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.

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.

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

The main object of this paper is to construct a new generating function of the ( q- ) Bernstein-type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the ( q- ) Bernstein-type polynomials. We also give relations between the ( q- ) 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- ) Bernstein-type polynomials. Moreover, we give some applications and questionson approximations of ( q- ) Bernstein-type polynomials, moments of some distributions inStatistics.

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.

We discuss the solvability of the fourth-order boundary value problem u ( 4 ) = f ( t , u , u ′′ ) , 0 ≤ t ≤ 1 , u ( 0 ) = u ( 1 ) = u ′′ ( 0 ) = u ′′ ( 1 ) = 0 , which models a statically bending elastic beam whose two ends are simply supported, where f : [ 0,1 ] × R 2 → 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.

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-ordernonlinear differential equation with impulses.

We answer the question: for α , β , γ ∈ ( 0,1 ) with α + β + γ = 1 , what are the greatest value p and the least value q , such that the double inequality L p ( a , b ) < A α ( a , b ) G β ( a , b ) H γ ( a , b ) < L q ( a , b ) holds for all a , b > 0 with a ≠ 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 investigate the blowup properties of the positive solutions for a semilinearreaction-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.

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 ofthe variational inequality for an α-inverse-strongly monotone, and the set of solutions of the generalizedequilibrium problem in the framework of a real Banach space. We obtain a strong convergence theoremfor 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.

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

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

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

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.

We study the following difference equation x n + 1 = ( p + x n - 1 ) / ( q x n + x n - 1 ) , n = 0,1 , … , where p , q ∈ ( 0 , + ∞ ) and the initial conditions x - 1 , x 0 ∈ ( 0 , + ∞ ) . 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.

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

We consider the Friedrichs self-adjoint extension for a differentialoperator A of the form A= A 0 +q( x )⋅ , which is defined on a boundeddomain Ω⊂ ℝ n , n≥1 (for n=1 we assume that Ω=( a,b ) is a finiteinterval). Here A 0 = A 0 ( x,D ) is a formally self-adjoint and a uniformly elliptic differential operator of order 2m with bounded smoothcoefficients and a potential q( x ) is a real-valued integrable functionsatisfying the generalized Kato condition. Under these assumptionsfor the coefficients of A and for positive λ large enough we obtain theexistence of Green's function for the operator A+λI and its estimatesup to the boundary of Ω. These estimates allow us to prove the absolute and uniform convergence up to the boundary of Ω of Fourierseries in eigenfunctions of this operator. In particular, these resultscan be applied for the basis of the Fourier method which is usuallyused in practice for solving some equations of mathematical physics.

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

A study of the convergence of weak solutions of the nonstationary micropolar fluids, inbounded 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.

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 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 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.

Let A denote the operator generated in L 2 ( R + ) by the Sturm-Liouville problem: - y ′′ + q ( x ) y = λ 2 y , x ∈ R + = [ 0 , ∞ ) , ( y ′ / y ) ( 0 ) = ( β 1 λ + β 0 ) / ( α 1 λ + α 0 ) , where q is a complex valued function and α 0 , α 1 , β 0 , β 1 ∈ C , with α 0 β 1 - α 1 β 0 ≠ 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.

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

Let 픹 denote the open unit ball of ℂ n . For a holomorphic self-map φ of 픹 and a holomorphic function g in 픹 with g ( 0 ) = 0 , we define the following integral-type operator: I φ g f ( z ) = ∫ 0 1 ℜ f ( φ ( t z ) ) g ( t z ) ( d t / t ) , z ∈ 픹 . Here ℜ f denotes the radial derivative of a holomorphic function f in 픹 . We study the boundedness and compactness of the operator between Bloch-type spaces ℬ ω and ℬ μ , where ω is a normal weight function and μ is a weight function. Also we consider the operator between the little Bloch-type spaces ℬ ω , 0 and ℬ μ , 0 .

We introduce the notion of regularized quasi-semigroupof bounded linear operators on Banach spaces and its infinitesimal generator, as a generalization of regularized semigroups ofoperators. 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 evolution equations will be considered. Next some elementary perturbation results on regularized quasi-semigroups will be discussed.

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

Let ℙ = ( P t ) t > 0 be a C 0 -contraction semigroup on a real Banach space ℬ . A ℙ -exit law is a ℬ -valued function t ∈ ] 0 , ∞ [ → φ t ∈ ℬ satisfying the functional equation: P t φ s = φ t + s , s , t > 0 . Let β be a Bochner subordinator and let ℙ β be the subordinated semigroup of ℙ (in the Bochner sense) by means of β . Under some regularity assumption, it is proved in this paper that each ℙ β -exit law is subordinated to a unique ℙ -exit law.

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

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 ⊂ 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 ( z ) defined in the disk, we will prove Chuang's inequality, which is to compare the relationship between T ( r , f ) and T ( r , f ′ ) . Consequently, we obtain that the order and the lower order of f ( z ) and its derivative f ′ ( z ) are the same.

We study necessary and sufficient conditions for the oscillation of thethird-order nonlinear ordinary differential equation with damping term and deviating argument x ‴ ( t )+q( t ) x ′ ( t )+r( t )f( x( φ( t ) ) )=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( t ) x ′ is oscillatory.

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.

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.

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 consider the generalized shift operator, associated with the Dunkl operator Λ α ( f ) ( x ) = ( d / d x ) f ( x ) + ( ( 2 α + 1 ) / x ) ( ( f ( x ) - f ( - x ) ) / 2 ) , α > - 1 / 2 . We study some embeddings into the Morrey space ( D -Morrey space) L p , λ , α , 0 ≤ λ < 2 α + 2 and modified Morrey space (modified D -Morrey space) L ̃ p , λ , α associated with the Dunkl operator on ℝ . As applications we get boundedness of the fractional maximal operator M β , 0 ≤ β < 2 α + 2 , associated with the Dunkl operator (fractional D -maximal operator) from the spaces L p , λ , α to L ∞ ( ℝ ) for p = ( 2 α + 2 - λ ) / β and from the spaces L ̃ p , λ , α ( ℝ ) to L ∞ ( ℝ ) for ( 2 α + 2 - λ ) / β ≤ p ≤ ( 2 α + 2 ) / β .

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.

For p ∈ 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 ) = ( ( a p + b p ) / 2 ) 1 / p , for p ≠ 0 and M p ( a , b ) = a b , for p = 0 , L ( a , b ) = ( b - a ) / ( log ⁡ b - log ⁡ a ) , for a ≠ 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 α ∈ ( 0,1 ) , what are the greatest value p and the least value q , such that the double inequality M p ( a , b ) ≤ α A ( a , b ) + ( 1 - α ) L ( a , b ) ≤ M q ( a , b ) holds for all a , b > 0 ?

The existence of multiple periodic solutions of the following differential delay equation x ′ ( t ) = − f ( x ( t − r ) ) is established by applying variational approaches directly, where x ∈ ℝ , f ∈ C ( ℝ , ℝ ) 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 establish the Weyl-Titchmarsh theory for singular linearHamiltonian dynamic systems on a time scale 핋, which allows one to treat both continuousand discrete linear Hamiltonian systems as special cases for 핋 = ℝ and 핋 = ℤ within onetheory and to explain the discrepancies between these two theories. This paper extends theWeyl-Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includesthe following: (i) M( λ ) theory for singular Hamiltonian systems, (ii) on the spectrumof Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.