Search

Estimates on Solutions of a New Quadruple Integral Inequalities

Author(s)

Yusong Lu, Meijin Luo*

Affiliation(s)

School of Mathematics and Physics, Hechi University, Hechi, Guangxi, China *Corresponding Author.

Abstract

With the development of science and technology and the progress of society, differential equations have played important roles in natural science, social science, engineering technology, etc. By investigating the properties of differential equation solutions, some phenomena could be explained, future development trends could be predicted, and reference and theoretical bases for decision-making could be provided. However, in several cases, explicit solutions cannot be derived for differential equations, but using appropriate integral inequality techniques, one could explore the existence, uniqueness, vibration, stability and other qualitative properties of differential equation solutions and estimate solution sizes of differential equations. Therefore, researchers are investigating integral inequalities and constantly enriching the existing achievements. Based on previous research works, a class of four-fold nonlinear integral inequalities with unknown derivative functions was established in this research. By using various inequality analysis methods such as variable substitution, amplification, differentiation and integration, estimations were provided for unknown functions in quadruple integral inequalities. This type of inequality could be applied to evaluate the estimation of corresponding differential equation solutions, so as to provide an effective mathematical tool to solve practical problems.

Keywords

Integral Inequality; Quadruple Integral; Differential Equation; Estimation

References

[1]Gronwall T H. Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. vol. 1919: 20, 292 pages, 1919. [2]Bellman R. The stability of solutions of linear differential equations, Duke Math. J. vol. 1943:10, 643 pages, 1943. [3]Zheng B. (G'/G)-expansion method for solving fractional partial differential equations in the theory of mathematical physics, Commun Theor Phys. vol. 2012:58, 623 pages, 2012. [4]Wang W S. Some generalized nonlinear retarded integral inequalities with application-s, J Inequal Appl. vol. 2012:31, 14 pages, 2012. [5]Lu Y S. Wang Wu-sheng. A class of Volterra-Fredholm-type nonlinear iterated integral inequalities with p power, Journal of Southwest University (Natural Science Edition). vol. 2015:37, 76 pages, 2015. [6]Pachpatte B G. Inequalities for differential and integral equations, New York: Academic Press, 1998:45. [7]Zareen A K. On some fundamental integro differential inequalities, Applied Mathematics, 2014, 5: 2968-2973. [8]Huang X S, Wang W S, Luo R C. Estimation of unknown function of a class of double integral inequalities, Journal of South China Normal University (Natural Science Edition). vol. 2019:51, 108pages, 2019. [9]Huang X S, Wang W S, Luo R C. Estimation of unknown function of a class of triple integral inequalities with unknown deriative function, Journal of Southwest China Normal University (Natural Science Edition). vol. 2019:44, 9 pages, 2019. [10]Lu Yu S, Huang X S. Estimation of Unknown Function of a Class of Triple Integral Inequalities, Advances in Applied Mathematics. vol.2020:9, 1200 pages, 2020.