Abstract
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.
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1 Introduction
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.
Lemma 1 (Gronwall [1]). Let u(t) be a continuous function defined on the interval [a, a + h], a, h are nonnegative constants and
Then, 0 ≤ u(t) ≤ ah exp(bh), ∀t ∈ [a,a + h].
Lemma 2 (Bellman [2]). Let f, u ∈ C([0, h], [0, ∞)), h, c are positive constants. If u satisfy the inequality
Then, .
Lemma 3 (Lipovan [3]). Let u, f ∈ C([t0,T), R+). Further, let α ∈ C1([t0,T),[t0,T)) be nondecreasing with α(t) ≤ t on [t0,T), and let c be a nonnegative constant. Then the inequality
implies that .
Lemma 4 (Abdeldaim and Yakout [4]). We assume that u(t) and f(t) are nonnegative real-valued continuous functions defined on I and satisfy the inequality
where u 0 be a positive constant. Then
where
such that .
Lemma 5 (Abdeldaim and Yakout [4]). We assume that u(t) and f(t) are nonnegative real-valued continuous functions defined on I and satisfy the inequality
for all t ∈ I, where u0 > 0, p ∈ (0,1), are constants. Then
where
for all t ∈ I.
Lemma 6 (see [5]). Let φ ∈ C(R+,R+) be a increasing function, u,a,f ∈ C([t0,T),R+), a(t) be a increasing function, and α ∈ C1([t0,T), [t0,T)) be nondecreasing with α(t) ≤ t on [t0,T) where T ∈ (0,∞) is a constant. Then the inequality
implies that
where
W -1 is the reverse function of W, T 1 is the largest number such that
There can be found a lot of generalizations of Gronwall-Bellman inequalities in various cases from literature (e.g., [3–13]).
In this article, we discuss some retarded nonlinear integral inequalities, where linear case u(t) in integral functions in [4] is changed into the nonlinear case ϕ(u(t)), and the non-retarded case t in [4] is changed into retarded case α(t), and give upper bound estimation of the unknown function by integral inequality technique.
2 Main result
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
for all t ∈ I. If , then
where
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
Differentiating z(t) with respect to t, we have
Using (2.5), we obtain
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
By w(t) > 0, we have
Let v(t) = w-1(t), from (2.9) we have
Consider ordinary differential equation
The solution of Equation (2.11) is
By (2.10), (2.11), and (2.13), we obtain
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
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
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
for all t ∈ I. Then
where
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
Differentiating ψ(z(t)) with respect to t, we have
Using (2.19) and the relation ψ'(z(t)) = φ(z(t)), from (2.20) we obtain
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
By w(t) > 0, we have
Integrating (2.23) from 0 to t, we have
for all t ∈ I. Using Lemma 6 and the Definition (2.18) of Ξ, we obtain
Using the relation u(t) ≤ z(t) ≤ w(t), we can obtain the estimation (2.17) of (2.16).
3 Application
In this section, we apply our result to the following nonlinear differential equation [4]
where x0 is a constant, F, K ∈ C(I × I, R), H ∈ C(I3, R), satisfy the following conditions
where f(t) is nonnegative real-valued continuous function defined on I.
Corollary 1. Consider nonlinear system (3.26) and suppose that F,K, H satisfy the conditions (3.27) and (3.28). Let φ,φ', α ∈ C1(I, I) be increasing functions with φ'(t) ≤ k,α(t) ≤ t, α(0) = 0,∀t ∈ I, k be positive constants; then all solutions of Equation (3.26) exist on I and satisfy the following estimation
where
Proof. Integrating both sides of the Equation (3.26) from 0 to t, we get
From (3.27), (3.28), and (3.32) we obtain
Applying Theorem 1 to (3.33), we get the estimation (3.29). This completes the proof of the Corollary 1.
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Acknowledgements
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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Wang, WS. Some retarded nonlinear integral inequalities and their applications in retarded differential equations. J Inequal Appl 2012, 75 (2012). https://doi.org/10.1186/1029-242X-2012-75
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DOI: https://doi.org/10.1186/1029-242X-2012-75