Recently, the max-min approach was systematically studied in the review article (Ji-Huan, 2012). This paper concludes that He's max-min approach is also a very much effective method for nonlinear oscillators with discontinuous terms.

We discuss the nonexistence of nontrivial solutions for the Chern-Simons-Higgs and Chern-Simons-Schrödinger equations. The Derrick-Pohozaev type identities are derived to prove it.

Wu and Liu (2012) presented some results for the existence and uniqueness of the periodic solutions for the hematopoiesis model. This paper gives a simple approach to find an approximate period of the model.

As we know, intuitionistic fuzzy sets are extensions of the standard fuzzy sets. Now, in this paper, the basic definitions and properties of intuitionistic fuzzy Γ-hyperideals of a Γ-semihyperring are introduced. A few examples are presented. In particular, some characterizations of Artinian and Noetherian Γ-semihyperring are given.

Smoking subject is an interesting area to study. The aim of this paper is to derive and analyze a model taking into account light smokers compartment, recovery compartment, and two relapses in the giving up smoking model. Stability of the model is obtained. Some numerical simulations are also provided to illustrate our analytical results and to show the effect of controlling the rate of relapse on the giving up smoking model.

Motivated by the method of Su and Pu (2009), we present an improved nonmonotone filter trust region algorithm for solving nonlinear equality constrained optimization. In our algorithm a modified nonmonotone filter technique is proposed and the restoration phase is not needed. At every iteration, in common with the composite-step SQP methods, the step is viewed as the sum of two distinct components, a quasinormal step and a tangential step. A more relaxed accepted condition for trial step is given and a crucial criterion is weakened. Under some suitable conditions, the global convergence is established. In the end, numerical results show our method is effective.

By using a specific way of choosing the indexes, we introduce an up-to-date iterative algorithm for approximating common fixed points of a countable family of generalized quasi-ϕ-asymptotically nonexpansive mappings and obtain a strong convergence theorem under some suitable conditions. As application, an iterative solution to a system of generalized mixed equilibrium problems is studied. The results extend those of other authors, in which the involved mappings consist of just finite families.

We study some properties of umbral calculus related to the Appell sequence. From those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.

We introduce a modified Mann’s iterative procedure by using the hybrid projection method for solving the common solution of the system of equilibrium problems for a finite family of bifunctions satisfying certain condition, the common solution of fixed point problems for two finite families of quasi-$\varphi$-nonexpansive mappings, and the common solution of variational inequality problems for a finite family of continuous monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by some mild conditions. Our result presented in this paper improves and generalizes some well-known results in the literature.

The concept of quasivariational inequality problem on proximally smooth sets is studied. Some sufficient conditions for solving the existence of solutions of such a problem are provided; also some interesting cases are discussed. Of course, due to the significance of proximally smooth sets, the results which are presented in this paper improve and extend many important results in the literature.

The Dirichlet problem for the Stokes system in a multiply connected domain of ${ℝ}^n(n\ge 2)$ is considered in the present paper. We give the necessary and sufficient conditions for the representability of the solution by means of a simple layer hydrodynamic potential, instead of the classical double layer hydrodynamic potential.

The method of spectral decimation is applied to an infinite collection of self-similar fractals. The sets considered are a generalization of the Sierpinski Gasket to higher dimensions; they belong to the class of nested fractals and are thus very symmetric. An explicit construction is given to obtain formulas for the eigenvalues of the Laplace operator acting on these fractals.

We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

This paper deals with the existence and uniqueness of mild solutions for a second order evolution equation initial value problem in a Banach space, which can model an elastic system with structural damping. The discussion is based on the operator semigroups theory and fixed point theorem. In addition, an example is presented to illustrate our theoretical results.

Korovkin type approximation theorems are useful tools to check whether a given sequence ${(L_n)}_{n\ge 1}$ of positive linear operators on $C[\mathrm{0,1}]$ of all continuous functions on the real interval $[\mathrm{0,1}]$ is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, $x$, and $x^2$ in the space $C[\mathrm{0,1}]$ as well as for the functions 1, cos, and sin in the space of all continuous 2$\pi$-periodic functions on the real line. In this paper, we use the notion of $B$-statistical $A$-summability to prove the Korovkin second approximation theorem. We also study the rate of $B$-statistical $A$-summability of a sequence of positive linear operators defined from $C_{2\pi }(ℝ)$ into $C_{2\pi }(ℝ)$.

We consider the problem of image recovery by the metric projections in a real Banach space. For a countable family of nonempty closed convex subsets, we generate an iterative sequence converging weakly to a point in the intersection of these subsets. Our convergence theorems extend the results proved by Bregman and Crombez.

In this paper we find the best possible lower power mean bounds for the Neuman-Sándor mean and present the sharp bounds for the ratio of the Neuman-Sándor and identric means.

The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, the generalized nonlinear Schrodinger (GNLS) equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.

Let $E$ be a reflexive Banach space having a weakly sequentially continuous duality mapping $J_{\phi }$ with gauge function $\phi$, $C$ a nonempty closed convex subset of $E$, and $T:C\to 𝒦(E)$ a multivalued nonself-mapping such that $P_T$ is nonexpansive, where $P_T(x)=\{u_x\in Tx:\parallel x-u_x\parallel =d(x,Tx)\}$. Let $f:C\to C$ be a contraction with constant $k$. Suppose that, for each $v\in C$ and $t\in (\mathrm{0,1})$, the contraction defined by $S_{t}x=t{P}_{T}x+(1-t)v$ has a fixed point $x_t\in C$. Let $\{{\alpha }_n\}$, $\{{\beta }_n\},$ and $\{{\gamma }_n\}$ be three sequences in $(\mathrm{0,1})$ satisfying approximate conditions. Then, for arbitrary $x_0\in C$, the sequence $\{x_n\}$ generated by $x_n\in {\alpha }_{n}f(x_{n-1})+{\beta }_n{x}_{n-1}+{\gamma }_n{P}_T(x_n)$ for all $n\ge 1$ converges strongly to a fixed point of $T$.

