A retarded Gronwall-like inequality and its applications. (English) Zbl 0974.26007

Lipovan, Olivia

J. Math. Anal. Appl. 252, No. 1, 389-401 (2000).

In the present paper a retarded Gronwall-like inequality is proved and some applications are given to show its usefulness. The main result given here can be stated as follows.
Let \(u,f\in C([t_0,T), R_+)\). Moreover, let \(w\in C(R_+, R_+)\) be nondecreasing with \(w(u)> 0\) on \((0,\infty)\) and \(\alpha\in C^1([t_0, T),[t_0,T))\) be nondecreasing with \(\alpha(t)\leq t\) on \([t_0, T)\). If \[ u(t)\leq k+ \int^{\alpha(t)}_{\alpha(t_0)} f(s) w(u(s)) ds,\quad t_0\leq t< T, \] where \(k\) is a nonnegative constant, then, for \(t_0\leq t< t_1\), \[ u(t)\leq G^{-1}\Biggl(G(k)+ \int^{\alpha(t)}_{\alpha(t_0)} f(s) ds\Biggr), \] where \(G(r)= \int^r_1{ds\over w(s)}\), \(r>0\), and \(t_1\in (t_0, T)\) is chosen so that \[ G(k)+ \int^{\alpha(t)}_{\alpha(t_0)} f(s) ds\in \text{Dom}(G^{-1}), \] for all \(t\) lying in the interval \([t_0,t_1)\).

Reviewer: B.G.Pachpatte (Aurangabad)

MSC:

26D10

Inequalities involving derivatives and differential and integral operators

Keywords:

retarded Gronwall-like inequality

References:

[1]	Bellman, R., The stability of solutions of linear differential equations, Duke Math. J., 10, 643-647 (1943) · Zbl 0061.18502
[2]	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
[3]	Constantin, A., Global existence of solutions for perturbed differential equations, Ann. Mat. Pura Appl., 168, 237-299 (1995)
[4]	Corduneanu, C., Integral Equations and Applications (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0714.45002
[5]	Driver, R., Existence and continuous dependence of solutions of neutral functional differential equations, Arch. Rational Mech. Anal., 19, 149-166 (1965) · Zbl 0148.05703
[6]	Gronwall, T., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 20, 292-296 (1919) · JFM 47.0399.02
[7]	Györi, I.; Stavroulakis, I., Positive solutions of functional differential equations, Boll. Un. Mat. Ital. B, 3, 185-198 (1989) · Zbl 0683.34035
[8]	Hale, J., Theory of Functional Differential Equations (1977), Springer-Verlag: Springer-Verlag New York
[9]	Hara, T.; Yoneyama, T.; Sugie, J., Continuability of solutions of perturbed differential equations, Nonlinear Anal., 8, 963-975 (1984) · Zbl 0575.34046
[10]	Iserles, A., On the generalised pantograph functional differential equation, European J. Appl. Math., 4, 1-38 (1993) · Zbl 0767.34054
[11]	Pachpatte, B. G., Inequalities for Differential and Integral Equations (1998), Academic Press: Academic Press New York · Zbl 1032.26008
[12]	Philos, Ch, Oscillations of first order linear retarded differential equations, J. Math. Anal. Appl., 157, 17-33 (1991) · Zbl 0731.34080
[13]	Sengadir, T., Existence and stability of nonlinear functional differential equations, J. Math. Anal. Appl., 197, 890-907 (1996) · Zbl 0853.34060
[14]	Sugie, J., Continuability of solutions of the generalized Liénard system with time lag, Proc. Japan Acad., 60, 357-360 (1984) · Zbl 0567.34061
[15]	Yan, J.; Zhao, A., Oscillation and stability of linear impulsive delay differential equations, J. Math. Anal. Appl., 227, 187-194 (1998) · Zbl 0917.34060