This paper is concerned with a generalization of the $q$-Bernstein polynomials and Stancu operators, where the function is evaluated at intervals which are in geometric progression. It is shown that these polynomials can be generated by a de Casteljau algorithm, which is a generalization of that relating to the classical case and $q$-Bernstein case.

Functions$f\left(z\right)=z+{{\displaystyle \sum }}_2^{\infty }{a}_n{z}^n$ that are analytic in the unit disk and satisfy the differential equation $f\text{'}\left(z\right)+\alpha \mathrm{zf}\text{'}\text{'}\left(z\right)+\gamma z^{2}f\text{'}\text{'}\text{'}\left(z\right)=g\left(z\right)$ are considered, where $g$ is subordinated to a normalized convex univalent function $h$. These functions $f$ are given by a double integral operator of the form $f\left(z\right)={{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^1\mathrm{?}G\left(z{t}^{\mu }{s}^{\nu }\right)t^{-\mu }{s}^{-\nu }\text{i}tdsdt$ $f\left(z\right)={{\displaystyle \int }}_0^1{{\displaystyle \int }}_0^1\mathrm{?}G\left(z{t}^{\mu }{s}^{\nu }\right)t^{-\mu }{s}^{-\nu }\text{i}tdsdt$ with $G\text{'}$ subordinated to $h$. The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function $h$.

The authors introduce two new subclasses denoted by ${\Im }^{*}(\Omega ,\lambda ,p,\alpha )$ and ${\Im }_{ϵ}^{*}(\Omega ,\lambda ,p,\alpha )$ of $p$-valent analytic functions. They obtain coefficient inequality for the class ${\Im }^{*}(\Omega ,\lambda ,p,\alpha )$. They investigate various properties of classes ${\Im }^{*}(\Omega ,\lambda ,p,\alpha )$ and ${\Im }_{ϵ}^{*}(\Omega ,\lambda ,p,\alpha )$. Furthermore, they derive partial sums associated with the class ${\Im }_{ϵ}^{*}(\Omega ,\lambda ,p,\alpha )$.

The task scheduling problem has been widely studied for assigning resources to tasks in heterogeneous grid environment. Effective task scheduling is an important issue for the performance of grid computing. Meanwhile, the task scheduling problem is an NP-complete problem. Hence, this investigation introduces a named “standard“ particle swarm optimization (PSO) metaheuristic approach to efficiently solve the task scheduling problems in grid. Meanwhile, two promising heuristics based on multimode project scheduling are proposed to help in solving interesting scheduling problems. They are the best performance resource heuristic and the latest finish time heuristic. These two heuristics applied to the PSO scheme are for speeding up the search of the particle and improving the capability of finding a sound schedule. Moreover, both global communication topology and local ring communication topology are also investigated for efficient study of proposed scheme. Simulation results demonstrate that the proposed approach in this investigation can successfully solve the task-resource assignment problems in grid computing and similar scheduling problems.

Using a specially constructed cone and the fixed point index theory, this work shows existence and nonexistence results of positive solutions for fourth-order boundary value problem with two different parameters in Banach spaces.

Let $C^0\left(I\right)$ be the set of all continuous self-maps of the closed interval $\mathrm{I}$, and $P(u,v)=\{f\in C^0\left(I\right):f$ has a cycle with rotation pair $(u,v)\}$ for any positive integer $v>u$. In this paper, we prove that if $(2^{m}ns,2^{m}nt)⊣(\gamma ,\lambda )$, then $P(2^{m}ns,2^{m}nt)\subset \text{int}P(\gamma ,\lambda )$, where $m\ge 0$ is integer, $n\ge 1$ odd, $1\le s<t$ with $s,t$ coprime, and $1\le \gamma <\lambda$.

We study some aspects of the fixed point theory for a class of generalized nonexpansive mappings, which among others contain the class of generalized nonexpansive mappings recently defined by Suzuki in 2008.

In recent decades, many researchers have investigated the ecological models with three and more species to understand complex dynamical behaviors of ecological systems in nature. However, when they studied the models with three species, they have just considered the functional responses between prey and mid-predator and between mid-predator and top predator as the same type. However, in the paper, in order to describe more realistic ecological world, a three-species food chain system with two types of functional response, Holling type and Beddington-DeAngelis type, is considered. It is shown that this system is dissipative. Also, the local and global stability of equilibrium points of the system is established. In addition, conditions for the persistence of the system are found according to the existence of limit cycles. Some numerical examples are given to substantiate our theoretical results. Moreover, we provide numerical evidence of the existence of chaotic phenomena by illustrating bifurcation diagrams of system and by calculating the largest Lyapunov exponent.

Using fixed point methods, we prove the superstability and generalized Hyers-Ulam stability of ring homomorphisms on non-Archimedean Banach algebras. Moreover, we investigate the superstability of ring homomorphisms in non-Archimedean Banach algebras associated with the Jensen functional equation.

A semilinear elliptic problem $\left(E_{\lambda }\right)$ with concave-convex nonlinearities and multiple Hardy-type terms is considered. By means of a variational method, we establish the existence and multiplicity of positive solutions for problem $\left(E_{\lambda }\right)$.