Consider a sequence of positive integers in arithmetic progression $u_k=u_0+kr$ with $(u_0,r)=1$. Denote the least common multiple of $u_0,\dots ,u_n$ by $L_n$. We show that if $n⩾r^{\mathrm{2}}+r$, then $L_{\mathrm{n}}⩾u_{\mathrm{0}}{r}^{r+\mathrm{1}}\left(r+\mathrm{1}\right)$, and we obtain optimum result on $n$ in some cases for such estimate. Besides, for quadratic sequences $m^2+\mathrm{c},{\left(m+1\right)}^{\mathrm{2}}+c,\dots ,n^2+c$, we also show that the least common multiple is at least $2^n$ when $m⩽⌈n/2⌉$, which sharpens a recent result of Farhi.

This paper deals with the oscillation of third-order nonlinear impulsive equations with delay. The results in this paper improve and extend some results for the equations without impulses. Some examples are given to illustrate the main results.

By using fractional power of operators and Sadovskii fixed point theorem, we study the complete controllability of fractional neutral differential systems in abstract space without involving the compactness of characteristic solution operators introduced by us.

We present a new reversed version of a generalized sharp Hölder's inequality which is due to Wu and then give a new refinement of Hölder's inequality. Moreover, the obtained result is used to improve the well-known Popoviciu-Vasić inequality. Finally, we establish the time scales version of Beckenbach-type inequality.

We investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with local superquadratic potential by using the Mountain Pass Theorem and the Fountain Theorem, respectively.

We prove that the cardinality of the space $\mathscr{H}𝒦([a,b])$ is equal to the cardinality of real numbers. Based on this fact we show that there exists a norm on $\mathscr{H}𝒦([a,b])$ under which it is a Banach space. Therefore if we equip $\mathscr{H}𝒦([a,b])$ with the Alexiewicz topology then $\mathscr{H}𝒦([a,b])$ is not K-Suslin, neither infra-($u$) nor a webbed space.

Some new oscillation criteria for a general class of second-order differential equations with nonlinear damping are shown. Except some general structural assumptions on the coefficients and nonlinear terms, we additionally assume only one sufficient condition (of Fite-Wintner-Leighton type). It is different compared to many early published papers which use rather complex sufficient conditions. Our method contains three items: classic Riccati transformations, a pointwise comparison principle, and a blow-up principle for sub- and supersolutions of a class of the generalized Riccati differential equations associated to any nonoscillatory solution of the main equation.

This paper is concerned with solvability of the second-order nonlinear neutral delay difference equation ${\mathrm{\Delta }}^2(x_n+a_n{x}_{n-\tau })+\mathrm{\Delta }h(n,x_{h_{1n}},x_{h_{2n}},\dots ,x_{h_{kn}})+f(n,x_{f_{1n}},x_{f_{2n}},\dots ,x_{f_{kn}})=b_n,\forall n\ge n_0$. Utilizing the Banach fixed point theorem and some new techniques, we show the existence of uncountably many unbounded positive solutions for the difference equation, suggest several Mann-type iterative schemes with errors, and discuss the error estimates between the unbounded positive solutions and the sequences generated by the Mann iterative schemes. Four nontrivial examples are given to illustrate the results presented in this paper.

Stage-structured predator-prey models with disease in the prey are constructed. For the purpose of integrated pest management, two types of impulsive control strategies (impulsive release of infective prey and impulsive release of predator) are used. For Case 1, infective prey applications are more frequent than releases of predator (natural enemies). For Case 2, predator (natural enemies) releases are more frequent than infective prey applications. In both cases, we get the sufficient conditions for the global attractivity of the susceptible prey-eradication periodic solution. In addition, the persistence of the systems is also discussed. At last, the results are discussed and some possible future work is put forward.

We establish the existence and uniqueness of a positive solution $u$ for the fractional boundary value problem $D^{\alpha }u(x)=-a(x)u^{\sigma }(x)$, $x\in (\mathrm{0,1})$ with the condition ${\lim }_{x\to 0}{D}^{\alpha -1}u(x)=0,u(1)=0$, where , $\sigma \in (-\mathrm{1,1}),$ and $a$ is a nonnegative continuous function on $(\mathrm{0,1})$ that may be singular at $x=0$ or $x=1$.

The aim of the present paper is to study an infinite horizon optimal control problem in which the controlled state dynamics is governed by a stochastic delay evolution equation in Hilbert spaces. The existence and uniqueness of the optimal control are obtained by means of associated infinite horizon backward stochastic differential equations without assuming the Gâteaux differentiability of the drift coefficient and the diffusion coefficient. An optimal control problem of stochastic delay partial differential equations is also given as an example to illustrate our results.

A competitive economic equilibrium model integrated with exchange, consumption, and production is considered. Our goal is to give an existence result when the utility functions are concave, proper, and upper semicontinuous. To this aim we are able to characterize the equilibrium by means of a suitable generalized quasi-variational inequality; then we give the existence of equilibrium by using the variational approach.

The purpose of this paper is to introduce an iterative algorithm for finding a common element of the set of fixed points of quasi-nonexpansive mappings and the solution of split feasibility problems (SFP) and systems of equilibrium problems (SEP) in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge weakly to a common element of the fixed points set of quasi-nonexpansive mappings and the solution of split feasibility problems and systems of equilibrium problems under mild conditions. Our main result improves and extends the recent ones announced by Ceng et al. (2012) and many others.

We study the existence and asymptotic stability in pth moment of a mild solution to a class of nonlinear fractional neutral stochastic differential equations with infinite delays in Hilbert spaces. A set of novel sufficient conditions are derived with the help of semigroup theory and fixed point technique for achieving the required result. The uniqueness of the solution of the considered problem is also studied under suitable conditions. Finally, an example is given to illustrate the obtained theory.

A class of constrained nonsmooth nonconvex optimization problems, that is, piecewise $C^2$ objectives with smooth inequality constraints are discussed in this paper. Based on the $𝒱𝒰$-theory, a superlinear convergent $𝒱𝒰$-algorithm, which uses a nonconvex redistributed proximal bundle subroutine, is designed to solve these optimization problems. An illustrative example is given to show how this convergent method works on a Second-Order Cone programming problem.

The necessary of classification research on common formula of $D_n$ group (dihedral group) cycle decomposition expression is illustrated. It includes the reflection and rotation conversion, which derived six common formulae on cycle decomposition expressions of $D_n$ group; it designed the generation algorithm on the cycle decomposition expressions of $D_n$ group, which is based on the method of replacement conversion and the classification formula; algorithm analysis and the results of the process show that the generation algorithm which is based on the classification formula is outperformed by the general algorithm which is based on replacement conversion; it has great significance to solve the enumeration of the necklace combinational scheme, especially the structural problems of combinational scheme, by using group theory and computer.

