The purpose of this paper is to examine oscillatory properties of the third-order neutral delay differential equation $[a(t)(b(t)(x(t)+p(t)x(\sigma (t)))\prime )\prime ]\prime +q(t)x(\tau (t))=0$. Some oscillatory and asymptotic criteria are presented. These criteria improve and complement those results in the literature. Moreover, some examples are given to illustrate the main results.

The paper investigates a dynamic equation $\Delta y\left(t_n\right)=\beta \left(t_n\right)\left[y\left(t_{n-j}\right)-y\left(t_{n-k}\right)\right]$ for $n\to \infty$, where $k$ and $j$ are integers such that $k>j\ge 0$, on an arbitrary discrete time scale $\mathbb{T}:=\left\{t_n\right\}$ with $t_n\in ℝ$, $n\in {\mathbb{Z}}_{n_0-k}^{\infty }=\{n_0-k,n_0-k+1,\dots \}$, $n_0\in \mathbb{N}$, , $\Delta y\left(t_n\right)=y\left(t_{n+1}\right)-y\left(t_n\right)$, and ${\text{lim}}_{n\to \infty }{t}_n=\infty$. We assume $\beta :\mathbb{T}\to \left(0,\infty \right)$. It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent for $n\to \infty$. The results are presented as inequalities for the function $\beta$. Examples demonstrate that the criteria obtained are sharp in a sense.

We first present two convergence results about the second-order quadratic variations of the subfractional Brownian motion: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we combine these results and concentration inequalities to build confidence intervals for the self-similarity parameter associated with one-dimensional subfractional Brownian motion.

We establish some stability results over $p$-adic fields for the generalized quadratic functional equation ${\sum }_{k=2}^n{\sum }_{i_1=2}^k{\sum }_{i_2=i_1+1}^{k+1}\cdots {\sum }_{i_{n-k+1}=i_{n-k}+1}^{n}f({\sum }_{i=1,i\ne i_1,\dots ,i_{n-k+1}}^n{x}_i-{\sum }_{r=1}^{n-k+1}{x}_{i_r})+f\left({\sum }_{i=1}^n{x}_i\right)=2^{n-1}{\sum }_{i=1}^{n}f(x_i),$ where $n\in \mathbb{N}$ and $n\ge 2$.

Badora (2002) proved the following stability result. Let $\epsilon$ and $\delta$ be nonnegative real numbers, then for every mapping $f$ of a ring $\mathcal{R}$ onto a Banach algebra $\mathcal{B}$ satisfying $\left|\right|f(x+y)-f\left(x\right)-f\left(y\right)\left|\right|\le \epsilon$ and $\left|\right|f(x\cdot y)-f\left(x\right)f\left(y\right)\left|\right|\le \delta$ for all $x,y\in \mathcal{R}$, there exists a unique ring homomorphism $h:\mathcal{R}\to \mathcal{B}$ such that $\left|\right|f\left(x\right)-h\left(x\right)\left|\right|\le \epsilon ,x\in \mathcal{R}$. Moreover, $b\cdot \left(f\right(x)-h(x\left)\right)=0,\left(f\right(x)-h(x\left)\right)\cdot b=0$, for all $x\in \mathcal{R}$ and all $b$ from the algebra generated by $h\left(\mathcal{R}\right)$. In this paper, we generalize Badora's stability result above on ring homomorphisms for Riesz algebras with extended norms.

The present paper deles with a fractional differential equation $z^{\alpha }{D}_z^{\alpha }u\left(z\right)+zu\text{'}\left(z\right)+(z^2-a^2)u\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^{n+\alpha }$, , where in sense of Srivastava-Owa fractional operators. The existence and uniqueness of holomorphic solutions are established. Ulam stability for the approximation and holomorphic solutions are suggested.

The purpose of this paper is to investigate some properties of $q$-Euler numbers and polynomials with weight 0. From those $q$-Euler numbers with weight 0, we derive some identities on the $q$-Euler numbers and polynomials with weight 0.

We firstly construct a concrete semi-invex set which is not invex. Basing on concept of semi-invex set, we introduce some kinds of generalized convex functions, which include semi-$(B,G)$-preinvex functions, strictly semi-$(B,G)$-preinvex functions and explicitly semi-$(B,G)$-preinvex functions. Moreover, we establish relationships between our new generalized convexity and generalized convexity introduced in the literature. With these relationships and the well-known results pertaining to common generalized convexity, we obtain results for our new generalized convexities. We extend the existing results in the literature.

It is shown that occasionally $\mathcal{J}\mathcal{H}$ operators as well as occasionally weakly biased mappings reduce to weakly compatible mappings in the presence of a unique point of coincidence (and a unique common fixed point) of the given maps.

Let $\mathcal{F}$ be a family of meromorphic functions defined in $D$, let $\psi \left(\cancel{\not\equiv }0\right)$, $a_0,a_1,...,a_{k-1}$ be holomorphic functions in $D$, and let $k$ be a positive integer. Suppose that, for every function $f\in \mathcal{F}$, $f\ne 0$, $P\left(f\right)=f^{\left(k\right)}+a_{k-1}{f}^{(k-1)}+\cdots +a_{1}f\text{'}+a_{0}f\ne 0$ and, for every pair functions $(f,g)\in \mathcal{F}$, $P\left(f\right)$, $P\left(g\right)$ share $\psi$, then $\mathcal{F}$ is normal in $D$.

We study multiplicative isometries on the following $F$-algebras of holomorphic functions: Smirnov class $N_{*}(X)$, Privalov class $N^p(X)$, Bergman-Privalov class $A{N}_{\alpha }^p(X),$ and Zygmund $F$-algebra $N{\log }^{\beta }N(X),$ where $X$ is the open unit ball ${𝔹}_n$ or the open unit polydisk ${𝔻}^n$ in ${ℂ}^n$.