We give a generalization of g-frame in Hilbert $C^{*}$-modules that was introduced by Khosravies then investigated some properties of it by Xiao and Zeng. This generalization is a natural generalization of continuous and discrete g-frames and frame in Hilbert space too. We characterize continuous g-frame g-Riesz in Hilbert $C^{*}$-modules and give some equality and inequality of these frames.

A mathematical model for growth of tumors with two discrete delays is studied. The delays, respectively, represent the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis and kill of cells by the inhibitor. We show the influence of time delays on the Hopf bifurcation when one of delays is used as a bifurcation parameter.

We introduce the new difference sequence space $a_p^r\left(\Delta \right)$ . Further, it is proved that the space $a_p^r\left(\Delta \right)$ is the BK-space including the space $b{v}_p$, which is the space of sequences of pbounded variation. We also show that the spaces $a_p^r\left(\Delta \right)$, and ${\ell }_p$ are linearly isomorphic for $1\le p<\infty$. Furthermore, the basis and the $\alpha \text{-}$, $\beta \text{-}$ and $\gamma \text{-}$ duals of the space $a_p^r\left(\Delta \right)$ are determined. We devote the final section of the paper to examine some geometric properties of the space $a_p^r\left(\Delta \right)$.

This paper is devoted to the investigation of the nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic systems involving delayed dynamics with point delays. The obtained results are independent of the sizes of the delays.

We investigate the extent to which the study of quasimultipliers can be made beyond Banach algebras. We will focus mainly on the class of $F$-algebras, in particular on complete $k$-normed algebras, $0\lt k\le 1$, not necessarily locally convex. We include a few counterexamples to demonstrate that some of our results do not carry over to general $F$-algebras. The bilinearity and joint continuity of quasimultipliers on an $F$-algebra $A$ are obtained under the assumption of strong factorability. Further, we establish several properties of the strict and quasistrict topologies on the algebra $QM\left(A\right)$ of quasimultipliers of a complete $k$-normed algebra $A$ having a minimal ultra-approximate identity.

We introduce two general hybrid iterative approximation methods (one implicit and one explicit) for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu (2006), Wangkeeree et al. (in press), and Ceng et al. (2009).

This paper is concerned with a pair of second-order mixed symmetric dual programs involving nondifferentiable functions. Weak, strong, and converse duality theorems are proved for aforementioned pair using the notion of second-order $F$-convexity/pseudoconvexity assumptions.

The long-term behavior of solutions of the following difference equation: $x_{n+1}=x_n{x}_{n-2}-1$, $n\in {\mathbb{N}}_0$, where the initial values $x_{-2}$, $x_{-1}$, $x_0$ are real numbers, is investigated in the paper.

This paper introduces multiresolution analyses with composite dilations (AB-MRAs) and addresses frame multiresolution analyses with composite dilations in the setting of reducing subspaces of $L^2\left({?}^n\right)$ (AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspace $L^2{\left(S\right)}^{\vee }$. For a general expansive matrix, we obtain the characterizations for a scaling function to generate an AB-RMRA, and the main theorems generalize the classical results. Finally, some examples are provided to illustrate the general theory.

Bacterial Foraging Optimization (BFO) is a recently developed nature-inspired optimization algorithm, which is based on the foraging behavior of E. coli bacteria. Up to now, BFO has been applied successfully to some engineering problems due to its simplicity and ease of implementation. However, BFO possesses a poor convergence behavior over complex optimization problems as compared to other nature-inspired optimization techniques. This paper first analyzes how the run-length unit parameter of BFO controls the exploration of the whole search space and the exploitation of the promising areas. Then it presents a variation on the original BFO, called the adaptive bacterial foraging optimization (ABFO), employing the adaptive foraging strategies to improve the performance of the original BFO. This improvement is achieved by enabling the bacterial foraging algorithm to adjust the run-length unit parameter dynamically during algorithm execution in order to balance the exploration/exploitation tradeoff. The experiments compare the performance of two versions of ABFO with the original BFO, the standard particle swarm optimization (PSO) and a real-coded genetic algorithm (GA) on four widely-used benchmark functions. The proposed ABFO shows a marked improvement in performance over the original BFO and appears to be comparable with the PSO and GA.

We investigate some interesting properties of the $q$-Euler polynomials. The purpose of this paper is to give some relationships between Bernstein and $q$-Euler polynomials, which are derived by the $p$-adic integral representation of the Bernstein polynomials associated with $q$-Euler polynomials.

We characterize the centre of the Banach lattice of Banach lattice $E$-valued continuous functions on the Alexandroff duplicate of a compact Hausdorff space $K$ in terms of the centre of $C(K,E)$, the space of $E$-valued continuous functions on $K$. We also identify the centre of $C{D}_{0}(Q,E)=C(Q,E)+{c}_{0}(Q,E)$ whose elements are the sums of $E$-valued continuous and discrete functions defined on a compact Hausdorff space $Q$ without isolated points, which was given by Alpay and Ercan (2000).