Some new existence theorems concerning approximate coincidence point property and approximate fixed point property for nonlinear maps in metric spaces without global completeness are established in this paper. By exploiting these results, we prove some new coincidence point and fixed point theorems which generalize and improve Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, Kikkawa-Suzuki's fixed point theorem, and some well known results in the literature. Moreover, some applications of our results to the existence of coupled coincidence point and coupled fixed point are also presented.

A deterministic model with variable human population for the transmission dynamics of malaria disease, which allows transmission by the recovered humans, is first developed and rigorously analyzed. The model reveals the presence of the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon may arise due to the reinfection of host individuals who recovered from the disease. The model in an asymptotical constant population is also investigated. This results in a model with mass action incidence. A complete global analysis of the model with mass action incidence is given, which reveals that the global dynamics of malaria disease with reinfection is completely determined by the associated reproduction number. Moreover, it is shown that the phenomenon of backward bifurcation can be removed by replacing the standard incidence function with a mass action incidence. Graphical representations are provided to study the effect of reinfection rate and to qualitatively support the analytical results on the transmission dynamics of malaria.

We deal with the existence of mild solutions of a class of nonlocal impulsive second-order functional differential equations with finite delay in a real Banach space $X$. An existence result on the mild solution is obtained by using the theory of the measures of noncompactness. An example is presented.

We study the stability and stabilization of complex fractional Lorenz system. The fractional calculus are taken in sense of the Caputo derivatives. The technique is based on stability theory of fractional-order systems. Numerical solutions are imposed.

The classical growth has been characterized in terms of approximation errors for a continuous function on $[-\mathrm{1,1}]$ by Reddy (1970), and a compact $K$ of positive capacity by Nguyen (1982) and Winiarski (1970) with respect to the maximum norm. The aim of this paper is to give the general growth ($(p,q)$-growth) of entire functions in ${ℂ}^n$ by means of the best polynomial approximation in terms of $L^p$-norm, with respect to the set ${\mathrm{\Omega }}_r=\{z\in {\mathbf{C}}^n;\exp V_K(z)\le r\}$, where $V_K=\sup \{\mathrm{(1/}\mathrm{d)}\log |P_d|,P_d\text{polynomial}\text{of}\text{degree}\le d,\parallel P_d{\parallel }_K\le 1\}$ is the Siciak's extremal function on an $L$-regular nonpluripolar compact $K$ is not pluripolar.

The problem of global exponential stability for recurrent neural networks with time-varying delay is investigated. By dividing the time delay interval [$0,\tau (t)$] into $K+1$ dynamical subintervals, a new Lyapunov-Krasovskii functional is introduced; then, a novel linear-matrix-inequality (LMI-) based delay-dependent exponential stability criterion is derived, which is less conservative than some previous literatures (Zhang et al., 2005; He et al., 2006; and Wu et al., 2008). An illustrate example is finally provided to show the effectiveness and the advantage of the proposed result.

We estimate the growth of the meromorphic solutions of some complex $q$-difference equations and investigate the convergence exponents of fixed points and zeros of the transcendental solutions of the second order $q$-difference equation. We also obtain a theorem about the $q$-difference equation mixing with difference.

We present an existence result for generalized solutions of initial value problems obtained through the order completion method. The solutions we obtain satisfy the initial condition in a suitable extended sense, and each such solution may be represented in a canonical way through its generalized partial derivatives as nearly finite normal lower semicontinuous function.

We establish some new Opial-type inequalities involving higher order delta derivatives on time scales. These extend some known results in the continuous case in the literature and provide new estimates in the setting of time scales.

We study the incompressible limit of weak solutions for the compressible flows of liquid crystals under strong stratification on bounded domains.

We investigate the controllability of impulsive neutral functional differential inclusions in Banach spaces. Our main aim is to find an effective method to solve the controllability problem of impulsive neutral functional differential inclusions with multivalued jump sizes in Banach spaces. Based on a fixed point theorem with regard to condensing map, sufficient conditions for the controllability of the impulsive neutral functional differential inclusions in Banach spaces are derived. Moreover, a remark is given to explain less conservative criteria for special cases, and work is improved in the previous literature.

The purpose of this paper is to investigate the existence of standing waves for a generalized Davey-Stewartson system. By reducing the system to a single Schrödinger equation problem, we are able to establish some existence results for the system by variational methods.

New explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Li soliton hierarchy are obtained. Then, the nonlinear integrable couplings of Li soliton hierarchy with self-consistent sources are established. Finally, we present the infinitely many conservation laws for the nonlinear integrable coupling of Li soliton hierarchy.

We will first prove the existence of $(V,V)$- and $(V,H^2)$-absorbing sets for the three-dimensional primitive equations of large-scale atmosphere in log-pressure coordinate and then prove the existence of $(V,H^3)$-absorbing set by the use of the elliptic regularity theory. Finally, we obtain the existence of $(V,V)$- and $(V,H^2)$-global attractors for the three-dimensional viscous primitive equations of large-scale atmosphere in log-pressure coordinate by using the Sobolev compactness embedding theory.

We prove that the Chern-Simons-Schrödinger system, under the condition of a Coulomb gauge, has a unique local-in-time solution in the energy space $H^1({ℝ}^2)$. The Coulomb gauge provides elliptic features for gauge fields $A_0,A_j$. The Koch- and Tzvetkov-type Strichartz estimate is applied with Hardy-Littlewood-Sobolev and Wente's inequalities.

This paper is to present a common fixed point theorem for two R-weakly commuting self-mappings satisfying nonlinear contractive type condition defined using a Φ-function, defined on fuzzy metric spaces. Some comments on previously published results and some examples are given.

This paper is a counterpart of Bi et al., 2011. For a locally optimal solution to the nonlinear second-order cone programming (SOCP), specifically, under Robinson’s constraint qualification, we establish the equivalence among the following three conditions: the nonsingularity of Clarke’s Jacobian of Fischer-Burmeister (FB) nonsmooth system for the Karush-Kuhn-Tucker conditions, the strong second-order sufficient condition and constraint nondegeneracy, and the strong regularity of the Karush-Kuhn-Tucker point.

The controllability issues for discrete-time linear systems with delay in state and control are addressed. By introducing a new concept, the controllability realization index (CRI), the characteristic of controllability is revealed. An easily testable necessary and sufficient condition for the controllability of discrete-time linear systems with state and control delay is established.

