Existence of Mild Solutions for the Elastic Systems with Structural Damping in Banach Spaces

Abstract

This paper deals with the existence and uniqueness of mild solutions for a second order evolution equation initial value problem in a Banach space, which can model an elastic system with structural damping. The discussion is based on the operator semigroups theory and fixed point theorem. In addition, an example is presented to illustrate our theoretical results.

1. Introduction

Our aim in this paper is to study the existence and uniqueness of mild solutions for the semilinear elastic system with structural damping 𝑢̈(𝑡)+𝜌𝒜𝑢̇(𝑡)+𝒜²𝑢(𝑡)((𝑡𝑢(0)=𝑓𝑡,𝑢)),0<𝑡<𝑎,=𝑥₀,𝑢̇(0)=𝑦₀ (1) in a Banach space 𝕏, where ⋅ means 𝑑/𝑑𝑡, 𝜌≥2 is a constant; 𝒜∶𝐷(𝒜)⊂𝕏→𝕏 is a closed linear operator and −𝒜 generates a 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) on 𝕏; 𝑓∈𝐶([0,𝑎]×𝕏,𝕏), 𝑥₀∈𝐷(𝒜), 𝑦₀∈𝕏.

In 1982, Chen and Russell [1] investigated the following linear elastic system described by the second order equation𝑢̈(𝑡)+𝐵𝑢̇(𝑡)+𝐴𝑢(𝑡)=0 (2) in a Hilbert space 𝐻 with inner (⋅,⋅), where 𝐴 (the elastic operator) and 𝐵 (the damping operator) are positive definite selfadjoint operators in 𝐻. They reduced (2) to the first order equation in 𝐻×𝐻𝑑𝐴𝑑𝑡^1/2𝑢𝑢̇=0𝐴^1/2−𝐴^1/2𝐴−𝐵^1/2𝑢𝑢̇. (3) Let 𝑉=𝒟(𝐴^1/2), ℋ=𝑉×𝐻 with the naturally induced inner products. Then, (2) is equivalent to the first order equation in ℋ𝑑𝑢𝑢̇𝑑𝑡=𝒜_𝐵𝑢𝑢̇, (4) where 𝒜_𝐵=,𝐷𝒜0𝐼−𝐴−𝐵_𝐵=𝐷(𝐴)×𝐷𝐴^1/2∩𝐷(𝐵). (5) Chen and Russell [1] conjectured that 𝒜_𝐵 is the infinitesimal generator of an analytic semigroup on ℋ if 𝐷𝐴^1/2⊂𝐷(𝐵) (6) and either of the following two inequalities holds for some 𝛽₁,𝛽₂>0: 𝛽₁𝐴^1/2𝑣,𝑣≤(𝐵𝑣,𝑣)≤𝛽₂𝐴^1/2𝑣,𝑣,𝑣∈𝐷𝐴^1/2;𝛽₁(𝐴𝑣,𝑣)≤𝐵²𝑣,𝑣≤𝛽₂(𝐴𝑣,𝑣),𝑣∈𝐷(𝐴). (7) In the same paper they obtained some results in this direction. The complete proofs of the two conjectures were given by Huang [2, 3]. Then, other sufficient conditions for 𝒜_𝐵 or its closure 𝒜_𝐵 to generate an analytic or differentiable semigroup on ℋ were discussed in [4–10], 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.

However, so far as we know, among the previous works, little is concerned with an elastic system with structural damping in a Banach space. Motivated by previous works, in this paper, we investigate the existence and uniqueness of mild solutions for the elastic system (1) in a frame of Banach spaces. To this end, we firstly introduce the concept of mild solutions for system (1), which is based on the discussion about associated linear system. Secondly, we prove the existence and uniqueness of mild solutions for the semilinear elastic system (1) in a Banach space 𝕏.

The paper is organized as follows. In Section 2, we discuss the associated linear elastic system and give its definition of mild solutions. In Section 3, we study the existence and uniqueness of mild solutions for the semilinear elastic system (1). An example to illustrate our theoretical results is given in Section 4.

