The purpose of this paper is to introduce a new hybrid projection method for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of the variational inequality for an $\alpha$-inverse-strongly monotone, and the set of solutions of the generalized equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Base on this result, we also get some new and interesting results. The results in this paper generalize, extend, and unify some well-known strong convergence results in the literature.

We study necessary and sufficient conditions for the oscillation of the third-order nonlinear ordinary differential equation with damping term and deviating argument $x^{‴}\left(t\right)+q\left(t\right)x^{\prime }\left(t\right)+r\left(t\right)f\left(x\left(\phi \left(t\right)\right)\right)=0$. Motivated by the work of Kiguradze (1992), the existence and asymptotic properties of nonoscillatory solutions are investigated in case when the differential operator $ℒx=x^{‴}+q\left(t\right)x^{\prime }$ is oscillatory.

The existence of multiple periodic solutions of the following differential delay equation $x^{\prime }\left(t\right)=-f\left(x\left(t-r\right)\right)$ is established by applying variational approaches directly, where $x\in ℝ$, $f\in C\left(ℝ,ℝ\right)$ and $r>0$ is a given constant. This means that we do not need to use Kaplan and Yorke's reduction technique to reduce the existence problem of the above equation to an existence problem for a related coupled system. Such a reduction method introduced first by Kaplan and Yorke in (1974) is often employed in previous papers to study the existence of periodic solutions for the above equation and its similar ones by variational approaches.

We introduce new spaces that are extensions of the Hardy spaces and we investigate the continuity of the point evaluations as well as the boundedness and the compactness of the composition operators on these spaces.

The mixed viscosity approximation is proposed for finding fixed points of nonexpansive mappings, and the strong convergence of the scheme to a fixed point of the nonexpansive mapping is proved in a real Banach space with uniformly Gâteaux differentiable norm. The theorem about Halpern type approximation for nonexpansive mappings is shown also. Our theorems extend and improve the correspondingly results shown recently.

The qualitative behavior of a perturbed fractional-order differential equation with Caputo's derivative that differs in initial position and initial time with respect to the unperturbed fractional-order differential equation with Caputo's derivative has been investigated. We compare the classical notion of stability to the notion of initial time difference stability for fractional-order differential equations in Caputo's sense. We present a comparison result which again gives the null solution a central role in the comparison fractional-order differential equation when establishing initial time difference stability of the perturbed fractional-order differential equation with respect to the unperturbed fractional-order differential equation.

This paper develops some new Razumikhin-type theorems on global exponential stability of impulsive functional differential equations. Some applications are given to impulsive delay differential equations. Compared with some existing works, a distinctive feature of this paper is to address exponential stability problems for any finite delay. It is shown that the functional differential equations can be globally exponentially stabilized by impulses even if it may be unstable itself. Two examples verify the effectiveness of the proposed results.

We consider the Friedrichs self-adjoint extension for a differential operator $A$ of the form $A=A_0+q\left(x\right)\cdot$, which is defined on a bounded domain $\Omega \subset {ℝ}^n$, $n\ge 1$ (for $n=1$ we assume that $\Omega =\left(a,b\right)$ is a finite interval). Here $A_0=A_0\left(x,D\right)$ is a formally self-adjoint and a uniformly elliptic differential operator of order 2m with bounded smooth coefficients and a potential $q\left(x\right)$ is a real-valued integrable function satisfying the generalized Kato condition. Under these assumptions for the coefficients of $A$ and for positive $\lambda$ large enough we obtain the existence of Green's function for the operator $A+\lambda I$ and its estimates up to the boundary of $\Omega$. These estimates allow us to prove the absolute and uniform convergence up to the boundary of $\Omega$ of Fourier series in eigenfunctions of this operator. In particular, these results can be applied for the basis of the Fourier method which is usually used in practice for solving some equations of mathematical physics.

Let $A$ denote the operator generated in $L_2\left({\mathcal{R}}_{+}\right)$ by the Sturm-Liouville problem: $-y^{\mathrm{\prime \prime }}+q\left(x\right)y={\lambda }^{2}y$, $x\in {\mathcal{R}}_{+}=[0,\infty )$, $(y^{\prime }/y)\left(0\right)=({\beta }_1\lambda +{\beta }_0)/({\alpha }_1\lambda +{\alpha }_0)$, where $q$ is a complex valued function and ${\alpha }_0,{\alpha }_1,{\beta }_0,{\beta }_1\in \mathcal{C}$, with ${\alpha }_0{\beta }_1-{\alpha }_1{\beta }_0\ne 0.$ In this paper, using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of $A$. In particular, we obtain the conditions on $q$ under which the operator $A$ has a finite number of the eigenvalues and the spectral singularities.

This paper investigates a class of impulsive pulse-width sampler systems and its steadystate control in the infinite dimensional spaces. Firstly, some definitions of pulse-width sampler systems with impulses are introduced. Then applying impulsive evolution operator and fixed point theorem, some existent results of steady-state of infinite dimensional linear and semilinear pulse-width sampler systems with impulses are obtained. An example to illustrate the theory is presented in the end.

By using the Leray-Schauder fixed point theorem and differential inequality techniques, several new sufficient conditions are obtained for the existence and global exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with discrete and distributed delays. The model in this paper possesses two characters: nonlinear behaved functions and all coefficients are time varying. Hence, our model is general and applicable to many known models. Moreover, our main results are also general and can be easily deduced to many simple cases, including some existing results. An example and its simulation are employed to illustrate our feasible results.