The criteria for nonsquareness in the classical Orlicz function spaces have been given already. However, because of the complication of Musielak-Orlicz-Bochner function spaces, at present the criteria for nonsquareness have not been discussed yet. In the paper, the criteria for nonsquareness of Musielak-Orlicz-Bochner function spaces are given. As a corollary, the criteria for nonsquareness of Musielak-Orlicz function spaces are given.

A Jacobi dual-Petrov-Galerkin (JDPG) method is introduced and used for solving fully integrated reformulations of third- and fifth-order ordinary differential equations (ODEs) with constant coefficients. The reformulated equation for the $J$th order ODE involves $n$-fold indefinite integrals for $n=1,\dots ,J$. Extension of the JDPG for ODEs with polynomial coefficients is treated using the Jacobi-Gauss-Lobatto quadrature. Numerical results with comparisons are given to confirm the reliability of the proposed method for some constant and polynomial coefficients ODEs.

We use star functions to determine the integral means for starlike log-harmonic mappings. Moreover, we include the upper bound for the arc length of starlike log-harmonic mappings.

We introduce a new iterative scheme and a new mapping generated by infinite family of nonexpansive mappings and infinite real number. By using both of these ideas, we obtain strong convergence theorem for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of fixed-point problems of infinite family of nonexpansive mappings. Moreover, we apply our main result to obtain strong convergence theorems for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of common fixed point of pseudocontractive mappings.

We give some sufficient conditions for mappings defined on vector ultrametric spaces to be modular locally constant.

An implicit iterative process is considered. Strong and weak convergence theorems of common fixed points of a finite family of asymptotically pseudocontractive mappings in the intermediate sense are established in a real Hilbert space.

Let $A$ and $B$ be nonempty subsets of a metric space $X$ and also $T:A\cup B\to A\cup B$ and $T\left(A\right)\subseteq B$, $T\left(B\right)\subseteq A$. We are going to consider element $x\in A$ such that $d(x,Tx)\le d(A,B)+?$ for some $?>0$. We call pair $(A,B)$ an approximate best proximity pair. In this paper, definitions of approximate best proximity pair for a map and two maps, their diameters, $T$-minimizing a sequence are given in a metric space.

We deal with the existence and multiplicity of solutions for the periodic boundary value problem $x\text{'}\text{'}\left(t\right)+a\left(t\right)x\left(t\right)=\lambda g\left(t\right)f\left(x\right)+c\left(t\right)$, $x\left(0\right)=x\left(T\right)$, $x\text{'}\left(0\right)=x\text{'}\left(T\right)$, where $\lambda$ is a positive parameter. The function $f:(0,\infty )\to (0,\infty )$ is allowed to be singular, and the related Green's function is nonnegative and can vanish at some points.

Using the equivalence relation between $K$-functional and modulus of smoothness, we establish a strong direct theorem and an inverse theorem of weak type for multivariate Bernstein-Durrmeyer operators with Jacobi weights on a simplex in this paper. We also obtain a characterization for multivariate Bernstein-Durrmeyer operators with Jacobi weights on a simplex. The obtained results not only generalize the corresponding ones for Bernstein-Durrmeyer operators, but also give approximation order of Bernstein-Durrmeyer operators.

We offer computational formula of subdivision depth for tensor product $n$-ary $(n⩾2)$ volumetric models based on error bound evaluation technique. This formula provides and error control tool in subdivision schemes over regular hexahedron lattice in higher-dimensional spaces. Moreover, the error bounds of Mustafa et al. (2006) are special cases of our bounds.

We construct a new Halpern type iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized $f$-projection operator. Using this result, we discuss strong convergence theorem concerning general $H$-monotone mappings. Our results extend many known recent results in the literature.

We study the time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection. Applying methods from the theory of univalent functions, we prove the invariance in time of $\Phi$-likeness property (a geometric property which includes starlikeness and spiral-likeness) for two basic cases: the inner problem and the outer problem. We study both zero and nonzero surface tension models. Certain particular cases are also presented.

The paper is devoted to integro-differential operators, which correspond to nonlocal reaction-diffusion equations considered on the whole axis. Their Fredholm property and properness will be proved. This will allow one to define the topological degree.

We give a partition of the critical strip, associated with each partial sum $1+2^z+\cdots +n^z$ of the Riemann zeta function for Re $z<-1$ formed by infinitely many rectangles for which a formula allows us to count the number of its zeros inside each of them with an error, at most, of two zeros. A generalization of this formula is also given to a large class of almost-periodic functions with bounded spectrum.

Existence and uniqueness of the solution for a time-fractional diffusion equation with Robin boundary condition on a bounded domain with Lyapunov boundary is proved in the space of continuous functions up to boundary. Since a Green matrix of the problem is known, we may seek the solution as the linear combination of the single-layer potential, the volume potential, and the Poisson integral. Then the original problem may be reduced to a Volterra integral equation of the second kind associated with a compact operator. Classical analysis may be employed to show that the corresponding integral equation has a unique solution if the boundary data is continuous, the initial data is continuously differentiable, and the source term is Hölder continuous in the spatial variable. This in turn proves that the original problem has a unique solution.