We consider the existence of the periodic solutions in the neighbourhood of equilibria for $C^{\infty }$ equivariant Hamiltonian vector fields. If the equivariant symmetry $S$ acts antisymplectically and $S^2=I$, we prove that generically purely imaginary eigenvalues are doubly degenerate and the equilibrium is contained in a local two-dimensional flow-invariant manifold, consisting of a one-parameter family of symmetric periodic solutions and two two-dimensional flow-invariant manifolds each containing a one-parameter family of nonsymmetric periodic solutions. The result is a version of Liapunov Center theorem for a class of equivariant Hamiltonian systems.

Compact subsets of a topological space are used to define coc-open sets as new generalized open sets, and then coc-open sets are used to define (coc) ^∗ -open sets as another type of generalized open sets. Several results and examples related to them are obtained; particularly a decomposition of open sets is given. Also, coc-open sets and (coc) ^∗ -open sets are used to introduce coc-continuity and (coc) ^∗ -continuity, respectively. As a main result, a decomposition theorem of continuity is obtained.

The asymptotic behavior of nonoscillatory solutions of the superlinear dynamic equation on time scales ${\left(r\right(t\left)x^{\Delta }\right(t\left)\right)}^{\Delta }+p\left(t\right){\left|x\left(\sigma \right(t\left)\right)\right|}^{\gamma }\text{sgn}x\left(\sigma \right(t\left)\right)=0$, $\gamma >\mathrm{1,}$ is discussed under the condition that $P\left(t\right)={\text{lim}}_{\tau \to \infty }{\displaystyle {\int }_t^{\tau }p\left(s\right)\Delta s}$ exists and $P(t)\ge 0$ for large $t$.

We study a $q$-analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by Malek in (2011). First, we construct solutions defined in open $q$-spirals to the origin. By means of a $q$-Gevrey version of Malgrange-Sibuya theorem we show the existence of a formal power series in the perturbation parameter which turns out to be the $q$-Gevrey asymptotic expansion (of certain type) of the actual solutions.

A Lotka-Volterra-type predator-prey system with state-dependent feedback control is investigated in both theoretical and numerical ways. Using the Poincaré map and the analogue of the Poincaré criterion, the sufficient conditions for the existence and stability of semitrivial periodic solutions and positive periodic solutions are obtained. In addition, we show that there is no positive periodic solution with period greater than and equal to three under some conditions. The qualitative analysis shows that the positive period-one solution bifurcates from the semitrivial solution through a fold bifurcation. Numerical simulations to substantiate our theoretical results are provided. Also, the bifurcation diagrams of solutions are illustrated by using the Poincaré map, and it is shown that the chaotic solutions take place via a cascade of period-doubling bifurcations.

By using a linear scalarization method, we establish sufficient conditions for the Hölder continuity of the solution mappings to a parametric generalized vector quasiequilibrium problem with set-valued mappings. These results extend the recent ones in the recent literature, (e.g., Li et al. (2009), Li et al. (2011)). Furthermore, two examples are given to illustrate the obtained result.

We investigate the stability problems for a functional equation $2f\left({\sum }_{j=1}^n{x}_j\right)+$ ${\sum }_{1\le i,j\le n,i\ne j}f(x_i-x_j)=(n+1){\sum }_{j=1}^{n}f\left(x_j\right)+(n-1){\sum }_{j=1}^{n}f(-x_j)$ by using the fixed point method.

We proved a fixed point theorem for a class of maps that satisfy Ćirić's contractive condition dependent on another function. We presented an example to show that our result is a real generalization.

We investigate the generalized Drazin inverse of $A-CB$ over Banach spaces stemmed from the Drazin inverse of a modified matrix and present its expressions under some conditions.

By applying formal asymptotic analysis and Laplace transformation, we obtain two-dimensional nonlinear viscoelastic shells model satisfied by the leading term of asymptotic expansion of the solution to the three-dimensional equations.

We deal with the quasi-periodic solutions of the following second-order Hamiltonian systems $\ddot{x}$ $\left(t\right)=\nabla F\left(t,x\left(t\right)\right)$, where $x\left(t\right)=\left(x_1\left(t\right),\dots ,x_N\left(t\right)\right)$, and we present a new approach via variational methods and Minmax method to obtain the existence of quasi-periodic solutions to the above equation.

Some inequalities of the Slater type for convex functions defined on general linear spaces are given. Applications for norm inequalities and f-divergence measures are also provided.

We present some conditions for the existence and uniqueness of almost periodic solutions of $N$th-order neutral differential equations with piecewise constant arguments of the form ${\left(x\left(t\right)+px\left(t-1\right)\right)}^{\left(N\right)}=qx\left(\left[t\right]\right)+f\left(t\right)$, here $[\cdot ]$ is the greatest integer function, $p$ and $q$ are nonzero constants, $N$ is a positive integer, and $f\left(t\right)$ is almost periodic.

In this paper, by using the potential theory we prove the existence of filling discs dealing with multiple values of an algebroid function of finite order defined in the unit disc.

We study the univalence conditions for two integral operators to be univalent in the open unit disk. Many known univalence conditions are written to prove our main results.

The purpose of this paper is to give some properties of the modified $q$-Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonic $p$-adic $q$-integral on ${\mathbb{Z}}_p$, we derive new identities of $q$-Bernoulli numbers and polynomials with weight.

In this paper, we present the existence of extremal solutions of set-valued differential equations with feedback control on semilinear Hausdorff space under Hukuhara derivative which is developed under the form $D_{H}X\left(t\right)=F\left(t,X\left(t\right),H(t,X(t))\right)$, $X(0)=X_0$, for all $t\in \left[0,T\right]$ with the monotone iterative technique and we will verify that monotone sequence of approximate solutions converging uniformly to the solution of the problem, that is useful for optimization problems.

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomials $L\left(f\right)=d_{2}f″+d_1{f}^{\prime }+d_{0}f$ generated by solutions of the above equation, where $d_0\left(z\right),d_1\left(z\right),$ and $d_2\left(z\right)$ are entire functions that are not all equal to zero.

Using the Lyapunov direct method, the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference is studied. The conditions for uniform stability, uniform asymptotic stability, and uniform global stability are discussed.

