THE METHOD OF LOWER AND UPPER SOLUTIONS FOR DAMPED ELASTIC SYSTEMS IN BANACH SPACES∗

In this paper, we are concerned with the initial value problem of a class of damped elastic systems in an order Banach spaces E. By employing the method of lower and upper solutions, we discuss the existence of extremal mild solutions between lower and upper mild solutions for such problem with the associated semigroup is equicontinuous. In addition, two examples are given to illustrate our results.


Introduction
In this article, we use a monotone iterative method in the presence of lower and upper mild solutions to discuss the existence of extremal mild solutions for the semilinear damped elastic systems in an order Banach spaces E: In 1982, Chen and Russell [2] investigated the following linear elastic system described by the second order equation Chen and Russell [2] conjectured that A B is the infinitesimal generator of an analytic semigroup on H if and either of the following two inequalities holds for some β 1 , β 2 > 0: The complete proofs of the two conjectures were given by Huang [12,13]. Then, other sufficient conditions for A B or its closure A B to generate an analytic or differentiable semigroup on H were discussed in [4,[14][15][16][17]19], by choosing B to be an operator comparable with A α for 0 < α ≤ 1, based on an explicit matrix representation of the resolvent operator of A B or A B . In [7][8][9], Fan et al. studied the existence, the asymptotic stability of solutions and the analyticity and exponential stability of associated semigroups for the following elastic system with structural damping given by    u ′′ (t) + ρAu ′ (t) + A 2 u(t) = f (t, u(t)), 0 < t < a, where A : D(A) ⊂ E → E is a sectorial linear operator on a complex Banach space E and ρ > 0 is a constants.
In [22], the authors considered nonlinear evolution equations of second order in Banach spaces          u ′′ (t) + ρAu ′ (t) + A 2 u(t) = f (t, u(t), u t ), t ∈ I = [0, T ], u(s) = φ(s), s ≤ 0, where u is the unknown function defined on I and taking values in E, u t is the history state defined by u t : (−∞, 0] → E, u t (s) = u(t + s), t ∈ I. By means of the fixed point for condensing maps, they proved the existence and exponential decay of mild solutions. In [23,27], the authors discussed the polynomial stability of elastic systems, the discussion is based on the operator semigroups theory and some fixed point theorem. In [6], T. Diagana studied the well-posedness and existence of bounded solutions to the linear elastic systems with damping On the other hand, the monotone iterative method based on lower and upper solutions is an effective and flexible mechanism. It yields monotone sequences of lower and upper approximate solutions that converge to the minimal and maximal solutions between the lower and upper solutions. Lately, the monotone iterative method has been extended to evolution equations in ordered Banach spaces by Li [21].
In [10], Fan and Li used a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of extremal mild solutions and positive mild solutions to the initial value problem of second order semilinear evolution equations in ordered Banach space However, motivated by the above works, ideas and methods based on the paper [6], in this paper, we give the expression of the solution of problem (1.1), which is different from the expression given in article [6]. Moreover, we obtain the existence of the minimal and maximal mild solution, and the mild solutions between the minimal and maximal mild solution of the problem (1.1) through the monotone iterative and measure of noncompactness. Our results presented in this paper is differential from [10]. First of all, the equations (1.1) we consider is different the equations (1.4) from [10]. Our research is extensive, which contains the equations (1.4). When B = A 1 2 , equations (1.1) is transformed into equations (1.4); then our results improve and generalize many classical results [6][7][8][9][10].
The paper is organized as follows: In Section 2, we introduce some notations and recall some basic known results. In Section 3 we present the existence of extremal mild solutions for damped elastic systems (1.1) in order Banach space. In Section 4, we give an example to illustrate our results.

Preliminaries
Let E be an ordered complex Banach space with the norm ∥ · ∥ and partial order ≤, whose positive cone P = {x ∈ E : x ≥ 0} is normal with normal constant N . Let α(·) denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [1,5]. For any B ⊂ C(J, E) and t ∈ J, set Now we introduce some basic definitions and properties about Kuratowski measure of noncompactness that will be used in sequel.
is a nondecreasing strict α-set-contraction operator such that v 0 ≤ Qv 0 and Qw 0 ≤ w 0 . The Q has a minimal fixed point u and a maximal fixed point u in [v 0 , w 0 ]; Moreover, v n → u and w n → u, where v n = Qv n−1 and w n = Qw n−1 (n = 1, 2, . . .) which satisfy v 0 ≤ v 1 ≤ · · · ≤ v n ≤ · · · ≤ u ≤ u ≤ · · · ≤ w n ≤ · · · ≤ w 1 ≤ w 0 . Lemma 2.5 (Sadovskii's fixed point theorem). Let E be a Banach space and Ω 0 be a nonempty bounded convex closed set in E. If Q : Ω 0 → Ω 0 is a condensing mapping, then Q has a fixed point in Ω 0 . Lemma 2.6 ( [26]). Assume f ∈ C(J, E) and that A is the infinitesimal generator of C 0 -semigroup (T (t)) t≥0 . Then the inhomogeneous Cauchy problem has a mild solution u given by Thoughts and methods based on paper [9]. We consider the following linear For the second order evolution equation it was rewritten as That is, It follows from (2.3) and (2.5) that In order to study the existence to Eq.(1.1), we will make use of the above linear operator which links both A and B: In the following discussion, we will focus on the following cases: Obviously, C(ρ) = 0 corresponds to the case studied in papers [7,8].