We propose and study the permanence of the following periodic Holling III predator-prey system with stage structure for prey and both two predators which consume immature prey. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

We define the so-called box convolution product and study their properties in order to present the approximate solutions for the general coupled matrix convolution equations by using iterative methods. Furthermore, we prove that these solutions consistently converge to the exact solutions and independent of the initial value.

We answer the question: for $\alpha ,\beta ,\gamma \in \left(\mathrm{0,1}\right)$ with $\alpha +\beta +\gamma =1$, what are the greatest value $p$ and the least value $q$, such that the double inequality $L_p(a,b)<A^{\alpha }(a,b)G^{\beta }(a,b)H^{\gamma }(a,b)<L_q(a,b)$ holds for all $a,b>0$ with $a\ne b$? Here $L_p(a,b)$, $A(a,b)$, $G(a,b)$, and $H(a,b)$ denote the generalized logarithmic, arithmetic, geometric, and harmonic means of two positive numbers $a$ and $b$, respectively.

We consider the generalized shift operator, associated with the Dunkl operator ${\Lambda }_{\alpha }\left(f\right)\left(x\right)=(d/dx)f\left(x\right)+\left(\right(2\alpha +1)/x)\left(\right(f\left(x\right)-f(-x))/2)$, $\alpha >-1/2$. We study some embeddings into the Morrey space ($D$-Morrey space) $L_{p,\lambda ,\alpha }$, $0\le \lambda <2\alpha +2$ and modified Morrey space (modified $D$-Morrey space) ${\tilde{L}}_{p,\lambda ,\alpha }$ associated with the Dunkl operator on $\mathbb{ℝ}$. As applications we get boundedness of the fractional maximal operator $M_{\beta }$, $0\le \beta <2\alpha +2$, associated with the Dunkl operator (fractional $D$-maximal operator) from the spaces $L_{p,\lambda ,\alpha }$ to $L_{\infty }\left(\mathbb{ℝ}\right)$ for $p=(2\alpha +2-\lambda )/\beta$ and from the spaces ${\tilde{L}}_{p,\lambda ,\alpha }\left(\mathbb{ℝ}\right)$ to $L_{\infty }\left(\mathbb{ℝ}\right)$ for $(2\alpha +2-\lambda )/\beta \le p\le (2\alpha +2)/\beta$.

The main object of this paper is to construct a new generating function of the ($q\text{-}$) Bernstein-type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the ($q\text{-}$) Bernstein-type polynomials. We also give relations between the ($q\text{-}$) Bernstein-type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and the second-kind Stirling numbers. By applying Mellin transformation to this generating function, we define interpolation of the ($q\text{-}$) Bernstein-type polynomials. Moreover, we give some applications and questions on approximations of ($q\text{-}$) Bernstein-type polynomials, moments of some distributions in Statistics.

In some interesting applications in control and system theory, linear descriptor (singular) matrix differential equations of higher order with time-invariant coefficients and (non-) consistent initial conditions have been used. In this paper, we provide a study for the solution properties of a more general class of the Apostol-Kolodner-type equations with consistent and nonconsistent initial conditions.

We establish the Weyl-Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale $𝕋$, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for $𝕋$ $=ℝ$ and $𝕋=\mathbb{Z}$ within one theory and to explain the discrepancies between these two theories. This paper extends the Weyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i) $M\left(\lambda \right)$ theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.

We give a Bäcklund transformation in a unified form for each member in the Burgers hierarchy. By applying the Bäcklund transformation to the trivial solutions, we generate some solutions of the Burgers hierarchy.

We present a new method to study analytic inequalities. As for its applications, we prove the well-known Hölder inequality and establish several new analytic inequalities.

The regular system of differential equations with convolution terms solved by Sumudu transform.

For $p\in ℝ$, the power mean $M_p(a,b)$ of order $p$, logarithmic mean $L(a,b)$, and arithmetic mean $A(a,b)$ of two positive real values $a$ and $b$ are defined by $M_p(a,b)=\left(\right(a^p+b^p)/2{)}^{1/p}$, for $p\ne 0$ and $M_p(a,b)=\sqrt{ab}$, for $p=0$, $L(a,b)=(b-a)/(\log b-\log a)$, for $a\ne b$ and $L(a,b)=a$, for $a=b$ and $A(a,b)=(a+b)/2$, respectively. In this paper, we answer the question: for $\alpha \in \left(\mathrm{0,1}\right)$, what are the greatest value $p$ and the least value $q$, such that the double inequality $M_p(a,b)\le \alpha A(a,b)+(1-\alpha )L(a,b)\le M_q(a,b)$ holds for all $a,b>0$.

The necessary and sufficient conditions for Schur geometrical convexity of the four-parameter means are given. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means.

A study of the convergence of weak solutions of the nonstationary micropolar fluids, in bounded domains of ${ℝ}^n$, when the viscosities tend to zero, is established. In the limit, a fluid governed by an Euler-like system is found.

A new concept of quasi slowly oscillating continuity is introduced. Furthermore, it is shown that this kind of continuity implies ordinary continuity, but the converse is not always true.

We investigate the existence and uniqueness of positive solution for system of nonlinear fractional differential equations in two dimensions with delay. Our analysis relies on a nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorem in a cone.

Using fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation $f\left(x+2y\right)+f\left(x-2y\right)=4f\left(x+y\right)+4f\left(x-y\right)-6f\left(x\right)+f\left(2y\right)+f\left(-2y\right)-4f\left(y\right)-4f\left(-y\right)$ in non-Archimedean Banach spaces.

