BOUNDEDNESS IN NONLINEAR FUNCTIONAL DIFFERENTIAL SYSTEMS VIA t_∞-SIMILARITY

Abstract

In this paper, we investigate bounds for solutions of nonlinear functional differential systems using the notion of t_∞-similarity.

1. INTRODUCTION

Integral inequalities play a vital role in the study of boundedness and other qualitative properties of solutions of differential equations. In particular, Bihari’s integral inequality is continuous to be an effective tool to study sophisticated problems such as stability, boundedness, and uniqueness of solutions. The behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. There are two useful methods for showing the qualitative behavior of the solutions of perturbed nonlinear system : the use of integral inequalities, the method of variation of constants formula.

The notion of h-stability (hS) was introduced by Pinto [13,14] with the intention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential asymptotic stability) under some perturbations. He obtained a general variational h-stability and some properties about asymptotic behavior of solutions of differential systems called h-systems. Choi and Ryu [3], and Choi et al. [4] investigated h-stability and bounds for solutions of the perturbed functional differential systems. Also, Goo [6,7,8] and Goo et al. [9] studied boundedness of solutions for the perturbed functional differential systems.

The aim of this paper is to obtain boundedness for solutions of nonlinear functional differential systems under suitable conditions on perturbed term.

 

2. PRELIMINARIES

We consider the nonlinear nonautonomous differential system

where f ∈ C(ℝ+× ℝn, ℝn), ℝ+ = [0, ∞) and ℝn is the Euclidean n-space. We assume that the Jacobian matrix fx = ∂f /∂x exists and is continuous on ℝ+× ℝn and f(t, 0) = 0. Also, consider the perturbed functional differential systems of (2.1)

where g ∈ C(ℝ+ × ℝn, ℝn), h ∈ C(ℝ+ × ℝn × ℝn, ℝn), g(t, 0) = 0, h(t, 0, 0) = 0, and T : C(ℝ+, ℝn) → C(ℝ+, ℝn) is a continuous operator.

For x ∈ ℝn, let . For an n × n matrix A, define the norm |A| of A by |A| = sup|x|≤1|Ax|.

Let x(t, t0, x0) denote the unique solution of (2.1) with x(t0, t0, x0) = x0, existing on [t0, ∞). Then we can consider the associated variational systems around the zero solution of (2.1) and around x(t), respectively,

and

The fundamental matrix Φ(t, t0, x0) of (2.4) is given by

and Φ(t, t0, 0) is the fundamental matrix of (2.3).

We recall some notions of h-stability [14].

Definition 2.1. The system (2.1) (the zero solution x = 0 of (2.1)) is called an h-system if there exist a constant c ≥ 1, and a positive continuous function h on ℝ+ such that

for t ≥ t0 ≥ 0 and |x0| small enough

Definition 2.2. The system (2.1) (the zero solution x = 0 of (2.1)) is called (hS) h-stable if there exists δ > 0 such that (2.1) is an h-system for |x0| ≤ δ and h is bounded.

Let M denote the set of all n × n continuous matrices A(t) defined on ℝ+ and N be the subset of M consisting of those nonsingular matrices S(t) that are of class C1 with the property that S(t) and S−1(t) are bounded. The notion of t∞-similarity in M was introduced by Conti [5].

Definition 2.3. A matrix A(t) ∈ M is t∞-similar to a matrix B(t) ∈ M if there exists an n × n matrix F(t) absolutely integrable over ℝ+, i.e.,

such that

for some S(t) ∈ N .

The notion of t∞-similarity is an equivalence relation in the set of all n × n continuous matrices on ℝ+, and it preserves some stability concepts [5, 10].

In this paper, we investigate bounds for solutions of the nonlinear differential systems using the notion of t∞-similarity.

We give some related properties that we need in the sequal.

Lemma 2.4 ([14]). The linear system

where A(t) is an n × n continuous matrix, is an h-system (respectively h-stable) if and only if there exist c ≥ 1 and a positive and continuous (respectively bounded) function h defined on ℝ+ such that

for t ≥ t0 ≥ 0, where ϕ(t, t0) is a fundamental matrix of (2.6).