We introduce a general composite algorithm for finding a common element of the set of solutions of a general equilibrium problem and the common fixed point set of a finite family of asymptotically nonexpansive mappings in the framework of Hilbert spaces. Strong convergence of such iterative scheme is obtained which solving some variational inequalities for a strongly monotone and strictly pseudocontractive mapping. Our results extend the corresponding recent results of Yao and Liou (2010).

This paper characterizes the boundedness and compactness of the product of extended Cesàro operator and composition operator from Lipschitz space to $F(p,q,s)$ space on the unit ball of ${ℂ}^n$.

Navigation safety has a huge impact on the world economy and our everyday lives. One navigation safety simulation model in ECDIS based on international standard format (S-57) is put forward, which is mainly involved in route plan and route monitoring. The universal kriging interpolation is used in the route planning and to compute the water depth of any place in the sea bottom. The man-machine conversation method is taken to amend planned route to obtain autodeciding of feasibility according to ECDIS information, and the route monitoring algorithm is improved by enhancing its precision caused by screen coordinate conversion. The DCQA (distance close quarters situation of approach) model and TCQA (time close quarters situation of approach) model are adopted to judge if the close quarters situation or the risk of collision between own ship and target ship is emerging. All these methods are proven to be reliable through the navigation simulator made by Dalian Maritime University which is certified by DNV to class A.

We prove the Hyers-Ulam stability of power series equation ${{\displaystyle \sum }}_{n=0}^{\infty }{a}_n{x}^n=0$, where $a_n$ for $n=0,1,2,3,\dots$ can be real or complex.

The Sharma-Tasso-Olver (STO) equation is investigated. The Painlevé analysis is efficiently used for analytic study of this equation. The Bäcklund transformations and some new exact solutions are formally derived.

This paper presents an investigation of asymptotic properties of a stochastic predator-prey model with modified Leslie-Gower response. We obtain the global existence of positive unique solution of the stochastic model. That is, the solution of the system is positive and not to explode to infinity in a finite time. And we show some asymptotic properties of the stochastic system. Moreover, the sufficient conditions for persistence in mean and extinction are obtained. Finally we work out some figures to illustrate our main results.

We will prove some dynamic inequalities of Opial type on time scales which not only extend some results in the literature but also improve some of them. Some discrete inequalities are derived from the main results as special cases.

This paper is devoted to the study of the stochastic stability of a class of Cohen-Grossberg neural networks, in which the interconnections and delays are time-varying. With the help of Lyapunov function, Burkholder-Davids-Gundy inequality, and Borel-Cantell's theory, a set of novel sufficient conditions on $p$th moment exponential stability and almost sure exponential stability for the trivial solution of the system is derived. Compared with the previous published results, our method does not resort to the Razumikhin-type theorem and the semimartingale convergence theorem. Results of the development as presented in this paper are more general than those reported in some previously published papers. An illustrative example is also given to show the effectiveness of the obtained results.

On utilising the spectral representation of selfadjoint operators in Hilbert spaces, some trapezoidal inequalities for various classes of continuous functions of such operators are given.

We consider the operator $L$ generated in $L^2\left({?}_{+}\right)$ by the differential expression $l\left(y\right)=-y^{?}+q\left(x\right)y$, $x\in {?}_{+}:=[0,\infty )$ and the boundary condition $y^{\prime }\left(0\right)/y\left(0\right)={\alpha }_0+{\alpha }_1\lambda +{\alpha }_2{\lambda }^2$, where $q$ is a complex-valued function and ${\alpha }_i\in ?$, $i=0,1,2$ with ${\alpha }_2\ne 0$. In this paper we obtain the properties of the principal functions corresponding to the spectral singularities of $L$.

A second-order semilinear Volterra integrodifferential equation involving fractional time derivatives is considered. We prove existence and uniqueness of mild solutions and classical solutions in appropriate spaces.

We prove that some classes of summable sequences of positive real numbers satisfy several selection principles related to a special kind of convergence.

In the study by Papanastassiou and Papachristodoulos, 2009 the notion of $p$-convergence in measure was introduced. In a natural way $p$-convergence in measure induces an equivalence relation on the space $M$ of all sequences of measurable functions converging in measure to zero. We show that the quotient space $ℳ$ is a complete but not compact metric space.

New characterizations of partial isometries and EP elements in Banach algebra are presented.

Let $A$, $B$ be two unital $C^{*}$-algebras. We prove that every almost unital almost linear mapping $h$ : $A\to B$ which satisfies $h(3^{n}uy+3^{n}yu)=h\left(3^{n}u\right)h\left(y\right)+h\left(y\right)h\left(3^{n}u\right)$ for all $u\in U\left(A\right)$, all $y\in A$, and all $n=0,1,2,\dots$, is a Jordan homomorphism. Also, for a unital $C^{*}$-algebra $A$ of real rank zero, every almost unital almost linear continuous mapping $h:A\to B$ is a Jordan homomorphism when $h(3^{n}uy+3^{n}yu)=h\left(3^{n}u\right)h\left(y\right)+h\left(y\right)h\left(3^{n}u\right)$ holds for all $u\in I_1$ ($A_{\text{sa}}$), all $y\in A$, and all $n=0,1,2,\dots$. Furthermore, we investigate the Hyers- Ulam-Aoki-Rassias stability of Jordan $*$-homomorphisms between unital $C^{*}$-algebras by using the fixed points methods.