We presented some maximal and minimal fixed point theorems of set-valued monotone mappings with respect to a partial order introduced by a vector functional in cone metric spaces. In addition, we proved not only the existence of maximal and minimal fixed points but also the existence of the largest and the least fixed points of single-valued increasing mappings. It is worth mentioning that the results on single-valued mappings in this paper are still new even in the case of metric spaces and hence they indeed improve the recent results.

We prove some new characterizations of strongly continuous lattices using two new intrinsic topologies and a class of convergences. Lastly we show that the category of strongly continuous lattices and Scott continuous mappings is cartesian closed.

The $(G^{\prime }/G)$-expansion method is used for the first time to find traveling wave solutions for thin film equations, where it is found that the related balance numbers are not the usual positive integers. The closed-form solution obtained via this method is in good agreement with the previously obtained solutions of other researchers. It is also noted that, for appropriate parameters, new solitary waves solutions are found.

Solutions and weakly compact uniform attractor for the nonautonomous long-short wave equations with translation compact forces were studied in a bounded domain. We first established the existence and the uniqueness of the solution to the system by using Galerkin method and then obtained the uniform absorbing set and the weakly compact uniform attractor of the problem by applying techniques of constructing skew product flow in the extended phase space.

This paper is concerned with the study of a class of variational inequalities with multivalued $D_J$-pseudomonotone mappings in reflexive Banach spaces by using the $D_J$-antiresolvent technique. An application to the multivalued nonlinear $D_J$-complementarity problem is also presented. The results coincide with the corresponding results announced by many others for the gradient state.

We study the $q$-extension of the $p$-adic gamma function ${\mathrm{\Gamma }}_{p,q}$. We give a new identity for the $q$-extension of the $p$-adic gamma ${\mathrm{\Gamma }}_{p,q}$ in the case $p=2$. Also, we derive some properties and new representations of the $q$-extension of the $p$-adic gamma ${\mathrm{\Gamma }}_{p,q}$ in general case.

Recently, the concept of $N_{\theta }$-ward continuity was introduced and studied. In this paper, we prove that the uniform limit of $N_{\theta }$-ward continuous functions is $N_{\theta }$-ward continuous, and the set of all $N_{\theta }$-ward continuous functions is a closed subset of the set of all continuous functions. We also obtain that a real function $f$ defined on an interval $E$ is uniformly continuous if and only if ($f$(${\alpha }_k$)) is $N_{\theta }$-quasi-Cauchy whenever (${\alpha }_k$) is a quasi-Cauchy sequence of points in $E$.

We suggest and analyze relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of a general system of variational inequalities, the solution set of a split feasibility problem, and the fixed point set of a strictly pseudocontractive mapping defined on a real Hilbert space. Here the relaxed extragradient methods with regularization are based on the well-known successive approximation method, extragradient method, viscosity approximation method, regularization method, and so on. Strong convergence of the proposed algorithms under some mild conditions is established. Our results represent the supplementation, improvement, extension, and development of the corresponding results in the very recent literature.

Rough set theory was introduced by Pawlak in 1982 to handle imprecision, vagueness, and uncertainty in data analysis. Our aim is to generalize rough set theory by introducing concepts of ${\bigwedge }_{\beta }$-lower and ${\bigwedge }_{\beta }$-upper approximations which depends on the concept of ${\bigwedge }_{\beta }$-sets. Also, we study some of their basic properties.

We characterize a broad class of semilinear dense range operators $G_H:W\to Z$ given by the following formula, $G_{H}w=Gw+H(w),w\in W$, where $Z$, $W$ are Hilbert spaces, $G\in L(W,Z)$, and $H:W\to Z$ is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator $G$ to have dense range. Second, under some condition on the nonlinear term $H$, we prove the following statement: If $\overline{\text{R}\text{a}\text{n}\text{g}(G)}=Z$, then $\overline{\text{R}\text{a}\text{n}\text{g}(G_H)}=Z$ and for all $z\in Z$ there exists a sequence given by $w_{\alpha }=G^{*}(\alpha I+G{G}^{*}{)}^{-1}(z-H(w_{\alpha }))$, such that $\text{l}\text{i}\text{m}\alpha \to 0^{+}\{G{u}_{\alpha }+H(u_{\alpha })\}=z$. Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: $z^{\prime }=Az+Bu(t)+F(t,z,u(t)),z\in Z,u\in U,t>0$, where $Z$, $U$ are Hilbert spaces, $A:D(A)\subset Z\to Z$ is the infinitesimal generator of strongly continuous compact semigroup $\{T(t){\}}_{t\ge 0}$ in $Z,B\in L(U,Z)$, the control function $u$ belongs to $L^2(0,\tau ;U)$, and $F:[0,\tau ]\times Z\times U\to Z$ is a suitable function. As a particular case we consider the controlled semilinear heat equation.

The initial-boundary value problem for the density-dependent flow of nematic crystals is studied in a 2-D bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is proved for the global strong solution with the large initial velocity $u_0$ and small $\nabla d_0$. We also give a regularity criterion of the problem with the Dirichlet boundary condition $u=0$, $d=d_0$ on $\partial \mathrm{\Omega }$.

We will study a maximal solution of the time-space fractional heat equation in complex domain. The fractional time is taken in the sense of the Riemann-Liouville operator, while the fractional space is assumed in the Srivastava-Owa operator. Here we employ some properties of the univalent functions in the unit disk to determine the upper bound of this solution. The maximal solution is illustrated in terms of the generalized hypergeometric functions.

Shunting inhibitory cellular neural networks (SICNNs) are considered with the introduction of continuously distributed delays in the leakage (or forgetting) terms. By using the Lyapunov functional method and differential inequality techniques, some sufficient conditions for the existence and exponential stability of almost periodic solutions are established. Our results complement with some recent ones.

We studied the iterative algorithm in Tung et al. (2010) to find out that their assertion is questionable. We derived two new relations between safe factor and order quantity so that we can execute the iterative algorithm proposed by Wu and Ouyang (2001). We have proved that three generated sequences indeed converge to provide a theoretical validation for their iterative procedure.

