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Some new nonlinear integral inequalities and their applications in the qualitative analysis of differential equations
Journal of Inequalities and Applications volume 2011, Article number: 20 (2011)
Abstract
In this paper, some new nonlinear integral inequalities are established, which provide a handy tool for analyzing the global existence and boundedness of solutions of differential and integral equations. The established results generalize the main results in Sun (J. Math. Anal. Appl. 301, 265-275, 2005), Ferreira and Torres (Appl. Math. Lett. 22, 876-881, 2009), Xu and Sun (Appl. Math. Comput. 182, 1260-1266, 2006) and Li et al. (J. Math. Anal. Appl. 372, 339-349 2010).
MSC 2010: 26D15; 26D10
1 Introduction
During the past decades, with the development of the theory of differential and integral equations, a lot of integral inequalities, for example [1–12], have been discovered, which play an important role in the research of boundedness, global existence, stability of solutions of differential and integral equations.
In [9], the following two theorems for retarded integral inequalities were established.
Theorem A: R+ = [0, ∞). Let u, f, g be nondecreasing continuous functions defined on R+ and let c be a nonnegative constant. Moreover, let ω ∈ C(R+, R+) be nondecreasing with ω(u) > 0 on (0, ∞) and α ∈ C1(R+, R+) be nondecreasing with α(t) ≤ t on R+. m, n are constants, and m > n > 0. If
then for t ∈ [0, ξ]
where , r > 0,Ω - 1 is the inverse of Ω, Ω (∞) = ∞, and ξ ∈ R+ is chosen so that .
Theorem B: Under the hypothesis of Theorem B, if
then for t ∈ [0, ξ]
Recently, in [10], the author provided a more general result.
Theorem C: , R+ = (0, ∞). Let f(t, s) and be nondecreasing in t for every s fixed. Moreover, let be a strictly increasing function such that and suppose that is a nondecreasing function. Further, let be nondecreasing with {η, ω}(x) > 0 for x ∈ (0, ∞) and , with x0 defined as below. Finally, assume that is nondecreasing with α(t) ≤ t. If satisfies
then there exists τ ∈ R+ so that for all t ∈ [0, τ] we have
and
where
with x ≥ c(0) > x0 > 0 if and x ≥ c(0) > x0 ≥ 0 if
Here G -1 and ψ -1 are inverse functions of G and ψ, respectively.
In [11], Xu presented the following two theorems:
Theorem D: R+ = [0, ∞). Let u, f, g be real-valued nonnegative continuous functions defined for x ≥ 0, y ≥ 0 and let c be a nonnegative constant. Moreover, let ω ∈ C(R+, R+) be nondecreasing with ω(u) > 0 on (0, ∞) and α, β, ∈ C1(R+, R+) be nondecreasing with α(x) ≤ x, β(y) ≤ y on R+. m, n are constants, and m > n > 0. If
then for x ∈ [0, ξ], y ∈ [0, η]
where
Ω is defined as in Theorem A, and ξ, η are chosen so that
Theorem E: Under the hypothesis of Theorem D, if
then
where
In this paper, motivated by the above work, we will prove more general theorems and establish some new integral inequalities. Also we will give some examples so as to illustrate the validity of the present integral inequalities.
2 Main results
In the rest of the paper we denote the set of real numbers as R, and R+ = [0, ∞) is a subset of R. Dom(f) and Im(f) denote the definition domain and the image of f, respectively.
Theorem 2.1: Assume that x, a ∈ C(R+, R+) and a(t) is nondecreasing. f i , g i , h i , ∂ t f i , ∂ t g i , ∂ t h i ∈ C(R+ × R+, R+), i = 1, 2. Let ω ∈ C(R+, R+) be nondecreasing with ω(u) > 0 on (0, ∞). p, q are constants, and p > q > 0. If α ∈ C1(R+, R+) is nondecreasing with α(t) ≤ t on R+, and
then there exists such that for
where
, r > 0. Ω -1 is the inverse of Ω, and Ω (∞) = ∞.
Proof: The proof for the existence of can be referred to Remark 1 in [10]. We notice (3) obviously holds for t = 0. Now given an arbitrary number , for t ∈ (0, T], we have
Let the right-hand side of (4) be z(t), then xp(t) ≤ z(t) and xp(α(t)) ≤ z(α (t)) ≤ z(t). So
Then
An integration for (5) from 0 to t, considering z(0) = a(T), yields
Then
Let the right-hand side of (7) be y(t). Then we have , , and
that is
Integrating (9) from 0 to t, considering y(0) = H(T), it follows
So
Taking t = T in (11), then
Considering is arbitrary, substituting T with t, and then the proof is complete.
Remark 1 : We note that the right-hand side of (2) is well defined since Ω (∞) = ∞.
Remark 2 : If we take p = 2, q = 1, ω(u) = u, h1(s, t) = h2(s, t) ≡ 0 or p = 2, q = 1, h1(s, t) = h2(s, t) ≡ 0, respectively, then our Theorem 2.1 reduces to [12, Theorems 2.1, 2.2].