We study slant submanifolds of a cosymplectic manifold. It is shown that a totally umbilical slant submanifold $M$ of a cosymplectic manifold $\bar{M}$ is either an anti-invariant submanifold or a 1−dimensional submanifold. We show that every totally umbilical proper slant submanifold of a cosymplectic manifold is totally geodesic.

A hybrid system with state-based switchings is proposed to describe the fed-batch production of 1,3-propandiol from glycerol in our previous work. However, the on-off switching of alkali is too frequent, which greatly increases the computational cost of the numerical solution to the system so as to locate the state-based switchings in strict time order and implement the correct mode changes. To deal with this problem, we consider the switching of alkali pump as an impulsive event and present a nonlinear impulsive switching system to describe the fed-batch culture. It is proved that the impulsive switching system is non-Zeno. Some basic properties of solutions to the impulsive switching system are also explored. In order to overcome the discontinuities of the system, the Skorohod topology is induced and a specific form of λ is constructed to prove the main theorem. Additionally, a numerical simulation is carried out to show that the proposed system can describe the fed-batch culture properly and the essential difference with the previous work.

Let A be a ternary Banach algebra. We prove that if A has a left-bounded approximating set, then A has a left-bounded approximate identity. Moreover, we show that if A has bounded left and right approximate identities, then A has a bounded approximate identity. Hence, we prove Altman’s Theorem and Dixon’s Theorem for ternary Banach algebras.

We study the existence of solutions for the boundary value problem $-{\Delta }^{\nu }{y}_1\left(t\right)=f\left(y_1\right(t+\nu -1),y_2(t+\mu -1\left)\right)$, $-{\Delta }^{\mu }{y}_2\left(t\right)=g\left(y_1\right(t+\nu -1),y_2(t+\mu -1\left)\right)$, $y_1(\nu -2)=\Delta y_1(\nu +b)=0$, $y_2(\mu -2)=\Delta y_2(\mu +b)=0$, where , $f,g:\mathbb{R}\{\backslash times\}\mathbb{R}\to \mathbb{R}$ are continuous functions, $b\in {\mathbb{N}}_0$. The existence of solutions to this problem is established by the Guo-Krasnosel'kii theorem and the Schauder fixed-point theorem, and some examples are given to illustrate the main results.

A previous paper (2011), Pitea and Postolache, considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problem, subject to PDE and/or PDI constraints. They have chosen the suitable framework offered by the second-order jet bundle, and initiated an optimization theory for this class of problems by introducing necessary conditions. As natural continuation of these results, the present work introduces a dual program theory, the general setting, and the theory which is new as a whole, containing our results.

We discuss a two-group SEIR epidemic model with distributed delays, incorporating random fluctuation around the endemic equilibrium. Our research shows that the endemic equilibrium of the model with distributed delays and random perturbation is stochastically asymptotically stable in the large. In addition, a sufficient stability condition is obtained by constructing suitable Lyapunov function.

Using methods from the theory of commutative graded Banach algebras, we obtain a generalization of the two-dimensional Borsuk-Ulam theorem as follows. Let $\varphi :S^2\to S^2$ be a homeomorphism of order $n$, and let $\lambda \ne 1$ be an $n$th root of the unity, then, for every complex valued continuous function $f$ on $S^2$, the function ${\sum }_{i=0}^{n-1}{\lambda }^{i}f\left({\varphi }^i\right(x\left)\right)$ must vanish at some point of $S^2$. We also discuss some noncommutative versions of the Borsuk-Ulam theorem.

A Banach space $E$ is said to have (D) property if every bounded linear operator $T:F\to E^{\mathrm{\{\backslash ast\}}}$ is weakly compact for every Banach space $F$ whose dual does not contain an isomorphic copy of $l_{\infty }$. Studying this property in connection with other geometric properties, we show that every Banach space whose dual has (V ^∗) property of Pełczyński (and hence every Banach space with (V) property) has (D) property. We show that the space $L^1\left(v\right)$ of real functions, which are integrable with respect to a measure $v$ with values in a Banach space $X$, has (D) property. We give some other results concerning Banach spaces with (D) property.

This paper deals with sequential Riesz-Caputo fractional variational problems with and without the presence of delay in the state variables and their derivatives. In both cases the necessary conditions for the optimal control are reported.

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem: $‖(\begin{smallmatrix}AXB\\[5pt] CXD\end{smallmatrix})-(\begin{smallmatrix}E\\[5pt] F\end{smallmatrix})‖=$ min over generalized reflexive matrix $X$. For any initial generalized reflexive matrix ${X}_{1}$, by the iterative algorithm, the generalized reflexive solution ${X}^{\ast}$ can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution ${X}^{\ast}$ can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution $\widehat{X}$ to a given matrix ${X}_{0}$ in Frobenius norm can be derived by finding the least-norm generalized reflexive solution ${\stackrel{~}{X}}^{\ast}$ of a new corresponding minimum Frobenius norm residual problem: $\text{min}‖(\begin{smallmatrix}A\stackrel{~}{X}B\\[5pt] C\stackrel{~}{X}D\end{smallmatrix})-(\begin{smallmatrix}\stackrel{~}{E}\\[5pt] \stackrel{~}{F}\end{smallmatrix})‖$ with $\stackrel{~}{E}=E-A{X}_{0}B$, $\stackrel{~}{F}=F-C{X}_{0}D$. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.

The problem of exponential stabilization of neutral-type neural networks with various activation functions and interval nondifferentiable and distributed time-varying delays is considered. The interval time-varying delay function is not required to be differentiable. By employing new and improved Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, the stabilizability criteria are formulated in terms of a linear matrix inequalities. Numerical examples are given to illustrate and show the effectiveness of the obtained results.

We determine the conditions for some matrix transformations from $n(\varphi )$, where the sequence space $n(\varphi )$, which is related to the ${\ell }_p$ spaces, was introduced by Sargent (1960). We also obtain estimates for the norms of the bounded linear operators defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.