The fine spectra of lower triangular triple-band matrices have been examined by several authors (e.g., Akhmedov (2006), Başar (2007), and Furken et al. (2010)). Here we determine the fine spectra of upper triangular triple-band matrices over the sequence space ${\ell }_p$. The operator $A(r,s,t)$ on sequence space on ${\ell }_p$ is defined by $A(r,s,t)x=(r{x}_k+s{x}_{k+1}+t{x}_{k+2}{)}_{k=0}^{\infty }$, where $x=(x_k)\in {\ell }_p$, with . In this paper we have obtained the results on the spectrum and point spectrum for the operator $A(r,s,t)$ on the sequence space ${\ell }_p$. Further, the results on continuous spectrum, residual spectrum, and fine spectrum of the operator $A(r,s,t)$ on the sequence space ${\ell }_p$ are also derived. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator $A(r,s,t)$ over the space ${\ell }_p$ and we give some applications.

We propose a new variable-coefficient Riccati subequation method to establish new exact solutions for nonlinear differential-difference equations. For illustrating the validity of this method, we apply it to the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new and generalized traveling wave solutions including hyperbolic function solutions, trigonometric function solutions, and rational function solutions are obtained.

We consider the problem of existence of conformal metrics with prescribed Q-curvature on standard sphere $S^n,n\ge 5$. Under the assumption that the order of flatness at critical points of prescribed Q-curvature function $K(x)$ is $\mathrm{\beta \hspace{0.17em}}\in \hspace{0.17em}]1,n-4]$, we give precise estimates on the losses of the compactness, and we prove new existence and multiplicity results through an Euler-Hopf type formula.

In several situations the notion of uniform continuity can be strengthened to strong uniform continuity to produce interesting properties, especially in constrained problems. The same happens in the setting of proximity spaces. While a parallel theory for uniform and strong uniform convergence was recently developed, and a notion of proximal convergence is present in the literature, the notion of strong proximal convergence was never considered. In this paper, we propose several possible convergence notions, and we provide complete comparisons among these concepts and the notion of strong uniform convergence in uniform spaces. It is also shown that in particularly meaningful classes of functions these notions are equivalent and can be considered as natural definitions of strong proximal convergence. Finally we consider a function acting between two proximity spaces and we connect its continuity/strong continuity to convergence in the respective hyperspaces of a natural functor associated to the function itself.

We deduce formulas for the Fréchet derivatives of cost functionals of several inverse problems for a parabolic integrodifferential equation in a weak formulation. The method consists in the application of an integrated convolutional form of the weak problem and all computations are implemented in regular Sobolev spaces.

In this paper, the problem of the stability for mutually delay-coupled semiconductor lasers system is investigated. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated. The new phenomena such as stability switch is found. The $Z_2$ equivariant property and the existence of multiple periodic solutions is also discussed. Numerical simulations are presented to illustrate the results in the paper.

We study a stochastic logistic model with diffusion between two patches in this paper. Using the definition of stationary distribution, we discuss the effect of dispersal in detail. If the species are able to have nontrivial stationary distributions when the patches are isolated, then they continue to do so for small diffusion rates. In addition, we use some examples and numerical experiments to reflect that diffusions are capable of both stabilizing and destabilizing a given ecosystem.

We study a generalized KdV equation of neglecting the highest order infinitesimal term, which is an important water wave model. Some exact traveling wave solutions such as singular solitary wave solutions, semiloop soliton solutions, dark soliton solutions, dark peakon solutions, dark loop-soliton solutions, broken loop-soliton solutions, broken wave solutions of U-form and C-form, periodic wave solutions of singular type, and broken wave solution of semiparabola form are obtained. By using mathematical software Maple, we show their profiles and discuss their dynamic properties. Investigating these properties, we find that the waveforms of some traveling wave solutions vary with changes of certain parameters.

We consider additive perturbation theorems for subgenerators of (a, k)-regularized C-resolvent families. A major part of our research is devoted to the study of perturbation properties of abstract time-fractional equations, primarily from their importance in modeling of various physical phenomena. We illustrate the results with several examples.

This paper deals with state estimation problem for a class of fractional-order complex dynamical networks with parametric uncertainty. The parametric uncertainty is assumed to be of linear fractional form. Firstly, based on the properties of Kronecker product and the stability of fractional-order system, a sufficient condition is derived for robust asymptotic stability of linear fractional-order augmented system. Secondly, state estimation problem is then studied for the same fractional-order complex networks, where the purpose is to design a state estimator to estimate the network state through available output measurement, the existence conditions of designing state estimator are derived using matrix's singular value decomposition and LMI techniques. These conditions are in the form of linear matrix inequalities which can be readily solved by applying the LMI toolbox. Finally, two numerical examples are provided to demonstrate the validity of our approach.

Positive solutions for a kind of third-order multipoint boundary value problem under the nonresonant conditions and the resonant conditions are considered. In the nonresonant case, by using the Leggett-Williams fixed point theorem, the existence of at least three positive solutions is obtained. In the resonant case, by using the Leggett-Williams norm-type theorem due to O’Regan and Zima, the existence result of at least one positive solution is established. It is remarkable to point out that it is the first time that the positive solution is considered for the third-order boundary value problem at resonance. Some examples are given to demonstrate the main results of the paper.

By introducing the fractional derivative in the sense of Caputo and combining the pretreatment technique to deal with long nonlinear items, the generalized two-dimensional differential transform method is proposed for solving the time-fractional Hirota-Satsuma coupled KdV equation and coupled MKdV equation. The presented method is a numerical method based on the generalized Taylor series expansion which constructs an analytical solution in the form of a polynomial. The numerical results show that the generalized two-dimensional differential transform method is very effective for the fractional coupled equations.

The positive definite solutions for the system of nonlinear matrix equations $X+A^{\ast }{Y}^{-n}A=I,Y+B^{\ast }{X}^{-m}B=I$ are considered, where $n,\hspace{0.17em}m$ are two positive integers and A, B are nonsingular complex matrices. Some sufficient conditions for the existence of positive definite solutions for the system are derived. Under some conditions, an iterative algorithm for computing the positive definite solutions for the system is proposed. Also, the estimation of the error is obtained. Finally, some numerical examples are given to show the efficiency of the proposed iterative algorithm.

We give several sufficient conditions under which the first-order nonlinear discrete Hamiltonian system $\mathrm{\Delta }x(n)=\alpha (n)x(n+1)+\beta (n)|y(n){|}^{\mu -2}y(n),\mathrm{\Delta }y(n)=-\gamma (n)|x(n+1){|}^{\nu -2}x(n+1)-\alpha (n)y(n)$ has no solution $(x(n),y(n))$ satisfying condition , where $\mu ,\nu >1$ and $1/\mu +1/\nu =1$ and $\alpha (n),\beta (n),$ and $\gamma (n)$ are real-valued functions defined on $\mathbb{Z}$.