2. Preliminaries on Linear Elastic Systems

Let 𝕏 be a Banach space, we consider the linear elastic system with structural damping 𝑢̈(𝑡)+𝜌𝒜𝑢̇(𝑡)+𝒜²𝑢(𝑡)(𝑡)𝑢(0)=ℎ,0<𝑡<𝑎,=𝑥₀,𝑢̇(0)=𝑦₀, (8) where ⋅ means 𝑑/𝑑𝑡, 𝜌≥2 is a constant; 𝒜∶𝐷(𝒜)⊂𝕏→𝕏 is a closed linear operator, and −𝒜 generates a 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) on 𝕏; ℎ∶[0,𝑎]→𝕏, 𝑥₀∈𝐷(𝒜), 𝑦₀∈𝕏.

For the second order evolution equation 𝑢̈(𝑡)+𝜌𝒜𝑢̇(𝑡)+𝒜²𝑢(𝑡)=ℎ(𝑡), (9) it has the following decomposition 𝜕𝜕𝑡+𝜎₁𝒜𝜕𝜕𝑡+𝜎₂𝒜𝑢=ℎ(𝑡). (10) That is, 𝜕²𝑢𝜕𝑡²+𝜎₁+𝜎₂𝒜𝜕𝑢𝜕𝑡+𝜎₁𝜎₂𝒜²𝑢=ℎ(𝑡). (11) It follows from (9) and (11) that 𝜎₁+𝜎₂=𝜌,𝜎₁𝜎₂=1. (12) By (12), we have (i)if𝜌>2,then𝜎₁=𝜌+𝜌²−42,𝜎₂=𝜌−𝜌²−42,(ii)if𝜌=2,then𝜎₁=𝜎₂=1. (13) Let 𝜕𝑢𝜕𝑡+𝜎₂𝒜𝑢=𝑣(𝑡),0≤𝑡≤𝑎, (14) which means 𝑣₀∶=𝑣(0)=𝑦₀+𝜎₂𝒜𝑥₀. (15) So we reduce the linear elastic system (8) to the following two abstract Cauchy problems in Banach space 𝕏: 𝜕𝑣𝜕𝑡+𝜎₁(𝑡)𝑣(0)𝒜𝑣=ℎ,0<𝑡<𝑎,=𝑣₀,𝜕𝑢𝜕𝑡+𝜎₂(𝑡)𝑢(0)𝒜𝑢=𝑣,0<𝑡<𝑎,=𝑥₀. (16)(17) It is clear that (16) and (17) are linear inhomogeneous initial value problems for −𝜎₁𝒜 and −𝜎₂𝒜, respectively. Since −𝒜 is the infinitesimal generator of 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0). Furthermore, for any 𝜌≥2, (13) yield 𝜎₁>0, 𝜎₂>0. Thus, by operator semigroups theory [11], −𝜎₁𝒜 and −𝜎₂𝒜 are infinitesimal generators of 𝐶₀-semigroups, which implies initial value problems (16) and (17) are well-posed.

Throughout this paper, we assume that −𝜎₁𝒜 and −𝜎₂𝒜 generate 𝐶₀-semigroups 𝑆₁(𝑡)(𝑡≥0) and 𝑆₂(𝑡)(𝑡≥0) on 𝕏, respectively. Note that 𝜎₁>0, 𝜎₂>0 and −𝒜 is the infinitesimal generator of 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0). It follows that 𝑆₁(𝑡)=𝑇𝜎₁𝑡,𝑆₂(𝑡)=𝑇𝜎₂𝑡,𝑡≥0. (18)

It is well known [12, Chapter 4], when ℎ∈𝐿¹([0,𝑎],𝕏), the linear initial value problem (16) has a mild solution 𝑣 given by 𝑣(𝑡)=𝑆₁(𝑡)𝑣₀+^𝑡₀𝑆₁(𝑡−𝑠)ℎ(𝑠)𝑑𝑠. (19) Similarly, if 𝑣∈𝐶([0,𝑎],𝕏), then the mild solution of the linear initial value problem (17) expressed by 𝑢(𝑡)=𝑆₂(𝑡)𝑥₀+^𝑡₀𝑆₂(𝑡−𝑠)𝑣(𝑠)𝑑𝑠. (20) Substituting (19) into (20), we get 𝑢(𝑡)=𝑆₂(𝑡)𝑥₀+^𝑡₀𝑆₂(𝑡−𝑠)𝑆₁(𝑠)𝑣₀+𝑑𝑠_𝑡0𝑠0𝑆₂(𝑡−𝑠)𝑆₁(𝑠−𝜏)ℎ(𝜏)𝑑𝜏𝑑𝑠. (21) From the argument above, we obtain the following corollary.

