We construct global smooth approximate solution to a scalar conservation law with arbitrary smooth monotonic initial data. Different kinds of singularities interactions which arise during the evolution of the initial data are described as well. In order to solve the problem, we use and further develop the weak asymptotic method, recently introduced technique for investigating nonlinear waves interactions.

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in $C^{\ast }$-algebras and Lie $C^{\ast }$-algebras and also of derivations on$C^{\ast }$-algebras and Lie $C^{\ast }$-algebras for the Jensen-type functional equation$f\left(\left(x+y\right)/2\right)+f\left(\left(x-y\right)/2\right)=f\left(x\right)$.

The paper is concerned with a two-dimensional Landau-Lifshitz equation which was first raised by A. DeSimone and F. Otto, and so fourth, when studying thin film micromagnetics. We get the existence of a local weak solution by approximating it with a higher-order equation. Penalty approximation and semigroup theory are employed to deal with the higher-order equation.

By using the ordinary fermionic $p$-adic invariant integral on ${\mathbb{Z}}_p$, we derive some interesting identities related to the Frobenius-Euler polynomials.

Let $H\left(B\right)$ denote the space of all holomorphic functions on the unit ball $B$. Let $u\in H\left(B\right)$ and $\phi$ be a holomorphic self-map of $B$. In this paper, we investigate the boundedness and compactness of the weighted composition operator $u{C}_{\phi }$ from the general function space $F\left(p,q,s\right)$ to the weighted-type space $H_{\mu }^{\infty }$ in the unit ball.

Here we introduce the $n$th weighted space on the upper half-plane in the complex plane $ℂ$. For the case $n=2$, we call it the Zygmund-type space, and denote it by $𝒵\left({\Pi }_{+}\right)$. The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator $C_{\phi }f\left(z\right)=f\left(\phi \left(z\right)\right)$ from the Hardy space $H^p\left({\Pi }_{+}\right)$ on the upper half-plane, to the Zygmund-type space, where $\phi$ is an analytic self-map of the upper half-plane.

This paper is devoted to studying growth of solutions of linear differential equations of type $f^{\left(k\right)}+A_{k-1}\left(z\right)f^{\left(k-1\right)}+\cdots +A_1\left(z\right)f^{\prime }+A_0\left(z\right)f=H\left(z\right)$ where $A_j\left(j=0,\dots ,k-1\right)$ and $H$ are entire functions of finite order.

In the theorems of Galambos-Bojanić-Seneta's type, the asymptotic behavior of the functions $c_{\left[x\right]},\hspace{0.17em}\hspace{0.17em}x\ge 1$, for $x\to +\infty$, is investigated by the asymptotic behavior of the given sequence of positive numbers $\left(c_n\right)$, as $n\to +\infty$ and vice versa. The main result of this paper is one theorem of such a type for sequences of positive numbers $\left(c_n\right)$ which satisfy an asymptotic condition of the Karamata type ${\underset{¯}{\lim }}_{n\to \infty }$ $c_{\left[\lambda n\right]}/c_n>1$, for .

This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics including Volterra class dynamics. The proofs are based on a Perron-type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation.

We study global behavior of the following max-type difference equation $x_{n+1}=\max \left\{1/x_n,A_n/x_{n-1}\right\}$, $n=0,1,\dots ,$ where ${\left\{A_n\right\}}_{n=0}^{\infty }$ is a sequence of positive real numbers with $0\le \inf A_n\le \sup A_n<1$. The special case when ${\left\{A_n\right\}}_{n=0}^{\infty }$ is a periodic sequence with period $k$ and $A_n\in \left(0,1\right)$ for every $n\ge 0$ has been completely investigated by Y. Chen. Here we extend his results to the general case.

We consider the one-dimensional viscoelastic Porous-Thermo-Elastic system. We establish a general decay results. The usual exponential and polynomial decay rates are only special cases.

Based on the Gateaux differential on time scales, we investigate and establish necessary conditions for Lagrange optimal control problems on time scales. Moreover, we present an economic model to demonstrate the effectiveness of our results.

A new 4-point $C^3$ quaternary approximating subdivision scheme with one shape parameter is proposed and analyzed. Its smoothness and approximation order are higher but support is smaller in comparison with the existing binary and ternary 4-point subdivision schemes.

We obtain sufficient conditions which guarantee the global attractivity of solutions for nonlinear delay survival red blood cells model. Then, some criteria are established for the existence, uniqueness and global attractivity of positive almost periodic solutions of the almost periodic system.

