## Some new nonlinear inequalities and applications to boundary value problems.(English)Zbl 1094.26011

The author establishes some new retarded Gronwall-Bellman-Ou-Iang type inequalities in two variables with explicit bounds on unknown functions. These results on the one hand generalize several known inequalities and on the other hand furnich a handy tool for the study of qualitative as well as quantitative properties of solutions of differential and integral equations. The obtained new inequalities are applied to study the boundedness, uniqueness and continuous dependence of the solutions of certain initial value problems for hyperbolic partial differential equations.

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B35 Stability in context of PDEs
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### References:

 [1] Bainov, D.; Simeonov, P., Integral inequalities and applications, (1992), Kluwer Academic Publishers Dordrecht · Zbl 0759.26012 [2] Beckenbach, E.F.; Bellman, R., Inequalities, (1961), Springer New York · Zbl 0206.06802 [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 generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta math. acad. sci. hungar., 7, 71-94, (1956) · Zbl 0070.08201 [5] Cheung, W.S., On some new integrodifferential inequalities of the Gronwall and Wendroff type, J. math. anal. appl., 178, 438-449, (1993) · Zbl 0796.26007 [6] W.S. Cheung, Some retarded Gronwall-Bellman-Ou-Iang-type inequalities and applications to initial boundary value problems, preprint. [7] W.S. Cheung, Q.H. Ma, Nonlinear retarded integral inequalities for functions in two variables, J. Concrete Appl. Math., to appear. · Zbl 1091.26012 [8] Dafermos, C.M., The second law of thermodynamics and stability, Arch. rational mech. anal., 70, 167-179, (1979) · Zbl 0448.73004 [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] H. Haraux, Nonlinear Evolution Equation: Global Behavior of Solutions, Lecture Notes in Mathematics, vol. 841, Springer, Berlin, 1981. · Zbl 0461.35002 [11] Lipovan, O., A retarded Gronwall-like inequality and its applications, J. math. anal. appl., 252, 389-401, (2000) · Zbl 0974.26007 [12] Ma, Q.M.; Yang, E.H., On some new nonlinear delay integral inequalities, J. math. anal. appl., 252, 864-878, (2000) · Zbl 0974.26015 [13] Mitrinović, D.S., Analytic inequalities, (1970), Springer New York · Zbl 0199.38101 [14] Mitrinović, D.S.; Pečarić, J.E.; Fink, A.M., Inequalities involving functions and their integrals and derivatives, (1991), Kluwer Academic Publishers Dordrecht · Zbl 0744.26011 [15] Ou-Iang, 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] Pachpatte, B.G., Explicit bounds on certain integral inequalities, J. math. anal. appl., 267, 48-61, (2002) · Zbl 0996.26008
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