This paper describes an index for judging the severity of transient events of power systems in simulation. The proposed transient stability index, known as the area-based COI-referred rotor angle index, is developed by considering the fact that a large-sized power system is divided into several areas according to the coherency of generators in a particular area. It can be assumed that an equivalent single large machine can represent all the generators in that area. Thus, the assessment of rotor angles for all generators can be simplified by only assessing the index of areas in a power system. The effectiveness of the proposed index in assessing the stability of power systems and its ability in pinpointing the weakest area in the power system is analyzed. Furthermore, this paper developed an emergency control scheme known as the combined UFLS and generator tripping in order to stabilize the system when unstable faults occurred in a power system. The proposed index is used to identify the generator to be tripped when the developed emergency control scheme operates. The performance of the proposed index and the combined UFLS and generator tripping scheme are evaluated on the IEEE 39-bus test system.

Using fixed point methods, we establish the stability of $n$-Lie homomorphisms and Jordan $n$-Lie homomorphisms on $n$-Lie algebras associated to the following generalized Jensen functional equation $\mu f({\sum }_{i=1}^n{x}_i/n)+\mu {\sum }_{j=2}^{n}f({\sum }_{i=1,i\ne j}^n{x}_i-(n-1)x_j/n)=f\left(\mu x_1\right)(n\ge 2)$.

We introduce an iterative method for finding a common fixed point of a countable family of multivalued quasi-nonexpansive mapping $\left\{T_i\right\}$ in a uniformly convex Banach space. We prove that under certain control conditions, the iterative sequence generated by our method is an approximating fixed point sequence of each $T_i$. Some strong convergence theorems of the proposed method are also obtained for the following cases: all $T_i$ are continuous and one of $T_i$ is hemicompact, and the domain $K$ is compact.

We consider the fourth-order differential equation with middle-term and deviating argument $x^{(4)}(t)+q(t)x^{(2)}(t)+r(t)f(x(\phi (t)))=0$, in case when the corresponding second-order equation $h^{″}+q(t)h=0$ is oscillatory. Necessary and sufficient conditions for the existence of bounded and unbounded asymptotically linear solutions are given. The roles of the deviating argument and the nonlinearity are explained, too.

We will give the Lebesgue-Radon-Nikodym theorem with respect to weighted $p$-adic $q$-measure on ${\mathbb{Z}}_p$. In special case, $q=1$, we can derive the same result as Kim, 2012; Kim et al., 2011.

For zero-balanced Gaussian hypergeometric functions $F(a,b;a+b;x)$, $a,b>0$, we determine maximal regions of $ab$ plane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each $x\in \left(\mathrm{0,1}\right)$. Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.

New coupled coincidence point and coupled fixed point results in ordered partial metric spaces under the contractive conditions of Geraghty, Rakotch, and Branciari types are obtained. Examples show that these results are distinct from the known ones.

We prove the equivalence and the strong convergence of (1) the modified Mann iterative process and (2) the modified Ishikawa iterative process for asymptotically $\varphi$-strongly pseudocontractive mappings in a uniformly smooth Banach space.

We establish some delay integral inequalities on time scales, which on one hand provide a handy tool in the study of qualitative as well as quantitative properties of solutions of certain delay dynamic equations on time scales and on the other hand unify some known continuous and discrete results in the literature.

We first give a solution to a key problem concerning the completeness of the space of weighted pseudo almost-periodic functions and then establish a new composition theorem with respect to these functions. Some important remarks with concrete examples are also presented. Moreover, we prove an existence theorem for the weighted pseudo almost-periodic mild solution to the semilinear evolution equation: $x^{\prime }\left(t\right)=Ax\left(t\right)+f\left(t,x\left(t\right)\right)$, $t\in \mathrm{\backslash Bbb\; R}$, where $A$ is the infinitesimal generator of an exponentially stable $C_0$-semigroup. An application is also given to illustrate the abstract existence theorem.

This paper applies an ancient Chinese algorithm to differential-difference equations, and a solitary-solution formulation is obtained. The discrete mKdV lattice equation is used as an example to elucidate the solution procedure.

This paper deals with the problems of integrability and linearizable conditions at degenerate singular point in a class of quasianalytic septic polynomial differential system. We solve the problems by an indirect method, that is, we transform the quasianalytic system into an analytic system firstly, and the degenerate singular point into an elementary singular point. Then we calculate the singular values at the origin of the analytic system by the known classical methods. We obtain the center conditions and isochronous center conditions. Accordingly, integrability and pseudolinearizable conditions at degenerate singular point in the quasianalytic system are obtained. Especially, when $\lambda =1$, the system has been studied in Wu and Zhang (2010).

By variational methods and some analysis techniques, the multiplicity of positive solutions is obtained for a class of weighted quasilinear elliptic equations with critical Hardy-Sobolev exponents and concave-convex nonlinearities.

We investigate relations between solutions, their derivatives of differential equation $f^{\left(k\right)}+A_{k-1}{f}^{\left(k-1\right)}+\cdots +A_1{f}^{’}+A_{0}f=0$, and functions of small growth, where $A_j\left(j=0,1,\dots ,k-1\right)$ are entire functions of finite order. By these relations, we see that every transcendental solution and its derivative of above equation have infinitely many fixed points.

We investigate the stability and superstability of ternary homomorphisms between $C^{\mathrm{\{\backslash ast\}}}$-ternary algebras and derivations on $C^{\mathrm{\{\backslash ast\}}}$-ternary algebras, associated with the following functional equation $f\left(\right(x_2-x_1)/3)+f\left(\right(x_1-3x_3)/3)+f\left(\right(3x_1+3x_3-x_2)/3)=f\left(x_1\right)$.

We study the abstract Cauchy problem for a class of integrodifferential equations in a Banach space with nonlinear perturbations and nonlocal conditions. By using MNC estimates, the existence and continuous dependence results are proved. Under some additional assumptions, we study the topological structure of the solution set.