We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappings in a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for viscosity iterative scheme are given and strong convergence of viscosity iterative scheme to a solution of a ceratin variational inequality is established.

In 2006, C. Park proved the stability of homomorphisms in $C^{\ast }$-ternary algebras and of derivations on $C^{\ast }$-ternary algebras for the following generalized Cauchy-Jensen additive mapping: $2f\left(\left({\sum }_{j=1}^p{x}_j/2\right)+{\sum }_{j=1}^d{y}_j\right)={\sum }_{j=1}^{p}f\left(x_j\right)+2{\sum }_{j=1}^{d}f\left(y_j\right)$. In this note, we improve and generalize some results concerning this functional equation.

We provide a representation of elements of the space $l^p\left(A,X\right)$ for a locally convex space $X$ and $1\le p<\infty$ and determine its continuous dual for normed space $X$ and $1<p<\infty$. In particular, we study the extension and characterization of isometries on $l^p\left(\text{N},X\right)$ space, when $X$ is a normed space with an unconditional basis and with a symmetric norm. In addition, we give a simple proof of the main result of G. Ding (2002).

A Chemostat system incorporating hepatocellular carcinomas is discussed. The model generalizes the classical Chemostat model, and it assumes that the Chemostat is an increasing function of the concentration. The asymptotic behavior of solutions is determined. Sufficient conditions for the local and global asymptotic stability of equilibrium and numerical simulation are obtained, which is used to select the disease control tactics.

By using the well-known Schauder fixed point theorem and upper and lower solution method, we present some existence criteria for positive solution of an $m$-point singular $p$-Laplacian dynamic equation on time scales with the sign changing nonlinearity. These results are new even for the corresponding differential ($𝕋=ℝ$) and difference equations ($𝕋=\mathbb{Z}$), as well as in general time scales setting. As an application, an example is given to illustrate the results.

All solutions of a fourth-order nonlinear delay differential equation are shown to converge to zero or to oscillate. Novel Riccati type techniques involving third-order linear differential equations are employed. Implications in the deflection of elastic horizontal beams are also indicated.

We prove the generalized Hyers-Ulam stability of the following quadratic functional equations $2f\left((x+y)/2\right)+2f\left((x-y)/2\right)=f\left(x\right)+f\left(y\right)$ and $f\left(ax+ay\right)+\left(ax-ay\right)=2a^{2}f\left(x\right)+2a^{2}f\left(y\right)$ in fuzzy Banach spaces for a nonzero real number $a$ with $a\ne \pm 1/2$.

The boundedness and compactness of the extended Cesàro operatorfrom logarithmic-type spaces to Bloch-type spaces on the unit ball arecompletely characterized in this paper.

The boundedness of the motions of the dynamical system described by a differential inclusion with control vector is studied. It is assumed that the right-hand side of the differential inclusion is upper semicontinuous. Using positionally weakly invariant sets, sufficient conditions for boundedness of the motions of a dynamical system are given. These conditions have infinitesimal form and are expressed by the Hamiltonian of the dynamical system.

In (2006) and (2009), Kim defined new generating functions of the Genocchi, Nörlund-type $q$-Euler polynomials and their interpolation functions. In this paper, we give another definitionof the multiple Hurwitz type $q$-zeta function. This function interpolates Nörlund-type $q$-Euler polynomials at negative integers. We also give some identities related to these polynomialsand functions. Furthermore, we give some remarks about approximations of Bernoulli andEuler polynomials.

This paper is concerned with evolution equations of fractional order ${\mathbf{\text{D}}}^{\alpha }u\left(t\right)=Au\left(t\right)$; $u\left(0\right)=u_0,u^{\prime }\left(0\right)=0$, where $A$ is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than $\pi$ and $1<\alpha <2$. We show that such equations are well posed in the sense that there always exists an $\alpha$-times resolvent family for the operator $A$.

We study the total stability in nonlinear discrete Volterra equations with unbounded delay, as a discrete analogue of the results for integrodifferential equations by Y. Hamaya (1990).

We achieve the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities for quadratic functional equations $f\left(ax+by\right)+f\left(ax-by\right)=\left(b\left(a+b\right)/2\right)f\left(x+y\right)+$ $\left(b\left(a+b\right)/2\right)f\left(x-y\right)+$ $\left(2a^2-ab-b^2\right)f\left(x\right)+$ $\left(b^2-ab\right)f\left(y\right)$ where $\mathrm{a,\; b}$ are nonzero fixed integers with $b\ne \pm a,-3a$, and $f\left(ax+by\right)+f\left(ax-by\right)=2a^{2}f\left(x\right)+2b^{2}f\left(y\right)$ for fixed integers $\mathrm{a,\; b}$ with $a,b\ne 0$ and $a\pm b\ne 0$.

