A class of nonlinear delay integral inequalities for two-variable functions and their applications in Volterra integral equations
Authors
S. Kriket
- Department of Mathematics and Computer Science, University of Ghardaia, Ghardaia, Algeria.
A. Boudeliou
- Department of Mathematics, University of Constantine 1, BP 325, Ain El Bey Street, Constantine 25017, Algeria.
Abstract
In this paper, we establish some new nonlinear integral inequalities with delay in two independent variables which generalize some known inequalities recently obtained. These results can be used as handy tools to study the boundedness of solutions of Volterra integral equations. An application is given to illustrate how our results can be applied to study the boundedness of the solutions of certain Volterra equations.
Share and Cite
ISRP Style
S. Kriket, A. Boudeliou, A class of nonlinear delay integral inequalities for two-variable functions and their applications in Volterra integral equations, Journal of Mathematics and Computer Science, 32 (2024), no. 1, 1--12
AMA Style
Kriket S., Boudeliou A., A class of nonlinear delay integral inequalities for two-variable functions and their applications in Volterra integral equations. J Math Comput SCI-JM. (2024); 32(1):1--12
Chicago/Turabian Style
Kriket, S., Boudeliou, A.. "A class of nonlinear delay integral inequalities for two-variable functions and their applications in Volterra integral equations." Journal of Mathematics and Computer Science, 32, no. 1 (2024): 1--12
Keywords
- Delay integral inequality
- two-variable functions
- properties of solutions
- Volterra integral equation
MSC
References
-
[1]
A. Abdeldaim, Nonlinear retarded integral inequalities of Gronwall-Bellman type and applications, J. Math. Inequal., 10 (2016), 285–299
-
[2]
A. Abdeldaim, M. Yakout, On some new integral inequalities of Gronwall-Bellman-Pachpatte type, Appl. Math. Comput., 217 (2011), 7887–7899
-
[3]
R. P. Agarwal, S. Deng, W. Zhang, Generalization of a retarded Gronwall-like inequality and its applications, Appl. Math. Comput., 165 (2005), 599–612
-
[4]
D. Ba˘ınov, P. Simeonov, Integral inequalities and applications, Kluwer Academic Publishers Group, Dordrecht (1992)
-
[5]
R. Bellman, The stability of solutions of linear differential equations, Duke Math. J., 10 (1943), 643–647
-
[6]
A. Boudeliou, On certain new nonlinear retarded integral inequalities in two independent variables and applications, Appl. Math. Comput., 335 (2018), 103–111
-
[7]
A. Boudeliou, H. Khellaf, On some delay nonlinear integral inequalities in two independent variables, J. Inequal. Appl., 2015 (2015), 14 pages
-
[8]
K. Boukerrioua, B. Kilani, I. Meziri, Refinements of some retarded integral inequalities of Gronwall-Bellman-Bihari type and their applications, Surv. Math. Appl., 15 (2020), 233–255
-
[9]
A. A. El-Deeb, R. G. Ahmed, On some explicit bounds on certain retarded nonlinear integral inequalities with applications, Adv. Inequal. Appl., 2016 (2016), 19 pages
-
[10]
T. H. Gronwall, Note on the derivatives with respect to a parameter of solutions of a system of differential equations, Ann. of Math. (2), 2 (1919), 292–296
-
[11]
F. Jiang, F. Meng, Bounds on some new nonlinear integral inequalities with delay, J. Comput. Appl. Math., 205 (2007), 479–486
-
[12]
V. Lakshmikantham, S. Leela, Differential and integral inequalities theory and applications, Academic Press, New York (1969)
-
[13]
Z. Li, W.-S. Wang, Some nonlinear Gronwall-Bellman type retarded integral inequalities with power and their applications, Appl. Math. Comput., 347 (2019), 839–852
-
[14]
O. Lipovan, A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl., 252 (2000), 389–401
-
[15]
B. G. Pachpatte, Inequalities for differential and integral equations, Academic Press, San Diego (1998)
-
[16]
B. G. Pachpatte, Integral and finite difference inequalities and applications, Elsevier Science B.V., Amsterdam (2006)
-
[17]
A. Shakoor, I. Ali, M. Azam, A. Rehman, M. Z. Iqbal, Further nonlinear retarded integral inequalities for Gronwall- Bellman type and their applications, Iran. J. Sci. Technol. Trans. A Sci., 43 (2019), 2559–2568
-
[18]
W.-S. Wang, Some retarded nonlinear integral inequalities and their applications in retarded differential equations, J. Inequal. Appl., 2012 (2012), 8 pages