Corollary 1. If ℎ∈𝐿¹([0,𝑎],𝕏), then the initial value problem (8) has at most one solution. If it has a solution, this solution is given by (21).

For every ℎ∈𝐿¹([0,𝑎],𝕏), the right-hand side of (21) is a continuous function on [0,𝑎]. It is natural to consider it as a generalized solution of (8) even it is not differentiable and dose not strictly satisfy the equation. We therefore define the following.

Definition 2. Let −𝒜 is the infinitesimal generator of 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0). Then a continuous solution 𝑢(𝑡) of the integral equation 𝑢(𝑡)=𝑆₂(𝑡)𝑥₀+^𝑡₀𝑆₂(𝑡−𝑠)𝑆₁(𝑠)𝑣₀+𝑑𝑠_𝑡0𝑠0𝑆₂(𝑡−𝑠)𝑆₁(𝑠−𝜏)ℎ(𝜏)𝑑𝜏𝑑𝑠 (22) is said to be a mild solution of the initial value problem (8). Where 𝑆₁(𝑡)(𝑡≥0), 𝑆₂(𝑡)(𝑡≥0) were defined in (18) and 𝑣₀ was specified in (15).

3. Main Results

Let 𝐶(𝐽,𝕏) be the Banach space of all continuous functions 𝑢∶𝐽→𝕏 with norm ‖𝑢‖_𝐶=max_𝑠∈𝐽‖𝑢(𝑠)‖, 𝐽=[0,𝑎]. Let ℒ(𝕏) be the Banach space of all linear and bounded operators on 𝕏. Note that 𝑆₁(𝑡)(𝑡≥0) and 𝑆₂(𝑡)(𝑡≥0) are 𝐶₀-semigroups on 𝕏. Thus, there exist 𝑀₁≥1 and 𝑀₂≥1 such that 𝑀₁=sup_𝑡∈𝐽𝑆₁(𝑡)_ℒ(𝕏),𝑀₂=sup_𝑡∈𝐽𝑆₂(𝑡)_ℒ(𝕏). (23)

In what follows, we firstly give the definition of a mild solution for the initial value problem (1) below.

Definition 3. Let −𝒜 is the infinitesimal generator of 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0). Then a continuous solution 𝑢(𝑡) of the integral equation 𝑢(𝑡)=𝑆₂(𝑡)𝑥₀+^𝑡₀𝑆₂(𝑡−𝑠)𝑆₁(𝑠)𝑣₀+𝑑𝑠_𝑡0𝑠0𝑆₂(𝑡−𝑠)𝑆₁(𝑠−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠 (24) is said to be a mild solution of the initial value problem (1). Where 𝑆₁(𝑡)(𝑡≥0), 𝑆₂(𝑡)(𝑡≥0) were defined in (18) and 𝑣₀ was specified in (15).

Secondly, we consider the existence and uniqueness of mild solutions for (1). To this end, we make the following assumptions: (𝐻1)𝑓∶[0,𝑎]×𝕏→𝕏 be continuous and there exists 𝐿>0, such that 𝑓𝑡,𝑢₂−𝑓𝑡,𝑢₁≤𝐿𝑢₂−𝑢₁,𝑡∈[0,𝑎],𝑢₁,𝑢₂∈𝕏. (25)(𝐻2)𝑓∶[0,𝑎]×𝕏→𝕏 be continuous and there exists a positive function 𝜇∈𝐿^∞(𝐽,ℝ⁺)(ℝ⁺=[0,+∞)) such that 𝑓(𝑡,𝑢)≤𝜇(𝑡),𝑡∈[0,𝑎],𝑢∈𝕏. (26)(𝐻3) The 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) is compact for 𝑡>0.