We use the theory of normal families to prove the following. Let $Q_1\left(z\right)=a_1{z}^p+a_{1,p-1}{z}^{p-1}+\cdots +a_{1,0}$ and $Q_2\left(z\right)=a_2{z}^p+a_{2,p-1}{z}^{p-1}+\cdots +a_{2,0}$ be two polynomials such that $\mathrm{deg}{Q}_1=\mathrm{deg}{Q}_2=p$ (where $p$ is a nonnegative integer) and $a_1,a_2\left(a_2\ne 0\right)$ are two distinct complex numbers. Let $f\left(z\right)$ be a transcendental entire function. If $f\left(z\right)$ and $f^{\prime }\left(z\right)$ share the polynomial $Q_1\left(z\right)\text{\hspace{0.17em}CM}$ and if $f\left(\text{z}\right)=Q_2\left(z\right)$ whenever $f^{\prime }\left(z\right)=Q_2\left(z\right)$, then $f\equiv f^{\prime }$. This result improves a result due to Li and Yi.

By the properties of $p$-adic invariant integral on ${\mathbb{Z}}_p$,we establish various identities concerning the generalized Bernoulli numbers andpolynomials. From the symmetric properties of $p$-adic invariant integral on ${\mathbb{Z}}_p$, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.

Let $3\le n$, and $3\le k\le n$ be positive integers. Let $A$ be an algebra and let $X$ be an $A$-bimodule. A $ℂ$-linearmapping $d:A\to X$ is called a generalized $\left(n,k\right)$-derivation ifthere exists a $\left(k-1\right)$-derivation $\delta :A\to X$ such that$d\left(a_1{a}_2\cdots a_n\right)=\delta \left(a_1\right)a_2\cdots a_n+a_1\delta \left(a_2\right)a_3\cdots a_n+\cdots +a_1{a}_2\cdots a_{k-2}\delta \left(a_{k-1}\right)a_k\cdots a_n+a_1{a}_2\cdots a_{k-1}d\left(a_k\right)a_{k+1}\cdots a_n+a_1{a}_2\cdots a_{k}d\left(a_{k+1}\right)a_{k+2}\cdots a_n+a_1{a}_2\cdots a_{k+1}d\left(a_{k+2}\right)a_{k+3}\cdots a_n+\cdots +a_1\cdots a_{\mathrm{n}-1}d\left(a_n\right)$ for all $a_1,a_2,\dots ,a_n\in A$. The main purpose of this paper is to prove the generalized Hyers-Ulam stability of the generalized $\left(n,k\right)$-derivations.

We study some existence results in a Banach space for a nonlocal boundary value problem involving a nonlinear differential equation of fractional order $q$ given by ${}^c{D}^{q}x\left(t\right)=f\left(t,x\left(t\right)\right)$, $0<t<1$, $q\in \left(m-1,m\right]$, $m\in \mathbb{N}$, $m\ge 2$, $x\left(0\right)=0$, $x^{\prime }\left(0\right)=0,\hspace{0.17em}{x}^{\prime \prime }\left(0\right)=0,\dots ,x^{\left(m-2\right)}\left(0\right)=0$, $x\left(1\right)=\alpha x\left(\eta \right)$. Our results are based on the contraction mapping principle and Krasnoselskii's fixed point theorem.

We study the boundedness and compactness of the products of composition and differentiation operators from ${𝒬}_K\left(p,q\right)$ spaces to Bloch-type spaces and little Bloch-type spaces.

In this paper, we give some new Wilker-type inequalities for circular and hyperbolic functions in exponential form by using generalizations of Cusa-Huygens inequality and Cusa-Huygens-type inequality.

For $0<p<\infty$ the unit vector basis of ${\ell }_p$ has the property of perfect homogeneity: it is equivalent to all its normalized block basic sequences, that is, perfectly homogeneous bases are a special case of symmetric bases. For Banach spaces, a classical result of Zippin (1966) proved that perfectly homogeneous bases are equivalent to either the canonical $c_0$-basis or the canonical ${\ell }_p$-basis for some $1\le p<\infty$. In this note, we show that (a relaxed form of) perfect homogeneity characterizes the unit vector bases of ${\ell }_p$ for $0<p<1$ as well amongst bases in nonlocally convex quasi-Banach spaces.

