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In this note, we point out several gaps in Gurudwan and Sharma (Appl. Math. Comput. 217(15):6748-6754, 2011) and Yang (Appl. Math. Comput. 218(21):10367-10369, 2012) and give the main results under weaker conditions. MSC: 47H10; 47H09; 46B20

The paper deals with systems of linear differential equations with coefficients depending on the Markov process. Equations for particular density and the moment equations for given systems are derived and used in the investigation of solvability of initial problems and stability. Results are illustrated by examples. The paper deals with systems of linear differential equations with coefficients depending on the Markov process. Equations for particular density and the moment equations for given systems are derived and used in the investigation of solvability of initial problems and stability. Results are illustrated by examples.

In this paper we have given conditions on exponential polynomials $$P_{n}(s)$$ of Dirichlet type to be attained the equality between each of two pairs of bounds, called essential bounds, $$a_{P_{n}(s)}$$ , $$\rho _{N}$$ and $$b_{P_{n}(s)}$$ , $$\rho _{0}$$ associated with $$P_{n}(s)$$ . The reciprocal question has been also treated. The bounds $$a_{P_{n}(s)}$$ , $$b_{P_{n}(s)}$$ are defined as the end-points of the minimal closed and bounded real interval $$I= [ a_{P_{n}(s)},b_{P_{n}(s)} ] $$ such that all the zeros of $$P_{n}(s)$$ are contained in the strip $$I\times {\mathbb {R}}$$ of the complex plane $${\mathbb {C}}$$ . The bounds $$\rho _{N}$$ , $$\rho _{0}$$ are defined as the unique real solutions of Henry equations of $$P_{n}(s)$$ . Some applications to the partial sums of the Riemann zeta function have been also showed.

Let $${\mathbb {F}}$$ a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action $${(T_{g})_{{g} \in \mathbb {F}^{\omega}}}$$ of the infinite abelian group $${\mathbb {F}^{\omega}}$$ on a probability space $${X = (X, \mathcal {B}, \mu)}$$ is generated by phase polynomials $${\phi : X \to S^{1}}$$ of degree less than C(k) on X, where C(k) depends only on k. In the case where $${k \leq {\rm char}(\mathbb {F})}$$ we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for $${\mathbb {Z}}$$ by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case $${k \leq {\rm char}(\mathbb {F})}$$ , with a partial result in low characteristic.

The purpose of this paper is to introduce and study the strong convergence problem of the explicit iteration process for a Lipschitzian and demicontraction semigroups in arbitrary Banach spaces. The main results presented in this paper not only extend and improve some recent results announced by many authors, but also give an affirmative answer for the open questions raised by Suzuki (2003) and Xu (2005).

The main purpose of this paper is to introduce an explicit iterative algorithm to study the existence problem and the approximation problem of solution to the quadratic minimization problem. Under suitable conditions, some strong convergence theorems for a family of nonexpansive mappings are proved. The results presented in the paper improve and extend the corresponding results announced by some authors.

Let M be a shift invariant subspace in the vector-valued Hardy space \({H_{E}^{2}(\mathbb{D})}\). The Beurling–Lax–Halmos theorem says that M can be completely characterized by \({\mathcal{B}(E)}\)-valued inner function \({\Theta}\). When \({E = H^{2}(\mathbb{D}),\,H_{E}^{2}(\mathbb{D})}\) is the Hardy space on the bidisk \({H^{2}(\mathbb{D}^2)}\). Recently, Qin and Yang (Proc Am Math Soc, 2013) determines the operator valued inner function \({\Theta(z)}\) for two well-known invariant subspaces in \({H^{2}(\mathbb{D}^{2})}\). This paper generalizes the \({\Theta(z)}\) by Qin and Yang (Proc Am Math Soc, 2013) and deal with the structure of \({M = {\Theta}(z)H^{2}(\mathbb{D}^{2})}\) when M is an invariant subspace in \({H^{2}(\mathbb{D}^{2})}\). Unitary equivalence, spectrum of the compression operator and core operator are studied in this paper.

The purpose of this paper is (1) a general nonlinear mixed set-valued inclusion framework for the over-relaxed -proximal point algorithm based on the ( , )-accretive mapping is introduced, and (2) it is applied to the approximation solvability of a general class of inclusions problems using the generalized resolvent operator technique due to Lan-Cho-Verma, and the convergence of iterative sequences generated by the algorithm is discussed in -uniformly smooth Banach spaces. The results presented in the paper improve and extend some known results in the literature.