Fractional calculus techniques and methods started to be applied successfully during the last decades in several fields of science and engineering. In this paper we studied the stability of fractional-order nonlinear time-delay systems for Riemann-Liouville and Caputo derivatives and we extended Razumikhin theorem for the fractional nonlinear time-delay systems.

Some results of (Ćirić, 1974) on a nonunique fixed point theorem on the class of metric spaces are extended to the class of cone metric spaces. Namely, nonunique fixed point theorem is proved in orbitally $T$ complete cone metric spaces under the assumption that the cone is strongly minihedral. Regarding the scalar weight of cone metric, we are able to remove the assumption of strongly minihedral.

The purpose of this paper is to establish the first and second fundamental theorems for an $E$-valued meromorphic mapping from a generic domain $D\subset ℂ$ to an infinite dimensional complex Banach space $E$ with a Schauder basis. It is a continuation of the work of C. Hu and Q. Hu. For $f\left(z\right)$ defined in the disk, we will prove Chuang's inequality, which is to compare the relationship between $T(r,f)$ and $T(r,f^{\prime })$. Consequently, we obtain that the order and the lower order of $f\left(z\right)$ and its derivative $f^{\prime }\left(z\right)$ are the same.

We discuss the solvability of the fourth-order boundary value problem $u^{\left(4\right)}=f(t,u,u^{\mathrm{\prime \prime }})$, $0\le t\le 1$, $u\left(0\right)=u\left(1\right)=u^{\mathrm{\prime \prime }}\left(0\right)=u^{\mathrm{\prime \prime }}\left(1\right)=0$, which models a statically bending elastic beam whose two ends are simply supported, where $f:\left[\mathrm{0,1}\right]\times {\mathbb{R}}^2\to \mathbb{R}$ is continuous. Under a condition allowing that $f(t,u,v)$ is superlinear in $u$ and $v$, we obtain an existence and uniqueness result. Our discussion is based on the Leray-Schauder fixed point theorem.

We study the existence of positive solutions for a boundary value problem of fractional-order functional differential equations. Several new existence results are obtained.

A fixed-point theorem is proved under noncompact setting of general topological spaces. By applying the fixed-point theorem, several new existence theorems of solutions for equilibrium problems are proved under noncompact setting of topological spaces. These theorems improve and generalize the corresponding results in related literature.

We investigate the blowup properties of the positive solutions for a semilinear reaction-diffusion system with nonlinear nonlocal boundary condition. We obtain some sufficient conditions for global existence and blowup by utilizing the method of subsolution and supersolution.

Let $𝔹$ denote the open unit ball of ${ℂ}^n$. For a holomorphic self-map $\phi$ of $𝔹$ and a holomorphic function $g$ in $𝔹$ with $g\left(0\right)=0$, we define the following integral-type operator: $I_{\phi }^{g}f\left(z\right)={\int }_0^1\Re f\left(\phi \right(tz\left)\right)g\left(tz\right)(dt/t)$, $z\in 𝔹$. Here $\Re f$ denotes the radial derivative of a holomorphic function $f$ in $𝔹$. We study the boundedness and compactness of the operator between Bloch-type spaces ${ℬ}_{\omega }$ and ${ℬ}_{\mu }$, where $\omega$ is a normal weight function and $\mu$ is a weight function. Also we consider the operator between the little Bloch-type spaces ${ℬ}_{\omega ,0}$ and ${ℬ}_{\mu ,0}$.

We study the following difference equation $x_{n+1}=(p+x_{n-1})/(q{x}_n+x_{n-1})$, $n=\mathrm{0,1},\dots ,$ where $p,q\in (0,+\infty )$ and the initial conditions $x_{-1},x_0\in (0,+\infty )$. We show that every positive solution of the above equation either converges to a finite limit or to a two cycle, which confirms that the Conjecture 6.10.4 proposed by Kulenović and Ladas (2002) is true.

We introduce the notion of regularized quasi-semigroup of bounded linear operators on Banach spaces and its infinitesimal generator, as a generalization of regularized semigroups of operators. After some examples of such quasi-semigroups, the properties of this family of operators will be studied. Also some applications of regularized quasi-semigroups in the abstract evolution equations will be considered. Next some elementary perturbation results on regularized quasi-semigroups will be discussed.

Two sequences of distinct periodic solutions for second-order Hamiltonian systems with sublinear nonlinearity are obtained by using the minimax methods. One sequence of solutions is local minimum points of functional, and the other is minimax type critical points of functional. We do not assume any symmetry condition on nonlinearity.

Let $\mathbb{\mathbb{P}}=(P_t{)}_{t>0}$ be a $C_0$-contraction semigroup on a real Banach space $\mathcal{ℬ}$. A $\mathbb{\mathbb{P}}$-exit law is a $\mathcal{ℬ}$-valued function $t\in ]0,\infty [\to {\phi }_t\in \mathcal{ℬ}$ satisfying the functional equation: $P_t{\phi }_s={\phi }_{t+s}$, $s,t>0$. Let $\beta$ be a Bochner subordinator and let ${\mathbb{\mathbb{P}}}^{\beta }$ be the subordinated semigroup of $\mathbb{\mathbb{P}}$ (in the Bochner sense) by means of $\beta$. Under some regularity assumption, it is proved in this paper that each ${\mathbb{\mathbb{P}}}^{\beta }$-exit law is subordinated to a unique $\mathbb{\mathbb{P}}$-exit law.

This paper shows that all positive solutions of a higher-order nonlinear difference equation are bounded, extending some recent results in the literature.

