We introduce the class of (local) ( a,k )-regularized C-resolvent families anddiscuss its basic structural properties. In particular, our analysis covers subjectslike regularity, perturbations, duality, spectral properties and subordinationprinciples. We apply our results in the study of the backwards fractionaldiffusion-wave equation and provide several illustrative examples of differentiable ( a,k )-regularized C-resolvent families.

A new 4-point C 3 quaternary approximating subdivision scheme with one shape parameter is proposed and analyzed. Its smoothness and approximation order are higher but support is smaller in comparison with the existing binary and ternary 4-point subdivision schemes.

In the theorems of Galambos-Bojanić-Seneta's type, the asymptotic behavior of the functions c [ x ] ,  x≥1, for x→+∞ , is investigated by the asymptotic behavior of the given sequence of positivenumbers ( c n ), as n→+∞ and vice versa. The main result of this paper is one theorem of such a type for sequencesof positive numbers ( c n ) which satisfy an asymptotic condition of the Karamata type lim¯n→∞  c [ λn ] / c n >1, for λ>1.

The paper is concerned with a two-dimensional Landau-Lifshitz equation which was first raised by A. DeSimone and F. Otto, and so fourth, when studying thin film micromagnetics. We get the existence of a local weak solution by approximating it with a higher-order equation. Penalty approximation and semigroup theory are employed to deal with the higher-order equation.

This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics including Volterra class dynamics. The proofs are based on a Perron-type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation.

All solutions of a fourth-order nonlinear delay differential equation are shown to converge to zero or to oscillate. Novel Riccati type techniques involving third-order linear differential equations are employed. Implications in the deflection of elastic horizontal beams are also indicated.

We consider the one-dimensional viscoelastic Porous-Thermo-Elastic system. We establish a general decay results. The usual exponential and polynomial decay rates are only special cases.

The boundedness of the motions of the dynamical system described by a differential inclusion with control vector is studied. It is assumed that the right-hand side of the differential inclusion is upper semicontinuous. Using positionally weakly invariant sets, sufficient conditions for boundedness of the motions of a dynamical system are given. These conditions have infinitesimal form and are expressed by the Hamiltonian of the dynamical system.

Here we introduce the nth weighted space on the upper half-plane Π + ={ z∈ℂ:Im z>0 } in the complex plane ℂ . For the case n=2 , we call it the Zygmund-type space, and denote it by 풵( Π + ) . The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator C φ f( z )=f( φ( z ) ) from the Hardy space H p ( Π + ) on the upper half-plane, to the Zygmund-type space, where φ is an analytic self-map of the upper half-plane.

This paper deals with the existence and uniqueness of periodic solutions for the first-order functional differential equation y ′ ( t )=−a( t )y( t )+ f 1 ( t,y( t−τ( t ) ) )+ f 2 ( t,y( t−τ( t ) ) ) with periodic coefficients and delays. We choose the mixed monotone operator theory to approachour problem because such methods, besides providing the usual existence results, may also sometimes provide uniqueness as well as additional numerical schemes for the computation of solutions.

By using the well-known Schauder fixed point theorem and upper and lower solution method, we present some existence criteria for positive solution of an m-point singular p-Laplacian dynamic equation on time scales with the sign changing nonlinearity. These results are new even for the corresponding differential ( 핋=ℝ) and difference equations ( 핋=ℤ), as well as in general time scales setting. As an application, an example is given to illustrate the results.

This article is a short elementary review of the exponential polynomials (alsocalled single-variable Bell polynomials) from the point of view of analysis. Some new propertiesare included, and several analysis-related applications are mentioned. At the end of the paper oneapplication is described in details—certain Fourier integrals involving Γ( a+it ) and Γ( a+it )Γ( b−it ) are evaluated in terms of Stirling numbers.

The boundedness and compactness of the extended Cesàro operatorfrom logarithmic-type spaces to Bloch-type spaces on the unit ball arecompletely characterized in this paper.