We study common fixed point theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.

The purpose of this paper is to study the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces. The results presented in this paper extend and improve the corresponding results of Chang and Cho (2003), Xu and Ori (2001), and Zhou and Chang (2002).

The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping and a finite family of nonexpansive mappings , respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and others.

This is the third part in a series of papers concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part focuses on applications of general theory, developed in the authors' paper "Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory," Trans. Amer. Math. Soc. 365 (2013), pp. 5329-5365, to positive continuous-time random dynamical systems on infinite dimensional ordered Banach spaces arising from random parabolic equations and random delay systems. It is shown under some quite general assumptions that measurable linear skew-product semidynamical systems generated by random parabolic equations and by cooperative systems of linear delay differential equations admit measurable families of generalized principal Floquet subspaces, and generalized principal Lyapunov exponents., Comment: 41 pages

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly -Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend the corresponding results announced by Chang (2001), Cho et al. (2005), Ofoedu (2006), Schu (1991) and Zeng (2003 and 2005), and many others.

In the previous paper by the first and the third authors, we present six algorithms for determining whether a given symmetric matrix is strictly copositive, copositive (but not strictly), or not copositive. The algorithms for matrices of order are not guaranteed to produce an answer. It also shows that for 1000 symmetric random matrices of order 8, 9, and 10 with unit diagonal and with positive entries all being less than or equal to 1 and negative entries all being greater than or equal to , there are 8, 6, and 2 matrices remaing undetermined, respectively. In this paper we give two more algorithms for and our experiment shows that no such matrix of order 8 or 9 remains undetermined; and almost always no such matrix of order 10 remains undetermined. We also do some discussion based on our experimental results.

In a recent paper, Sun et al. (2007) studied the existence of positive solutions of nonhomogeneous multipoint boundary value problems for a second-order differential equation. It is the purpose of this paper to show that the shooting method approach proposed in the recent paper by the first author can be extended to treat this more general problem.

By defining a new terminology, scatter degree, as the supremum of graininess functional value, this paper studies the existence of solutions for a nonlinear two-point dynamic boundary value problem on time scales. We do not need any growth restrictions on nonlinear term of dynamic equation besides a barrier strips condition. The main tool in this paper is the induction principle on time scales.

The purpose of this paper is to use the modified block iterative method to propose an algorithm for solving the convex feasibility problems for an infinite family of quasi- -asymptotically nonexpansive mappings. Under suitable conditions some strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces with Kadec-Klee property. The results presented in the paper improve and extend some recent results.

For star bodies, the dual affine quermassintegrals were introduced and studied in several papers. The aim of this paper is to study them further. In this paper, some inequalities for dual affine quermassintegrals are established, such as the Minkowski inequality, the dual Brunn-Minkowski inequality, and the Blaschke-Santaló inequality.

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem and a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem.

In the paper by Olaleru and Akewe (2007), the authors tried to generalize Gregus fixed point theorem. In this paper we give a counterexample on their main statement.

In this paper, we study the existence of nontrivial solution for a class of quasilinear Schrodinger equations in ${\mathbb{R}}$ with the nonlinearity asymptotically linear and, furthermore, the potential indefinite in sign. The tool used in this paper is the direct variation method.

In this paper, we investigate the symmetric mixed isoperimetric deficit $\Delta_{2}(K_{0},K_{1})$ of domains $K_{0}$ and $K_{1}$ in the Euclidean plane $\mathbb{R}^{2}$ . Via the known kinematic formulae of Poincare and Blaschke in integral geometry, we obtain some Bonnesen-style symmetric mixed inequalities. These new Bonnesen-style symmetric mixed inequalities are known as Bonnesen-style inequalities if one of the domains is a disc. Some inequalities obtained in this paper strengthen the known Bonnesen-style inequalities.