We discuss thermodynamical restrictions for a linear constitutive equation containing fractional derivatives of stress and strain of different orders. Such an equation generalizes several known models. The restrictions on coefficients are derived by using entropy inequality for isothermal processes. In addition, we study waves in a rod of finite length modelled by a linear fractional constitutive equation. In particular, we examine stress relaxation and creep and compare results with the quasistatic analysis.

The spaces with a random variable exponent $L^{p\left(\omega \right)}(D\times \Omega )$and $W^{k,p\left(\omega \right)}(D\times \Omega )$ are introduced. After discussing the properties of the spaces $L^{p\left(\omega \right)}(D\times \Omega )$ and $W^{k,p\left(\omega \right)}(D\times \Omega )$, we give an application of these spaces to the stochastic partial differential equations with random variable growth.

We investigate the existence and multiplicity of periodic solutions for a class of second-order differential systems with impulses. By using variational methods and critical point theory, we obtain such a system possesses at least one nonzero, two nonzero, or infinitely many periodic solutions generated by impulses under different conditions, respectively. Recent results in the literature are generalized and significantly improved.

We also introduce forward curvature of a curve and give some formulas to calculate forward curvature of a curve on time scales which may be an arbitrary closed subsets of the set of all real numbers. We also introduce the length of a curve parametrized by a time scale parameter in ${ℝ}^3$.

Suppose that $X$ is a separable normed space and the operators $A$ and $Q$ are bounded on $X$. In this paper, it is shown that if $AQ=QA$, $A$ is an isometry, and $Q$ is a nilpotent then the operator $A+Q$ is neither supercyclic nor weakly hypercyclic. Moreover, if the underlying space is a Hilbert space and $A$ is a co-isometric operator, then we give sufficient conditions under which the operator $A+Q$ satisfies the supercyclicity criterion.

Using the generalized variational principle and the Riccati technique, new oscillation criteria are established for the forced second-order nonlinear differential equation, which improves and generalizes some of the new results in literature.

For $1<r<+\infty$, we find the least value $\alpha$ and the greatest value $\beta$ such that the inequality $H_{\alpha }(a,b)<A_r(a,b)<H_{\beta }(a,b)$ holds for all $a,b>0$ with $a\ne b$. Here, $H_{\omega }(a,b)$ and $A_r(a,b)$ are the generalized Heronian and the power means of two positive numbers $a$ and $b$, respectively.

We adapt the part metric and use it in studying positive solutions of a certain family of discrete dynamic systems. Some examples are presented, and we also compare some results in the literature.

We investigate an m-point boundary value problem for nonlinear fractional differential equations. The associated Green function for the boundary value problem is given at first, and some useful properties of the Green function are obtained. By using the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed point theorem, the existence of multiple positive solutions is obtained.

Let $f_n\left(z\right)=z+{{\displaystyle \sum }}_{k=2}^n{a}_k{z}^k$ be the sequence of partial sums of the analytic function $f\left(z\right)=z+{{\displaystyle \sum }}_{k=2}^{\infty }{a}_k{z}^k$. In this paper, we determine sharp lower bounds for $\Re \{f\left(z\right)/f_n\left(z\right)\},\Re \{f_n\left(z\right)/f\left(z\right)\},\Re \{f^{\prime }\left(z\right)/f_n^{\prime }\left(z\right)\}$, and $\Re \{f_n^{\prime }\left(z\right)/f^{\prime }\left(z\right)\}$. The usefulness of the main result not only provides the unification of the results discussed in the literature but also generates certain new results.

We established the existence of a positive solution of nonlinear fractional differential equations $𝔏\left(D\right)[x\left(t\right)-x\left(0\right)]=f(t,x_t)$, $t\in (0,b]$ with finite delay $x\left(t\right)=\omega \left(t\right)$, $t\in [-\tau ,0]$, where ${\lim }_{t\to 0}f(t,x_t)=+\infty$, that is, $f$ is singular at $t=0$ and $x_t\in C([-\tau ,0],{ℝ}^{\ge 0})$$. The operator of $𝔏\left(D\right)$ involves the Riemann-Liouville fractional derivatives. In this problem, the initial conditions with fractional order and some relations among them were considered. The analysis rely on the alternative of the Leray-Schauder fixed point theorem, the Banach fixed point theorem, and the Arzela-Ascoli theorem in a cone.

The concepts of $\alpha$-well-posedness, $\alpha$-well-posedness in the generalized sense, L-$\alpha$-well-posedness and L-$\alpha$-well-posedness in the generalized sense for mixed quasi variational-like inequality problems are investigated. We present some metric characterizations for these well-posednesses.

