb{R}^n) \cap RH_{1+1/\alpha}(\mathbb{R}^n)$ and $q \in [1,\infty]$, where $\varphi$ is an appropriate Schwartz function, $t_{B}$ denotes the radius of the ball $B$ and $\varphi_{t_{B}}(\cdot) \equiv t_{B}^{-n} \varphi(t_{B}^{-1}\cdot)$.

J. Arhippainen and V. Müller showed that $\|a + \lambda e\|_1 \leq 3\|a + \lambda e\|_{\operatorname{op}}$ for all $a + \lambda e \in \tilde{A}$, where $\tilde{A}$ is the unitization of an arbitrary non-unital Banach algebra $A$ with regular norm. In this note, we present a necessary and sufficient condition for the equality of this estimate and its related problem.

The aim of this note is to establish almost sure Marcinkiewicz-Zygmund type result for a field of negatively associated random variables indexed by $\mathbb{Z}_+^d$ ($d \geq 2$), the positive $d$-dimensional lattice points.

Let $f, \omega : \mathbb{R}_+^n \to \mathbb{C}$ and $T_{\omega} f(x)$ denote the weighted mean of $f$ at $x$ with respect to the weight function $\omega$. We prove that the conditions of slow oscillation and slow decrease are Tauberian conditions for the implications: $f(x) \stackrel{st}{\rightarrow} l \Longrightarrow f(x) \rightarrow l$ and $T_{\omega} f(x) \stackrel{st}{\rightarrow} l \Longrightarrow f(x) \rightarrow l$. We also prove that the statistical version of the conditions of deferred means are Tauberian conditions for the implication: $T_{\omega} f(x) \stackrel{st}{\rightarrow} l \Longrightarrow f(x) \stackrel{st}{\rightarrow} l$. These generalize several well-known results.

In this paper we consider the existence of at least three solutions for the Dirichlet problem $$\left\{\begin{array}{ll} u^{i\upsilon} + \alpha u'' + \beta u = \lambda f(x,u) + \mu g(x,u), \hspace{1cm} x \in (0,1) \\ u(0) = u(1) = 0, \\ u''(0) = u''(1) = 0 \end{array}\right.$$ where $\alpha,\ \beta$ are real constants, $f,g: [0,1] \times R \to R$ are $L^{2}$-Carathéodory functions and $\lambda,\mu \gt 0$. The approach is based on variational methods and critical points.

Some complementarity among a firm’s activities is an important source of its profits. In this paper, we focus on the way to prove complementarity. Though there are many studies on complementarity as supermodularity or the increasing differences of a function, we introduce the notion of self increasing differences with respect to a single activity, which is an essence of convexity from the viewpoint of complementarity, and investigate some interrelations among these three notions of complementarity. Mathematically, we give a sufficient condition for a composite function to have self increasing differences. This proposition is deeply related to Topkis’ (1998) Lemma 2.6.4 on sufficient conditions for a composite function to be supermodular. Both propositions are combined and applied to yield and/or strengthen complementarity in an organization, which will also disclose the functional structure of an organization’s activities.

We study maximal regularity in periodic Besov spaces $B_{p,q}^s(\mathbb T, X)$ for the integral equations ($P$): $u(t) = A \int^{t}_{-\infty} a(t-s) u(s) ds) + B \int^{t}_{-\infty} b(t-s) u(s) ds + f(t)$ on $[0,2\pi]$ with periodic boundary condition $u(0) = u(2\pi)$, where $A$ and $B$ are closed operators in a Banach space $X$, $a, \ b \in L^1(\mathbb R_+)$ and $f$ is a given function defined on $[0,2\pi]$ with values in $X$. Under suitable assumptions on the kernels $a, \ b$ and the closed operators $A, \ B$, we completely characterize $B_{p,q}^s$-maximal regularity of ($P$).

The main object of this paper is to give $q$-extensions of several explicit relationships of H. M. Srivastava and Á. Pintér [Appl. Math. Lett. 17 (2004), 375-380] between the Bernoulii and Euler polynomials. We also derive several other formulas in series of Carlitz's $q$-Stirling numbers of the second kind.

Some fundamental properties of resolvents of monotone operators in Banach spaces are investigated. Using them, we study the asymptotic behavior of the sequences generated by two modifications of the proximal point algorithm for monotone operators satisfying a range condition defined in Banach spaces.

The main object of this paper is to introduce and investigate a new generalization of the family of Euler polynomials by means of a suitable generating function. We establish several interesting properties of these general polynomials and derive explicit representations for them in terms of a certain generalized Hurwitz-Lerch Zeta function and in series involving the familiar Gaussian hypergeometric function. Finally, we propose an analogous generalization of the closely-related Genocchi polynomials and show how we can fruifully exploit some potentially useful linear connections of all these three important families of generalized Bernoulli, Euler and Genocchi polynomials with one another.

In this paper, we establish fixed point theorems for a family of multivalued maps defined on the product space of topological semilattice spaces. By using our fixed point theorems, we derive a result on the nonempty intersection of sets without convex structure and equilibrium existence theorems for generalized abstract economies with two constraint correspondences. We present some special cases of our results which generalize several known results in the literature. We consider systems of generalized vector quasi-equilibrium problems and their special cases. As an application of our equilibrium existence theorems, we establish some existence results for solutions of systems of generalized vector quasi-equilibrium problems and their special cases. The results of this paper improve and extend several results in the literature.

We establish criteria on entire series with finite logarithmic order in terms of several growth scale.

2-metric space is an interesting nonlinear generalization of metric space which was conceived and studied in details by Gahler. In this paper, for the first time, we establish Cantor’s intersection theorem and Baire category theorem in 2-metric spaces. As a departure from normal practice we then apply Cantor’s theorem to establish some fixed point theorems in such spaces.

In this paper, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitz continuous mapping in Hilbert spaces. We obtain some strong convergence theorems and weak convergence theorems. The results in this paper generalize, improve and unify some well-known convergence theorems in the literature.

The main purpose of this paper is the study of sufficient conditions for the existence of solutions of vector equilibrium problems. Our main results permit to obtain some existence results already established for the scalar case and, to provide some applications in the area of vector optimization and vector saddle point problems.

We offer an alternative definition for the $H_1$-integral of Garces, Lee, and Zhao.

We consider the oscillation of a class of first order neutral differential equations with nine real parameters. This relatively difficult problem is completely solved by applying the Cheng-Lin envelope method to find the exact conditions for the absence of real roots of the associated characteristic function. Several specific examples are also included to illustrate these conditions.