The split equality problem has board applications in many areas of applied mathematics. Many researchers studied this problem and proposed various algorithms to solve it. From the literature we know that most algorithms for the split equality problems came from the idea of the projected Landweber algorithm proposed by Byrne and Moudafi (Working paper UAG, 2013), and few algorithms came from the idea of the alternating CQ-algorithm given by Moudafi (Nonlinear Anal. 79:117-121, 2013). Hence, it is important and necessary to give new algorithms from the idea of the alternating CQ-algorithm. In this paper, we first present a hybrid projected Landweber algorithm to study the split equality problem. Next, we propose a hybrid alternating CQ-algorithm to study the split equality problem. As applications, we consider the split feasibility problem and linear inverse problem. Finally, we give numerical results for the split feasibility problem to demonstrate the efficiency of the proposed algorithms.

The purpose of this paper is to introduce and consider a new hybrid shrinking projection method for finding a common element of the set EP of solutions of a generalized equilibrium problem, the common fixed point set F of finite uniformly closed families of countable Bregman quasi-Lipschitz mappings in reflexive Banach spaces. It is proved that under appropriate conditions, the sequence generated by the hybrid shrinking projection method converges strongly to some point in $\mathit{EP} \cap F$ . Relative examples are given. Strong convergence theorems are proved. The application for Bregman asymptotically quasi-nonexpansive mappings is also given. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable Bregman quasi-Lipschitz mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable Bregman quasi-Lipschitz mappings are given in classical Banach spaces $l^{2}$ and $L^{2}$ ; (3) the application for Bregman asymptotically quasi-nonexpansive mappings is also given; (4) because the main theorems do not need the boundedness of the domain of mappings, so a corresponding technique for the proof is given. This new results improve and extend the previously known ones in the literature.

Calvert and Gupta’s results concerning the perturbations on the ranges of m-accretive mappings have been employed widely in the discussion of the existence of solutions of nonlinear elliptic differential equation with Neumann boundary. In this paper, we shall focus our attention on certain hyperbolic differential equation with mixed boundaries. By defining some suitable nonlinear mappings, we shall demonstrate that Calvert and Gupta’s results can be applied to hyperbolic equations, in addition to its wide usage in elliptic equations. Due to the differences between hyperbolic and elliptic equations, some new techniques have been developed in this paper, which can be regarded as the complement and extension of the previous work.

WOS: 000378992800001, This paper is devoted to the optimal control problem for stochastic linear switching systems with a quadratic cost functional. A necessary and sufficient condition of optimality for mentioned linear control systems under endpoint constraints is obtained. A linear quadratic controller is simply constructed via a set of stochastic backward Riccati equations., Anadolu University, Turkey [1505F202], The author thanks the two anonymous reviewers whose comments and suggestions helped improve this manuscript. The research underlying this paper is supported by the Scientific Research Project No. 1505F202 of Anadolu University, Turkey.

In this paper, we are interested in the eigenvalues and its algebraic multiplicities of a fractional linear boundary value problem with mixed set of Neumann and Dirichlet boundary conditions. The research results are then applied to consider the sign-changing solutions of the corresponding nonlinear problem by fixed point index and Leray-Schauder degree. To date, no paper has appeared in the literature which discusses sign-changing solutions of fractional boundary value problems. This paper attempts to fill this gap in the literature.

The present paper is devoted to the investigation of a parabolic equation with moving boundaries arising in ductal carcinoma in situ (DCIS) model. Approximation solution of this problem is implemented by Ritz-Galerkin, which is a first attempt at tackling such problem. In process of dealing with this moving boundary condition, we use a trick of introducing two transformations to convert moving boundary to nonclassical boundary that can be handled with Ritz-Galerkin method. Also, existence and uniqueness are proved. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper.

This paper discusses an autonomous competitive Lotka-Volterra model in random environments. The contributions of this paper are as follows. (a) Some sufficient conditions for partial permanence and extinction on this system are established; (b) By using some novel techniques, the conditions imposed on permanence and extinction of one-species are weakened. Finally, a numerical experiment is conducted to validate the theoretical findings.

This paper is concerned with the existence of universal attractors in $H_{+}^{i}$ ( $i=1,2$ ) for one-dimensional compressible and radiative magnetohydrodynamics equations in a bounded domain $\Omega:=(0,1)$ . In this paper, the author extends the results in (Qin et al. in J. Differ. Equ. 253:1439-1488, 2012).