A semigroup theory for a differential equation with delayed and advanced arguments is developed, with a detailed description of the infinitesimal generator. This in turn allows to study the exact controllability of the equation, by rewriting it as a classical Cauchy problem.

We establish the oscillation and asymptotic criteria for the second-order neutral delay differential equations with positive and negative coefficients having the forms $\left[x\right(t)+{\sum }_{i\in R}{c}_i(t\left)x\right({\alpha }_i\left(t\right)){]}^{\mathrm{\prime \prime }}+r(t\left)\right[x\left(t\right)+{\sum }_{i\in R}{c}_i\left(t\right)x\left({\alpha }_i\right(t\left)\right){]}^{\prime }+{\sum }_{i\in P}{p}_i\left(t\right)x\left({\beta }_i\right(t\left)\right)-{\sum }_{i\in Q}{q}_i\left(t\right)x\left({\gamma }_i\right(t\left)\right)=0$ and $\left[x\right(t)+{\sum }_{i\in R}{c}_i(t\left)x\right({\alpha }_i\left(t\right)){]}^{\mathrm{\prime \prime }}+r(t\left)\right[x\left(t\right)+{\sum }_{i\in R}{c}_i\left(t\right)x\left({\alpha }_i\right(t\left)\right){]}^{\prime }+{\sum }_{i\in P}{p}_i\left(t\right)x\left({\beta }_i\right(t\left)\right)-{\sum }_{i\in Q}{q}_i\left(t\right)x\left({\gamma }_i\right(t\left)\right)=f\left(t\right)$. The obtained new oscillation criteria extend and improve the recent results given in the paperof B. Karpuz et al. (2009).

The complex zeros of the orthogonal Laguerre polynomials $L_n^{\left(a\right)}\left(x\right)$ for $a<-n$, ultraspherical polynomials $P_n^{\left(\lambda \right)}\left(x\right)$ for $\lambda <-n$, Jacobi polynomials $P_n^{(a,\beta )}\left(x\right)$ for $a<-n$, $\beta <-n$, $a+\beta <-2(n+1)$, orthonormal Al-Salam-Carlitz II polynomials $P_n^{\left(a\right)}(x;q)$ for $a<0$, $0<q<1$, and $q$-Laguerre polynomials $L_n^{\left(a\right)}(x;q)$ for $a<-n$, $0<q<1$ are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results.

We consider the generalized shift operator, associated with the Dunkl operator ${\Lambda }_{\alpha }\left(f\right)\left(x\right)=(d/dx)f\left(x\right)+\left(\right(2\alpha +1)/x)\left(\right(f\left(x\right)-f(-x))/2)$, $\alpha >-1/2$. We study the boundedness of the Dunkl-type fractional maximal operator $M_{\beta }$ in the Dunkl-type Morrey space $L_{p,\lambda ,\alpha }\left(ℝ\right)$, $0\le \lambda <2\alpha +2$. We obtain necessary and sufficient conditions on the parameters for the boundedness $M_{\beta }$, $0\le \beta <2\alpha +2$ from the spaces $L_{p,\lambda ,\alpha }\left(ℝ\right)$ to the spaces $L_{q,\lambda ,\alpha }\left(ℝ\right)$, $1<p\le q<\infty$, and from the spaces $L_{1,\lambda ,\alpha }\left(ℝ\right)$ to the weak spaces $W{L}_{q,\lambda ,\alpha }\left(ℝ\right)$, $1<q<\infty$. As an application of this result, we get the boundedness of $M_{\beta }$ from the Dunkl-type Besov-Morrey spaces $B_{p\theta ,\lambda ,\alpha }^s\left(ℝ\right)$ to the spaces $B_{q\theta ,\lambda ,\alpha }^s\left(ℝ\right)$, $1<p\le q<\infty$, $0\le \lambda <2\alpha +2$, $1/p-1/q=\beta /(2\alpha +2-\lambda )$, $1\le \theta \le \infty$, and $0<s<1$.

We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation $-\Delta u+u=\lambda a\left(x\right)|u{|}^{q-2}u+b(x\left)\right|u{|}^{p-2}u$in ${ℝ}^N$, $u\in H^1\left({ℝ}^N\right)$, where $\lambda >0$ , $1<q<2<p<2^{*}(2^{*}=2N/(N-2)$ if $N\ge 3$, $2^{*}=\infty$ if $N=\mathrm{1,2}$), $a\left(x\right)$, $b\left(x\right)$ satisfy suitable conditions, and $a\left(x\right)$ may change sign in ${ℝ}^N$.

We obtain an approximation of the solution of the nonlinear Volterra integral equation of the second kind, by means of a new method for its numerical resolution. The main tools used to establish it are the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.

We present an iterative method for fixed point problems, generalized mixed equilibrium problems, and variational inequality problems. Our method is based on the so-called viscosity hybrid steepest descent method. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of generalized mixed equilibrium problems, and the set of solutions of variational inequality problems for a relaxed cocoercive mapping in a real Hilbert space. Then, we prove the strong convergence of the proposed iterative scheme to the unique solution of variational inequality. The results presented in this paper generalize and extend some well-known strong convergence theorems in the literature.

Motivated by Iričanin and Stević's paper (2006) in which for the first time were considered some cyclic systems of difference equations, here we study the global attractivity of some nonlinear k-dimensional cyclic systems of higher-order difference equations. To do this, we use the transformation method from Berenhaut et al. (2007) and Berenhaut and Stević (2007). The main results in this paper also extend our recent results in the work of (Liu and Yang 2010, in press).