Theorem 4. Assume that (𝐻1) holds, −𝒜 is the infinitesimal generator of 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0). Then for every 𝑥₀∈𝐷(𝒜), 𝑦₀∈𝕏 and 𝜌≥2, the initial value problem (1) has a unique mild solution 𝑢∈𝐶([0,𝑎],𝕏).

Proof. Define the operator 𝑄∶𝐶(𝐽,𝕏)→𝐶(𝐽,𝕏) by (𝑄𝑢)(𝑡)=𝑆₂(𝑡)𝑥₀+^𝑡₀𝑆₂(𝑡−𝑠)𝑆₁(𝑠)𝑣₀+𝑑𝑠_𝑡0𝑠0𝑆₂(𝑡−𝑠)𝑆₁(𝑠−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠. (27)
It is obvious that the mild solution of the initial value problem (1) is equivalent to the fixed point of 𝑄.
For any 𝑢₁,𝑢₂∈𝐶(𝐽,𝕏), (23), (27), and (𝐻1) yield 𝑄𝑢₂(𝑡)−𝑄𝑢₁(𝑡≤_𝑡0𝑠0)𝑆₂(𝑡−𝑠)_ℒ(𝕏)𝑆₁(𝑠−𝜏)_ℒ(𝕏)×𝑓𝜏,𝑢₂(𝜏)−𝑓𝜏,𝑢₁(𝜏)𝑑𝜏𝑑𝑠≤𝐿𝑀₁𝑀₂^𝑡₀^𝑠₀𝑢₂(𝜏)−𝑢₁(𝜏)𝑑𝜏𝑑𝑠≤𝐿𝑀₁𝑀₂^𝑡₀^𝑠₀𝑢₂−𝑢_1𝐶≤𝑑𝜏𝑑𝑠𝐿𝑀₁𝑀₂𝑎²2𝑢₂−𝑢_1𝐶. (28) Using (27), (28), and induction on 𝑛 it follows easily that 𝑄^𝑛𝑢₂(𝑡)−𝑄^𝑛𝑢₁(𝑡)≤𝐿𝑀₁𝑀₂𝑎^2𝑛()!𝑢2𝑛²−𝑢_1𝐶. (29) Hence 𝑄^𝑛𝑢₂−𝑄^𝑛𝑢_1𝐶≤𝐿𝑀₁𝑀₂𝑎^2𝑛()!𝑢2𝑛²−𝑢_1𝐶. (30) Since 𝐿𝑀₁𝑀₂𝑎^2𝑛(2𝑛)!→0as𝑛→∞. (31) Thus, for 𝑛 large enough (𝐿𝑀₁𝑀₂𝑎²)^𝑛/(2𝑛)!<1 and by well known extension of the contraction mapping principle, 𝑄 has a unique fixed point 𝑢∈𝐶([0,𝑎],𝕏). This fixed point is the desired solution of the integral equation (24).

Theorem 5. Suppose that assumptions (𝐻2) and (𝐻3) hold. Then for every 𝑥₀∈𝐷(𝒜), 𝑦₀∈𝕏 and 𝜌≥2, the initial value problem (1) has at least one mild solution 𝑢∈𝐶([0,𝑎],𝕏).

