Some new nonlinear Volterra–Fredholm-type discrete inequalities and their applications

Some new explicit bounds on solutions to a class of new 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 indicated.     

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.     

Some new nonlinear difference inequalities and their applications

In this paper, we establish some new nonlinear difference inequalities in two independent variables, which can be used as handy tools in the study of qualitative properties of solutions of certain classes of difference equations.     

Estimates on solutions of some new nonlinear retarded Volterra–Fredholm type integral inequalities

Some new explicit bounds on solutions to a class of new nonlinear retarded Volterra–Fredholm type integral inequalities are established, which can be used as effective tools in the study of certain integral equations. Applications examples are also indicated.     

Explicit bounds on some new nonlinear integral inequalities with delay

The purpose of the present note is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions and generalize some results of Li et al. [Some new delay integral inequalities and their applications, J. Comput. Appl. Math. 180 (2005) 191–200]. The inequalities given here can be used to investigate the qualitative properties of certain delay differential equations and delay integral equations.     

About some new integral inequalities of Wendroff type for discontinuous functions

In this article, we obtain some new nonlinear integral inequalities for discontinuous functions of two independent variables (Wendroff type) by including also inequalities with delay. We deduce new generalizations of earlier results given by R.P. Agarwal, R. Bellman, I. Bihari, B.K. Bondge, V. Lakshmikantham, S. Leela, B.G. Pachpatte for continuous and discrete functions. Furthermore, generalizations of some results for integro-sum inequalities are obtained as well.     

Some new nonlinear inequalities and applications to boundary value problems

In this paper, we establish some new nonlinear integral inequalities of the Gronwall–Bellman–Ou-Iang-type in two variables. These on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations. We illustrate this by applying our new results to certain boundary value problem.     

About new analogies of Gronwall–Bellman–Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems

In this article we present new integral Gronwall–Bellman–Bihari type inequalities for discontinuous functions (integro-sum inequalities). As applications, we investigate estimated solutions for impulsive differential systems, conditions of boundedness, stability, practical stability.     

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.     

Jensen inequalities for functions with higher monotonicities

Summary We investigate generalizations of the classical Jensen and Chebyshev inequalities. On one hand, we restrict the class of functions and on the other we enlarge the class of measures which are allowed. As an example, consider the inequality (J)(f(x) d) A (f(x) d, d d = 1. Iff is an arbitrary nonnegativeL _x function, this holds if 0, is convex andA = 1. Iff is monotone the measure need not be positive for (J) to hold for all convex withA = 1. If has higher monotonicity, e.g., is also convex, then we get a version of (J) withA < 1 and measures that need not be positive.     

A note on certain integral inequalities with delay

In this paper we establish some new integral inequalities with delay, which can be used as tools in the theory of some new classes of differential and integral equations. An application to obtain a bound on the solution of a certain integral equation is also given.     

On discrete inequalities of the Poincaré type

The aim of the present paper is to establish some new discrete inequalities of the Poincaré type involving functions ofn independent variables and their first order forward differences. The proofs given here are quite elementary and our results provide new estimates on this type of discrete inequalities.     

Polynomial inequalities on measurable sets and their applications

We study Pólya-and Remez-type inequalities for univariate and multivariate polynomials and discuss their applications to Nikolskii-type inequalities and upper estimates of trigonometric integrals.     

Chebyshev type inequalities involving permanents and their applications

We extend the well-known Chebyshev’s inequality to some new cases involving permanents under the proper hypotheses. Our main results are

     

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.     

On integral inequalities of the Sobolev type

Summary In this paper some new integral inequalities of the Sobolev type involving many functions of many variables are established. These in turn can be used to serve as generators of other integral inequalities.