a time scales version of a wirtinger type inequality and applications in this paper we derive a time scales version of a wirtinger type inequality the result is applied to derive nonoscillation of a certain second order euler type dynamic equation on time scales

the n n disconjugacy of a 2nth order linear difference equation abstract we give a formulation of generalized zeros and n n disconjugacy for even order formally self adjoint scalar difference equations positive definiteness of an associated quadratic functional is shown to imply n n disconjugacy this study is motivated by the open question of existence of a reid roundabout theorem for singular discrete linear hamiltonian systems

asymptotic properties of solutions of nonlinear difference equations using discrete inequalities and schauder s fixed point theorem we study the problem of asymptotic equilibrium for difference equations

asymptotic equivalence between two difference systems abstract in this paper we study the asymptotic equivalence between the linear system x n a n x n and its perturbation y n a n y n g n y n by using the comparison principle and supplementary projections furthermore we establish some asymptotic properties for the nonlinear system x n f n x n

factorization of disconjugate higher order sturm liouville difference operators abstract using a recently proved equivalence between disconjugacy of the 2nth order difference equation l y k n v 0 n 1 v v k v v y k n v 0 and solvability of the corresponding riccati matrix difference equation it is shown that the equation l y 0 is disconjugate on a given interval if and only if the operator l admits the factorization of the form l y k n m c k m y k k n where m and its adjoint m are certain nth order difference operators and ck is a sequence of positive numbers

nonoscillation of second order linear difference systems with varying coefficients abstract this paper deals with nonoscillation problem about the non autonomous linear difference system x n a n x n 1 n 1 2 where a n is a 2 2 variable matrix that is nonsingular for n n in the special case that a is a constant matrix it is well known that all non trivial solutions are nonoscillatory if and only if all eigenvalues of a are positive real numbers namely det a 0 tr a 0 and det a tr a 2 1 4 the well known result can be said to be an analogy of ordinary differential equations the results obtained in this paper extend this analogy result in other words this paper clarifies the distinction between difference equations and ordinary differential equations our results are explained with some specific examples in addition figures are attached to facilitate understanding of those examples

convergence of fourier series with complex quasimonotone coefficients and coefficients of bounded variation of order m abstract l1 convergence theorems for fourier series with complex quasimonotone coefficients are generalized the class of null sequences of bounded variation of order m is extended and the point wise theorem for fourier series with such coefficients is proved

asymptotic property for linear integro differential systems abstract we study the asymptotic property of linear integro differential systems by means of the resolvent matrices and the useful equivalent system of rao and srinivas in m r m rao p srinivas asymptotic behavior of solutions of volterra integro differential equations proc amer math soc 94 1985 55 60 11

equivalence of discrete euler equations and discrete hamiltonian systems abstract erbe and yan recently presented a discrete linear hamiltonian system their system is a special case of the discrete hamiltonian system y n l h z n y n z n l z n l h y n y n z n l where y n 1 y n y n 1 under certain implicit solvability hypotheses these systems are equivalent to the discrete euler equation y n yn yn l r n yn yn l a reid roundabout theorem for linear recurrence relations knyn 1 bnyn ktn 1yn 1 0 is shown to imply the corresponding result obtained by erbe and yan for discrete linear hamiltonian systems furthermore discrete linear hamiltonian systems are shown to have a symplectic transition matrix

l 1 approximation of fourier series of complex valued functions v b stanojevic suggested in her recent paper that it would be of interest to prove a corresponding l 1 convergence theorem for fourier series with complex o regularly varying quasimonotonc coefficients the present paper will discuss this question and establish l 1 convergence and furthermore l 1 approximation theorems for complex valued integrable functions

discrete linear hamiltonian eigenvalue problems abstract this paper introduces general discrete linear hamiltonian eigenvalue problems and characterizes the eigenvalues assumptions are given among them the new notion of strict controllability of a discrete system that imply isolatedness and lower boundedness of the eigenvalues due to the quite general assumptions discrete sturm liouville eigenvalue problems of higher order are included in the presented theory

remarks on convergence of trigonometric series with special varying coefficients abstract we give an essential generalization of telyakovskii s results on the convergence of trigonometric series with rarely changing coefficients in the norm of l 1

strong asymptotic equivalence and inversion of functions in the class kc abstract in this paper we consider the class of functions kc introduced by w matuszewska 1964 studia math 24 271 279 and w matuszewska and w orlicz 1965 studia math 26 11 24 as a main result we describe in terms of the class kc when two strictly increasing functions as well as their inverse functions are asymptoticaly equivalent in the strong sense this result also gives a proper characterization of the class kc

variationally stable difference systems we characterize the h stability in variation for nonlinear difference systems via n similarity and lyapunov functions furthermore using lyapunov s method and comparison principle we obtain some results related to stability for the perturbations of nonlinear difference systems

asymptotic equivalence between two linear volterra difference systems abstract we study asymptotic equivalence between the solutions of linear volterra difference system x n 1 a n x n s n 0 n b n s x s and its perturbed system y n 1 a n y n s n 0 n b n s y s f n