We consider the uniform attractors for the three-dimensional nonautonomous Camassa-Holm equations in the periodic box Ω= [ 0,L ] 3 . Assuming f=f( x,t )∈ L loc 2 ( ( 0,T );D( A −1/2 ) ) , we establish the existence of the uniform attractors in D( A 1/2 ) and D( A ) . The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

This paper is concerned with evolution equations of fractional order D α u( t )=Au( t ); u( 0 )= u 0 , u ′ ( 0 )=0, where A is a differential operator corresponding to a coercive polynomial taking values in a sector of angle less than π and 1<α<2 . We show that such equations are well posed in the sense that there always exists an α-times resolvent family for the operator A.

We prove the generalized Hyers-Ulam stability of the following quadratic functional equations 2 f ( ( x + y ) / 2 ) + 2 f ( ( x − y ) / 2 ) = f ( x ) + f ( y ) and f ( a x + a y ) + ( a x − a y ) = 2 a 2 f ( x ) + 2 a 2 f ( y ) in fuzzy Banach spaces for a nonzero real number a with a ≠ ± 1 / 2 .

We study global behavior of the following max-type difference equation x n+1 =max⁡{ 1/ x n , A n / x n−1 } , n=0,1,…, where { A n } n=0 ∞ is a sequence of positive real numbers with 0≤inf⁡ A n ≤sup⁡ A n <1 . The special case when { A n } n=0 ∞ is a periodic sequence with period k and A n ∈( 0,1 ) for every n≥0 has been completely investigated by Y. Chen. Here we extend his results to the general case.

This paper is devoted to studying growth of solutions of linear differentialequations of type f ( k ) + A k−1 ( z ) f ( k−1 ) +⋯+ A 1 ( z ) f ′ + A 0 ( z )f=H( z ) where A j   ( j=0,…,k−1 ) and H are entire functions of finite order.

In 2006, C. Park proved the stability of homomorphisms in C ∗ -ternary algebras and of derivations on C ∗ -ternary algebras for the followinggeneralized Cauchy-Jensen additive mapping: 2f( ( ∑ j=1 p x j /2 )+ ∑ j=1 d y j )= ∑ j=1 p f( x j ) +2 ∑ j=1 d f( y j ) . In this note, we improve and generalize some results concerning this functional equation.

We prove a weak convergence theorem of the modified Mann iteration process for a uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mapping in a uniformly convex Banach space. We also introduce two kinds of new monotone hybrid methods and obtain strong convergence theorems for an infinitely countable family of uniformly Lipschitzian and generalized asymptotically quasi-nonexpansive mappings in a Hilbert space. The results improve and extend the corresponding ones announced by Kim and Xu (2006) and Nakajo and Takahashi (2003).

Based on the Gateaux differential on time scales, we investigate and establish necessary conditions for Lagrange optimal control problems on time scales. Moreover, we present an economic model to demonstrate the effectiveness of our results.

In (2006) and (2009), Kim defined new generating functions of the Genocchi, Nörlund-type q-Euler polynomials and their interpolation functions. In this paper, we give another definitionof the multiple Hurwitz type q-zeta function. This function interpolates Nörlund-type q-Euler polynomials at negative integers. We also give some identities related to these polynomialsand functions. Furthermore, we give some remarks about approximations of Bernoulli andEuler polynomials.