Lemma 2.7. Assume that there exists a densely defined closed linear operator L(ρ)
: 2) has a unique solution given by So we reduce the linear elastic system (2.2) to the following two abstract Cauchy problems in Banach space E: and It is clear that (2.9) and (2.10) are linear inhomogeneous initial value problems for −E 1 (ρ) and −E 2 (ρ) respectively. Thus, by operator semigroups theory [26],−E 1 (ρ) and −E 2 (ρ) are infinitesimal generators of C 0 -semigroups, which implies initial value problems (2.9) and (2.10) are well-posed. Thus using Lemma 2.6, if h ∈ C(J, E), the Eq. (2.9) has a mild solution v given by Similarly, if v ∈ C(J, E), then the mild solution of the Eq. (2.10) expressed by Substituting (2.11) into (2.12), we get Based on the above discussion, motivated by the definition of mild solutions in [9], we give the definition of mild solution of the problem (1.1) as follows.

Remark 2.2.
In the case C(ρ) = −L 2 (ρ), the expression of mild solution for the problem (1.1) and the conclusion of Theorem 3.1 are correct and meaningful in complex Banach spaces. For more detail to see [6].
we call it a lower solution of the problem (1.1); if all the inequalities of (2.13) are reversed, we call it an upper solution of the problem (1.1).
we call it a lower mild solution of the problem (1.1); if the inequalities of (2.14) are reversed, we call it an upper mild solution of the problem (1.1), where E 1 (ρ), E 2 (ρ) were defined in (2.7).
is continuous by operator norm for every t > 0.
x ∈ E and t ≥ 0.

Main results
For For the convenience of discussion, we define the mapping Q : Suppose that the following conditions are satisfied: (H2) There exists a constant 0 < L 1 < 1 4M1M2a 2 (1+2K0) , such that for ∀t ∈ J, and equicontinuous countable and increasing or decreasing mono- Then for every u 1 ∈ E, the problem (1.1) has a minimal mild solution u and a maximal mild solution u in Proof. Define the mapping Q : [v 0 , w 0 ] → C(J, E) is given by (3.1). By Definition 2.2, it is obvious that the mild solution of the problem (1.1) is equivalent to the fixed point of Q. First, we prove that Q is continuous in C(J, E). To this end, let u n ∈ C(J, E) be a sequence such that u n → u in C(J, E). By the continuity of nonlinear term f with respect to the second variable, for each s ∈ J, we have f (s, u n (s), Gu n (s)) → f (s, u(s), Gu(s)), n → ∞, (3.2) that is for all ϵ > 0, there exists N, when n > N, we have ∥f (s, u n (s), Gu n (s)) − f (s, u(s), Gu(s))∥ ≤ ϵ.
So, when n > N , we have By the positivity of operators T 1 (t) and T 2 (t), thus Hence from (3.1) we see that Q(u 1 ) ≤ Q(u 2 ), which means that Q is a increase operator.
By the definition of lower mild solution and upper mild solution, we can conclude that v 0 ≤ Q(v 0 ) and Q(w 0 ) ≤ w 0 , respectively. So, Q : In the following, we demonstrate that the operator Q : In fact, we only need to check I 1 , I 2 , I 3 , I 4 and I 5 tend to 0 independently of u ∈ Since T 1 (t)(t ≥ 0) is a equicontinuous C 0 semigroup, thus, T 1 (t)u 0 is uniformly continuous on J and thus lim t ′′ →t ′ I 1 = 0.
Since T 2 (t)(t ≥ 0) is a equicontinuous C 0 semigroup, for I 2 , we have for t ∈ J allows us to conclude that lim t ′′ →t ′ I 2 = 0. By the normality of the cone P , there exists M > 0 such that For I 4 , we have Consequent, lim t ′′ →t ′ I 4 = 0.
For I 3 , I 5 , we have Now, we show that the operator Q is a α-set-contractive. For any bounded D ⊂ [v 0 , w 0 ], Q(D) is bounded and equicontinuous. Therefore, by Lemma 2.1, we know that there exists a countable set D 0 = {u n } ⊂ D, such that α(Q(D)) ≤ 2α(Q(D 0 )). (