The paper is concerned with the existence and stability of weak (variational) solutions for the problem of the quasistatic evolution of a viscoelastic material under mixed inhomogenous Dirichlet-Neumann boundary conditions. The main novelty of the paper relies in dealing with continuous-in-time weak solutions and allowing nonconvolution and weak-singular Volterra's relaxation kernels.

Let $V,W$ be real vector spaces. It is shown that an odd mapping $f:V\to W$ satisfies ${\displaystyle {\sum }_{i-1}^{2n}f\left(x_i-1/2n{\displaystyle {\sum }_{j=1}^{2n}{x}_j}\right)=}{\displaystyle {\sum }_{i=1}^{2n}f\left(x_i\right)-2nf\left(1/2n{\displaystyle {\sum }_{i=1}^{2n}{x}_i}\right)}$ for all $x_1,\dots ,x_{2n}\in V$ if and only if the odd mapping $f:V\to W$ is Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

We investigate the existence of positive solution to nonlinear fractional differential equation three-point singular boundary value problem: $D^{q}u\left(t\right)+f\left(t,u\left(t\right)\right)=0$, $0<t<1$, $u\left(0\right)=0$, $u\left(1\right)=\alpha D^{(q-1)/2}u\left(t\right){|}_{t=\xi }$, where $1<q\le 2$ is a real number, $\xi \in \left(0,1/2\right]$, $\alpha \in \left(0,+\infty \right)$ and $\alpha \Gamma \left(q\right){\xi }^{\left(q-1\right)/2}<\Gamma \left((q+1)/2\right),D^q$ is the standard Riemann-Liouville fractional derivative, and $f\in C\left(\left(0,1\right]\times \left[0,+\infty \right),\left[0,+\infty \right)\right),{\lim }_{t\to +0}f\left(t,\cdot \right)=+\infty$ (i.e., $f$ is singular at $t=0$). By using the fixed-point index theory, the existence result of positive solutions is obtained.

A generalized version of the Leslie-Gower predator-prey model that incorporates the prey structure and predator dispersal in two-patch environments is introduced. The focus is on the study of the boundedness of solution, permanence, and extinction of the model. Sufficient conditions for global asymptotic stability of the positive equilibrium are derived by constructing a Lyapunov functional. Numerical simulations are also presented to illustrate our main results.

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

We consider the uniform attractors for the three-dimensional nonautonomous Camassa-Holm equations in the periodic box $\Omega ={\left[0,L\right]}^3$. Assuming $f=f\left(x,t\right)\in L_{\mathrm{loc}}^2\left(\left(0,T\right);D\left(A^{-1/2}\right)\right)$, we establish the existence of the uniform attractors in $D\left(A^{1/2}\right)$ and $D\left(A\right)$. The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

We obtain a representation for the norm of certain compact weighted composition operator $C_{\psi ,\phi }$ on the Hardy space $H^2$, whenever $\phi \left(z\right)=az+b$ and $\psi \left(z\right)=az-b$. We also estimate the norm and essential norm of a class of noncompact weighted composition operators under certain conditions on $\phi$ and $\psi$. Moreover, we characterize the norm and essential norm of such operators in a special case.

Suppose that $X$ is a Banach space and $C$ is aninjective operator in $B\left(X\right)$, the space of all bounded linearoperators on $X$. In this note, a two-parameter $C$-semigroup (regularized semigroup) of operators is introduced, and some of its properties are discussed. As an application we show that the existence and uniqueness of solution of the 2-abstract Cauchy problem $(\partial /\left(\partial t_i\right))u\left(t_1,t_2\right)=H_{i}u\left(t_1,t_2\right),i=1,2$, , $u\left(0,0\right)=x$, $x\in C\left(D\left(H_1\right)\cap D\left(H_2\right)\right)$ is closely related to the two-parameter $C$-semigroups of operators.

We introduce the class of (local) $\left(a,k\right)$-regularized $C$-resolvent families and discuss its basic structural properties. In particular, our analysis covers subjects like regularity, perturbations, duality, spectral properties and subordination principles. We apply our results in the study of the backwards fractional diffusion-wave equation and provide several illustrative examples of differentiable $\left(a,k\right)$-regularized $C$-resolvent families.

