Sharp discrete inequalities and applications to discrete variational problems

Some new discrete inequalities involving monotonic or convex functions are obtained. While these are interesting inequalities in their own right, they can be applied to solving certain types of discrete variational problems effectively.     

Nonlinear discrete inequalities of Bihari-type and applications

Discrete Bihari-type inequalities with n nonlinear terms are discussed, which generalize some known results and may be used in the analysis of certain problems in the theory of difference equations. Examples to illustrate the boundedness of solutions of a difference equation are also given.     

On some new nonlinear discrete inequalities and their applications

In this paper, some new discrete inequalities in two independent variables which provide explicit bounds on unknown functions are established. The inequalities given here can be used as tools in the qualitative theory of certain finite difference equations.     

Rearrangement inequalities in the discrete setting and some applications

We prove extended Hardy–Littlewood, Riesz and Pólya–Szeg? inequalities in the discrete case. We also establish cases of equality in these inqualities and give some applications of our result.     

Nonlinear discrete inequalities with two variables and their applications

An error in the proof of Theorem 1 by Zheng et al. [K. Zheng, Y. Wu, S. Zhong, Discrete nonlinear integral inequalities in two variables and their applications, Appl. Math. Comput. 207 (2009) 140-147] is reported. This paper gives the right proof under some additional condition. Application examples to show boundedness and uniqueness of solutions of a Volterra type difference equation are also given.     

Estimates on some power nonlinear Volterra-Fredholm type discrete inequalities and their applications

Some explicit bounds on solutions to a class of new power nonlinear Volterra-Fredholm type discrete inequalities are established, which can be used as effective tools in the study of certain sum-difference equations. Application examples are also given.     

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.     

New generalization of perturbed Ostrowski type inequalities and applications

Generalizations of perturbed Ostrowski type inequality for functions of Lipschitzian type are established. Applications in numerical integration and cumulative distribution functions are also given.     

On discrete and continuous wirtinger inequalities

We shall present several generalizations of discrete Wirtinger's inequality, and establish their continuous analogs.     

On some new discrete inequalities

Some new discrete inequalities involving higher order differences have been obtained here. These inequalities can be used in the analysis of a class of summary difference equations as handy tools. Some applications are also given.     

Correlation inequalities and their applications

Correlation inequalities and their applications to the question of the presence or absence of phase transitions is classical lattice systems are considered. A major part of the known inequalities is obtained as a corollary of a single general theorem.Translated from Itogi Nauki i Tekhniki. Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 16, pp. 5–37, 1979.     

Discrete Ingham inequalities and applications

In this Note we prove a discrete version of the classical Ingham inequality for nonharmonic Fourier series whose exponents satisfy a gap condition. Time integrals are replaced by discrete sums on a discrete mesh. We prove that, as the mesh becomes finer and finer the limit of the discrete Ingham inequality is the classical continuous one. This analysis is partially motivated by control-theoretical applications. As an application we analyze the control/observation properties of numerical approximation schemes of the 1-d wave equation. The discrete Ingham inequality provides observability and controllability results which are uniform with respect to the mesh size in suitable classes of numerical solutions in which the high frequency components have been filtered. To cite this article: M. Negreanu, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Higher-order symmetrization inequalities and applications

We prove new extended forms of the Pólya-Szegö symmetrization principle. As a consequence new sharp embedding theorems for generalized Sobolev and Besov spaces are proved.     

Bifurcation problems for discrete variational inequalities

The buckling of a beam or a plate which is subject to obstacles is typical for the variational inequalities that are considered here. Birfurcation is known to occur from the first eigenvalue of the linearized problem. For a discretization the bifurcation point and the bifurcating branches may be obtained by solving a constrained optimization problem. An algorithm is proposed and its convergence is proved. The buckling of a clamped beam subject to point obstacles is considered in the continuous case and some numerical results for this problem are presented.     

Higher dimensional tautological inequalities and applications

We study the degeneracy of holomorphic mappings tangent to holomorphic foliations on projective manifolds. Using Ahlfors currents in higher dimension, we obtain several strong degeneracy statements such as the proof of a generalized Green-Griffiths–Lang conjecture for threefolds with holomorphic foliations of codimension one.     

Lexicographic variational inequalities with applications

In this article, we consider equivalence properties between various kinds of lexicographic variational inequalities. By employing various concepts of monotonicity, we show that the usual sequential variational inequality is equivalent to the direct lexicographic variational inequality or to the dual lexicographic variational inequality. We establish several existence results for lexicographic variational inequalities. Also, we introduce the lexicographic complementarity problem and establish its equivalence with the lexicographic variational inequality. We illustrate our approach by several examples of applications to vector transportation and vector spatial equilibrium problems.     

Some multivariate inequalities with applications

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>{\cal {X}}_{n} =(X_1,\ldots,X_n)$ be a random vector. Suppose that the random variables $(X_i)_{1\leq i\leq n}$ are stationary and fulfill a suitable dependence criterion. Let $f$ be a real valued function defined on $\mathbbm{R}^n$ having some regular properties. Let ${\cal {Y}}_{n}$ be a random vector, independent of ${\cal {X}}_{n}$, having independent and identically distributed components. We control $\left|\mathbbm{E}(f({\cal {X}}_{n}))-\mathbbm{E} (f({\cal {Y}}_{n}))\right|$. Suitable choices of the function $f$ yield, under minimal conditions, to rates of convergence in the central limit theorem, to some moment inequalities or to bounds useful for Poisson approximation. The proofs are derived from multivariate extensions of Taylor's formula and of the Lindeberg decomposition. In the univariate case and in the mixing setting the method is due to Rio (1995).