This paper investigates the causality properties of a class of linear time-delay systems under constant point delays which possess a finite set of distinct linear time-invariant parameterizations (or configurations) which, together with some switching function, conform a linear time-varying switched dynamic system. Explicit expressionsare given to define pointwisely the causal and anticausal Toeplitz and Hankel operators from the set of switching time instants generated from the switching function. The case of the auxiliary unforced system defined by the matrix of undelayed dynamics being dichotomic (i.e., it has no eigenvalue on the complex imaginary axis) is considered in detail. Stability conditions as well as dual instability ones are discussed for this case which guarantee that the whole system is either stable, or unstable but no configuration of the switched system has eigenvalues within some vertical strip including the imaginary axis. It is proved that if the system is causal and uniformly controllable and observable, then it is globally asymptotically Lyapunov stable independent of the delays, that is, for any possibly values of such delays, provided that a minimum residence time in-between consecutive switches is kept or if all the set of matrices describing the auxiliary unforced delay—free system parameterizations commute pairwise.

We achieve the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities for quadratic functional equations f ( a x + b y ) + f ( a x − b y ) = ( b ( a + b ) / 2 ) f ( x + y ) + ( b ( a + b ) / 2 ) f ( x − y ) + ( 2 a 2 − a b − b 2 ) f ( x ) + ( b 2 − a b ) f ( y ) where a, b are nonzero fixed integers with b ≠ ± a , − 3 a , and f ( a x + b y ) + f ( a x − b y ) = 2 a 2 f ( x ) + 2 b 2 f ( y ) for fixed integers a, b with a , b ≠ 0 and a ± b ≠ 0 .

We obtain a representation for the norm of certain compact weightedcomposition operator C ψ,φ on the Hardy space H 2 , whenever φ( z )=az+b and ψ( z )=az−b . We also estimate the norm and essential norm of a class of noncompact weighted composition operators under certain conditions on φ and ψ. Moreover, we characterize the norm and essential norm of such operators in a special case.

Suppose that X is a Banach space and C is aninjective operator in B( X ), the space of all bounded linearoperators on X. In this note, a two-parameter C-semigroup(regularized semigroup) of operators is introduced, and some of itsproperties are discussed. As an application we show that theexistence and uniqueness of solution of the 2- Cauchyproblem ( ∂/ ( ∂ t i ))u( t 1 , t 2 )= H i u( t 1 , t 2 ) , i=1,2, t i >0, u( 0,0 )=x, x∈C( D( H 1 )∩D( H 2 ) ) is closely related to the two-parameter C-semigroups ofoperators.

Sufficient conditions for permanence of a semi-ratio-dependent predator-prey system with nonmonotonic functional response and time delay x ˙ 1 ( t )= x 1 ( t )[ r 1 ( t )− a 11 ( t ) x 1 ( t−τ( t ) )− a 12 ( t ) x 2 ( t ) / ( m 2 + x 1 2 ( t ) ) ],    x ˙ 2 ( t )= x 2 ( t )[ r 2 ( t )− a 21 ( t ) x 2 ( t ) / x 1 ( t ) ], are obtained, where x 1 ( t ) and x 2 ( t ) stand for the density of the prey and the predator, respectively, and m≠0 is a constant. τ( t )≥0 stands for the time delays due to negative feedback of the prey population.

We study the permanence of periodic predator-prey system with general nonlinear functional responses and stage structure for both predator and prey and obtain that the predator and the prey species are permanent.

This paper is devoted to the following question: how to characterize convex nowhere dense subsets of normed linear spaces in terms of porosity? The motivation for this study originates from papers of V. Olevskii and L. Zajíček, where it is shown that convex nowhere dense subsets of normed linear spaces are porous in some strong senses.

We construct global smooth approximate solution to a scalar conservation law with arbitrary smooth monotonic initial data. Different kinds of singularities interactions which arise during the evolution of the initial data are described as well. In order to solve the problem, we use and further develop the weak asymptotic method, recently introduced technique for investigating nonlinear waves interactions.

We obtain the general solution and the generalized Ulam-Hyers stability of the mixed type cubic and quartic functional equation f( x+2y )+f( x−2y )=4( f( x+y )+f( x−y ) )−24f( y )−6f( x )+3f( 2y ) in quasi-Banach spaces.

