Some New Volterra-Fredholm Type Dynamic Integral Inequalities With Two Variables On Time Scales

When it comes to the equations,we know that not all of them can get the exact solutions.This phenomenon can cause problems in the relevant field and make us have some difficulties in learning about the relevant properties of the equations.A new mathematical tool is needed to do some research on the solutions of the equations as an effective indirective transition.The integral inequality,which is widely used in mathematical analysis,can be an excellent way for us to deal with these problems.By assuming the solution has existed first,we can analyze the nature of the solution without solving it,as in the reference [2,22,7-9,16-20,27-34].The time-scale dynamic theory,introduced by Hilger[1],constructs a way to deal with both continuous and discrete equations,has attracted wide attention.On the one hand,it is treated as a bridge between the differential and difference equations.On the other hand,the theory is also available in biology,economics,electrical engineering and digital control It is more practical to describe the population quantity,the electricity consumption peak.Many discoveries has been achieved by a lot of scholars in the The time-scale dynamic theory.These documents below are mainly about some vital details:In [2] the methods of integral inequality has been provided.In [3] we set up a linear Volterra-Fredholm dynamic integral inequality with one variable.In [4] we set up a nolinear Volterra-Fredholm dynamic integral inequality with one variable on time scale.In [5-7] we set up various time scale continusous and discreet nolinear Volterra-Fredholm dynamic integral inequality.In [8-10] we set up various time scale continusous and discreet linear VolterraFredholm dynamic integral inequality.In [11-12] we study some properties of VolterraFredholm dynamic integral inequality.In [13-14] we get some generalized Volterra-Fredholm integral inequality.However,the attention to the the nonlinear Volterra-Fredholm dynamic integral inequality with two variables is a little less.Based on the research of the former scholars,the article is mainly about increasing the number of variable and terms in the inequalities,and by means of variable substitution,derived function and inverse function The Volterra-Fredholm dynamic integral inequality with two variables on time scales can be dealt with.As the next step,the asymptotic estimation of the inequalities can be evaluated Furthermore,the qualitative analysis is logical,which can be used in the research of the equation solutions.Many research can emphasize on one variable or low dimension vector.It can be predicted that the topic concerning about high dimension vector has more common quality.This article plays an effective role in the supplement and popularization of the time-scaledynamic equations.