We prove the existence of new-type solutions of the modified Fischer-Kolmogorov equation with slow/fast diffusion and with possibly nonsmooth double-well potential. We show that a certain relation between the rate of the diffusion and the smoothness of the potential may originate new type solutions which do not occur in the classical Fischer-Kolmogorov equation. The main focus of this paper is to show the sensitivity of the mathematical modelling with respect to the chosen form of the diffusion term and the shape of the double-well potential.

We present a general result concerning the limit of the iterates of positive linear operators acting on continuous functions defined on a compact set. As applications, we deduce the asymptotic behaviour of the iterates of almost all classic and new positive linear operators.

We study a number of fixed point results for the two weakly increasing mappings $f$ and $g$ with respect to partial ordering relation $\le$ in generalized metric spaces. An application to integral equation is given.

Let $\left\{z_n\right\}$, $\left\{w_n\right\}$, and $\left\{v_n\right\}$ be bounded sequences in a metric space of hyperbolic type $(X,d)$, and let $\left\{{\alpha }_n\right\}$ be a sequence in $[0,1]$ with $0<{\lim \inf }_n{\alpha }_n\le {\lim \sup }_n{\alpha }_n<1$. If $z_{n+1}={\alpha }_n{w}_n\oplus (1-{\alpha }_n)v_n$ for all $n\in \mathbb{N}$, ${\lim }_{n}d(z_n,v_n)=0$ , and ${\lim \sup }_n(d(w_{n+1},w_n)-d(z_{n+1},z_n\left)\right)\le 0$ , then ${\lim }_{n}d(w_n,z_n)=0$. This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.

We study the membership of inner functions in Besov, Lipschitz, and Hardy-Sobolev spaces, finding conditions that enable an inner function to be in one of these spaces. Several results in this direction are given that complement or extend previous works on the subject from different authors. In particular, we prove that the only inner functions in either any of the Hardy-Sobolev spaces $H_{\alpha }^p$ with $1/p\le \alpha <\mathrm{\infty }$ or any of the Besov spaces $B_{\alpha }^{p,\mathrm{q}}$ with $0<p,q\le \mathrm{\infty }$ and $\alpha \ge 1/p$, except when $p=\mathrm{\infty }$, $\alpha =0$, and $2<q\le \mathrm{\infty }$ or when $0<p<\mathrm{\infty }$, $q=\mathrm{\infty }$, and $\alpha =1/p$ are finite Blaschke products. Our assertion for the spaces $B_0^{\mathrm{\infty },q}$, $0<q\le 2$, follows from the fact that they are included in the space $\text{i}tVMOA$. We prove also that for $2<q<\mathrm{\infty }$, $\text{i}tVMOA$ is not contained in $B_0^{\mathrm{\infty },q}$ and that this space contains infinite Blaschke products. Furthermore, we obtain distinct results for other values of $\alpha $ relating the membership of an inner function $I$ in the spaces under consideration with the distribution of the sequences of preimages $\left\{I^{-1}\left(a\right)\right\}$ , $\left|a\right|<1$. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.

We investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem (BVP) $a_{n-1}{y}_{n-1}+b_n{y}_n+a_n{y}_{n+1}=\lambda y_n$ , $n\in \mathbb{N}$ and $({\gamma }_0+{\gamma }_1\lambda )y_1+({\beta }_0+{\beta }_1\lambda )y_0=0$, where $\left(a_n\right)$ and $\left(b_n\right)$ are complex sequences, $\lambda$ is an eigenparameter, and ${\gamma }_i$,${\beta }_i\in ℂ$ for $i=0$, 1.

By making use of the Dziok-Srivastava operator, we introduce a new class of meromorphically multivalent functions. Some inclusion properties of functions belonging to this class are derived.

We present the results of the theoretical researches of some characteristics of the generalized hyperspherical function with two degrees as independent dimensions. Primarily are given the answers on the quantification of the dimensional potentials (fluxes) of this function in the domain of the integer natural numbers. Beside them, we have got the solutions for some continual fluxes of the contour hypercylindrical (HY) functions on the basis. The symbolical evaluation and numerical verification of the series values and integrals are realized through the program packages Mathcad Professional and Mathematica.

We study the existence of solutions for deviated-advanced nonlocal and integral condition problems for the differential inclusion $x^1\left(t\right)\in F(t,x(t\left)\right)$.

We give an extension of Jensen's inequality for $n$-tuples of self-adjoint operators, unital $n$-tuples of positive linear mappings, and real-valued continuous convex functions with conditions on the operators' bounds. We also study operator quasiarithmetic means under the same conditions.

A new nonlinear mapping is introduced. Hybrid projection algorithms are considered for the class of new nonlinear mappings. Strong convergence theorems are established in a real Banach space.

A class of two-neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.

We present some topics about random spaces. The main purpose of this paper is to study topological structure of random normed spaces and random functional analysis. These subjects are important to the study of nonlinear analysis in random normed spaces.

We study general solutions and generalized Hyers-Ulam-Rassias stability of the following $n$-dimensional functional equation $f\left({{\displaystyle \sum }}_{i=1}^k{x}_i\right)+(k-2){{\displaystyle \sum }}_{i=1}^{k}f\left(x_i\right)={{\displaystyle \sum }}_{i=1}^k{{\displaystyle \sum }}_{j=1,j>i}^{k}f(x_i+x_j)$, $k\ge 3$, on non-Archimedean normed spaces.