In this paper, we establish some boundedness conditions for the multidimensional Hausdorff operator on the homogeneous Hardy-Morrey and on the Besov-Morrey space, and we extend some results in the recent papers by Jia and Wang, and by Mazzucato, respectively. The main tool we implement in the study is the decomposition of the given function spaces in terms of atoms (smooth atoms for Besov-Morrey space) concentrated on dyadic cubes. The atomic decomposition of the classical Hardy space and Besov space is our study model, however, our case is a quite different one. Particularly, we will combine the Calderón reproducing formula with the atomic decomposition when we establish the boundedness of the Hausdorff operator on the Besov-Morrey space.

In this paper, we are concerned with nonlinear elliptic equations of the $p(x)$ -Laplace type operators $$\textstyle\begin{cases} -\operatorname{div}(a(x,\nabla u))+\vert u\vert ^{p(x)-2}u=\lambda f(x,u) &\mbox{in } \Omega, \\ a(x,\nabla u)\frac{\partial u}{\partial n} = \lambda\theta g(x,u) &\mbox{on } \partial\Omega, \end{cases} $$ which are subject to nonlinear Neumann boundary conditions. Here the function $a(x,v)$ is of type $\vert v\vert ^{p(x)-2}v$ with a continuous function $p: \overline{\Omega} \to(1,\infty)$ and the functions $f, g$ satisfy a Caratheodory condition. The main purpose of this paper is to establish the existence of at least three weak solutions of the above problem by applying an abstract three critical points theorem which is inspired by the work of Ricceri (Nonlinear Anal. 74:7446-7454, 2011) Furthermore, we determine two intervals of λ’s precisely such that the first is where the given problem admits only the trivial solution, and the second is where the given problem has at least two nontrivial solutions as considering the positive principal eigenvalue for the $p(x)$ -Laplacian Neumann problems and an estimate of the Sobolev trace embedding’s constant.

In this paper, we obtain a maximum principle for controlled fractional Fokker-Planck equations. We prove the well-posedness of a stochastic differential equation driven by an α-stable process. We give some estimates of the solutions by fractional calculus. A linear-quadratic example is given at the end of the paper.

The purpose of this paper is to propose an algorithm to solve the split common fixed point problems for total asymptotically strict pseudocontraction mappings in Hilbert spaces. Without the assumption of semi-compactness on the mappings, the iterative scheme is shown to converge strongly to a split common fixed point of such mappings. The results presented in the paper improve and extend some recent corresponding results.

This paper is concerned with fuzzy bidirectional associative memory (BAM) Cohen-Grossberg neural networks with mixed delays and impulses. By constructing an appropriate Lyapunov function and a new differential inequality, we obtain some sufficient conditions which ensure the existence and global exponential stability of a periodic solution of the model. The results in this paper extend and complement the previous publications. An example is given to illustrate the effectiveness of our obtained results.

Based on a specific decomposition of discrete singular systems, in this paper, we study the problem of state tracking control by using PD-type algorithm of iterative learning control. The convergence conditions and theoretical analysis of the PD-type algorithm are presented in detail. An illustrative example supporting the theoretical results and the effectiveness of the PD-type iterative learning control algorithm for discrete singular systems is shown at the end of the paper.

Sufficient oscillation conditions involving $\limsup $ and $\liminf $ for first-order differential equations with several non-monotone deviating arguments and nonnegative coefficients are obtained. The results are based on the iterative application of the Gr\"{o}nwall inequality. Examples illustrating the significance of the results are also given., Comment: The paper was originally open access. The purpose of this publication is correct the original mistake that occurred in Theorems 6 and 10 (in particular, the variables under the integral in (2.11) and (2.27)) in the published paper

In this paper, we introduce a new class of multivalued contractions with respect to b-generalized pseudodistances and prove a best proximity point theorem for this class of non-self mappings. In this way, we improve and extend several existing results in the literature. Examples are given to support our main results. This work is a continuation of studies on the use of a new type of distances in the fixed point theory. The pioneering effort in direction of defining distance is inter alia paper of O. Kada, T. Suzuki and W. Takahashi.

In this paper we use variational methods and generalized mountain pass theorem to investigate the existence of periodic solutions for some second-order delay differential systems with impulsive effects. To the authors’ knowledge, there is no paper about periodic solution of impulses delay differential systems via critical point theory. Our results are completely new.

In this paper, we consider a perturbed compound Poisson risk model with stochastic premiums and constant interest force. We obtain the upper bound and Lundberg-Cramér approximation for the infinite-time ruin probability, and consider the asymptotic formula for the finite-time ruin probability when the claim size is heavy-tailed. We show that the model in our paper has similar results to the classical risk process and some existing generalized models.