We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-$\varphi$-asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results.

A second order of accuracy difference scheme for the approximate solution of the abstract nonlocal boundary value problem $-d^{2}u\left(t\right)/d{t}^2+Au\left(t\right)=g\left(t\right)$, $\left(0\le t\le 1\right)$, $du\left(t\right)/dt-Au\left(t\right)=f\left(t\right)$, $\left(-1\le t\le 0\right)$, $u\left(1\right)=u\left(-1\right)+\mu$ for differential equations in a Hilbert space $H$ with a self-adjoint positive definite operator $A$ is considered. The well posedness of this difference scheme in Hölder spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained and a numerical example is presented.

We prove existence theorems for integro-differential equations $x^{\Delta }\left(t\right)=f(t,x(t),{\int }_0^{t}k(t,s,x\left(s\right)\left)\Delta s\right)$, $x\left(0\right)=x_0$, $t\in I_a=[0,a]\cap T$, $a\in R_{+}$, where $T$ denotes a time scale (nonempty closed subset of real numbers $R$), and $I_a$ is a time scale interval. The functions $f,k$ are weakly-weakly sequentially continuous with values in a Banach space $E$, and the integral is taken in the sense of Henstock-Kurzweil-Pettis delta integral. This integral generalizes the Henstock-Kurzweil delta integral and the Pettis integral. Additionally, the functions $f$ and $k$ satisfy some boundary conditions and conditions expressed in terms of measures of weak noncompactness. Moreover, we prove Ambrosetti's lemma.

The boundedness and compactness of the weighted differentiation composition operator from the mixed-norm space to the nth weighted-type space on the unit disk are characterized.

Our goal in this paper is to investigate the long-term behavior of solutions of the following difference equation: $x_{n+1}=x_n{x}_{n-1}-1$, $n=0,1,2,\dots ,$ where the initial conditions $x_{-1}$ and $x_0$ are real numbers. We examine the boundedness of solutions, periodicity of solutions, and existence of unbounded solutions and how these behaviors depend on initial conditions.

We will consider a pole assignment problem for a class of linear neutral functional differential equations in Banach spaces. We will think of the neutral system studied as that of involving no time delays and reduce the study of adjoint semigroups and spectral properties of neutral equations to those of Cauchy problems. Under the assumption that both the control and eigenspace of pole are finite dimensional, we establish the rank conditions for finite pole assignability.

We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize some well-known results in the literature.

This paper deals with the existence and uniqueness of mild solutions for the initial value problems of abstract impulsive evolution equations in an ordered Banach space $E:u^{\prime }\left(t\right)+Au\left(t\right)=f(t,u(t),Gu(t\left)\right)$, $t\in [0,a]$, $t\ne t_k$, $\Delta u{|}_{t=t_k}=I_k\left(u\right(t_k\left)\right)$, $0<t_1<t_2<\cdots <t_m<a$, $u\left(0\right)=u_0$, where $A:D\left(A\right)\subset E\to E$ is a closed linear operator, and $f:[0,a]\times E\times E\to E$ is a nonlinear mapping. Under wide monotone conditions and measure of noncompactness conditions of nonlinearity $f$, some existence and uniqueness results are obtained by using a monotone iterative technique in the presence of lower and upper solutions.

We present the normal and osculating planes of the curves parameterized by a compact subinterval of a time scale.

The fixed point theorem of cone expansion and compression of norm type for a strict set contraction operator is generalized by replacing the norms with a convex functional satisfying certain conditions. We then show how to apply our theorem to prove the existence of a positive solution to a second-order differential equation with integral boundary conditions in an ordered Banach space. An example is worked out to demonstrate the main results.

The existence of three solutions for elliptic Neumann problems with a perturbed nonlinear term depending on two real parameters is investigated. Our approach is based on variational methods.

The method of the quasilinearization technique in Caputo's sense fractional-order differential equation is applied to obtain lower and upper sequences in terms of the solutions of linear Caputo's sense fractional-order differential equations. It is also shown that these sequences converge to the unique solution of the nonlinear Caputo's sense fractional-order differential equation uniformly and semiquadratically with less restrictive assumptions.

Fractional calculus started to play an important role for analysis of the evolution of the nonlinear dynamical systems which are important in various branches of science and engineering. In this line of taught in this paper we studied the stability of fractional order nonlinear time-delay systems for Caputo's derivative, and we proved two theorems for Mittag-Leffler stability of the fractional nonlinear time delay systems.

The singular semilinear elliptic problem $-\Delta u+k\left(x\right)u^{-\gamma }=\lambda u^p$ in $\Omega$, $u>0$ in $\Omega$, $u=0$ on $\partial \Omega$, is considered, where $\Omega$ is a bounded domain with smooth boundary in $R^N$, $k\in C_{\text{loc}}^{\alpha }\left(\Omega \right)\cap C\left(\overline{\Omega }\right)$, and $\gamma ,p,\lambda$ are three positive constants. Some existence or nonexistence results are obtained for solutions of this problem by the sub-supersolution method.

Let $E$ be a real Banach space which is uniformly smooth and uniformly convex. Let $K$ be a nonempty, closed, and convex sunny nonexpansive retract of $E$, where $Q$ is the sunny nonexpansive retraction. If $E$ admits weakly sequentially continuous duality mapping $j$, path convergence is proved for a nonexpansive mapping $T:K\to K$. As an application, we prove strong convergence theorem for common zeroes of a finite family of $m$-accretive mappings of $K$ to $E$. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings from $K$ to $E$ under certain mild conditions.