This paper investigates the consensus problem in mean square for uncertain multiagent systems with stochastic measurement noises and symmetric or asymmetric time-varying delays. By combining the tools of stochastic analysis, algebraic graph theory, and matrix theory, we analyze the convergence of a class of distributed stochastic approximation type protocols with time-varying consensus gains. Numerical examples are also given to illustrate the theoretical results.

We use a method of investigation based on employing adequate variational calculus techniques in the study of some generalized Dieudonné-Rashevski problems. This approach allows us to state and prove optimality conditions for such kind of vector multitime variational problems, with mixed isoperimetric constraints. We state and prove efficiency conditions and develop a duality theory.

In this paper, the numerical range for two operators (both linear and nonlinear) have been studied in semi-inner product spaces. The inclusion relations between numerical range, approximate point spectrum, compression spectrum, eigenspectrum, and spectrum have been established for two linear operators. We also show the inclusion relation between approximate point spectrum and closure of the numerical range for two nonlinear operators. An approximation method for solving the operator equation involving two nonlinear operators is also established.

We study the following nonperiodic Hamiltonian system $\dot{z}=\mathcal{J}{H}_z(t,z)$, where $H\in C^1(\mathbb{R}\{\backslash times\}{\mathbb{R}}^{2N},\mathbb{R})$ is the form $H\left(t,z\right)=\left(1/2\right)B\left(t\right)z\cdot z+R(t,z)$. We introduce a new assumption on $B\left(t\right)$ and prove that the corresponding Hamiltonian operator has only point spectrum. Moreover, by applying a generalized linking theorem for strongly indefinite functionals, we establish the existence of homoclinic orbits for asymptotically quadratic nonlinearity as well as the existence of infinitely many homoclinic orbits for superquadratic nonlinearity.

The aim of this work is to characterize one-dimensional homogeneous diffusion process, under the assumption that marginal density of the process is Gaussian. The method considers the forward Kolmogorov equation and Fourier transform operator approach. The result establishes the necessary characteristic equation between drift and diffusion coefficients for homogeneous and nonhomogeneous diffusion processes. The equation for homogeneous diffusion process leads to characterize the possible diffusion processes that can exist. Two well-known examples using the necessary characteristic equation are also given.

A simple Cohen-Grossberg neural network with discrete delays is investigated in this paper. The existence of local Hopf bifurcations is first considered by choosing the appropriate bifurcation parameter, and then explicit formulas are given to determine the direction of Hopf bifurcation and stability of the periodic solutions. Moreover, a set of sufficient conditions are given to guarantee the global Hopf bifurcation. Numerical simulations are given to illustrate the obtained results.

We prove that a smooth solution of the 3D Cahn-Hilliard-Boussinesq system with zero viscosity in a bounded domain breaks down if a certain norm of vorticity blows up at the same time. Here, this norm is weaker than bmo-norm.

We study weak sharp minima for optimization problems with cone constraints. Some necessary conditions for weak sharp minima of higher order are established by means of upper Studniarski or Dini directional derivatives. In particular, when the objective and constrained functions are strict derivative, a necessary condition is obtained by a normal cone.

We investigate some properties and identities of Bernoulli and Euler polynomials. Further, we give some formulae on Bernoulli and Euler polynomials by using p-adic integral on ${\mathbb{Z}}_p$.

We give some initial properties of a subset of modular metric spaces and introduce some fixed-point theorems for multivalued mappings under the setting of contraction type. An appropriate example is as well provided. The stability of fixed points in our main theorems is also studied.

We investigate the generalized Hyers-Ulam stability of the following functional equation ${\sum }_{i=1}^{m}f(x_i)=1/2m\left[{\sum }_{i=1}^{m}f\left(m{x}_i+{\sum }_{j=1,j\ne i}^m{x}_j\right)+f\left({\sum }_{i=1}^m{x}_i\right)\right]$ for a fixed positive integer $m$ with $m\ge 2$ in quasi-Banach spaces.

We construct a novel reproducing kernel space and give the expression of reproducing kernel skillfully. Based on the orthogonal basis of the reproducing kernel space, an efficient algorithm is provided firstly to solve a three-point boundary value problem of parabolic equations with two-space integral condition. The exact solution of this problem can be expressed by the series form. The numerical method is supported by strong theories. The numerical experiment shows that the algorithm is simple and easy to implement by the common computer and software.

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equation $f\left(x+ky\right)+f\left(x-ky\right)=f\left(x+y\right)+f\left(x-y\right)+$ $\left(2\left(k+1\right)/k\right)f\left(ky\right)-2\left(k+1\right)f\left(y\right)$ for fixed integers $k$ with $k\ne 0,\pm 1$ in fuzzy Banach spaces.

We introduce a generalization of the Meir-Keeler-type contractions, referred to as generalized Meir-Keeler-type contractions, over partial metric spaces. Moreover, we show that every orbitally continuous generalized Meir-Keeler-type contraction has a fixed point on a 0-complete partial metric space.

We establish some stability results in 2-normed spaces for the radical quadratic functional equation $f\left(\sqrt{{\sum }_{i=1}^n(x_i+y_i{)}^2}\right)+f\left(\sqrt{{\sum }_{i=1}^n(x_i-y_i{)}^2}\right)=2{\sum }_{i=1}^n\left(f\left(x_i\right)+f\left(y_i\right)\right)$ and then use subadditive functions to prove its stability in $p$-2-normed spaces.

In this paper, the asymptotic stability of the analytic and numerical solutions for differential equations with piecewise continuous arguments is investigated by using Lyapunov methods. In particular, the linear equations with variable coefficients are considered. The stability conditions of the analytic solutions of those equations and the numerical solutions of the $\theta$-methods are obtained. Some examples are illustrated.

An analytical quadrilateral p element is developed for solving the free vibrations of piezoelectric-laminated plates. The formulations of the displacement and strain fields are based on first-order shear deformation plate theory. The coupling effect between the electrical and stress fields is also considered. The Legendre orthogonal polynomials are used as the element interpolation functions, and the analytical integration technique is adopted. It is found that the present p element method gives high numerical precision results, fast and monotonic convergence rate. In the numerical cases, the effects of the number of hierarchical terms and mesh size on the convergence rate are investigated. Examples of square plates with different displacement and potential boundary conditions are studied. In the comparisons, the solutions of the present element are in good agreement with those obtained from other classical and finite element methods.