We characterize the global and nonglobal solutions of the Timoshenko equation in a bounded domain. We consider nonlinear dissipation and a nonlinear source term. We prove blowup of solutions as well as convergence to the zero and nonzero equilibria, and we give rates of decay to the zero equilibrium. In particular, we prove instability of the ground state. We show existence of global solutions without a uniform bound in time for the equation with nonlinear damping. We define and use a potential well and positive invariant sets.

An optimal lower eigenvalue system is studied, and main theorems including a series of necessary and suffcient conditions concerning existence and a Lipschitz continuity result concerning stability are obtained. As applications, solvability results to some von-Neumann-type input-output inequalities, growth, and optimal growth factors, as well as Leontief-type balanced and optimal balanced growth paths, are also gotten.

We prove the eigenfunction expansion formula for a Dirichlet problem with explosive factor by two ways, first by standard method and second by proving a convergence in some metric space $L_2(0,\pi ;\rho (x\left)\right)$.

We consider a nonlinear equation $F(ϵ,\lambda ,u)=0$, where the parameter$ϵ$ is a perturbation parameter, $F$ is a differentiable mapping from $\text{b}fR\times \text{b}fR\times X$ to $\mathrm{Y}$and $X$, $Y$ are Banach spaces. We obtain an abstract bifurcation theorem by using the generalized saddle-node bifurcation theorem.

This paper is concerned with distribution of maps with transversal homoclinic orbits in a continuous map space, which consists of continuous maps defined in a closed and bounded set of a Banach space. By the transversal homoclinic theorem, it is shown that the map space contains a dense set of maps that have transversal homoclinic orbits and are chaotic in the sense of both Li-Yorke and Devaney with positive topological entropy.

Existence and multiplicity of positive solutions for the following semilinear elliptic equation: $-\Delta u+u=a\left(x\right){\left|u\right|}^{p-2}u+\lambda b\left(x\right){\left|u\right|}^{q-2}u$ in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, are established, where $\lambda >0,1<q<2<p<2^{*}(2^{*}=2N/(N-2)$ if $N\ge 3,2^{*}=\infty$ if $N=1,2)$, $a$, $b$ satisfy suitable conditions, and $b$ maybe changes sign in ${ℝ}^N$. The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.

Let $\psi$ be a holomorphic mapping on the upper half-plane ${\Pi }^{+}=\{z\in ℂ:𝔍z>0\}$ and $\varphi$ be a holomorphic self-map of ${\Pi }^{+}$. We characterize bounded weighted composition operators acting from the weighted Bergman space to the weighted-type space on the upper half-plane. Under a mild condition on $\psi$, we also characterize the compactness of these operators.

We improve the viscosity approximation process for approximation of a fixed point of a quasi-nonexpansive mapping in a Hilbert space proposed by Maingé (2010). An example beyond the scope of the previously known result is given.

We first establish the existence and uniqueness of a solution for a stochastic $p$-Laplacian-type equation with additive white noise and show that the unique solution generates a stochastic dynamical system. By using the Dirichlet forms of Laplacian and an approximation procedure, the nonlinear obstacle, arising from the additive noise is overcome when we make energy estimate. Then, we obtain a random attractor for this stochastic dynamical system. Finally, under a restrictive assumption on the monotonicity coefficient, we find that the random attractor consists of a single point, and therefore the system possesses a unique stationary solution.

We study some new $A$-sequence spaces using ideal convergence and an Orlicz function in 2-normed space and we give some relations related to these sequence spaces.

We investigate the oscillation of the following higher-order functional differential equation: $x^{\left(n\right)}\left(t\right)+q\left(t\right)|x(t-\tau ){|}^{\lambda -1}x(t-\tau )=e\left(t\right),$ where $q\left(t\right)$ and $e\left(t\right)$ are continuous functions on $[t_0,\mathrm{\infty })$, $1>\lambda >0$ and $\tau \ne 0$ are constants. Unlike most of delay-dependent oscillation results in the literature, two delay-independent oscillation criteria for the equation are established in both the case $\tau >0$ and the case $\tau <0$ under the assumption that the potentials $q\left(t\right)$ and $e\left(t\right)$ change signs on $[t_0,\mathrm{\infty })$.

The method of order completion provides a general and type-independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significant improvement upon the regularity of the solutions and provided new insight into the structure of solutions. In this paper, we show how this method may be adapted so as to allow for the infinite differentiability of generalized functions. Moreover, it is shown that a large class of smooth nonlinear PDEs admit generalized solutions in the space constructed here. As an indication of how the general theory can be applied to particular nonlinear equations, we construct generalized solutions of the parametrically driven, damped nonlinear Schrödinger equation in one spatial dimension.

We characterize the conditions under which approximately multiplicative functionals are near multiplicative functionals on weighted Hardy spaces.

