The main objective of this paper is to establish a class of new nonlinear Volterra-Fredholm type difference inequality, where the inequalities consist of multiple iterated sums. By technique of change of variable, difference and summation and inverse function, upper bound estimations of unknown functions are given. The derived results can be applied in the study of solutions of Volterra-Fredholm type difference equations.

The main objective of this paper is to establish a class of new nonlinear Volterra-Fredholm type difference inequality. Upper bound estimations of unknown functions are given by technique of change of variable, amplification method, difference and summation and inverse function. The derived results can be applied in the study of solutions of Volterra-Fredholm type difference equations.

Differential equations are important tools in studying of natural science, engineering technology, and the laws of social economic development. It is necessary to seek some new inequalities in order to study of boundedness, uniqueness, stability and boundary value problem of a differential equation. Motivated by Abdeldaim integral inequalities, in this paper, we establish a class of generalized retarded nonlinear Gronwall-Bellman-Type integral inequalities and give upper bound estimation of the unknown function by analysis skills. Finally we give an example to illustrate the effectiveness of our results in estimation of solutions of some differential equations with the initial conditions.

In this paper, we discuss a class of new Volterra-Fredholm weakly singular difference inequality. The explicit bounds for the unknown functions are given clearly by discrete Jensen inequality, Cauchy-Schwarz inequality, Gamma function, change of variable, the mean-value theorem for integrals and amplification method. The derived results can be applied in the study of fractional difference equations in engineering.

We discuss a class of generalized retarded nonlinear integral inequalities, which not only include nonlinear compound function of unknown function but also include retarded items, and give upper bound estimation of the unknown function by integral inequality technique. This estimation can be used as tool in the study of differential equations with the initial conditions.