By introducing subdifferentiability of lower semicontinuous convex function $\phi \left(x\right(t),x(t-\tau \left)\right)$ and its conjugate function, as well as critical point theory and operator equation theory, we obtain the existence of multiple subharmonic periodic solutions to the following second-order nonlinear nonautonomous neutral nonlinear functional differential equation $x^{″}\left(t\right)+x^{″}(t-2\tau )+f(t,x(t),x(t-\tau ),x(t-2\tau \left)\right)=0$, $x\left(0\right)=0.$

We present a sort of Korovkin-type result that provides a tool to obtain asymptotic formulae for sequences of linear positive operators.

Let $X$ and $Y$ be vector spaces. We show that a function $f:X\to Y$ with $f\left(0\right)=0$ satisfies $\Delta f(x_1,\dots ,x_n)=0$ for all $x_1,\dots ,x_n\in X$, if and only if there exist functions $C:X\times X\times X\to Y$, $B:X\times X\to Y$ and $A:X\to Y$ such that $f\left(x\right)=C(x,x,x)+B(x,x)+A\left(x\right)$ for all $x\in X$, where the function $C$ is symmetric for each fixed one variable and is additive for fixed two variables, $B$ is symmetric bi-additive, $A$ is additive and $\Delta f(x_1,\dots ,x_n)=$ ${\sum }_{k=2}^n({\sum }_{i_1=2}^k{\sum }_{i_2=i_1+1}^{k+1}\cdots {\sum }_{i_{n-k+1}=i_{n-k}+1}^n)f({\sum }_{i=1,i\ne i_1,\dots ,i_{n-k+1}}^n{x}_i-{\sum }_{r=1}^{n-k+1}{x}_{i_r})+$ $f\left({\sum }_{i=1}^n{x}_i\right)-2^{n-2}{\sum }_{i=2}^n\left(f\right(x_1+x_i)+f(x_1-x_i\left)\right)$ $+2^{n-1}(n-2)f\left(x_1\right)$ ($n\in \mathbb{N}$, $n\ge 3$) for all $x_1,\dots ,x_n\in X$. Furthermore, we solve the stability problem for a given function $f$ satisfying $\Delta f(x_1,\dots ,x_n)=0$, in the Menger probabilistic normed spaces.

Under arbitrary masses, in this paper, we discuss the existence of new families of spatial central configurations for the N + N + 2-body problem, $N\ge 2$. We study some necessary conditions and sufficient conditions for a families of spatial double pyramidical central configurations (d.p.c.c.), where 2N bodies are at the vertices of a nested regular N-gons ${\Gamma }_1\cup {\Gamma }_2$, and the other two bodies are symmetrically located on the straight line that is perpendicular to the plane that contains ${\Gamma }_1\cup {\Gamma }_2$ and passes through the geometric center of ${\Gamma }_1\cup {\Gamma }_2$. We prove that if the bodies are in a d.p.c.c., then the masses on each N-gon are equal, and the other two are also equal. And also we prove the existence and uniqueness of the central configurations for any given ratios of masses.

The purpose of this paper is to introduce new spaces $\widehat{f}$ and ${\widehat{f}}_0$ that consist of all sequences whose Riesz transforms of order one are in the spaces $f$ and $f_0$, respectively. We also show that $\widehat{f}$ and ${\widehat{f}}_0$ are linearly isomorphic to the spaces $f$ and $f_0$, respectively. The $\beta \text{-}$ and $\gamma \text{-}$duals of the spaces $\widehat{f}$ and ${\widehat{f}}_0$ are computed. Furthermore, the classes $(\widehat{f}:\mu )$ and $(\mu :\widehat{f})$ of infinite matrices are characterized for any given sequence space $\mu$ and determine the necessary and sufficient conditions on a matrix $A$ to satisfy $B_R-\text{c}\text{o}\text{r}\text{e}\left(Ax\right)\subseteq K-\text{c}\text{o}\text{r}\text{e}\left(x\right)$, $B_R-\text{c}\text{o}\text{r}\text{e}\left(Ax\right)\subseteq st-\text{c}\text{o}\text{r}\text{e}\left(x\right)$ for all $x\in {\mathcal{l}}_{\infty }$.

This paper develops the optimal fault-tolerant guaranteed cost control scheme for a batch process with actuator failures. Based on an equivalent two-dimensional Fornasini-Marchsini (2D-FM) model description of a batch process, the relevant concepts of the fault-tolerant guaranteed cost control are introduced. The robust iterative learning reliable guaranteed cost controller (ILRGCC), which includes a robust extended feedback control for ensuring the performances over time and an iterative learning control (ILC) for improving the tracking performance from cycle to cycle, is formulated such that it cannot only guarantee the closed-loop convergency along both the time and the cycle directions but also satisfy both the $H_{\infty }$ performance level and a cost function having upper bounds for all admissible uncertainties and any actuator failures. Conditions for the existence of the controller are derived in terms of linear matrix inequalities (LMIs), and a design procedure of the controller is presented. Furthermore, a convex optimization problem with LMI constraints is formulated to design the optimal guaranteed cost controller which minimizes the upper bound of the closed-loop system cost. Finally, an illustrative example of injection molding is given to demonstrate the effectiveness and advantages of the proposed 2D design approach.

We give some identities on the q-Bernoulli and q-Euler numbers by using p-adic integral equations on ${\mathbb{Z}}_p$.

We consider functionals derived from Petrović-type inequalities and establish their superadditivity, subadditivity, and monotonicity properties on the corresponding real n-tuples. By virtue of established results we also define some related functionals and investigate their properties regarding exponential convexity. Finally, the general results are then applied to some particular settings.