This paper investigates the causality properties of a class of linear time-delay systems under constant point delays which possess a finite set of distinct linear time-invariant parameterizations (or configurations) which, together with some switching function, conform a linear time-varying switched dynamic system. Explicit expressions are given to define pointwisely the causal and anticausal Toeplitz and Hankel operators from the set of switching time instants generated from the switching function. The case of the auxiliary unforced system defined by the matrix of undelayed dynamics being dichotomic (i.e., it has no eigenvalue on the complex imaginary axis) is considered in detail. Stability conditions as well as dual instability ones are discussed for this case which guarantee that the whole system is either stable, or unstable but no configuration of the switched system has eigenvalues within some vertical strip including the imaginary axis. It is proved that if the system is causal and uniformly controllable and observable, then it is globally asymptotically Lyapunov stable independent of the delays, that is, for any possibly values of such delays, provided that a minimum residence time in-between consecutive switches is kept or if all the set of matrices describing the auxiliary unforced delay—free system parameterizations commute pairwise.

The numerical range and normality of Toeplitz operator acting on the Bergman space and pluriharmonic Bergman space on the polydisk is investigated in this paper.

We consider hyperbolic equations with anisotropic elliptic part and some non-Lipschitz coefficients. We prove well-posedness of the corresponding Cauchy problem in some functional spaces. These functional spaces have finite smoothness with respect to variables corresponding to regular coefficients and infinite smoothness with respect to variables corresponding to singular coefficients.

We obtain the general solution and the generalized Ulam-Hyers stability of the mixed type cubic and quartic functional equation $f\left(x+2y\right)+f\left(x-2y\right)=4\left(f\left(x+y\right)+f\left(x-y\right)\right)-24f\left(y\right)-6f\left(x\right)+3f\left(2y\right)$ in quasi-Banach spaces.

Sufficient conditions for permanence of a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay ${\dot{x}}_1\left(t\right)=x_1\left(t\right)\left[r_1\left(t\right)-a_{11}\left(t\right)x_1\left(t-\tau \left(t\right)\right)-a_{12}\left(t\right)x_2\left(t\right)/\left(m^2+x_1^2\left(t\right)\right)\right],$  ${\dot{x}}_2\left(t\right)=x_2\left(t\right)\left[r_2\left(t\right)-a_{21}\left(t\right)x_2(t)/x_1\left(t\right)\right],$ are obtained, where $x_1\left(t\right)$ and $x_2\left(t\right)$ stand for the density of the prey and the predator, respectively, and $m\ne 0$ is a constant. $\tau \left(t\right)\ge 0$ stands for the time delays due to negative feedback of the prey population.

This paper deals with the existence and uniqueness of periodic solutions for the first-order functional differential equation $y^{\prime }\left(t\right)=-a\left(t\right)y\left(t\right)+f_1\left(t,y\left(t-\tau \left(t\right)\right)\right)+f_2\left(t,y\left(t-\tau \left(t\right)\right)\right)$ with periodic coefficients and delays. We choose the mixed monotone operator theory to approach our problem because such methods, besides providing the usual existence results, may also sometimes provide uniqueness as well as additional numerical schemes for the computation of solutions.

We introduce an $\ell \left(p\right)$-type new sequence space and investigate its some topological properties including $AK$ and $AD$ properties. Besides, we examine some geometric properties of this space concerning Banach-Saks type $p$ and Gurarii's modulus of convexity.

Consider the half-eigenvalue problem ${\left({\varphi }_p\left(x^{\prime }\right)\right)}^{\prime }+\lambda a\left(t\right){\varphi }_p\left(x_{+}\right)-\lambda b\left(t\right){\varphi }_p\left(x_{-}\right)=0$ a.e. $t\in \left[0,1\right]$, where $1<p<\infty$, ${\varphi }_p\left(x\right)={\left|x\right|}^{p-2}x$, $x_{\pm }(\cdot )=\max \left\{\pm x(\cdot ),0\right\}$ for $x\in {𝒞}^0:=C\left(\left[0,1\right],ℝ\right)$, and $a\left(t\right)$ and $b\left(t\right)$ are indefinite integrable weights in the Lebesgue space ${ℒ}^{\gamma }:=L^{\gamma }\left(\left[0,1\right],ℝ\right),1\le \gamma \le \infty$. We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in $\left(a,b\right)\in {\left({ℒ}^{\gamma },w_{\gamma }\right)}^2$, where $w_{\gamma }$ denotes the weak topology in ${ℒ}^{\gamma }$ space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in $\left(a,b\right)\in {\left({ℒ}^{\gamma },{\Vert \cdot \Vert }_{\gamma }\right)}^2$, where ${\Vert \cdot \Vert }_{\gamma }$ is the $L^{\gamma }$ norm of ${ℒ}^{\gamma }$.