Let $T_n(A,B,\gamma ,\alpha )(-1\le B<1,B<A,0<\gamma \le 1$ and $\alpha >0)$ denote the class of functions of the form $f\left(z\right)=z+{{\displaystyle \sum }}_{k=n+1}^{\mathrm{\infty }}{a}_k{z}^{\mathrm{k}}(n\in N=\{1,2,3,\dots \}),$ which are analytic in the open unit disk $U$ and satisfy the following subordination condition $f^{\prime }\left(z\right)+\alpha z{f}^{\prime \prime }\left(z\right)\prec {\left(\right(1+Az)/(1+Bz\left)\right)}^{\gamma }$ , for $(z\in U;A\le 1;0<\gamma <1),(1+Az)/(1+Bz)$, for $(z\in U;\gamma =1)$. We obtain sharp bounds on $\text{R}\text{e}f\text{'}\left(z\right),\text{R}\text{e}f\left(z\right)/z,\left|f\left(z\right)\right|$ , and coefficient estimates for functions $f\left(z\right)$ belonging to the class $T_n(A,B,\gamma ,\alpha )$. Conditions for univalency and starlikeness, convolution properties, and the radius of convexity are also considered.

We have presented an efficient spectral algorithm based on shifted Jacobi tau method of linear fifth-order two-point boundary value problems (BVPs). An approach that is implementing the shifted Jacobi tau method in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of fifth-order differential equations with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplify the problem. Shifted Jacobi collocation method is developed for solving nonlinear fifth-order BVPs. Numerical examples are performed to show the validity and applicability of the techniques. A comparison has been made with the existing results. The method is easy to implement and gives very accurate results.

We discuss a new concept of the $q$-extension of Bernoulli measure. From those measures, we derive some interesting properties on the generalized $q$-Bernoulli numbers with weight $\alpha$ attached to $\chi$.

This paper gives sufficient and necessary conditions for the classification of Sturm-Liouville differential equations with complex coefficients given by Brown et al. These conditions involve weighted Sobolev subspaces and the asymptotic behavior of elements in the maximal domain. The results of the present paper generalize the corresponding results for formally symmetric Sturm-Liouville differential equations to non-self-adjoint cases.

We find the least value $\lambda \in (0,1)$ and the greatest value $p=p\left(\alpha \right)$ such that $\alpha H(a,b)+(1-\alpha )L(a,b)>M_p(a,b)$ for $\alpha \in [\lambda ,1)$ and all $a,b>0$ with $a\ne b$, where $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ are the harmonic, logarithmic, and $p$-th power means of two positive numbers $a$ and $b$, respectively.

The main objective of this paper is to present some best proximity point theorems for K-cyclic mappings and C-cyclic mappings in the frameworks of metric spaces and uniformly convex Banach spaces, thereby furnishing an optimal approximate solution to the equations of the form $Tx=x$ where $T$ is a non-self mapping.

We are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional boundary value problem: $D_{0+}^{\alpha }u\left(t\right)+f(t,u(t\left)\right)=0$ , $0\le t\le 1$, $3<\alpha \le 4$, $u\left(0\right)=u^{\prime }\left(0\right)=u^{?}\left(0\right)=u^{?}\left(1\right)=0$, where $D_{0+}^{\alpha }$ denotes the standard Riemann-Liouville fractional derivative. Our analysis relies on a fixed point theorem in partially ordered sets. Some examples are also given to illustrate the results.

$2\times n$ or $m\times 2$ interval matrix games are considered, and a graphical method for solving such games is given. Interval matrix game is the interval generation of classical matrix games. Because of uncertainty in real-world applications, payoffs of a matrix game may not be a fixed number. Since the payoffs may vary within a range for fixed strategies, an interval-valued matrix can be used to model such uncertainties. In the literature, there are different approaches for the comparison of fuzzy numbers and interval numbers. In this work, the idea of acceptability index is used which is suggested by Sengupta et al. (2001) and Sengupta and Pal (2009), and in view of acceptability index, well-known graphical method for matrix games is adapted to interval matrix games.

We study univalence properties for certain subclasses of univalent functions $𝔎$, ${𝔎}_2$, ${𝔎}_3$, and $S\left(p\right)$, respectively. These subclasses are associated with a generalized integral operator. The extended Becker-typed univalence criteria will be studied for these subclasses.

The problem of robust exponential stabilization for dynamical nonlinear systems with uncertainties and time-varying delay is considered in the paper. By constructing the proposed Lyapunov-Krasovskii functional approach, continuous state feedback controllers are put forward, and the criteria which guarantee the exponential stabilization of the nonlinear systems with uncertainties and time-varying delay are established in terms of solutions to the standard Riccati differential equations. Furthermore, based on the Lyapunov method and the linear matrix inequality approach, the sufficient conditions of exponential stability for a class of uncertain systems with time-varying delays and nonlinear perturbations are derived. Finally, two numerical examples are given to demonstrate the validity of the results.

This paper is devoted to the study of mild solutions for the initial and boundary value problem of stochastic viscous Cahn-Hilliard equation driven by white noise. Under reasonable assumptions we first prove the existence and uniqueness result. Then, we show that the existence of a stochastic global attractor which pullback attracts each bounded set in appropriate phase spaces.