Proof. Define the operator 𝑄∶𝐶(𝐽,𝕏)→𝐶(𝐽,𝕏) as (27) and choose 𝑟>0 such that 𝑟≥𝑀₂𝑥₀+𝑀₁𝑀₂𝑎𝑣₀+𝑀₁𝑀₂𝑎²𝜇_{𝐿∞(𝐽,ℝ+)}. (32) Let 𝐵_𝑟={𝑢∈𝐶(𝐽,𝕏)∶‖𝑢‖_𝐶≤𝑟}. We proceed in two main steps.
Step 1. We show that 𝑄(𝐵_𝑟)⊂𝐵_𝑟. For that, let 𝑢∈𝐵_𝑟. Then for 𝑡∈𝐽, we have ‖(𝑄𝑢)(𝑡)‖≤𝑆₂(𝑡)𝑥₀+^𝑡₀𝑆₂(𝑡−𝑠)𝑆₁(𝑠)𝑣₀+𝑑𝑠_𝑡0𝑠0𝑆₂(𝑡−𝑠)𝑆₁(𝑠−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠, (33) which according to (𝐻2) and (23) gives ‖(𝑄𝑢)(𝑡)‖≤𝑀₂𝑥₀+𝑀₁𝑀₂𝑎𝑣₀+𝑀₁𝑀₂𝑎²𝜇_{𝐿∞(𝐽,ℝ+)}. (34) In view of the choice of 𝑟, we obtain ‖𝑄𝑢‖_𝐶≤𝑟. (35)
Step 2. We prove that 𝑄 is completely continuous. Note that 𝑓∶𝑢→𝑓(⋅,𝑢(⋅)) is a continuous mapping from 𝐵_𝑟 to 𝐶(𝐽,𝕏). Thus, 𝑄∶𝐵_𝑟→𝐵_𝑟 is continuous. Next, we show that 𝑄 is compact. To this end, we use the Ascoli-Arzela's theorem. For that, we first prove that {(𝑄𝑢)(𝑡)∶𝑢∈𝐵_𝑟} is relatively compact in 𝕏, for 𝑡∈𝐽. Obviously, {(𝑄𝑢)(0)∶𝑢∈𝐵_𝑟} is compact.
Let 𝑡∈(0,𝑎]. For each 𝜖∈(0,𝑡) and 𝑢∈𝐵_𝑟, we define the operator 𝑄_𝜖 by 𝑄_𝜖𝑢(𝑡)=𝑆₂(𝑡)𝑥₀+₀^𝑡−𝜖𝑆₂(𝑡−𝑠)𝑆₁(𝑠)𝑣₀+𝑑𝑠_{0𝑡−𝜖𝑠0}𝑆₂(𝑡−𝑠)𝑆₁(𝑠−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠=𝑆₂(𝑡)𝑥₀+𝑆₂(𝜖)₀^𝑡−𝜖𝑆₂(𝑡−𝜖−𝑠)𝑆₁(𝑠)𝑣₀𝑑𝑠+𝑆₂(𝜖)₀^{𝑡−𝜖𝑠}₀𝑆₂(𝑡−𝜖−𝑠)𝑆₁()((𝜏𝑠−𝜏×𝑓𝜏,𝑢))𝑑𝜏𝑑𝑠. (36) Then the sets {(𝑄_𝜖𝑢)(𝑡)∶𝑢∈𝐵_𝑟} are relatively compact in 𝕏 since by (𝐻3) and (18), the semigroup 𝑆₂(𝑡)(𝑡≥0) is compact for 𝑡>0 on 𝕏. Moreover, using (23) and (𝐻2) we have (𝑄𝑢)(𝑡)−𝑄_𝜖𝑢(𝑡≤_{𝑡𝑡−𝜀})𝑆₂(𝑡−𝑠)𝑆₁(𝑠)𝑣₀+𝑑𝑠_{𝑡𝑡−𝜀𝑠0}𝑆₂(𝑡−𝑠)𝑆₁(𝑠−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠≤𝑀₁𝑀₂𝑣₀𝜖+𝑀₁𝑀₂𝑎𝜇_{𝐿∞(𝐽,ℝ+)}𝜖. (37) Therefore, the set {(𝑄𝑢)(𝑡)∶𝑢∈𝐵_𝑟} is relatively compact in 𝕏 for all 𝑡∈(0,𝑎] and since it is compact at 𝑡=0 we have the relatively compactness in 𝕏 for all 𝑡∈𝐽.