This article is a short elementary review of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of analysis. Some new properties are included, and several analysis-related applications are mentioned. At the end of the paper one application is described in details—certain Fourier integrals involving $\Gamma \left(a+it\right)$ and $\Gamma \left(a+it\right)\Gamma \left(b-it\right)$ are evaluated in terms of Stirling numbers.

Let $L$ denote the operator generated in $L_2\left(R_{+}\right)$ by Sturm-Liouville equation $-y^{\mathrm{\prime \prime }}+q\left(x\right)y={\lambda }^{2}y$, $x\in R_{+}=\left[0,\infty \right)$, $y^{\prime }\left(0\right)/y\left(0\right)={\alpha }_0+{\alpha }_1\lambda +{\alpha }_2{\lambda }^2$, where $q$ is a complex-valued function and ${\alpha }_i\in ℂ$, $i=0,1,2$ with ${\alpha }_2\ne 0$. In this article, we investigate the eigenvalues and the spectral singularities of $L$ and obtain analogs ofNaimark and Pavlov conditions for $L$.

Existence of positive solutions has been studied by A. Babakhani and V. Daftardar-Gejji (2003) in caseof multiterm nonautonomous fractional differential equations with constantcoefficients. In the present paper we discuss existence of positive solutions incase of multiterm fractional differential equations of finite delay with polynomialcoefficients.

The multiple results of positive solutions for the following quasilinearelliptic equation: $-{\Delta }_{p}u=\lambda f\left(x\right){\left|u\right|}^{q-2}u+g\left(x\right){\left|u\right|}^{p^{\ast }-2}u$ in $\Omega$, $u=0$ on $\partial \Omega$, are established. Here, $0\in \Omega$ is a bounded smooth domain in ${ℝ}^N,{\Delta }_p$ denotes the $p$-Laplacian operator, $1\le q<p<N,p^{\ast }=Np/\left(N-p\right),\lambda$ is a positive real parameter, and $f,g$ are continuous functions on $\overline{\Omega }$ which are somewhere positive but which may change sign on $\Omega$. The study is based on the extraction of Palais-Smale sequences in the Nehari manifold.

We define the model of an abstract economy with differential (asymmetric) information and a measure space of agents. We generalize N. C. Yannelis's result (2007), considering that each agent is characterised by a random preference correspondence instead of having a random utility function. We establish two different equilibrium existence results.

We solve the inhomogeneous Euler differential equations of the form $x^2{y}^{\prime \prime }(x)+\alpha x{y}^{\prime }(x)+\beta y(x)={\sum }_{m=0}^{\infty }{a}_m{x}^m$ and apply this result to the approximation of analytic functions of a special type by the solutions of Euler differential equations.

We introduce a general iteration scheme for a finite familyof generalized asymptotically quasi-nonexpansive mappings in Banach spaces.The new iterative scheme includes the multistep Noor iterations with errors,modified Mann and Ishikawa iterations, three-step iterative scheme of Xu andNoor, and Khan and Takahashi scheme as special cases. Our results generalizeand improve the recent ones announced by Khan et al. (2008), H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006), and many others.

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then by using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self-adjoint differential operator with the periodic matrix coefficients is finite.

We introduce and study a new hybrid projection algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of relatively quasi-nonexpansive mappings, and theset of solutions of the variational inequality for an inverse-strongly-monotone operator in a Banach space. Under suitable assumptions, we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space. The results obtained in this paper extend and improve the several recent results in this area.

We study a Beddington-DeAngelis type predator-prey system with impulsive perturbation and seasonal effects. First, we numerically observe the influence of seasonal effects on the system without impulsive perturbations. Next, we find the conditions for the local and global stabilities of prey-free periodic solutions by using Floquet theory for the impulsive equation and small amplitude perturbation skills, and for the permanence of the system via comparison theorem. Finally, we show that seasonal effects and impulsive perturbation can give birth to various kinds of dynamical behavior of the system including chaotic phenomena by numerical simulations.

A class of second-order nonlinear differential equations with damping term$(r(t){\left|x^{\prime }(t)\right|}^{\sigma -1}{x}^{\prime }(t){)}^{\prime }+p(t){\left|x^{\prime }(t)\right|}^{\sigma -1}{x}^{\prime }(t)+q(t)f(x(t))=0$are investigated in this paper. By using a new method, we obtain some new sufficient conditions forthe oscillation of the above equation, and some references are extended in this paper. Examples areinserted to illustrate this result.