Using Riemann-Liouville fractional differential operator, a fractional extension of the Lagrange inversion theorem and related formulas are developed. The required basic definitions, lemmas, and theorems in the fractional calculus are presented. A fractional form of Lagrange's expansion for one implicitly defined independent variable is obtained. Then, a fractional version of Lagrange's expansion in more than one unknown function is generalized. For extending the treatment in higher dimensions, some relevant vectors and tensors definitions and notations are presented. A fractional Taylor expansion of a function of $N$-dimensional polyadics is derived. A fractional $N$-dimensional Lagrange inversion theorem is proved.

By using successive approximation, we prove existence and uniqueness result for a class of nonlinear stochastic differential equations. Moreover, it is shown that the solution of such equations is a diffusion process and its diffusion coefficients are found.

We prove that the double inequalities hold for all $a,b>0$ with $a\ne b$ if and only if ${\alpha }_1\ge 1/2$, ${\beta }_1\le \log [\sqrt{2}\log (1+\sqrt{2})]/(1-\log \sqrt{2})$, ${\alpha }_2\ge 5/7$, and ${\beta }_2\le \log [2\log (1+\sqrt{2})]$, where $I(a,b)$, $M(a,b)$, $Q(a,b)$, and $C(a,b)$ are the identric, Neuman-Sándor, quadratic, and contraharmonic means of $a$ and $b$, respectively.

We give the greatest values $r_{\mathrm{1}}$, $r_{\mathrm{2}}$ and the least values $s_{\mathrm{1}}$, $s_{\mathrm{2}}$ in (1/2, 1) such that the double inequalities and hold for any $\alpha \in (\mathrm{0,1})$ and all $a,b>\mathrm{0}$ with $a\ne b$, where $A(a,b)$, $M(a,b)$, $C(a,b),$ and $T(a,b)$ are the arithmetic, Neuman-Sándor, contraharmonic, and second Seiffert means of $a$ and $b$, respectively.

The problem of guaranteed cost control for exponential synchronization of cellular neural networks with interval nondifferentiable and distributed time-varying delays via hybrid feedback control is considered. The interval time-varying delay function is not necessary to be differentiable. Based on the construction of improved Lyapunov-Krasovskii functionals is combined with Leibniz-Newton's formula and the technique of dealing with some integral terms. New delay-dependent sufficient conditions for the exponential synchronization of the error systems with memoryless hybrid feedback control are first established in terms of LMIs without introducing any free-weighting matrices. The optimal guaranteed cost control with linear error hybrid feedback is turned into the solvable problem of a set of LMIs. A numerical example is also given to illustrate the effectiveness of the proposed method.

We study almost everywhere convergence for Riesz means related to Schrödinger operator with constant magnetic fields. Through researching the weighted norm estimates for the maximal operator with power-weight functions, we obtain the desired result, which is similar to the work given by Anthony Carbery, Jose L. Rubio de Francia, and Luis Vega.

A new model which allows both the effect of partial selfing selection and an exponential function of the expected payoff is considered. This combines ideas from genetics and evolutionary game theory. The aim of this work is to study the effects of partial selfing selection on the discrete dynamics of population evolution. It is shown that the system undergoes period doubling bifurcation, saddle-node bifurcation, and Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-3, 6 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, and the chaotic sets. These results reveal richer dynamics of the discrete model compared with the model in Tao et al., 1999. The analysis and results in this paper are interesting in mathematics and biology.

The spaces $H_{\alpha ,\delta ,\gamma }((a,b)\times (a,b),ℝ)$ and $H_{\alpha ,\delta }((a,b),ℝ)$ were defined in ((Hüseynov (1981)), pages 271–277). Some singular integral operators on Banach spaces were examined, (Dostanic (2012)), (Dunford (1988), pages 2419–2426 and (Plamenevskiy (1965)). The solutions of some singular Fredholm integral equations were given in (Babolian (2011), Okayama (2010), and Thomas (1981)) by numerical methods. In this paper, we define the sets $H_{\alpha ,\delta ,\gamma }((a,b)\times (a,b),X)$ and $H_{\alpha ,\delta }((a,b),X)$ by taking an arbitrary Banach space $X$ instead of $ℝ$, and we show that these sets which are different from the spaces given in (Dunford (1988)) and (Plamenevskiy (1965)) are Banach spaces with the norms ${\parallel ·\parallel }_{\alpha ,\delta ,\gamma }$ and ${\parallel ·\parallel }_{\alpha ,\delta }$. Besides, the bounded linear integral operators on the spaces $H_{\alpha ,\delta ,\gamma }((a,b)\times (a,b),X)$ and $H_{\alpha ,\delta }((a,b),X)$, some of which are singular, are derived, and the solutions of the linear Fredholm integral equations of the form $f(s)=\varphi (s)+\lambda {\int }_a^{b}A(s,t)f(t)dt,f(s)=\varphi (s)+\lambda {\int }_a^{b}A(t,s)f(t)dt$ and $f(s,t)=\varphi (s,t)+\lambda {\int }_a^{b}A(s,t)f(t,s)dt$ are investigated in these spaces by analytical methods.

Let $\{X_n,n\ge 1\}$ be a sequence of random variables satisfying the Rosenthal-type maximal inequality. Complete convergence is studied for linear statistics that are weighted sums of identically distributed random variables under a suitable moment condition. As an application, the Marcinkiewicz-Zygmund-type strong law of large numbers is obtained. Our result generalizes the corresponding one of Zhou et al. (2011) and improves the corresponding one of Wang et al. (2011, 2012).

In surface modeling a surface frequently encountered is a Coons patch that is defined only for a boundary composed of four analytical curves. In this paper we extend the range of applicability of a Coons patch by telling how to write it for a boundary composed of an arbitrary number of boundary curves. We partition the curves in a clear and natural way into four groups and then join all the curves in each group into one analytic curve by using representations of the unit step function including one that is fully analytic. Having a well-parameterized surface, we do some calculations on it that are motivated by differential geometry but give a better optimized and possibly more smooth surface. For this, we use an ansatz consisting of the original surface plus a variational parameter multiplying the numerator part of its mean curvature function and minimize with the respect to it the rms mean curvature and decrease the area of the surface we generate. We do a complete numerical implementation for a boundary composed of five straight lines, that can model a string breaking, and get about 0.82 percent decrease of the area. Given the demonstrated ability of our optimization algorithm to reduce area by as much as 23 percent for a spanning surface not close of being a minimal surface, this much smaller fractional decrease suggests that the Coons patch we have been able to write is already close of being a minimal surface.