Now, let us prove that 𝑄(𝐵_𝑟) is equicontinuous. For 0≤𝑡₁<𝑡₂≤𝑎, we have (𝑄𝑢)𝑡₂−(𝑄𝑢)𝑡₁≤𝑆₂𝑡₂𝑥₀−𝑆₂𝑡₁𝑥₀+^𝑡1₀𝑆₂𝑡₂−𝑠−𝑆₂𝑡₁−𝑠𝑆₁(𝑠)𝑣₀+𝑑𝑠_𝑡2𝑡1𝑆₂𝑡₂−𝑠𝑆₁(𝑠)𝑣₀+𝑑𝑠_𝑡10𝑠0𝑆₂𝑡₂−𝑠−𝑆₂𝑡₁−𝑠×𝑆₁()𝑓((𝜏+𝑠−𝜏𝜏,𝑢))𝑑𝜏𝑑𝑠_{𝑡2𝑡1𝑠0}𝑆₂𝑡₂−𝑠𝑆₁(𝑠−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠∶=𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅, (38) where 𝐼₁=𝑆₂𝑡₂𝑥₀−𝑆₁𝑡₁𝑥₀,𝐼₂=^𝑡1₀𝑆₂𝑡₂−𝑠−𝑆₂𝑡₁−𝑠𝑆₁(𝑠)𝑣₀,𝐼𝑑𝑠₃=^𝑡2_𝑡1𝑆₂𝑡₂−𝑠𝑆₁(𝑠)𝑣₀,𝐼𝑑𝑠₄=^𝑡1₀^𝑠₀𝑆₂𝑡₂−𝑠−𝑆₂𝑡₁−𝑠×𝑆₁()𝑓((𝜏,𝐼𝑠−𝜏𝜏,𝑢))𝑑𝜏𝑑𝑠₅=^𝑡2_𝑡1^𝑠₀𝑆₂𝑡₂−𝑠𝑆₁(𝑠−𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠. (39) In fact, 𝐼₁,𝐼₂,𝐼₃,𝐼₄ and 𝐼₅ tend to 0 independently of 𝑢∈𝐵_𝑟 when 𝑡₂−𝑡₁→0.
Note that the function 𝑆₂(𝑡)𝑥₀ is continuous for 𝑡≥0. Thus, 𝑆₂(𝑡)𝑥₀ is uniformly continuous on [0,𝑎] and thus lim_{𝑡2−𝑡1→0}𝐼₁=0.
From (23) and (𝐻2), we have 𝐼₂≤^𝑡1₀𝑆₂𝑡₂−𝑠−𝑆₂𝑡₁−𝑠_ℒ(𝕏)×𝑆₁(𝑠)_ℒ(𝕏)𝑣₀𝑑𝑠≤𝑀₁𝑣₀^𝑎₀𝑆₂(𝑡₂−𝑡₁+𝜏)−𝑆₂(𝜏)_ℒ(𝕏)𝐼𝑑𝜏.₄≤^𝑡1₀^𝑠₀𝑆₂𝑡₂−𝑠−𝑆₂𝑡₁−𝑠_ℒ(𝕏)×𝑆₁(𝑠−𝜏)_ℒ(𝕏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠≤𝑀₁𝑎𝜇_{𝐿∞(𝐽,ℝ+)}×^𝑎₀𝑆₂(𝑡₂−𝑡₁+𝜏)−𝑆₂(𝜏)_ℒ(𝕏)𝑑𝜏. (40) Let 𝜙(𝜏)=𝑆₂(𝑡₂−𝑡₁+𝜏)−𝑆₂(𝜏). By the compactness of 𝑇(⋅) and (18), we can easily conclude that 𝑆₂(⋅) is compact and therefore 𝑆₂(𝑡) is continuous in the uniform operator topology for 𝑡>0. Then, 𝜙(𝜏) is also continuous in the uniform operator topology on (0,𝑎]. Thus ‖𝑆₂(𝑡₂−𝑡₁+𝜏)−𝑆₂(𝜏)‖_ℒ(𝕏)→0 as 𝑡₂−𝑡₁→0. Meanwhile, 𝜙(𝜏) is bounded on [0,𝑎]. Hence, using Lebesgue dominated convergence theorem we deduce that lim_{𝑡2−𝑡1→0}𝐼₂=lim_{𝑡2−𝑡1→0}𝐼₄=0.
Moreover, from (23) we have 𝐼₃≤𝑀₁𝑀₂𝑣₀𝑡₂−𝑡₁,𝐼₅≤𝑀₁𝑀₂𝑎𝜇_{𝐿∞(𝐽,ℝ+)}𝑡₂−𝑡₁. (41) Hence, lim_{𝑡2−𝑡1→0}𝐼₃=lim_{𝑡2−𝑡1→0}𝐼₅=0.
In short, we have show, that 𝑄(𝐵_𝑟) is relatively compact for 𝑡∈𝐽, {𝑄𝑢∶𝑢∈𝐵_𝑟} is a family of equicontinuous functions. It follows from Ascoli-Arzela's theorem that 𝑄 is compact. By Schauder fixed point theorem 𝑄 has a fixed point 𝑢∈𝐵_𝑟, which obviously is a mild solution to (1).