This paper deals with the global existence and blow-up of the solution for a class of nonlinear reaction diffusion problems. The purpose of this paper is to establish conditions on the data to guarantee the blow-up of the solution at some finite time, and conditions to ensure that the solution remains global. In addition, an upper bound for the ‘blow-up time’, an upper estimate of the ‘blow-up rate’, and an upper estimate of the global solution are also specified. Finally, as applications of the obtained results, some examples are presented.

In this paper, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm and multi-step composite Mann-type viscosity iterative algorithm for finding a solution of triple hierarchical variational inequalities defined over the common set of solutions of mixed equilibrium problems, variational inclusions, variational inequalities, and fixed point problems. Under appropriate assumptions, we prove that the proposed algorithms converge strongly to a common element of the fixed point set of a strict pseudocontractive mapping, a solution set of finitely many generalized mixed equilibrium problems, a solution set of finitely many variational inclusions, and a solution set of a general system of variational inequalities. Such an element is a unique solution of a triple hierarchical variational inequality problem. In addition, we also consider as an application the proposed algorithm to solve a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, and a general system of variational inequalities. The results obtained in this paper improve and extend the corresponding results announced by many other authors.

The paper addresses the asymptotic properties of Camassa-Holm equation on the half-line. That is, using the method of asymptotic density, under the assumption that it is unique, the paper proves that the positive momentum density of the Camassa-Holm equation is a combination of Dirac measures supported on the positive axis. This means that as time goes to infinity, the momentum density concentrates in small intervals moving right with different constant speeds.

Consider an evolutionary equation related to the $p(x)$ -Laplacian: ${u_{t}}= \operatorname{div} ({\rho^{\alpha}}{ \vert {\nabla u} \vert ^{p(x) - 2}}\nabla u)+ {\frac{{\partial{b_{i}}(u,x,t)}}{{\partial {x_{i}}}}}$ , $(x,t) \in{Q_{T}} = \Omega \times(0,T)$ , which arises from electrorheological fluid mechanics. Since $\rho(x) = \operatorname{dist} (x,\partial\Omega)$ , the equation is degenerate on the boundary, one may expect that there is not flux across the boundary. The paper shows that the facts may be unexpected. The paper reviews Fichera-Oleinik theory, then uses the theory to discuss the boundary value condition related to the equation. If $p^{-}>2$ , the existence and the uniqueness of the solutions are researched. Finally, if $b_{i}\equiv0$ , the behavior of the solutions near the boundary is studied by the comparison theorem.

In this paper, a new iterative scheme for a finite family of $I_{i}$ -generalized asymptotically nonexpansive nonself-mappings $\{T_{i}\} _{i=1}^{r}$ is constructed in a uniformly convex hyperbolic space. We establish strong convergence theorems of this iterative scheme to a common fixed point of $\{T_{i}\}_{i=1}^{r}$ and $\{ I_{i}\}_{i=1}^{r}$ under certain conditions. Our results of this paper extend some results in the literature.

The paper studies the equation $$ {u_{t}}= \operatorname{div} \bigl(a(x)\vert {\nabla u} \vert ^{p(x) - 2}\nabla u \bigr), $$ with the boundary degeneracy coming from $a(x)\vert_{x\in \partial \Omega }=0$ . The paper introduces a new kind of weak solutions of the equation. One can study the stability of the new kind of weak solutions without any boundary value condition.

This paper is dedicated to the introduction a new class of equilibrium problems named generalized multivalued equilibrium-like problems which includes the classes of hemiequilibrium problems, equilibrium-like problems, equilibrium problems, hemivariational inequalities, and variational inequalities as special cases. By utilizing the auxiliary principle technique, some new predictor-corrector iterative algorithms for solving them are suggested and analyzed. The convergence analysis of the proposed iterative methods requires either partially relaxed monotonicity or jointly pseudomonotonicity of the bifunctions involved in generalized multivalued equilibrium-like problem. Results obtained in this paper include several new and known results as special cases.

This paper is concerned with a thermoelastic suspension bridge equations with memory effects. For the suspension bridge equations without memory, there are many classical results. However, the suspension bridge equations with both viscoelastic and thermal memories were not studied before. The object of the present paper is to provide a result on the global attractor to a thermoelastic suspension bridge equation with past history.