Corollary 2.1: Assume that x, a, α, ω, Ω are defined as in Theorem 2.1. f i , g i , h i ∈ C(R+, R+), m i , n i , l i ∈ C1(R+, R+), i = 1, 2. If
then we can find some such that for
where
Remark 3: If , m1(t) = n1(t) ≡ 1, l1(t) ≡ 0, m2(t) = n2(t) = l2(t) ≡ 0 for t ∈ R+, then Corollary 1 reduces to Theorem A [9, Theorem 2.1]. If , m1(t) ≡ 1, g1(t) ≡ 0, l1(t) ≡ 0, m2(t) ≡ 1, n2(t) = l2(t) ≡ 0 for t ∈ R+, then Corollary 2.1 reduces to Theorem B [9, Theorem 2.2].
Corollary 2:2: Assume that x, a, α, ω, Ω are defined as in Theorem 2.1. f, g, h, ∂ t f, ∂ t g, ∂ t h ∈ C(R+ × R+, R+). If
then for
where
Corollary 2:3: Assume that x, a, α, ω, Ω are defined as in Theorem 2.1. f, g, h ∈ C(R+, R+), m, n, l ∈ C1(R+, R+). If
then for
where
Motivated by Corollary 2.2 and Theorem C [10], we will give the following more general theorem:
Theorem 2:2: Assume that f(s, t), g(s, t), h(s, t) ∈ C(R+ × R+, R+) are nondecreasing in t for each s fixed, and ϕ ∈ C(R+, R+) is a strictly increasing function with . ψ, ω ∈ C(R+, R+) are nondecreasing with ψ (x) > 0, ω(x) > 0 for x ∈ (0, ∞) and , a(t), α(t) are defined as in Theorem 2.1, and a(0) > t0 > 0. If x ∈ C(R+, R+) satisfies the following integral inequality containing multiple integrals
then we can find some such that for
and
where
Proof: The proof for the existence of can be referred to Remark 1 in [10]. We notice (22) obviously holds for t = 0. Now given an arbitrary number T > 0, . Define
Then for t ∈ (0, T],
and
So
Integrating (27) from 0 to t, considering J is increasing, we can obtain
Define , then
and
that is,
Integrating (31) from 0 to t, considering and Y is increasing, it follows
Combining (25), (29) and (32) we have
Taking t = T in (33), it follows
Considering is arbitrary, substituting T with t we have completed the proof.
Remark 4: If h(s, t) ≡ 0, α1(t) = α2(t) = α(t), then Theorem 2.2 becomes Theorem C [10, Theorem 1].
Now we will apply the concept of establishing Theorem 2.2 to the situation with two independent variables.
Theorem 2:3: Assume that f i (x, y), g i (x, y), h i (x, y) ∈ C(R+ × R+, R+), i = 1, 2, and ϕ ∈ C(R+, R+) is a strictly increasing function with . a(x, y) ∈ C(R+ × R+, R+) is nondecreasing in x for every fixed y and nondecreasing in y for every fixed x. α(x), β(y) ∈ C1(R+, R+) are nondecreasing with α(x) ≤ x, β(y) ≤ y. ψ, ω ∈ C(R+, R+) are nondecreasing with ψ(x) > 0, ω(x) > 0 for x ∈ (0, ∞) and , where 0 < t0 < a(0, 0).
If u ∈ C(R+ × R+, R+) satisfies the following integral inequality containing multiple integrals
then we can find some , so that for all ,
and
where J, Y are defined as in Theorem 2.2, and
Proof: The process for seeking for , can also be referred to Remark 1 in [10].
If we take x = 0 or y = 0, then (35) holds trivially. Now fix ,, and x ∈ (0, x0], y ∈ (0, y0]. Let
Considering a(x, y) is nondecreasing, we have u(x, y) ≤ ϕ -1(z(x, y)) ≤ ϕ -1(z(x0, y)). Moreover,
So
Integrating (38) from 0 to y we have
Let
Then
Furthermore let
Then
and
that is,
Integrating (42) from 0 to y, considering we have
Then
and
Take x = x0, y = y0 and we have
Since , are arbitrary, substitute x0, y0 with x, y and the proof is complete.
Corollary 2.4: Assume that f(x, y), g(x, y), h(x, y) ∈ C(R+ × R+, R+), and a, ϕ, ψ, ω, α, β, J, Y are defined as in Theorem 2.3. If u ∈ C(R+ × R+, R+) satisfies the following integral inequality containing multiple integrals
then we can find some , such that for all ,
and
where
Remark 5: If we take h(x, y) ≡ 0, ψ (u(x, y)) = un(x, y), ϕ(x, y) = um(x, y), m > n > 0, then Corollary 2.4 reduces to Theorem D [11, Theorem 2.1].
Corollary 2.5: Assume that f i , g i (x, y) ∈ C(R+ × R+, R+), i = 1, 2, and a, ϕ, ψ, ω, J, Y are defined as in Theorem 2.3. If u ∈ C(R+ × R+, R+) satisfies the following integral inequality containing multiple integrals
then we can find some , such that for all ,
and
where
Remark 6: If we take f1(x, y) = f2(x, y) ≡ 0, ψ(u(x, y)) = un(x, y), ϕ(x, y) = um(x, y), m > n, then Corollary 8 reduces to Theorem E [11, Theorem 2.2].
