In this section, we discuss some retarded integral inequalities of Gronwall-Bellman type. Throughout this article, let I = [0, ∞).
Theorem 1. Let φ,φ',α ∈ C1(I, I) be increasing functions with φ'(t) ≤ k, α(t) ≤ t, α(0) = 0, ∀t ∈ I; k, u0 be positive constants, we assume that u(t) and f(t) are nonnegative real-valued continuous functions defined on I and satisfy the inequality
(2.1)
for all t ∈ I. If , then
(2.2)
where
(2.3)
(2.4)
Remark 1. If α(t) = t, φ(u(s)) = u(s), then Theorem 1 reduces Lemma 4.
Proof. Let z(t) denotes the function on the right-hand side of (2.1), which is a nonnegative and nondecreasing function on I with z(0) = u0. Then (2.1) is equivalent to
(2.5)
Differentiating z(t) with respect to t, we have
(2.6)
Using (2.5), we obtain
(2.7)
where , w is a nonnegative and nondecreasing function on I. By the monotonicity φ, φ',z, and α(t) ≤ t we have φ(z(α(t))) ≤ w(t), φ'(z(α(t))) ≤ k. Differentiating w(t) with respect to t, and using (2.7) we have
(2.8)
By w(t) > 0, we have
(2.9)
Let v(t) = w-1(t), from (2.9) we have
(2.10)
Consider ordinary differential equation
(2.11)
The solution of Equation (2.11) is
(2.12)
By (2.10), (2.11), and (2.13), we obtain
(2.13)
By the definition of B3(t) in (2.4) and the inequality (2.13), we have w(t) < B3(t),∀t ∈ I. From (2.7), we get
(2.14)
By taking t = s in the inequality (2.14) and integrating (2.14) from 0 to t, by the definition (2.3) of Φ we obtain
(2.15)
for all t ∈ I. This completes the proof of the Theorem 1.
Theorem 2. Let ψ(t),φ(t),φ(t)/t,α(t) ∈ C1(I,I) be increasing functions with ψ'(t) = φ(t),α(t) ≤ t, α(0) = 0, ∀t ∈ I; k, u0 be positive constants, we assume that u(t) and f(t) are nonnegative real-valued continuous functions defined on I and satisfy the inequality
(2.16)
for all t ∈ I. Then
(2.17)
where
(2.18)
Ξ-1,ψ-1 are the reverse function of Ξ, ψ respectively, T2 is the largest number such that
Remark 2. If α(t) = t,φ(u(t)) = up(t),ψ(u(t)) = up+1(t)/(p + 1), by Theorem 2, we can obtain the result similar to Lemma 5.
Proof. Let ψ(z(t)) denotes the function on the right-hand side of (2.16), then z(t) is a nonnegative and nondecreasing function on I with z(0) = ψ-1(u0). Then (2.16) is equivalent to
(2.19)
Differentiating ψ(z(t)) with respect to t, we have
(2.20)
Using (2.19) and the relation ψ'(z(t)) = φ(z(t)), from (2.20) we obtain
(2.21)
Let , then w(0) = z(0) = ψ-1(u0), z(t) ≤ w(t), w is a nonnegative and nondecreasing function on I. Differentiating w(t) with respect to t, and using (2.21) we have
(2.22)
By w(t) > 0, we have
(2.23)
Integrating (2.23) from 0 to t, we have
(2.24)
for all t ∈ I. Using Lemma 6 and the Definition (2.18) of Ξ, we obtain
(2.25)
Using the relation u(t) ≤ z(t) ≤ w(t), we can obtain the estimation (2.17) of (2.16).