We need Alekseev formula to compare between the solutions of (2.1) and the solutions of perturbed nonlinear system

where g ∈ C(ℝ+ × ℝn, ℝn) and g(t, 0) = 0. Let y(t) = y(t, t0, y0) denote the solution of (2.8) passing through the point (t0, y0) in ℝ+ × ℝn.

The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev [1].

Lemma 2.5 ([2]). Let x and y be a solution of (2.1) and (2.8), respectively. If y0 ∈ ℝn , then for all t ≥ t0 such that x(t, t0, y0) ∈ ℝn , y(t, t0, y0) ∈ ℝn ,

Theorem 2.6 ([3]). If the zero solution of (2.1) is hS, then the zero solution of (2.3) is hS.

Theorem 2.7 ([4]). Suppose that fx(t, 0) is t∞ -similar to fx(t, x(t, t0, x0)) for t ≥ t0 ≥ 0 and |x0| ≤ δ for some constant δ > 0. If the solution v = 0 of (2.3) is hS, then the solution z = 0 of (2.4) is hS.

Lemma 2.8. (Bihari − type inequality) Let u, λ ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u. Suppose that, for some c > 0,

Then

where , W−1(u) is the inverse of W(u), and

Lemma 2.9 ([6]). Let u, λ1, λ2, λ3, λ4, λ5 ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c > 0,

Then

, where W, W−1 are the same functions as in Lemma 2.8, and

We obtain the following corollary from Lemma 2.9.

Corollary 2.10. Let u, λ1, λ2, λ3 ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c > 0,

Then

where W, W−1 are the same functions as in Lemma 2.8, and

 

3. MAIN RESULTS

In this section, we investigate boundedness for solutions of the nonlinear perturbed differential systems via t∞-similarity.

For the proof we need the following lemma.

Lemma 3.1. Let u, λ1, λ2, λ3, λ4, λ5, λ6, λ7 ∈ C(ℝ+), w ∈ C((0, ∞)), and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c > 0 and 0 ≤ t0 ≤ t,

Then

t0 ≤ t < b1, where W, W−1 are the same functions as in Lemma 2.8 and

Proof. Defining

then we have z(t0) = c and

t ≥ t0, since z(t) and w(u) are nondecreasing, u ≤ w(u), and u(t) ≤ z(t). Therefore, by integrating on [t0, t], the function z satisfies

It follows from Lemma 2.8 that (3.2) yields the estimate (3.1). ☐

To obtain the bounded result, the following assumptions are needed:

(H1) fx(t, 0) is t∞ -similar to fx(t, x(t, t0, x0)) for t ≥ t0 ≥ 0 and |x0| ≤ δ for some constant δ > 0. (H2) The solution x = 0 of (1.1) is hS with the increasing function h. (H3) w(u) be nondecreasing in u such that u ≤ w(u) and for some v > 0.

Theorem 3.2. Let a, b, c, k, q, u, w ∈ C(ℝ+). Suppose that (H1), (H2), (H3), and g in (2.2) satisfies

and

where t ≥ t0 ≥ 0 and a, b, c, k, q ∈ L1(ℝ+). Then, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on [t0, ∞) and

where W, W−1 are the same functions as in Lemma 2.8, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. By Theorem 2.6, since the solution x = 0 of (2.1) is hS, the solution v = 0 of (2.3) is hS. Using (H1), by Theorem 2.7, the solution z = 0 of (2.4) is hS. By Lemma 2.4, Lemma 2.5, together with (3.3), and (3.4), we have

Applying (H2) and (H3), we obtain

Set u(t) = |y(t)‖h(t)|−1. Then, it follows from Lemma 3.1 that we have

where c = c1|y0| h(t0)−1. From the above estimation, we obtain the desired result.

Thus, the theorem is proved. ☐

Remark 3.3. Letting c(t) = 0 in Theorem 3.2, we obtain the similar result as that of Theorem 3.3 in [7].

