- Open Access
Some retarded nonlinear integral inequalities and their applications in retarded differential equations
© Wang; licensee Springer. 2012
- Received: 4 February 2012
- Accepted: 29 March 2012
- Published: 29 March 2012
In this article, we discuss some generalized retarded nonlinear integral inequalities, which not only include nonlinear compound function of unknown function but also include retard items, and give upper bound estimation of the unknown function by integral inequality technique. This estimation can be used as tool in the study of differential equations with the initial conditions.
2000 MSC: 26D10; 26D15; 26D20; 34A12; 34A40.
- integral inequality
- integral inequality technique
- Retarded differential equation
Gronwall-Bellman inequalities [1, 2] and their various generalizations can be used important tools in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential equations, integral equations, and integral-differential equations.
Then, 0 ≤ u(t) ≤ ah exp(bh), ∀t ∈ [a,a + h].
implies that .
such that .
for all t ∈ I.
In this article, we discuss some retarded nonlinear integral inequalities, where linear case u(t) in integral functions in  is changed into the nonlinear case ϕ(u(t)), and the non-retarded case t in  is changed into retarded case α(t), and give upper bound estimation of the unknown function by integral inequality technique.
In this section, we discuss some retarded integral inequalities of Gronwall-Bellman type. Throughout this article, let I = [0, ∞).
Remark 1. If α(t) = t, φ(u(s)) = u(s), then Theorem 1 reduces Lemma 4.
for all t ∈ I. This completes the proof of the Theorem 1.
Remark 2. If α(t) = t,φ(u(t)) = u p (t),ψ(u(t)) = up+1(t)/(p + 1), by Theorem 2, we can obtain the result similar to Lemma 5.
Using the relation u(t) ≤ z(t) ≤ w(t), we can obtain the estimation (2.17) of (2.16).
where f(t) is nonnegative real-valued continuous function defined on I.
Applying Theorem 1 to (3.33), we get the estimation (3.29). This completes the proof of the Corollary 1.
The authors are very grateful to the editor and the referees for their helpful comments and valuable suggestions. This research was supported by National Natural Science Foundation of China (Project No. 11161018), Guangxi Natural Science Foundation (Project No. 0991265), Scientific Research Foundation of the Education Department of Guangxi Province of China (Project No. 201106LX599), and the Key Discipline of Applied Mathematics of Hechi University of China (200725).
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