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Some new nonlinear integral inequalities and their applications in the qualitative analysis of differential equations

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 [112], 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 xc(0) > x0 > 0 if and xc(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

(1)

then there exists such that for

(2)

where

(3)

, 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

(4)

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

(5)

An integration for (5) from 0 to t, considering z(0) = a(T), yields

(6)

Then

(7)

Let the right-hand side of (7) be y(t). Then we have , , and

(8)

that is

(9)

Integrating (9) from 0 to t, considering y(0) = H(T), it follows

(10)

So

(11)

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

(12)

then we can find some such that for

(13)

where

(14)

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

(15)

then for

(16)

where

(17)

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

(18)

then for

(19)

where

(20)

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

(21)

then we can find some such that for

and

(22)

where

(23)
(24)

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],

(25)

and

(26)

So

(27)

Integrating (27) from 0 to t, considering J is increasing, we can obtain

(28)

Define , then

(29)

and

(30)

that is,

(31)

Integrating (31) from 0 to t, considering and Y is increasing, it follows

(32)

Combining (25), (29) and (32) we have

(33)

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

(34)

then we can find some , so that for all ,

and

(35)

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

(36)

Considering a(x, y) is nondecreasing, we have u(x, y) ≤ ϕ -1(z(x, y)) ≤ ϕ -1(z(x0, y)). Moreover,

(37)

So

(38)

Integrating (38) from 0 to y we have

(39)

Let

(40)

Then

Furthermore let

Then

(41)

and

that is,

(42)

Integrating (42) from 0 to y, considering we have

Then

and

(43)

Take x = x0, y = y0 and we have

(44)

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

(45)

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

(46)

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

(47)

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

(48)

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

(49)

where , are determined similar to the process in Theorem 2.3, and

Remark 7: we note that the methods in [112] 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.

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Acknowledgements

The authors thank the referees very much for their valuable suggestions on this paper.

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Correspondence to Bin Zheng.

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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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Keywords

  • integral inequality
  • global existence
  • integral equation
  • differential equation
  • bounded