This paper deals with a new iterative algorithm $\{x_n\}$ with a strongly positive operator A for a k-strict pseudo-contraction T and a non-self-Lipschitzian mapping S in Hilbert spaces. Under certain appropriate conditions, the sequence $\{x_n\}$ converges strongly to a fixed point of T, which solves some variational inequality. The results here improve and extend some recent related results.

We provide an iterative process which converges strongly to a common fixed point of finite family of asymptotically $k$-strict pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

Let $H$ be a real Hilbert space and $C\subset \mathrm{H\hspace{0.17em}}$a closed convex subset. Let $T:C\to C$ be a nonexpansive mapping with the nonempty set of fixed points $\text{F}\text{i}\text{x}(T)$. Kim and Xu (2005) introduced a modified Mann iteration $x_0=x\in C$, $y_n={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n$, $x_{n+1}={\beta }_{n}u+(1-{\beta }_n)y_n$, where $u\in C$ is an arbitrary (but fixed) element, and $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are two sequences in $(0,1)$. In the case where $0\in C$, the minimum-norm fixed point of $T$ can be obtained by taking $u=0$. But in the case where $0\notin C$, this iteration process becomes invalid because $x_n$ may not belong to $C$. In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of $\mathrm{\hspace{0.17em}T}$ and prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projection $P_C$, which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others.

Ji-Huan He systematically studied the inverse problem of calculus of variations. This note reveals that the semi-inverse method also works for a generalized KdV-mKdV equation with nonlinear terms of any orders.

The aim of this paper is to study generalized vector quasi-equilibrium problems (GVQEPs) by scalarization method in locally convex topological vector spaces. A general nonlinear scalarization function for set-valued mappings is introduced, its main properties are established, and some results on the existence of solutions of the GVQEPs are shown by utilizing the introduced scalarization function. Finally, a vector variational inclusion problem is discussed as an application of the results of GVQEPs.

We consider a degenerate equation with a memory condition at the boundary. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases.

The paper starts with a discussion involving the Sobolev constant on geodesic balls and then follows with a derivation of a lower bound for the first eigenvalue of the Laplacian on manifolds with small negative curvature. The derivation involves Moser iteration.

The object of the present paper is to study $(\text{LCS}{)}_n$-manifolds with vanishing quasi-conformal curvature tensor. $(\text{LCS}{)}_n$-manifolds satisfying Ricci-symmetric condition are also characterized.

We study the analytic integrability problem through the formal integrability problem and we show its connection, in some cases, with the existence of invariant analytic (sometimes algebraic) curves. From the results obtained, we consider some families of analytic differential systems in ${ℂ}^2$, and imposing the formal integrability we find resonant centers obviating the computation of some necessary conditions.

Based on stress-deflection variational formulation, we propose a family of local projection-based stabilized mixed finite element methods for Kirchhoff plate bending problems. According to the error equations, we obtain the error estimates of the approximation to stress tensor in energy norm. And by duality argument, error estimates of the approximation to deflection in H¹-norm are achieved. Then we design an a posteriori error estimator which is closely related to the equilibrium equation, constitutive equation, and nonconformity of the finite element spaces. With the help of Zienkiewicz-Guzmán-Neilan element spaces, we prove the reliability of the a posteriori error estimator. And the efficiency of the a posteriori error estimator is proved by standard bubble function argument.

The main purpose of this paper is to determine the fine spectrum with respect to Goldberg's classification of the operator defined by the lambda matrix over the sequence spaces $c_0$ and c. As a new development, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator $\mathrm{\Lambda }$ on the sequence spaces $c_0$ and c. Finally, we present a Mercerian theorem. Since the matrix $\mathrm{\Lambda }$ is reduced to a regular matrix depending on the choice of the sequence $({\lambda }_k)$ having certain properties and its spectrum is firstly investigated, our work is new and the results are comprehensive.

The basic assumption of ecological economics is that resource allocation exists social optimal solution, and the social optimal solution and the optimal solution of enterprises can be complementary. The mathematical methods and the ecological model are one of the important means in the study of ecological economics. In this paper, we study an ecological model arising from ecological economics by mathematical method, that is, study the existence of positive solutions for the fractional differential equation with $p$-Laplacian operator ${{\mathcal{D}}_{\mathit{t}}}^{\beta }({\phi }_p({{\mathcal{D}}_{\mathit{t}}}^{\alpha }x))(t)=f(t,x(t))$, $t\in (\mathrm{0,1})$, $x(\mathrm{0})=\mathrm{0}$, $x(\mathrm{1})=ax(\xi )$, ${{\mathcal{D}}_{\mathit{t}}}^{\alpha }x(\mathrm{0})=\mathrm{0}$, and ${{\mathcal{D}}_{\mathit{t}}}^{\alpha }x\left(1\right)=b{{\mathcal{D}}_{\mathit{t}}}^{\alpha }x\left(\eta \right)$, where ${{\mathcal{D}}_{\mathit{t}}}^{\alpha },{{\mathcal{D}}_{\mathit{t}}}^{\beta }$ are the standard Riemann-Liouville derivatives, $p$-Laplacian operator is defined as ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s,p>1$, and the nonlinearity $f$ may be singular at both $t=\mathrm{0,1}$ and $x=0.$ By finding more suitable upper and lower solutions, we omit some key conditions of some existing works, and the existence of positive solution is established.

By using the fixed point theorems and the theory of analytic semigroup, we investigate the existence of positive mild solutions to the Cauchy problem of Caputo fractional evolution equations in Banach spaces. Some existence theorems are obtained under the case that the analytic semigroup is compact and noncompact, respectively. As an example, we study the partial differential equation of the parabolic type of fractional order.

Here we show that the main results in the papers by Yalcinkaya (2008), Yalcinkaya and Cinar (2010), and Yalcinkaya, Cinar, and Simsek (2008), as well as a conjecture from the last mentioned paper, follow from a slight modification of a result by G. Papaschinopoulos and C. J. Schinas. We also give some generalizations of these results.

