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On a generalized retarded integral inequality with two variables. (English) Zbl 1151.45010

Summary: This paper improves B. G. Pachpatte’s results on linear integral inequalities with two variables [JIPAM, J. Inequal. Pure Appl. Math. 3, No. 3, Paper No. 47, 10 p., electronic only (2002; Zbl 1006.26012)], and gives an estimation for a general form of nonlinear integral inequality with two variables. This paper does not require monotonicity of known functions. The result of this paper can be applied to discuss on boundedness and uniqueness for an integrodifferential equation.

MSC:

45K05 Integro-partial differential equations
26D15 Inequalities for sums, series and integrals
45G10 Other nonlinear integral equations

Citations:

Zbl 1006.26012
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References:

[1] Bellman R: The stability of solutions of linear differential equations.Duke Mathematical Journal 1943,10(4):643-647. 10.1215/S0012-7094-43-01059-2 · Zbl 0061.18502 · doi:10.1215/S0012-7094-43-01059-2
[2] Gronwall TH: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations.The Annals of Mathematics 1919,20(4):292-296. 10.2307/1967124 · JFM 47.0399.02 · doi:10.2307/1967124
[3] Agarwal RP, Deng S, Zhang W: Generalization of a retarded Gronwall-like inequality and its applications.Applied Mathematics and Computation 2005,165(3):599-612. 10.1016/j.amc.2004.04.067 · Zbl 1078.26010 · doi:10.1016/j.amc.2004.04.067
[4] Bihari I: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations.Acta Mathematica Hungarica 1956, 7: 81-94. 10.1007/BF02022967 · Zbl 0070.08201 · doi:10.1007/BF02022967
[5] Cheung W-S: Some new nonlinear inequalities and applications to boundary value problems.Nonlinear Analysis: Theory, Methods & Applications 2006,64(9):2112-2128. 10.1016/j.na.2005.08.009 · Zbl 1094.26011 · doi:10.1016/j.na.2005.08.009
[6] Dafermos CM: The second law of thermodynamics and stability.Archive for Rational Mechanics and Analysis 1979,70(2):167-179. · Zbl 0448.73004 · doi:10.1007/BF00250353
[7] Dannan FM: Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear differential equations.Journal of Mathematical Analysis and Applications 1985,108(1):151-164. 10.1016/0022-247X(85)90014-9 · Zbl 0586.26008 · doi:10.1016/0022-247X(85)90014-9
[8] Medina R, Pinto M: On the asymptotic behavior of solutions of certain second order nonlinear differential equations.Journal of Mathematical Analysis and Applications 1988,135(2):399-405. 10.1016/0022-247X(88)90163-1 · Zbl 0668.34056 · doi:10.1016/0022-247X(88)90163-1
[9] Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and Its Applications. Volume 53. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xvi+587. · Zbl 0744.26011 · doi:10.1007/978-94-011-3562-7
[10] Pachpatte BG: Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering. Volume 197. Academic Press, San Diego, Calif, USA; 1998:x+611. · Zbl 1032.26008
[11] Pachpatte, BG, Bounds on certain integral inequalities (2002) · Zbl 1006.26012
[12] Wang, W-S, A generalized sum-difference inequality and applications to partial difference equations, No. 2008 (2008)
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