3 Applications
In this section, we will present two examples in order to illustrate the validity of the above results. In the first example, we will try to prove the global existence of the solutions of a delay differential equation, while in the second example, we will obtain the bound of the solutions of an integral equation.
For the sake of proving the global existence of solutions of differential equations, we first recall some basic facts. Consider the following equation
with X0 ∈ Rn, H ∈ C(R+ × R2n, Rn), α ∈ C1(R+, R+) satisfying α(t) ≤ t for t ≥ 0. A result in [13] guarantees that for every X0 ∈ Rn, Equation 45 has a solution, but the uniqueness of solutions cannot be guaranteed. Furthermore, every solution of (45) has a maximal time of existence T > 0, and if T < ∞, then .
Example 1: Consider the following differential equation group
where p is an even number. α(t) is a nondecreasing function, α(t) ∈ C1(R+, R+), α(t) ≤ t, ∀t ≥ 0. p > q > 0. Assume , , , where , , and v is nondecreasing.
Let up(t) = xp(t) + yp(t), then
If (x(t), y(t)) is a solution of (46) defined on the maximal existence interval [0, T), integrating (47) from 0 to t, we have
Then
where . From Theorem 2.1 we have
Obviously we have {|x(t)|, |y(t)|} ≤ |u(t)|. So x(t), y(t) do not blow up in finite time. Then T = ∞, and the solutions of (46) are global.
Example 2: Considering the following integral equation
where u ∈ C(R+ × R+, R+), |F (x, y, u(x, y))| ≤ f(x, y)u(x, y),
|G(x, y, u(x, y))| ≤ g(x, y)u2(x, y), f, g ∈ C(R+ × R+, R+), a(x, y), α(x), β(y) are defined as in Theorem 2.3.
Let ϕ(u) = uln(u + 1), ω (u) = u, η(u) = u. Then one can easily see the conditions of Theorem 2.3 are satisfied. So we can obtain the bound of u(x, y) as
where , are determined similar to the process in Theorem 2.3, and
Remark 7: we note that the methods in [1–12] are not available for the estimate of bound for the solution of Equation 48.
5 Authors'contributions
BZ carried out the main part of this article. Both of the two authors read and approved the final manuscript.
References
Li WN, Han MA, Meng FW: Some new delay integral inequalities and their applications. J Comput Appl Math 2005, 180: 191–200. 10.1016/j.cam.2004.10.011
Lipovan O: A retarded integral inequality and its applications. J Math Anal Appl 2003, 285: 436–443. 10.1016/S0022-247X(03)00409-8
Ma QH, Yang EH: Some new Gronwall-Bellman-Bihari type integral inequalities with delay. Period Math Hungar 2002,44(2):225–238. 10.1023/A:1019600715281
Yuan ZL, Yuan XW, Meng FW: Some new delay integral inequalities and their applications. Appl Math Comput 2009, 208: 231–237. 10.1016/j.amc.2008.11.043
Lipovan O: Integral inequalities for retarded Volterra equations. J Math Anal Appl 2006, 322: 349–358. 10.1016/j.jmaa.2005.08.097
Pachpatte BG: Explicit bounds on certain integral inequalities. J Math Anal Appl 2002, 267: 48–61. 10.1006/jmaa.2001.7743
Pachpatte BG: A note on certain integral inequalities with delay. Period Math Hungar 1995, 31: 234–299.
Pachpatte BG: On some new nonlinear retarded integral inequalities. J Inequal Pure Appl Math 2004, 5: 1–8. (Article 80)
Sun YG: On retarded integral inequalities and their applications. J Math Anal Appl 2005, 301: 265–275. 10.1016/j.jmaa.2004.07.020
Ferreira RAC, Torres DFM: Generalized retarded integral inequalities. Appl Math Lett 2009, 22: 876–881. 10.1016/j.aml.2008.08.022
Xu R, Sun YG: On retarded integral inequalities in two independent variables and their applications. Appl Math Comput 2006, 182: 1260–1266. 10.1016/j.amc.2006.05.013
Li LZ, Meng FW, He LL: Some generalized integral inequalities and their applications. J Math Anal Appl 2010, 372: 339–349. 10.1016/j.jmaa.2010.06.042
Driver R: Existence and continuous dependence of solutions of neutral functional differential equations. Arch Ration Mech Anal 1965, 19: 149–166. 10.1007/BF00282279
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The authors thank the referees very much for their valuable suggestions on this paper.
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The authors declare that they have no competing interests.
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Zheng, B., Feng, Q. Some new nonlinear integral inequalities and their applications in the qualitative analysis of differential equations. J Inequal Appl 2011, 20 (2011). https://doi.org/10.1186/1029-242X-2011-20
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DOI: https://doi.org/10.1186/1029-242X-2011-20