If $F$ is a continuous function on the real line and $f=F^{\prime }$ is its distributional derivative, then the continuous primitive integral of distribution $f$ is ${\int }_a^{b}f=F(b)-F(a)$. This integral contains the Lebesgue, Henstock-Kurzweil, and wide Denjoy integrals. Underthe Alexiewicz norm, the space of integrable distributions is a Banach space. We define theconvolution $f\ast g(x)={\int }_{-\infty }^{\infty }f(x-y)g(y)dy$ for $f$ an integrable distribution and $g$ a function of bounded variation or an $L^1$ function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For $g$ of bounded variation,$f\ast g$ is uniformly continuous and we have the estimate ${\Vert f\ast g\Vert }_{\infty }\le \Vert f\Vert {\Vert g\Vert }_{ℬ𝒱}$, where $\Vert f\Vert ={\text{sup}}_I\left|{\int }_{I}f\right|$ is the Alexiewicz norm. This supremum is taken over all intervals$I\subset ℝ$. When $g\in L^1$, the estimate is $\Vert f\ast g\Vert \le \Vert f\Vert {\Vert g\Vert }_1$. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.

The present paper discusses in the metric $L_p$ S. N. Bernstein type inequalities of the most general kind on very general accessible classes of curves in a complex plane. The obtained estimations, generally speaking, are not improvable.

We establish two optimal inequalities among power mean $M_p\left(a,b\right)={\left(a^p/2+b^p/2\right)}^{1/p}$, arithmetic mean $A\left(a,b\right)=\left(a+b\right)/2$, logarithmic mean $L\left(a,b\right)=\left(a-b\right)/\left(\log a-\log b\right)$, and geometric mean $G\left(a,b\right)=\sqrt{ab}$.

We establish strong convergence theorems for finding a common element ofthe zero point set of a maximal monotone operator and the fixed point set of two relativelynonexpansive mappings in a Banach space by using a new hybrid method. Moreover weapply our main results to obtain strong convergence for a maximal monotone operator andtwo nonexpansive mappings in a Hilbert space.

The author establishes some identities involving the $D$ numbers, Bernoulli numbers, and central factorial numbers of the first kind. A generating function and several computational formulas for $D$-Nörlund numbers are also presented.

We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi-$\varphi$-nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.

We investigate the stochastic 3D Navier-Stokes-$\alpha$ model which arises in the modelling of turbulent flows of fluids. Our model contains nonlinear forcing terms which do not satisfy the Lipschitz conditions. The adequate notion of solutions is that of probabilistic weak solution. We establish the existence of a such of solution. We also discuss the uniqueness.

The problem of robust stability for a class of neutral control systems with mixed delays is investigated. Based on Lyapunov stable theory, by constructing a new Lyapunov-Krasovskii function, some new stable criteria are obtained. These criteria are formulated in the forms of linear matrix inequalities (LMIs). Compared with some previous publications, our results are less conservative. Simulation examples are presented to illustrate the improvement of the main results.

We abstract the definition of the Costas property in the context of a group and study specifically denseCostas sets (named Costas clouds) in groups with the topological property that they are dense in themselves:as a result, we prove the existence of nowhere continuous dense bijections that satisfy the Costas property on ${\mathbb{Q}}^2$, ${ℝ}^2$, and ${ℂ}^2$, the latter two being based on nonlinear solutions of Cauchy's functional equation, as well as on$\mathbb{Q}$, $ℝ$, and $ℂ$, which are, in effect, generalized Golomb rulers. We generalize the Welch and Golomb constructionmethods for Costas arrays to apply on $ℝ$ and $ℂ$, and we prove that group isomorphisms on and tensor products of Costas sets result to new Costas sets with respect to an appropriate set of distance vectors. We also give two constructive examples of a nowhere continuous function that satisfies a constrained form of the Costas property (over rational or algebraic displacements only, i.e.), based on the indicator function of a dense subset of $ℝ$.

In this work we use analytical tools—Schauder bases and Geometric Series theorem—in order to develop a new method for the numerical resolution of the linear Volterra integral equation of the second kind.

A bounded linear operator $T$ on a Hilbert space $\mathscr{H}$, satisfying ${\Vert T^{2}h\Vert }^2+{\Vert h\Vert }^2\ge 2{\Vert Th\Vert }^2$ for every $h\in \mathscr{H}$, is called a convex operator. In this paper, we give necessary and sufficient conditions under which a convexcomposition operator on a large class of weighted Hardy spaces is an isometry. Also, we discuss convexity of multiplication operators.