We extend the recent results of the coupled fixed point theorems of Cho et al. (2012) by weakening the concept of the mixed monotone property. We also give some examples of a nonlinear contraction mapping, which is not applied to the existence of the coupled fixed point by the results of Cho et al. but can be applied to our results. The main results extend and unify the results of Cho et al. and many results of the coupled fixed point theorems.

This paper is concerned with strictly cyclic functionals of operator algebras on Banach spaces. It is shown that if X is a reflexive Banach space and $\mathcal{A}$ is a norm-closed semisimple abelian subalgebra of B(X) with a strictly cyclic functional $f\in X^{\ast }$, then $\mathcal{A}$ is reflexive and hereditarily reflexive. Moreover, we construct a semisimple abelian operator algebra having a strictly cyclic functional but having no strictly cyclic vectors. The hereditary reflexivity of an algbra of this type can follow from theorems in this paper, but does not follow directly from the known theorems that, if a strictly cyclic operator algebra on Banach spaces is semisimple and abelian, then it is a hereditarily reflexive algebra.

An integral equation of Volterra type with additional compact operator in Banach space is considered. A special case is an integral equation of contact problem that arises in theory of viscoelasticity of mixed Fredholm and Volterra type with spectral parameter depending on time. In case the initial value of the parameter coincides with some isolated point of the spectrum of compact operator, the conditions of solvability are established.

We generalize Jungck's theorem in Jungck (1976) to fuzzy metric spaces and prove common fixed point theorems for commutative mappings in fuzzy metric spaces.

Recently many mathematicians are working on Genocchi polynomials and Genocchi numbers. We define a new type of twisted q-Genocchi numbers and polynomials with weight $\alpha$ and weak weight $\beta$ and give some interesting relations of the twisted q-Genocchi numbers and polynomials with weight $\alpha$ and weak weight $\beta$. Finally, we find relations between twisted q-Genocchi zeta function and twisted Hurwitz q-Genocchi zeta function.

We establish a new LaSalle's invariance principle and discuss the asymptotic behavior of a class of first-order evolution variational inequalities.

This paper studies uniqueness of weak solutions to an electrohydrodynamics model in ${\mathrm{\backslash Bbb\; R}}^d\left(d=2,3\right)$. When $d=2$, we prove a uniqueness without any condition on the velocity. For $d=3$, we prove a weak-strong uniqueness result with a condition on the vorticity in the homogeneous Besov space.

Results on common fixed points of mappings in cone metric spaces under weak contractive conditions (B. S. Choudhury and N. Metiya (2010)) are unified and generalized. Also, cone metric versions of some other related results on weak contractions are proved. Examples show that our results are different than the existing ones.

We introduce new integral operators of analytic functions $f$ and $g$ defined in the open unit disk $\mathbb{U}$. For these operators, we discuss some univalence conditions.

We introduce a matrix form of differential transformation method (DTM) and apply for nonlinear second-order initial value problems (IVPs) and boundary value problems (BVPs) of matrix models which are given by ${\mathbf{u}}^{″}\left(t\right)=f\left(t,\mathbf{u}\left(t\right),{\mathbf{u}}^{\prime }\left(t\right)\right)$ and subject to initial conditions $\mathbf{u}\left(a\right)={\mathbf{u}}_0,{\mathbf{u}}^{\prime }\left(a\right)={\mathbf{u}}_1$ and boundary conditions $\mathbf{u}\left(a\right)={\mathbf{u}}_0,\mathbf{u}\left(b\right)={\mathbf{u}}_1$, where ${\mathbf{u}}_0,\hspace{0.17em}{\mathbf{u}}_1\in R^{r\times q}$. Also the convergence of present method is established. Several illustrative examples are given to demonstrate the effectiveness of the present method.

We present a numerical method to solve the linear Fredholm integro-differential equation in reproducing kernel space. A simple algorithm is given to obtain the approximate solutions of the equation. Through the comparison of approximate and true solution, we can find that the method can effectively solve the linear Fredholm integro-differential equation. At the same time the numerical solution of the equation is stable.

This paper investigates the mean-square exponential synchronization of stochastic complex networks with Markovian switching and time-varying delays by using the pinning control method. The switching parameters are modeled by a continuous-time, finite-state Markov chain, and the complex network is subject to noise perturbations, Markovian switching, and internal and outer time-varying delays. Sufficient conditions for mean-square exponential synchronization are obtained by using the Lyapunov-Krasovskii functional, Itö’s formula, and the linear matrix inequality (LMI), and numerical examples are given to demonstrate the validity of the theoretical results.

By means of the Galerkin method and by using a suitable version of the Brouwer fixed-point theorem, we establish the existence of at least one positive solution of a nonlocal elliptic N-dimensional system coupled with Dirichlet boundary conditions.

We investigate the growth of solutions of higher-order nonhomogeneous linear differential equations with meromorphic coefficients. We also discuss the relationship between small functions and solutions of such equations.

We obtain in this paper the solutions of the following recursive sequences $x_{n+1}=x_n{x}_{n-3}/x_{n-2}\left(\pm 1\pm x_n{x}_{n-3}\right)$, $n=0,1,\dots ,$ where the initial conditions are arbitrary real numbers and we study the behaviors of the solutions and we obtained the equilibrium points of the considered equations. Some qualitative behavior of the solutions such as the boundedness, the global stability, and the periodicity character of the solutions in each case have been studied. We presented some numerical examples by giving some numerical values for the initial values and the coefficients of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program Mathematica to confirm the obtained results.

We introduce and study the concepts of α-well-posedness and L-α-well-posedness for quasivariational inequality problems having a unique solution and the concepts of α-well-posedness in the generalized sense and L-α-well-posedness in the generalized sense for quasivariational inequality problems having more than one solution. We present some necessary and/or sufficient conditions for the various kinds of well-posedness to occur. Our results generalize and strengthen previously known results for quasivariational inequality problems.

This paper discusses dissipativity problem for system of linear state/input delay equations. Motivated by dissipativity theory of control systems, we choose a new quadratic supply rate. Using the concept of dissipativity, necessary and sufficient conditions for the linear state/input delay systems to be dissipative and exponentially dissipative are derived. The connection of dissipativity with stability is also considered. Finally, passivity and finite gain are explored, correspondingly. The positive-real and bounded-real lemmas are derived.