We consider a higher-order three-point boundary value problem on time scales.A new existence result is first obtained by using a fixed point theorem due to Krasnoselskii andZabreiko. Later, under certain growth conditions imposed on the nonlinearity, several sufficientconditions for the existence of a nonnegative and nontrivial solution are obtained by usingLeray-Schauder nonlinear alternative. Our conditions imposed on nonlinearity are all very easyto verify; as an application, some examples to demonstrate our results are given.

We introduce an ℓ( p ) -type new sequence space and investigate its some topological properties including AK and AD properties. Besides, we examine some geometric properties of this space concerningBanach-Saks type p and Gurarii's modulus of convexity.

By using the ordinary fermionic p -adic invariant integral on ℤ p , we derive some interesting identities related to the Frobenius-Euler polynomials.

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in C ∗ -algebras and Lie C ∗ -algebras and also of derivations on C ∗ -algebras and Lie C ∗ -algebras for the Jensen-type functional equation f( ( x+y ) /2 )+f( ( x−y ) /2 )=f( x ) .

A Chemostat system incorporating hepatocellular carcinomasis discussed. The model generalizes the classical Chemostat model, and it assumes that the Chemostat is an increasing function of the concentration. Theasymptotic behavior of solutions is determined. Sufficient conditions for thelocal and global asymptotic stability of equilibrium and numerical simulationare obtained, which is used to select the disease control tactics.

We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappingsin a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for viscosity iterative scheme are given and strong convergence of viscosity iterativescheme to a solution of a ceratin variational inequality is established.

The numerical range and normality of Toeplitz operator acting on the Bergman space and pluriharmonic Bergman space on the polydisk is investigated in this paper.

We study the total stability in nonlinear discrete Volterra equations with unbounded delay, as a discrete analogue of the results for integrodifferential equations by Y. Hamaya (1990).

We obtain sufficient conditions which guarantee the global attractivityof solutions for nonlinear delay survival red blood cells model. Then, some criteria are establishedfor the existence, uniqueness and global attractivity of positive almost periodic solutions of thealmost periodic system.

Consider the half-eigenvalue problem ( ϕ p ( x ′ ) ) ′ + λ a ( t ) ϕ p ( x + ) − λ b ( t ) ϕ p ( x − ) = 0 a.e. t ∈ [ 0 , 1 ] , where 1 < p < ∞ , ϕ p ( x ) = | x | p − 2 x , x ± ( ⋅ ) = max ⁡ { ± x ( ⋅ ) , 0 } for x ∈ 풞 0 : = C ( [ 0 , 1 ] , ℝ ) , and a ( t ) and b ( t ) are indefinite integrable weights in the Lebesguespace ℒ γ : = L γ ( [ 0 , 1 ] , ℝ ) , 1 ≤ γ ≤ ∞ . We characterize the spectra structure under periodic, antiperiodic, Dirichlet, and Neumann boundary conditions, respectively. Furthermore, all these half-eigenvalues are continuous in ( a , b ) ∈ ( ℒ γ , w γ ) 2 , where w γ denotes the weak topology in ℒ γ space. The Dirichlet and the Neumann half-eigenvalues are continuously Fréchet differentiable in ( a , b ) ∈ ( ℒ γ , ‖ ⋅ ‖ γ ) 2 , where ‖ ⋅ ‖ γ is the L γ norm of ℒ γ .

Let H( B ) denote the space of all holomorphic functions on the unit ball B . Let u∈H( B ) and φ be a holomorphic self-map of B . In this paper, we investigate the boundedness and compactness of the weighted composition operator u C φ from the general function space F( p,q,s ) to the weighted-type space H μ ∞ in the unit ball.

We consider hyperbolic equations with anisotropic elliptic part and some non-Lipschitz coefficients.We prove well-posedness of the corresponding Cauchy problem in some functional spaces. These functional spaces have finite smoothness with respect to variables corresponding to regular coefficients and infinite smoothnesswith respect to variables corresponding to singular coefficients.