The aim of this paper is to present new results related to the $q$-Bernstein polynomials $B_{n,q}(f;x)$ of unbounded functions in the case $q>1$ and to illustrate those results using numerical examples. As a model, the behavior of polynomials $B_{n,q}(f;x)$ is examined both theoretically and numerically in detail for functions on $[0,1]$ satisfying $f(x)~K{x}^{-\alpha }$ as $x\to 0^{+}$, where $\alpha >0$ and $K\ne 0$ are real numbers.

The sequence space $\ell (p)$ was introduced by Maddox (1967). Quite recently, the sequence space $\ell (\tilde{B},p)$ of nonabsolute type has been introduced and studied which is the domain of the double sequential band matrix $B(\tilde{r},\tilde{s})$ in the sequence space $\ell (p)$ by Nergiz and Başar (2012). The main purpose of this paper is to investigate the geometric properties of the space $\ell (\tilde{B},p)$, like rotundity and Kadec-Klee and the uniform Opial properties. The last section of the paper is devoted to the conclusion.

Erratum to “Some Geometric Properties of the Domain of the Double Sequential Band Matrix $B(\tilde{r},\tilde{s})$ in the Sequence Space $\ell (p)$” dx.doi.org/10.1155/2013/326762

In the present paper, we consider a subclass of $p$-valent analytic functions and obtain certain simple sufficiency criteria by using three different methods for the functions belonging to this class. Many known results appear as special consequences of our work.

In 1991, Chen and Ishikawa initially studied biharmonic marginally trapped surfaces in neutral pseudo-Euclidean 4-space. Recently, biharmonic and quasi-biharmonic marginally trapped Lagrangian surfaces in Lorentzian complex space forms were studied by Sasahara in 2007 and 2011, respectively. In this paper we extend Sasahara's results to the case of slant surfaces in Lorentzian complex space forms. By results, we completely classify biharmonic marginally trapped slant surfaces and quasi-biharmonic marginally trapped slant surfaces in Lorentzian complex space forms.

We consider a new chaotic system based on merging two well-known systems (the Lorentz and Rössler systems). Meanwhile, taking into account the effect of environmental noise, we incorporate whit-enoise in each equation. We prove the existence, uniqueness, and the moments estimations of the Lorentz-Rössler systems. Numerical experiments show the applications of our systems and illustrate the results.

We constructed three two-step semi-implicit hybrid methods (SIHMs) for solving oscillatory second order ordinary differential equations (ODEs). The first two methods are three-stage fourth-order and three-stage fifth-order with dispersion order six and zero dissipation. The third is a four-stage fifth-order method with dispersion order eight and dissipation order five. Numerical results show that SIHMs are more accurate as compared to the existing hybrid methods, Runge-Kutta Nyström (RKN) and Runge-Kutta (RK) methods of the same order and Diagonally Implicit Runge-Kutta Nyström (DIRKN) method of the same stage. The intervals of absolute stability or periodicity of SIHM for ODE are also presented.

We explore the nonautonomous fourth-order differential equation which has important applications in materials science. By variational approach, we find heteroclinic solutions of the equation. The conditions on the potential function $V(t,u)$ are mild enough to include a broad class of equations. We also consider a separate case where $V(t,u)$ is periodic in $t$.

The paper explores the optimal vibration control design problem for a half-car suspension working on in-vehicle networks in delta domain. First, the original suspension system with ECU-actuator delay and sensor-ECU delay is modeled. By using delta operators, the original system is transformed into an associated sampled-data system with time delays in delta domain. After model transformation, the sampled-data system equation is reduced to one without actuator delays and convenient to calculate the states with nonintegral time delay. Therefore, the sampled-data optimal vibration control law can be easily obtained deriving from a Riccati equation and a Stein equation of delta domain. The feedforward control term and the control memory terms designed in the control law ensure the compensation for the effects produced by disturbance and actuator delay, respectively. Moreover, an observer is constructed to implement the physical realizability of the feedforward term and solve the immeasurability problem of some state variables. A half-car suspension model with delays is applied to simulate the responses through the designed controller. Simulation results illustrate the effectiveness of the proposed controller and the simplicity of the designing approach.

We address the problem of asymptotic stability and region-of-attraction analysis of nonlinear dynamical systems. A hybrid symbolic-numeric method is presented to compute exact Lyapunov functions and exact estimates of regions of attraction of nonlinear systems efficiently. A numerical Lyapunov function and an estimate of region of attraction can be obtained by solving an (bilinear) SOS programming via BMI solver, then the modified Newton refinement and rational vector recovery techniques are applied to obtain exact Lyapunov functions and verified estimates of regions of attraction with rational coefficients. Experiments on some benchmarks are given to illustrate the efficiency of our algorithm.

We use the quadrature method to show the existence and multiplicity of positive solutions of the boundary value problems involving one-dimensional $p$-Laplacian ${\left(u^{\prime }\left(t\right){|}^{p-2}{u}^{\prime }\left(t\right)\right)}^{\prime }+\lambda f\left(u\left(t\right)\right)=0$, $t\in \left(\mathrm{0,1}\right)$, $u(0)=u(1)=0$, where $p\in (\mathrm{1,2}]$, $\lambda \in (0,\mathrm{\infty })$ is a parameter, $f\in C^1([0,r),[0,\mathrm{\infty }))$ for some constant $r>0$, $f(s)>0$ in $(0,r)$, and ${\lim }_{s\to r^{-}}(r-s{)}^{p-\mathrm{1}}f(s)=+\infty$.

Fixed-point theory in complex valued metric spaces has greatly developed in recent times. In this paper, we prove certain common fixed-point theorems for two single-valued mappings in such spaces. The mappings we consider here are assumed to satisfy certain metric inequalities with generalized fixed-point theorems due to Rouzkard and Imdad (2012). This extends and subsumes many results of other authors which were obtained for mappings on complex-valued metric spaces.

A new splitting method designed for the numerical solutions of stochastic delay Hopfield neural networks is introduced and analysed. Under Lipschitz and linear growth conditions, this split-step θ-Milstein method is proved to have a strong convergence of order 1 in mean-square sense, which is higher than that of existing split-step θ-method. Further, mean-square stability of the proposed method is investigated. Numerical experiments and comparisons with existing methods illustrate the computational efficiency of our method.