(3.5)
For t ∈ J, by Lemma 2.2, we get For every t ∈ J, by Lemma 2.2, the assumption (H3) and (3.2), we have Therefore, from (3.4) and (3.6) we know that Then from the monotonicity of Q, it follows that In what follows we prove that {v n } and {w n } are convergent in J. For convenience, let B = {v n : n ∈ N} and B 0 = {v n−1 : n ∈ N}. Then For t ∈ J, by Lemma 2.2, we get Thus, by Lemma 2.2, the assumption (H3) and (3.2), we have Hence by the Gronwall's inequality, φ(t) = 0, a.e. t ∈ J. So t 0 φ(s)ds ≡ 0, by the above inequality, φ(t) ≤ 0 , combing this with the property of noncompactness, Hence, for any t ∈ J, {v n (t)} is precompact, and {v n (t)}, {w n (t)} has a convergent subsequence. Combing this with the monotonicity (3.8), we easily prove that {v n (t)} itself is convergent, i.e., lim n→∞ v n (t) = u(t), t ∈ J. Similarly, lim n→∞ w n (t) = u(t), t ∈ J.
It follows from (3.7) and the Lebesgue dominated convergence theorem that and u = Qu, u = Qu.
Combing this with monotonicity (3.8), we see that v 0 ≤ u ≤ u ≤ w 0 . By the monotonicity of Q, it is easy to see that u and u are the minimal and maximal fixed points of Q in [v 0 , w 0 ]. Therefore, u and u are the minimal and maximal mild solutions of the problem (1.1) in [v 0 , w 0 ], and u and u can be obtained by the iterative scheme (3.7) starting from v 0 and w 0 , respectively. Now, we discuss the existence of the mild solution to the problem (1.1) between the minimal and maximal mild solutions u and u. If we replace the assumptions (H3) by the following assumptions: (H4) There exists a constant L 1 > 0 such that We will have the following existence result.

Theorem 3.3.
Let E be an ordered Banach space, whose positive cone P is normal, there exists a densely defined closed linear operator L(ρ) : For every t ∈ J, by Lemma 2.2, the assumption (H4) and (3.9), we have (3.11) Therefore, from (3.9) and (3.11) we know that is a condensing mapping. It follows from Lemma 2.5 that Q has at least one fixed point u in [v 0 , w 0 ], so u is the mild solution of the problem (1.1) By (i) and (3.12), the problem (1.1) has mild solution u 1 (t) in [0, t ′ 1 ]; Again by (i) and (3.10) Continuing such a process, the mild solution of the equation can be continuously extended to J. So, we obtain a mild solution u ∈ C(J, E) of the problem (1.1), which satisfies Therefore, the problem (1.1) at least has one mild solution between u and u.
Remark 3.1. The analytic semigroup and differentiable semigroup are equicontinuous semigroup [26]. In the application of partial differential equations, such as parabolic and strongly damped wave equations, the corresponding solution semigroup are analytic semigroup. Therefore, Theorem 3.2 and Theorem 3.3 have some broad applicability.
Proof. Since f (t, x, u(t, x), Gu(t, x)) = 1 10 sin u(t, x) + From (4.4), for u, v ∈ E, we have From all the assumptions, it is easily seen that the conditions in Theorem 3.1 are satisfied. Hence, by Theorem 3.1, the problem (4.1) has a mild solution u ∈ C(J, E), which means u is a mild solution for the problem (1.1).
Proof. Assumption (F1) implies that v 0 ≡ 0 and w 0 ≡ w(x, t) are lower and upper solutions of the problem (4.5), respectively, and from (F1) and (F2), it is easy to verify that all conditions (H1) are satisfied under the constant M 1 = M 2 = 1. So our conclusion follows from Theorem 3.2.

Conclusions
This paper investigates the existence of the extremal mild solutions for damped elastic systems in Banach spaces. By introducing a new concept of lower and upper mild solutions, we construct a new monotone iterative method for damped elastic systems and obtain the existence of extremal mild solutions between lower and upper mild solutions for the problem under the situation that the associated semigroup is equicontinuous. Here, we do not need the associated semigroup is compact. Our results presented in this paper improve and generalize many classical results [7][8][9].
For future work will be focused on investigate the asymptotic stability of solutions, and the analyticity and exponential stability of associated semigroup for damping elastic system in Banach spaces.