We will establish a new interval oscillation criterion for second-order half-linear dynamic equation $\left(r\right(t\left)\right[x^{\Delta }\left(t\right){]}^{\alpha }{)}^{\Delta }+p\left(t\right)x^{\alpha }\left(\sigma \right(t\left)\right)=f\left(t\right)$ on a time scale $\mathbb{T}$ which is unbounded, which is a extension of the oscillation result for second order linear dynamic equation established by Erbe et al. (2008). As an application, we obtain a sufficient condition of oscillation of the second-order half-linear differential equation $\left(\right[x^{\prime }\left(t\right){]}^{\alpha }{)}^{\prime }+c\sin t{x}^{\alpha }\left(t\right)=\cos t$, where $\alpha =p/q$, $p$, $q$ are odd positive integers.

With the help of the fixed point index theorem in cones, we get an existence theorem concerning the existence of positive solution for a second-order three-point eigenvalue problem $x^{\prime \prime }\left(t\right)+\lambda f\left(t,x\left(t\right)\right)=0$, $0\le t\le 1,x\left(0\right)=0,x\left(1\right)=x\left(\eta \right)$, where $\lambda$ is a parameter. An illustrative example is given to demonstrate the effectiveness of the obtained result.

Let $a\left(t\right),b\left(t\right)$ be continuous $T$-periodic functions with ${\int }_0^{T}b\left(t\right)dt=0$. We establish one stability criterion for the linear damped oscillator $x^{\mathrm{\prime \prime }}+b\left(t\right)x^{\prime }+a\left(t\right)x=0$. Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillator $x^{\mathrm{\prime \prime }}+b\left(t\right)x^{\prime }+a\left(t\right)x+c\left(t\right)x^{2n-1}+e(t,x)=0$, where $n\ge 2,c\left(t\right)$ is a continuous $T$-periodic function, $e(t,x)$ is continuous $T$-periodic in $t$ and dominated by the power $x^{2n}$ in a neighborhood of $x=0$.

A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered, $Lx=f(t,x,x^{\prime })$, $t\in (a,b)$, $g\left(x\right(a),x(b),x^{\prime }(a),x^{\prime }(b\left)\right)=0$, $x\left(b\right)=x\left(a\right)$ in which $L$ is a formally self-adjoint second-order differential operator. Under appropriate assumptions on $L$, $f$, and $g$, existence and uniqueness of solutions is established by the method of upper and lower solutions and Leray-Schauder degree theory. The general quasilinearization method is then applied to this problem. Two monotone sequences converging quadratically to the unique solution are constructed.

Let E be a smooth Banach space with a norm $\Vert \cdot \Vert$. Let $V\left(x,y\right)={\Vert x\Vert }^2+{\Vert y\Vert }^2-2\langle x,Jy\rangle$ for any $x,y\in E$, where $\langle \cdot ,\cdot \rangle$ stands for the duality pair and J is the normalized duality mapping. With respect to this bifunction $V\left(\cdot ,\cdot \right)$, a generalized nonexpansive mapping and a $V$-strongly nonexpansive mapping are defined in $E$. In this paper, using the properties of generalized nonexpansive mappings, we prove convergence theorems for common zero points of a maximal monotone operator and a $V$-strongly nonexpansive mapping.

It is well known that production, distribution, marketing, inventory control, and financing all/each have a positive impact on the performance of a supply chain. Despite the growing interest in the development of integrated inventory models, the interactions between these elements of a supply chain may not be efficiently included, resulting in a restricted supply chain model presentation. To incorporate this phenomenon, a mathematical model that tackles the interdependent relationships between these aforementioned elements is developed in this paper. This study considers the determination of the optimal pricing, ordering, and delivery policies of a profit-maximizing supply chain system, faced with (1) unit wholesale price of the supplier is set based on unit production cost, (2) unit production cost is taken as a function of demand rate and production rate, (3) the supplier's production rate is adjusted according to market demand, (4) market demand depends upon buyer's selling price, (5) a free freight is offered if the buyer's order exceeds a certain minimum requirement, and (6) a constant credit period is offered by the supplier to stimulate the demand of the buyer. Algorithm for computing the optimal policies is derived. The sensitivity of the optimal results with respect to those parameters which directly influence the production and transportation costs is also examined.

We introduce a new hybrid iterative scheme for finding a common element in the solutions set of a system of equilibrium problems and the common fixed points set of an infinitely countable family of relatively quasi-nonexpansive mappings in the framework of Banach spaces. We prove the strong convergence theorem by the shrinking projection method. In addition, the results obtained in this paper can be applied to a system of variational inequality problems and to a system of convex minimization problems in a Banach space.

We establish existence and multiplicity of solutions for an elliptic system which presents resonance at infinity of Landesman-Lazer type. In order to describe the resonance, we use an eigenvalue problem with indefinite weights. In all results, we use Variational Methods, Morse Theory and Critical Groups.

We study the higher-order neutral dynamic equation ${\left\{a\left(t\right){\left[{\left(x\left(t\right)-p\left(t\right)x\left(\tau \left(t\right)\right)\right)}^{{\Delta }^m}\right]}^{\alpha }\right\}}^{\Delta }+f\left(t,x\left(\delta \left(t\right)\right)\right)=0$ for $t\in {\left[t_0,\infty \right)}_{\mathbb{T}}$ and obtain some necessary and sufficient conditions for the existence of nonoscillatory bounded solutions for this equation.

