×

A generalized Gronwall inequality and its application to a fractional differential equation. (English) Zbl 1120.26003

The authors prove a statement about a Gronwall type inequality in the following form. Let \(\beta >0, a(t)\) be a nonnegative function locally integrable on \([0,T), \;T\leq\infty\), and \(g(t)\) be a nonnegative nondecreasing bounded continuous function on \([0,T)\). Suppose that \(u(t)\) is nonnegative and locally integrable on \([0,T)\) and satisfies the inequality \[ u(t)\leq a(t)+g(t)\int\limits_0^t(t-s)^{\beta-1}u(s)ds \] on this interval. Then \[ u(t)\leq a(t)+\int\limits_0^t\left[\sum\limits_{n=1}^\infty \frac{(g(t)\Gamma(\beta))^n}{\Gamma(n\beta)}(t-s)^{n\beta-1} a(s)\right]ds. \] An application of this statement is given to a fractional differential equation \(D^\alpha y(t)= f(t,y(t))\) with the Cauchy type condition \(\left. D^{\alpha-1}y(t)\right|_{t=0}=\eta\), where \(0<\alpha<1\).

MSC:

26A33 Fractional derivatives and integrals
26D15 Inequalities for sums, series and integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pachpatte, B. G., Inequalities for Differential and Integral Equations (1998), Academic Press: Academic Press New York · Zbl 1032.26008
[2] Pachpatte, B. G., On some generalizations of Bellman’s lemma, J. Math. Anal. Appl., 5, 141-150 (1975) · Zbl 0305.26010
[3] Lipovan, O., A retarded Gronwall-like inequality and its applications, J. Math. Anal. Appl., 252, 389-401 (2000) · Zbl 0974.26007
[4] 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
[5] Henry, D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840 (1981), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0456.35001
[6] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[7] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., 204, 609-625 (1996) · Zbl 0881.34005
[8] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003
[9] Heymans, N.; Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta, 37, 1-7 (2005)
[10] Corduneanu, C., Principles of Differential and Integral Equations (1971), Allyn and Bacon: Allyn and Bacon Boston · Zbl 0208.10701
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.