Theorem 3.4. Let a, b, c, q, u, w ∈ C(ℝ+). Suppose that (H1), (H2), (H3), and g in (2.2) satisfies

and

where a, b, c, q ∈ L1(ℝ+). Then, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on [t0, ∞) and it satisfies

t0 ≤ t < b1, where W, W−1 are the same functions as in Lemma 2.8, and

Proof. Let x(t) = x(t, t0, Y0) and y(t) = y(t, t0, Y0) be solutions of (2.1) and (2.2), respectively. By the same argument as in the proof in Theorem 3.2, the solution z = 0 of (2.4) is hS. Applying the nonlinear variation of constants formula, Lemma 2.4, together with (3.5), and (3.6), we have

By the assumptions (H2) and (H3), we obtain

Set u(t) = |y(t)‖h(t)|−1. Then, by Corollary 2.10, we have

where c = c1|Y0| h(t0)−1. Thus, any solution y(t) = y(t, t0, Y0) of (2.2) is bounded on [t0, ∞). This completes the proof. ☐

Remark 3.5. Letting b(t) = c(t) = 0 in Theorem 3.4, we obtain the same result as that of Theorem 3.2 in [9].

We need the following lemma for the proof of Theorem 3.7.

Lemma 3.6. Let u, λ1, λ2, λ3, λ4, λ5, λ6, λ7 ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u, u ≤ w(u). Suppose that, for some c ≥ 0, we have

Then

Proof. Define a function v(t) by the right member of (3.7). Then, we have v(t0) = c and

t ≥ t0, since v(t) is nondecreasing, u ≤ w(u), and u(t) ≤ v(t). Now, by integrating the above inequality on [t0, t] and v(t0) = c, we have

Thus, by Lemma 2.8, (3.9) yields the estimate (3.8). ☐

Theorem 3.7. Let a, b, c, k, q, u, w ∈ C(ℝ+). Suppose that (H1), (H2), (H3), and g in (2.2) satisfies

and

where t ≥ t0 ≥ 0 and a, b, c, k, q ∈ L1(ℝ+). Then, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on on [t0, ∞) and it satisfies

where W, W−1 are the same functions as in Lemma 2.8, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. By the same argument as in the proof in Theorem 3.2, the solution z = 0 of (2.4) is hS. Applying the nonlinear variation of constants formula , Lemma 2.4, together with (3.10), and (3.11), we have

Using the assumptions (H2) and (H3), we obtain

Set u(t) = |y(t)‖h(t)|−1. Then, by Lemma 3.6, we have

where c = c1|y0| h(t0)−1. The above estimation yields the desired result since the function h is bounded, and so the proof is complete. ☐

Remark 3.8. Letting c(t) = 0 in Theorem 3.7, we obtain the similar result as that of Theorem 3.7 in [8].

Theorem 3.9. Let a, b, c, k, q, u, w ∈ C(ℝ+). Suppose that (H1), (H2), (H3), and g in (2.2) satisfies

and

where s ≥ t0 ≥ 0 and a, b, c, k, q ∈ L1(ℝ+). Then, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on [t0, ∞) and it satisfies

Proof. Using the nonlinear variation of constants formula of Alekseev [1], any solution y(t) = y(t, t0, y0) of (2.2) passing through (t0, y0) is given by

By the same argument as in the proof in Theorem 3.2, the solution z = 0 of (2.4) is hS. Applying Lemma 2.4, together with (3.12), (3.13), and (3.14), we have

It follows from (H2) and (H3) that

Defining u(t) = |y(t)‖h(t)|−1, then, by Lemma 2.9, we have

where t0 ≤ t < b1 and c = c1|y0| h(t0)−1. The above estimation yields the desired result since the function h is bounded. Hence, the proof is complete. ☐

Remark 3.10. Letting c(s) = b(s) = 0 in Theorem 3.9, we obtain the same result as that of Theorem 3.2 in [9].

Remark 3.11. Letting c(s) = 0 in Theorem 3.9, we obtain the same result as that of Theorem 3.4 in [8].