×

A generalized retarded Gronwall-like inequality in two variables and applications to BVP. (English) Zbl 1193.26014

Summary: We establish a generalized retarded integral inequality of Gronwall-like type in two variables, which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals without assumption of monotonicity. Using our result we can solve both the integral inequality in [W.S. Cheung, Nonlinear Anal., Theory Methods Appl. 64, No. 9 (A), 2112–2128 (2006; Zbl 1094.26011)] and the one in [R.P. Agarwal, S. Deng, W. Zhang, Appl. Math. Comput. 165, No. 3, 599–612 (2005; Zbl 1078.26010)]. We apply our result to a boundary value problem of a partial differential equation for boundedness, uniqueness and continuous dependence.

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
35G30 Boundary value problems for nonlinear higher-order PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agarwal, R.P.; Deng, S.; Zhang, W., Generalization of a retarded Gronwall-like inequality and its applications, Appl. math. comput., 165, 599-612, (2005) · Zbl 1078.26010
[2] Bainov, D.; Simeonov, P., Integral inequalities and applications, (1992), Kluwer Academic Dordrecht · Zbl 0759.26012
[3] Bellman, R., The stability of solutions of linear differential equations, Duke math. J., 10, 643-647, (1943) · Zbl 0061.18502
[4] Bihari, I.A., A generalization of a lemma of Bellman and its application to uniqueness problem of differential equation, Acta math. acad. sci. hung., 7, 81-94, (1956) · Zbl 0070.08201
[5] Cheung, W.S., Some new nonlinear inequalities and applications to boundary value problems, Nonlinear anal., 64, 2112-2128, (2006) · Zbl 1094.26011
[6] Dafermos, C.M., The second law of thermodynamics and stability, Arch. rat. mech. anal., 70, 167-179, (1979) · Zbl 0448.73004
[7] Dannan, F., Integral inequalities of gronwall – bellman – bihari type and asymptotic behavior of certain second order nonlinear differential equations, J. math. anal. appl., 108, 151-164, (1985) · Zbl 0586.26008
[8] Dragomir, S.S.; Kim, Y.H., Some integral inequalities for functions of two variables, Electr. J. diff. eqns., 2003, 10, 1C13, (2003)
[9] Gronwall, T.H., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of math., 20, 292-296, (1919) · JFM 47.0399.02
[10] Kim, Y.H., On some new integral inequalities for functions in one and two variables, Acta math. sinica, 21, 423-434, (2005) · Zbl 1084.26013
[11] Lipovan, O., A retarded Gronwall-like inequality and its applications, J. math. anal. appl., 252, 389-401, (2000) · Zbl 0974.26007
[12] Massalitina, E.V., On the perow integro-summable inequality for functions of two variables, Ukrainian math. J., 56, 1864-1872, (2004)
[13] Medina, R.; Pinto, M., On the asymptotic behavior of solutions of a class of second order nonlinear differential equations, J. math. anal. appl., 135, 399-405, (1988) · Zbl 0668.34056
[14] Mitrinović, D.S.; Pečarić, J.E.; Fink, A.M., Inequalities involving functions and their integrals and derivatives, (1991), Kluwer Academic Dordrecht · Zbl 0744.26011
[15] Ou-Yang, L., The boundedness of solutions of linear differential equations \(y'' + A(t) y = 0\), Shuxue jinzhan, 3, 409-415, (1957)
[16] Pachpatte, B.G., On some new inequalities related to certain inequalities in the theory of differential equations, J. math. anal. appl., 189, 128-144, (1995) · Zbl 0824.26010
[17] Pachpatte, B.G., Inequalities for differential and integral equations, (1998), Academic Press New York · Zbl 1032.26008
[18] Pinto, M., Integral inequalities of bihari-type and applications, Funkcial. ekvac., 33, 387-430, (1990) · Zbl 0717.45004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.