The criteria for extreme point and rotundity of Musielak-Orlicz-Bochner function spaces equipped with Orlicz norm are given. Although criteria for extreme point of Musielak-Orlicz function spaces equipped with the Orlicz norm were known, we can easily deduce them from our main results.

We answer the question: for $\alpha \in \left(0,1\right)$, what are the greatest value $p$ and the least value $q$ such that the double inequality $M_p\left(a,b\right)<P^{\alpha }\left(a,b\right)G^{1-\alpha }\left(a,b\right)<M_q\left(a,b\right)$ holds for all $a,b>0$ with $a\ne b$. Here, $M_p\left(a,b\right)$, $P\left(a,b\right)$, and $G\left(a,b\right)$ denote the power of order $p$, Seiffert, and geometric means of two positive numbers $a$ and $b$, respectively.

We consider q-Euler numbers, polynomials, and q-Stirling numbers of first and second kinds. Finally, we investigate some interesting properties of the modified q-Bernstein polynomials related to q-Euler numbers and q-Stirling numbers by using fermionic p-adic integrals on ${\mathbb{Z}}_p$.

We show that every U-space and every Banach space X satisfying ${\delta }_X\left(1\right)>0$ are P(3)-convex, and we study the nonuniform version of P-convexity, which we call p-convexity.

New nonoscillation and oscillation criteria are derived for scalar delay differential equations $\dot{x}\left(t\right)+a\left(t\right)x\left(h\left(t\right)\right)=0,a\left(t\right)\ge 0,h\left(t\right)\le t,t\ge t_0,$ and $\dot{x}\left(t\right)+{\displaystyle {\sum }_{k=1}^m{a}_k\left(t\right)x\left(h_k\left(t\right)\right)=0},a_k\left(t\right)\ge 0,h_k\left(t\right)\le t,$ and $t\ge t_0,$ in the critical case including equations with several unbounded delays, without the usual assumption that the parameters $a,h,a_k$, and $h_k$ of the equations are continuous functions. These conditions improve and extend some known oscillation results in the critical case for delay differential equations.

The upper and lower solutions method is used to study the $p$-Laplacian fractional boundary value problem $D_{0+}^{\gamma }\left({\varphi }_p\right(D_{0+}^{\alpha }u\left(t\right)\left)\right)=f(t,u(t\left)\right)$, $0<t<1$, $u\left(0\right)=0$, $u\left(1\right)=au\left(\xi \right)$, $D_{0+}^{\alpha }u\left(0\right)=0$, and $D_{0+}^{\alpha }u\left(1\right)=b{D}_{0+}^{\alpha }u\left(\eta \right)$, where $1<\alpha ,\gamma ⩽2,0⩽a,b⩽1,0<\xi ,\eta <1$. Some new results on the existence of at least one positive solution are obtained. It is valuable to point out that the nonlinearity $f$ can be singular at $t=\mathrm{0,1}$ or $u=0$.

Let $\mu$ be a finite, positive measure on ${𝔻}^n$, the polydisc in ${\mathbb{ℂ}}^n$, and let ${\sigma }_n$ be 2n-dimensional Lebesgue volume measure on ${𝔻}^n$. For an Orlicz function $\phi$, a necessary and sufficient condition on $\mu$ is given in order that the identity map $J:L_a^{\phi }({𝔻}^n,{\sigma }_n)\to L^{\phi }({𝔻}^n,\mu )$ is bounded.

By using critical point theory, some new sufficient conditions for the existence of solutions of impulsive Duffing dynamic equations on time scales with Dirichlet boundary conditions are obtained. Some examples are also given to illustrate our results.

The purpose of this paper is to present a fixed point theorem using a contractive condition of rational type in the context of partially ordered metric spaces.

The paper is devoted to obtaining the sufficient conditions for Fredholm property for the general boundary value problem of the second-order linear integro-differential equation. Here, the boundary conditions corresponding with the boundary value problem contain both nonlocal and global terms.

We identify the blow-up set of solutions to the problem $u_t=\left(\right|u_x{|}^{p-2}{u}_x{)}_x$, $x>0$, $t\in (0,T)$, $-\left|u_x{|}^{p-2}{u}_x\right(0,t)=u^{p-1}(0,t)$, $t\in (0,T)$, and $u(x,0)=u_0\left(x\right)\ge 0$, $x>0$, where $p>2$. We obtain that the blow up set $B\left(u\right)$ satisfies $[0,p(p-1)/(p-2\left)\right)\subset B\left(u\right)\subset [0,p(p-1)/(p-2\left)\right]$. The proof is based on the analysis of the asymptotic behavior of self-similar representation and on the comparison methods.

Under a reasonable assumption, we derive an analytical approach that verifies uniqueness of the optimal solution for stochastic inventory models with defective items. Our approach implies a robust method to find the optimal solution.

In this paper, we study a general second-order $m$-point boundary value problem for nonlinear singular dynamic equation on time scales $u^{\Delta \nabla }\left(t\right)+a\left(t\right)u^{\Delta }\left(t\right)+b\left(t\right)u\left(t\right)+\lambda q\left(t\right)f(t,u(t\left)\right)=0$, $t\in (\mathrm{0,1}{)}_{\mathbb{𝕋}}$, $u\left(\rho \right(0\left)\right)=0$, $u\left(\sigma \right(1\left)\right)={\sum }_{i=1}^{m-2}{\alpha }_{i}u\left({\eta }_i\right)$. This paper shows the existence of multiple positive solutions if $f$ is semipositone and superlinear. The arguments are based upon fixed-point theorems in a cone.