linear hamiltonian difference systems disconjugacy and jacobi type conditions abstract we consider a linear hamiltonian difference system for the so called singular case so that discrete sturm liouville equations of higher order are included in our theory we introduce the concepts of focal points for matrix valued and generalized zeros for vector valued solutions of the system and define disconjugacy for linear hamiltonian difference systems we prove a reid roundabout theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional among them the strengthened jacobi s condition and a condition on a certain riccati difference equation the key to this theorem is a discrete version of picone s identity furthermore for the sake of generalization of our theorem we introduce controllability for linear hamiltonian difference systems and prove a reid roundabout theorem for a more general functional and more general boundary conditions

a class of sturm liouville difference equations non oscillation constants and property bd abstract in this paper we study oscillatory properties of the higher order sturm liouville difference equation 1 n n k n yk q k y k n r in particular we derive sufficient conditions for non oscillation of and compute explicitly its non oscillation constants the obtained results are used to derive necessary and sufficient conditions for discreteness and boundedness below the spectrum of associated difference operators we discuss also non oscillation of the euler type difference equation

criteria for c disfocality of a self adjoint vector difference equation abstract we are concerned with the vector difference equation p t 1 y t 1 q t y t 0 t a 1 h 1 where the n n matrices p t q t are hermitian and p t 0 on a b 1 this difference equation is said to be c disfocal on a b 2 provided that if y is a nontrivial solution with y b 1 0 then y has no generalized zeros in a b 2 one case of the main result is that the above difference equation is c disfocal if and only if a certain quadratic functional is positive definite our results will imply several comparison theorems further a necessary and sufficient condition in terms of the coefficients of the difference equation for c disfocality will be given

existence of homoclinic orbits for 2nth order nonlinear difference equations containing both many advances and retardations abstract by establishing a proper variational framework and using the critical point theory we establish some new existence criteria to guarantee the 2nth order nonlinear difference equation containing both many advances and retardations n r t n n u t n q t u t f t u t n u t u t n n z 3 t z has at least one or infinitely many homoclinic orbits where r q f are nonperiodic in t our conditions on the potential are rather relaxed and some existing results in the literature are improved

discrete spectra criteria for certain classes of singular differential and difference operators in this paper we consider singular differential and difference operators of the form m y 1 n w t r t y n n and b y k n 1 n w k delta n r k delta n y k as well as their full term analogies we establish criteria for discreteness and boudnedness bellow of their spectrum property bd difference operators are studied as the discrete counterparts of the continuous ones

a class of l 1 convergence it is proved that if the fourier coefficients an of e l 0 it satisfy n 2 k aa p o l for some 1 p 2 then j f o l if and only if a lg n o for cosine trigonometric series with coefficients of bounded variation and satisfying it is proved that a necessary and sufficient condition for the series to be a fourier series is an e c where q is the garrett stanojevic 4 class

representations of fourier coefficients in tauberian l1 convergence classes abstract it is shown that the fourier coefficients satisfying stanojevic s tauberian condition can be represented in terms of o regularly varying sequences

existence of periodic solutions for a 2nth order nonlinear difference equation abstract the authors consider the 2 n th order difference equation n r t n n x t n f t x t 0 n z 3 t z where f z r r is a continuous function in the second variable f t t z f t z for all t z z r r t t r t for all t z and t a given positive integer by the linking theorem some new criteria are obtained for the existence and multiplicity of periodic solutions of the above equation

qualitative properties of hamiltonian difference systems abstract we study some qualitative properties of the hamiltonian difference system y t b t y t 1 c t z t z t a t y t 1 b t z t the methods used involve both variational and riccati type arguments

disconjugacy for linear hamiltonian difference systems abstract in this paper we study the linear hamiltonian difference system y t b t y t 1 c t z t z t a t y t 1 b t z t where a t c t are hermitian matrices with i b t and c t invertible i is identity disconjugacy criteria analogous to those for linear hamiltonian differential systems are obtained by a discrete riccati method

periodicity and stability of linear volterra difference systems abstract we give sufficient conditions for a system of volterra difference equations to have a unique periodic solution then a criterion for uniform asymptotic stability is established

asymptotic property of linear volterra difference systems by using the resolvent matrix and the comparison principle we investigate the asymptotic behavior of linear volterra difference systems

on l1 convergence of fourier series abstract in this paper we consider the trigonometric fourier series with the general monotone coefficients necessary and sufficient conditions of l 1 convergence for such a series that is f s n o 1 are obtained in terms of coefficients

tauberian conditions for l 1 convergence of fourier series it is proved that fourier series with asymptotically even coefficients and satisfying lima 1lim supn exn jpi i ajf j p 0 for some i p 2 converge in l norm if and only if i1 f n en f n e n ii o l where en t k e k recent results of stanojevic 1 bojanic and stanojevic 2 and goldberg and stanojevic 3 are special cases of some corollaries to the main theorem