4. An Example

In order to illustrate our main results, we consider the following initial-boundary value problem, which is a model for elastic system with structural damping 𝜕²𝑢(𝑥,𝑡)𝜕𝑡²𝜕−𝜌³𝑢(𝑥,𝑡)𝜕𝑥²+𝜕𝜕𝑡⁴𝑢(𝑥,𝑡)𝜕𝑥⁴((,()∈[]×[],𝑢()()[],𝑢()(𝑥),𝜕=𝑓𝑥,𝑡,𝑢𝑥,𝑡))𝑥,𝑡0,10,𝑎0,𝑡=𝑢1,𝑡=0,𝑡∈0,𝑎𝑥,0=𝜑𝑢()(𝑥)[],𝜕𝑡𝑥,0=𝜓,𝑥∈0,1 (42) where 𝑎>0, 𝜌≥2 are all constants, 𝑓∶[0,1]×[0,𝑎]×ℝ→ℝ is continuous.

Let 𝕏=𝐿^𝑝([0,1],ℝ)(1<𝑝<+∞), we define the linear operator 𝒜 in 𝕏 by 𝜕𝒜𝑢=−²𝑢𝜕𝑥²,𝑢∈𝐷(𝒜)=𝑊^2,𝑝(0,1)∩𝑊₀^1,𝑝(0,1). (43) It is well known from [13] that −𝒜 is the infinitesimal generator of a 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) on 𝕏.

Let 𝑢(𝑡)=𝑢(⋅,𝑡), 𝑓(𝑡,𝑢(𝑡))=𝑓(⋅,𝑡,𝑢(⋅,𝑡)), then the initial-boundary value problem (42) can be reformulated as the following abstract second order evolution equation initial value problem in 𝕏: 𝑢̈(𝑡)+𝜌𝒜𝑢̇(𝑡)+𝒜²𝑢(𝑡)((𝑡𝑢(0)𝑢̇(0)=𝑓𝑡,𝑢)),0<𝑡<𝑎,=𝜑,=𝜓. (44)

In order to solve the initial-boundary value problem (42), we also need the following assumptions:(𝑏1)𝜑∈𝑊^2,𝑝(0,1)∩𝑊₀^1,𝑝(0,1), 𝜓∈𝐿^𝑝([0,1],ℝ).(𝑏2) The partial derivative 𝑓^′_𝑢(𝑥,𝑡,𝑢) is continuous.

Theorem 6. If the assumptions (𝑏1) and (𝑏2) are satisfied, then for any 𝜌≥2, the initial-boundary value problem (42) has a unique mild solution 𝑢∈𝐶([0,𝑎],𝐿^𝑝([0,1],ℝ)).

Proof. From the assumptions (𝑏1) and (𝑏2), it is easily seen that the conditions in Theorem 4 are satisfied. Hence, by Theorem 4, for any 𝜌≥2, the problem (44) has a unique mild solution 𝑢∈𝐶([0,𝑎],𝕏), which means 𝑢 is a mild solution for initial-boundary value problem (42).

Acknowledgments

The authors are grateful to the anonymous referee for his/her valuable comments and suggestions, which improve the presentation of the original paper. Research was supported by NNSFs of China (11261053, 11061031).