We introduce a new iterative scheme to approximate a common fixed point for a finite family of generalized asymptotically quasinonexpansive mappings. Several strong and weak convergence theorems of the proposed iteration in Banach spaces are established. The main results obtianed in this paper generalize and refine many known results in the current literature.

The purpose of this paper is to solve the minimization problem of finding $x^{\ast }$ such that $x^{\ast }=\arg {\min }_{x\in \Gamma }{\Vert x\Vert }^2$, where $\Gamma$ stands for the intersection set of the solution set of the equilibrium problem and the fixed points set of a nonexpansive mapping. We first present two new composite algorithms (one implicit and one explicit). Further, we prove that the proposed composite algorithms converge strongly to $x^{\ast }$.

The Schur and Hurwitz stability problems for a parametric polynomial family as well as the Schur stability problem for a compact set of real matrix family are considered. It is established that the Schur stability of a family of real matrices $\mathcal{A}$ is equivalent to the nonsingularity of the family $\left\{A^2-2tA+I:A\in \mathcal{A},t\in \left[-1,1\right]\right\}$ if $\mathcal{A}$ has at least one stable member. Based on the Bernstein expansion of a multivariable polynomial and extremal properties of a multilinear function, fast algorithms are suggested.

Inspired by the concept of U-spaces introduced by Lau, (1978), we introduced the class of semi-uniform Kadec-Klee spaces, which is a uniform version of semi-Kadec-Klee spaces studied by Vlasov, (1972). This class of spaces is a wider subclass of spaces with weak normal structure and hence generalizes many known results in the literature. We give a characterization for a certain direct sum of Banach spaces to be semi-uniform Kadec-Klee and use this result to construct a semi-uniform Kadec-Klee space which is not uniform Kadec-Klee. At the end of the paper, we give a remark concerning the uniformly alternative convexity or smoothness introduced by Kadets et al., (1997).

The study of dual Toeplitz operators was elaborated by Stroethoff and Zheng (2002), where various corresponding algebraic and spectral properties were established. In this paper, we characterize numerical ranges of certain classes of dual Toeplitz operators. Moreover, we introduce the analog of Halmos' fifth classification problem for quasinormal dual Toeplitz operators. In particular, we show that there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols which are not normal.

We prove the Hyers-Ulam stability of the polynomial equation $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0=0$. We give an affirmative answer to a problem posed by Li and Hua (2009).

We give sufficient conditions under which a weighted composition operator on a Hilbert space of analytic functions is not weakly supercyclic. Also, we give some necessary and sufficient conditions for hypercyclicity and supercyclicity of weighted composition operators on the space of analytic functions on the open unit disc.

Operator norm and essential norm of an integral-type operator, recently introduced by this author, from the Dirichlet space to the Bloch-type space on the unit ball in ${ℂ}^n$ are calculated here.

We introduce a new hybrid iterative scheme for finding a common element of the set of common fixed points of two countable families of relatively quasi-nonexpansive mappings, the set of the variational inequality for an $\alpha$-inverse-strongly monotone operator, the set of solutions of the generalized mixed equilibrium problem and zeros of a maximal monotone operator in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2 uniformly convex and uniformly smooth Banach space. The results presented in this paper improve and extend some recent results.

We deepen the study of some Morrey type spaces, denoted by $M^{p,\lambda }\left(\Omega \right)$, defined on an unbounded open subset $\Omega$ of ${ℝ}^n$. In particular, we construct decompositions for functions belonging to two different subspaces of $M^{p,\lambda }\left(\Omega \right)$, which allow us to prove a compactness result for an operator in Sobolev spaces. We also introduce a weighted Morrey type space, settled between the above-mentioned subspaces.

We consider error estimates of iterative algorithm in shift-invariant signal spaces. For the classical sampling and reconstruction algorithm, error estimate from its samples corrupted by white noises are widely studied, but the error analysis of noise with time jitter and iterative noise has not been given as much attention. In this paper, three types of error estimates are studied. In detail, we obtain the error estimate for reconstructing a signal from its noise samples, noise samples with time jitter, and iterative noise.

We modify the iterative method introduced by Kim and Xu (2006) for a countable family of Lipschitzian mappings by the hybrid method of Takahashi et al. (2008). Our results include recent ones concerning asymptotically nonexpansive mappings due to Plubtieng and Ungchittrakool (2007) and Zegeye and Shahzad (2008, 2010) as special cases.

Some counterparts of theorems of Phragmén-Lindelöf and of Ahlfors are proved for differential forms of $𝒲𝒯$-classes.

The boundedness and compactness of an integral-type operator recently introduced by the author from Zygmund-type spaces to the mixed-norm space on the unit ball are characterized here.

We study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type ${}^c𝒟{}_{0+}^{\delta }u\left(t\right)+f(t,u(t\left)\right)=0$, $t\in (0,1)$, $0<t<1$. $u\left(1\right)=\beta u\left(\eta \right)+{\lambda }_2$, $u^{\prime }\left(0\right)=\alpha u^{\prime }\left(\eta \right)-{\lambda }_1$, $u^{\prime \prime }\left(0\right)=0$, $u^{\prime \prime \prime }\left(0\right)=0\cdots u^{(n-1)}\left(0\right)=0$, where, $n-1<\delta <n$, $n(\ge 3)\in \mathbb{N}$, $0<\eta ,\alpha ,\beta <1$, the boundary parameters ${\lambda }_1,{\lambda }_2\in {ℝ}^{+}$ and ${}^{c}D{}_{0+}^{\delta }$ is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results.