We consider a higher-order three-point boundary value problem on time scales.A new existence result is first obtained by using a fixed point theorem due to Krasnoselskii andZabreiko. Later, under certain growth conditions imposed on the nonlinearity, several sufficientconditions for the existence of a nonnegative and nontrivial solution are obtained by usingLeray-Schauder nonlinear alternative. Our conditions imposed on nonlinearity are all very easyto verify; as an application, some examples to demonstrate our results are given.

We study the permanence of periodic predator-prey system with general nonlinear functional responses and stage structure for both predator and prey and obtain that the predator and the prey species are permanent.

This paper is devoted to the following question: how to characterize convex nowhere dense subsets of normed linear spaces in terms of porosity? The motivation for this study originates from papers of V. Olevskii and L. Zajíček, where it is shown that convex nowhere dense subsets of normed linear spaces are porous in some strong senses.

We prove a weak convergence theorem of the modified Mann iteration process for a uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two kinds of new monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings in a Hilbert space. The results improve and extend the corresponding ones announced by Kim and Xu (2006) and Nakajo and Takahashi (2003).

The passivity problem is investigated for a class of stochastic uncertain neural networks with time-varyingdelay as well as generalized activation functions. By constructing appropriate Lyapunov-Krasovskii functionals,and employing Newton-Leibniz formulation, the free-weighting matrix method, and stochastic analysis technique, adelay-dependent criterion for checking the passivity of the addressed neural networks is established in terms of linearmatrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. An examplewith simulation is given to show the effectiveness and less conservatism of the proposed criterion. It is noteworthy thatthe traditional assumptions on the differentiability of the time-varying delays and the boundedness of its derivative areremoved.

We show that the $p$-convexified Tsirelson space ${𝒯}^{\left(p\right)}$ for $0<p<1$ and all its complemented subspaces with unconditional basis have unique unconditional basis up to permutation. The techniquesinvolved in the proof are different from the methods that have been used in all the other uniqueness results in the nonlocally convex setting.

Lee, An and Park introduced the quadraticfunctional equation $f\left(2x+y\right)+f\left(2x-y\right)=8f\left(x\right)+2f\left(y\right)$ and proved the stability of the quadratic functional equation in the spirit of Hyers, Ulam and Th. M. Rassias. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadraticfunctional equation in Banach spaces.

We study the policy iteration algorithm (PIA) for continuous-time jump Markov decision processes in general state and action spaces. The corresponding transition rates are allowed to be unbounded, and the reward rates may have neither upper nor lower bounds. The criterion that we are concerned with is expected average reward. We propose a set of conditions under which we first establish the average reward optimality equation and present the PIA. Then under two slightly different sets of conditions we show that the PIA yields the optimal (maximum) reward, an average optimal stationary policy, and a solution to the average reward optimality equation.

This paper concerns solutions for the Hamiltonian system: $\dot{z}=𝒥H_z\left(t,z\right)$.Here $H\left(t,z\right)=\left(1/2\right)z\cdot Lz+W\left(t,z\right)$, $L$ is a $2N\times 2N$ symmetric matrix, and $W\in C^1\left(ℝ\times {ℝ}^{2N},ℝ\right)$. We consider the case that $0\in {\sigma }_c\left(-\left(𝒥\left(d/dt\right)+L\right)\right)$ and $W$ satisfies some superquadratic condition different from the type of Ambrosetti-Rabinowitz. We study this problem by virtue of some weak linking theorem recently developed and prove the existence of homoclinic orbits.

By the continuation theorem of coincidence degree and $M$-matrix theory, we obtain some sufficient conditions for the existence and exponential stability of periodic solutions for a class of generalized neural networks with arbitrary delays, which are milder and less restrictive than those of previous known criteria. Moreover our results generalize and improve many existing ones.

The purpose of this paper is to give a general and short principlefor proving some convergence results of certain types of iterative sequences.A small gap in the paper by Imnang and Suantai (2009) is discussed and corrected. Finally, we prove that the generalized asymptoticallyquasi-nonexpansive mappings in the sense of Lan (2006) are nothing but asymptotically quasi-nonexpansive. Hence several results concerning these mappings become a special case of the known ones.

It has been known for some time that proportional output feedback will stabilize MIMO, minimum-phase, linear time-invariant systems if the feedback gain is sufficiently large. High-gain adaptive controllers achieve stability by automatically driving up the feedback gain monotonically. More recently, it was demonstrated that sample-and-hold implementations of the high-gain adaptive controller also require adaptation of the sampling rate. In this paper, we use recent advances in the mathematical field of dynamic equations on time scales to unify and generalize the discrete and continuous versions of the high-gain adaptive controller. We prove the stability of high-gain adaptive controllers on a wide class of time scales.