The purpose of this paper is to introduce new types of asymptotically ($g,\phi$)-contractions which generalize the Binayak S. Choudhury type contraction on fuzzy metric spaces and prove some fixed-point theorems for single- and multivalued mappings on fuzzy metric spaces. Hence, our results can be viewed as a generalization and improvement of many recent results.

This paper presents some efficient spectral algorithms for solving linear sixth-order two-point boundary value problems in one dimension based on the application of the Galerkin method. The proposed algorithms are extended to solve the two-dimensional sixth-order differential equations. A family of symmetric generalized Jacobi polynomials is introduced and used as basic functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. The various matrix systems resulting from the proposed algorithms are carefully investigated, especially their condition numbers and their complexities. These algorithms are extensions to some of the algorithms proposed by Doha and Abd-Elhameed (2002) and Doha and Bhrawy (2008) for second- and fourth-order elliptic equations, respectively. Three numerical results are presented to demonstrate the efficiency and the applicability of the proposed algorithms.

We investigate the existence and dimension of the solution set for a nonlocal problem of semilinear fractional differential inclusions. The main tools of our study include some well-known results on multivalued maps.

We introduce two powerful methods to solve the generalized Zakharov equations; one is the homotopy perturbation method and the other is the homotopy analysis method. The homotopy perturbation method is proposed for solving the generalized Zakharov equations. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions; the homotopy analysis method is applied to solve the generalized Zakharov equations. HAM is a strong and easy-to-use analytic tool for nonlinear problems. Computation of the absolute errors between the exact solutions of the GZE equations and the approximate solutions, comparison of the HPM results with those of Adomian’s decomposition method and the HAM results, and computation the absolute errors between the exact solutions of the GZE equations with the HPM solutions and HAM solutions are presented.

Intuitionistic fuzzy normed space is defined using concepts of $t$-norm and $t$-conorm. The concepts of fuzzy completeness, fuzzy minimality, fuzzy biorthogonality, fuzzy basicity, and fuzzy space of coefficients are introduced. Strong completeness of fuzzy space of coefficients with regard to fuzzy norm and strong basicity of canonical system in this space are proved. Strong basicity criterion in fuzzy Banach space is presented in terms of coefficient operator.

We investigate the DGH equation. Analogous to the Camassa-Holm equation, this equation possesses the blow-up phenomenon. We establish a new blow up criterion on the initial data to guarantee the formulation of singularities in finite time.

We show that the system of three difference equations $x_{n+1}=a_n^{\left(1\right)}{x}_{n-2}/\left(b_n^{\left(1\right)}{y}_n{z}_{n-1}{x}_{n-2}+c_n^{\left(1\right)}\right)$, $y_{n+1}=a_n^{\left(2\right)}{y}_{n-2}/\left(b_n^{\left(2\right)}{z}_n{x}_{n-1}{y}_{n-2}+c_n^{\left(2\right)}\right)$, and $z_{n+1}=a_n^{\left(3\right)}{z}_{n-2}/\left(b_n^{\left(3\right)}{x}_n{y}_{n-1}{z}_{n-2}+c_n^{\left(3\right)}\right)$, $n\in {\mathbb{N}}_0$, where all elements of the sequences $a_n^{\left(i\right)}$, $b_n^{\left(i\right)}$, $c_n^{\left(i\right)}$, $n\in {\mathbb{N}}_0$, $i\in \{\mathrm{1,2},3\}$, and initial values $x_{-j}$, $y_{-j}$, $z_{-j}$, $j\in \{\mathrm{0,1},2\}$, are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.

We consider the best approximation by Jackson-Matsuoka polynomials in the weighted $L_p$ space on the unit sphere of ${\mathbb{R}}^d$. Using the relation between $K$-functionals and modulus of smoothness on the sphere, we obtain the direct and inverse estimate of approximation by these polynomials for the $h$-spherical harmonics.

We establish the local well-posedness and obtain a blow-up criterion of smooth solutions for the Boussinesq equations in the framework of Triebel-Lizorkin-Lorentz spaces. The main ingredients of our proofs are Littlewood-Paley decomposition and the paradifferential calculus.

The generalized practical synchronization (GPS) of uncertain Duffing-Holmes chaotic systems with parameter mismatching, unknown external excitation, plant uncertainties, and uncertain deadzone nonlinearities is investigated. Based on the composite control approach, a tracking control is derived to realize the GPS for the uncertain Duffing-Holmes chaotic systems with parameter mismatching, unknown external excitation, plant uncertainties, and uncertain deadzone nonlinearities. Besides, the guaranteed exponential decay rate, convergence radius, and desired scaling factor can be prespecified. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of the proposed GPS scheme.

The nonlinear $n$th-order differential equations are considered. By using inequality techniques and coincidence degree theory, some criteria are obtained to guarantee the existence and uniqueness of $T$-periodic solutions for the equations. The obtained results are also valid and new for the problem discussed in the previous literature. Moreover, two illustrative examples are provided to illustrate the effectiveness of our results.

We give some new identities on $q$-Euler numbers and polynomials by using the fermionic $p$-adic integral on ${\mathbb{Z}}_p$.

A variational homotopy perturbation method (VHPM) which is based on variational iteration method and homotopy perturbation method is applied to solve the approximate solution of the fractional initial boundary value problems. The nonlinear terms can be easily handled by the use of He's polynomials. It is observed that the variational iteration method is very efficient and easier to implements; illustrative examples are included to demonstrate the high accuracy and fast convergence of this new algorithm.

For a fourth-order differential equation, we will establish some new Lyapunov-type inequalities, which give lower bounds of the distance between zeros of a nontrivial solution and also lower bounds of the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardy’s inequality and some generalizations of Opial-Wirtinger-type inequalities involving higher-order derivatives. Some examples are considered to illustrate the main results.