Opial-type inequalities involving higher order differences

In this paper, we present very general discrete Opial-type inequalities involving higher order differences of one or more functions. The results obtained here are, in fact, the discrete analogs of our several recently established inequalities involving higher order derivatives.     

New discrete type inequalities and global stability of nonlinear difference equations

In this work, we introduce discrete type inequalities. On the basis of these inequalities, we derive new global stability conditions for nonlinear difference equations.     

SOME NEW DISCRETE INEQUALITIES OF OPIAL WITH TWO SEQUENCES

In this paper, we establish some new discrete inequalities of Opial-type with two sequences by making use of some classical inequalities. These results contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case.     

离散型Carlson's不等式的一些推广及改进

In this paper, we consider Carlson type inequalities and discuss their possible improvement. First, we obtain two different types of generalizations of discrete Carlson's inequality by using the Hlder inequality and the method of real analysis, then we combine the obtained results with a summation formula of infinite series and some Mathieu type inequalities to establish some improvements of discrete Carlson's inequality and some Carlson type inequalities which are equivalent to the Mathieu type inequalities. Finally, we prove an integral inequality that enables us to deduce an improvement of the Nagy-Hardy-Carlson inequality.     

Some new dynamic inequalities on time scales

In this paper, based on some known dynamic inequalities, we investigate certain new dynamic inequalities on time scales, which provide explicit bounds on unknown functions. Our results unify and extend some continuous inequalities and their corresponding discrete analogues.     

On several new Hilbert-type inequalities involving means operators

In this paper, we establish several new Hilbert-type inequalities with a homogeneous kernel, involving arithmetic, geometric, and harmonic mean operators in both integral and discrete case. Such inequalities are derived by virtue of some recent results regarding general Hilbert-type inequalities and some well-known classical inequalities. We also prove that the constant factors appearing in established inequalities are the best possible. As an application, we consider some particular settings and compare our results with previously known from the literature.     

Numerical solution of quasi-variational inequalities arising in stochastic game theory

We study the finite-difference approximation for the quasi-variational inequalities for a stochastic game involving discrete actions of the players and continuous and discrete payoff. We prove convergence of iterative schemes for the solution of the discretized quasi-variational inequalities, with estimates of the rate of convergence (via contraction mappings) in two particular cases. Further, we prove stability of the finite-difference schemes, and convergence of the solution of the discrete problems to the solution of the continuous problem as the discretization mesh goes to zero. We provide a direct interpretation of the discrete problems in terms of finite-state, continuous-time Markov processes.     

Error estimate of Fourier pseudo-spectral method for multidimensional nonlinear complex fractional Ginzburg–Landau equations

In this paper, we discuss the error estimate of Fourier pseudo-spectral method for multidimensional nonlinear complex space fractional Ginzburg-Landau equations. The continuous mass and energy inequalities as well as their discrete versions are presented. Moreover, by the discrete mass and energy inequalities, the error estimate of the Fourier pseudo-spectral scheme is established, and the scheme is proved to have the spectral accuracy.     

Inégalités d'entropie pour un schéma de relaxation

This work is devoted to the discrete entropy inequalities when considering relaxation schemes. After describing the numerical method, we propose a direct proof to establish the discrete entropy inequalities. In fact, we show that the considered relaxation model satisfies a minimum principle on the entropy. This principle implies the expected inequalities. The work is concluded when applying the above results to the 10 moment model. To cite this article: C. Berthon, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On some dynamic inequalities of Steffensen type on time scales

In the present article, we investigate some new inequalities of Steffensen type on an arbitrary time scale using the diamond‐α dynamic integrals, which are defined as a linear combination of the delta and nabla integrals. The obtained inequalities extend some known dynamic inequalities on time scales and unify and extend some continuous inequalities and their discrete analogues.     

Some inequalities in inner product spaces related to the generalized triangle inequality

In this paper we obtain some inequalities related to the generalized triangle and quadratic triangle inequalities for vectors in inner product spaces. Some results that employ the Ostrowski discrete inequality for vectors in normed linear spaces are also obtained.     

Discrete linear Hamiltonian systems: Lyapunov type inequalities,stability and disconjugacy criteria

In this paper, we first establish new Lyapunov type inequalities for discrete planar linear Hamiltonian systems. Next, by making use of the inequalities, we derive stability and disconjugacy criteria. Stability criteria are obtained with the help of the Floquet theory, so the system is assumed to be periodic in that case.     

Discrete Hilbert-type inequalities with general homogeneous kernels

The main objective of this paper is an unified treatment of discrete Hilbert-type inequalities with homogeneous kernels. At the beginning, we prove and discuss two equivalent general inequalities of such type. Further, we analyze the conditions which yield the best possible constant factors in obtained inequalities. The obtained results are then applied to various settings considering homogeneous functions of a negative real degree. In such a way, we obtain the generalizations of numerous results, previously known from the literature.     

Discrete Hardy-Hilbert''''s Inequalities in R~n

By using some auxiliary functions and introducing relative Lemmas, we establish multidimensional discrete Hardy-Hilbert's inequalities.     

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

In this paper, we establish several new Lyapunov-type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.     

Discrete non-linear inequalities and applications to boundary value problems

In this paper, we generalize some existing discrete Gronwall-Bellman-Ou-Iang-type inequalities to more general situations. These are in turn applied to study the boundedness, uniqueness, and continuous dependence of solutions of certain discrete boundary value problem for difference equations.     

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

In this paper, we establish several new Lyapunov type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.     

Explicit Bounds for Some Special Integral Inequalities on Time Scales

In this paper, we investigate some nonlinear integral inequalities on time scales, which provide explicit bounds on unknown functions. Our results, which can be used as handy tools to study the properties of certain dynamic equations on time scales, unify and extend some integral inequalities and their corresponding discrete analogues.     

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.     

Relations between regularity coefficients

For a bounded sequence of matrices defining a nonautonomous dynamics with discrete time, we obtain all possible relations between the regularity coefficients introduced by Lyapunov, Perron and Grobman. This includes considering general inequalities between the coefficients and showing that these inequalities are the best possible, in the sense that for any three nonnegative numbers satisfying them, and for no others, there exists a bounded sequence of matrices having the numbers respectively as Lyapunov, Perron and Grobman coefficients. Moreover, we establish inequalities between the three coefficients and some other regularity coefficients.