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Some nonlinear weakly singular integral inequalities with two variables and applications. (English) Zbl 1204.26046

Summary: Some nonlinear weakly singular integral inequalities in two variables which generalize some known results are discussed. The results can be used as powerful tools in the analysis of certain classes of differential equations, integral equations, and evolution equations. An example is presented to show boundedness of solution of a differential equation here.

MSC:

26D15 Inequalities for sums, series and integrals
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References:

[1] Henry D: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics. Volume 840. Springer, Berlin, Germany; 1981:iv+348. · Zbl 0456.35001
[2] Sano H, Kunimatsu N: Modified Gronwall’s inequality and its application to stabilization problem for semilinear parabolic systems. Systems & Control Letters 1994, 22(2):145-156. 10.1016/0167-6911(94)90109-0 · Zbl 0792.93095 · doi:10.1016/0167-6911(94)90109-0
[3] Medveď M: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. Journal of Mathematical Analysis and Applications 1997, 214(2):349-366. 10.1006/jmaa.1997.5532 · Zbl 0893.26006 · doi:10.1006/jmaa.1997.5532
[4] Medveď M: Nonlinear singular integral inequalities for functions in two and n independent variables. Journal of Inequalities and Applications 2000, 5(3):287-308. 10.1155/S102558340000014X · Zbl 0985.26010 · doi:10.1155/S102558340000014X
[5] Dauer JP, Mahmudov NI: Integral inequalities and mild solutions of semilinear neutral evolution equations. Journal of Mathematical Analysis and Applications 2004, 300(1):189-202. 10.1016/j.jmaa.2004.06.040 · Zbl 1074.34059 · doi:10.1016/j.jmaa.2004.06.040
[6] Furati KM, Tatar N: Power-type estimates for a nonlinear fractional differential equation. Nonlinear Analysis: Theory, Methods & Applications 2005, 62(6):1025-1036. 10.1016/j.na.2005.04.010 · Zbl 1078.34028 · doi:10.1016/j.na.2005.04.010
[7] Medveď M: Integral inequalities and global solutions of semilinear evolution equations. Journal of Mathematical Analysis and Applications 2002, 267(2):643-650. 10.1006/jmaa.2001.7798 · Zbl 1028.34055 · doi:10.1006/jmaa.2001.7798
[8] Ma QH, Yang EH: Estimates on solutions of some weakly singular Volterra integral inequalities. Acta Mathematicae Applicatae Sinica 2002, 25(3):505-515. · Zbl 1032.26009
[9] Ma Q-H, Pečarić J: Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations. Journal of Mathematical Analysis and Applications 2008, 341(2):894-905. 10.1016/j.jmaa.2007.10.036 · Zbl 1142.26015 · doi:10.1016/j.jmaa.2007.10.036
[10] Cheung, W-S; Ma, Q-H; Tseng, S., Some new nonlinear weakly singular integral inequalities of Wendroff type with applications, No. 2008, 13 (2008) · Zbl 1151.26014
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