APPROXIMATE CONTROLLABILITY OF SECOND-ORDER SEMILINEAR EVOLUTION SYSTEMS WITH STATE-DEPENDENT INFINITE DELAY∗

In this article, we study the problem of approximate controllability for a class of semilinear second-order control systems with state-dependent delay. We establish some sufficient conditions for approximate controllability for this kind of systems by constructing fundamental solutions and using the resolvent condition and techniques on cosine family of linear operators. Particularly, theory of fractional power operators for cosine families is also applied to discuss the problem so that the obtained results can be applied to the systems involving derivatives of spatial variables. To illustrate the applications of the obtained results, two examples are presented in the end.


Introduction
The concept of controllability has played a central role throughout the history of modern control theory. Moreover, approximately controllable systems are more prevalent and very often approximate controllability is completely adequate in applications. Therefore, various approximate controllability problems for different kinds of non-linear controlled systems represented by evolution equations have been extensively investigated in literature in these years, see [4,10,25,36], for example.
We are concerned with in this paper the approximate controllability of systems governed by a second-order semilinear functional evolution equation with statedependent delay of the form    d 2 dt 2 y(t) = −Ay(t) + L (y t ) + F (t, y ρ(t,yt) ) + Bu(t), t ∈ [0, T ], where the state y(·) and the control function u(·) take values respectively in Hilbert spaces X and U . The histories y t : (−∞, 0] → X, given in the usual way by y t (θ) = y(t + θ) for θ ∈ (−∞, 0], belong to the phase space B defined axiomatically. Here y ′ (0) denotes the right derivative of y(·) at zero. We assume that −A : D(−A) ⊂ X → X is the infinitesimal generator of a strongly continuous cosine family (C(t)) t∈R on X and L : B → X and B : U → X are bounded linear operators. F : [0, T ] × B α → X is a nonlinear continuous and uniformly bounded function to be specified later, and ρ : [0, T ] × B α → (−∞, T ] is also a continuous function. Eq. (1.1) is the abstract form of some semilinear second order partial functional differential equations (PFDEs) with state-dependent delay (for example delayed wave equations), which describe many physical, chemical and biological problems. An important way of studying semilinear second order PFDEs is to transfer them into this kind of abstract second order evolution equations. Indeed, in many cases it is more advantageous to treat the second-order abstract differential equations directly than converting them to the first order systems. The basic theory of abstract second-order evolution equations governed by generator of strongly continuous cosine family was initiated and developed respectively by Fattorini [12,13], Travis and Webb [44,45]. In the past decades, by means of the theory of cosine and sine operator families, existence, uniqueness, regularity, periodicity, stability and controllability of solutions for this kind of evolution equations have been much studied by various authors, see [3,5,7,17,26,32,37]. Particularly, the approximate controllability of second-order semilinear system with delay has also been discussed in some papers, see [11,22,31,40,42]. Henríquez and Hernández M [22] studied the approximate controllability of the following second-order neutral functional differential systems with infinite delay x 0 = ϕ ∈ B, x ′ (0) = w ∈ X, (1.2) where A, B, are defined as above. The authors established the approximate controllability results for Systems (1.2) under some assumptions of range type (also see Naito [36]). It seems that generally it is difficult in applications to verify such range conditions for evolution systems with delay. On the other hand, Numerous models that arise in applications are properly described using functional (partial) differential equations with state-dependent delay. We refer to the survey paper of Hartung et al. [16], for many examples. For this reason, the theory of functional differential equations with state-dependent delay has become, in recent years, an attractive area of research. We mention the works [1,6,8,21,28,33] and the references therein for information on recent results in this area. The problem of existence of solutions for the first-order and secondorder partial functional differential equations with state-dependent delay have been studied in [17][18][19][20]29,30], while qualitative properties of solutions of these equations with state-dependent delay have been investigated in [2,9,24].
There is no doubt that the control problems are also very interesting and challenging topics for functional evolution systems with state-dependent delay. Actually, some work on the approximate controllability of first and second order functional differential equations with state-dependent delay can be found in [11, 14, 39-41, 43, 47]. In [14] the authors considered the approximate controllability of systems represented in the following semilinear neutral functional differential equations with state-dependent delay x ρ(t,xt) ), t ∈ [0, T ], (1.3) By concerning analytic semigroup theory and the so-called resolvent condition introduced in [4], sufficient conditions of approximate controllability for System (1.3) were formulated and proved there. While in [40], Sakthivel et al. obtained sufficient conditions for the approximate controllability of second-order functional differential equation with state-dependent delay described by the equation In their paper, the authors also used the resolvent condition to obtain the sufficient conditions for the approximate controllability of (1.4). Note that in recent years this resolvent condition, which is equivalent to the linear system is approximately controllable, was extensively employed to investigate the approximate controllability of many deterministic or stochastic evolution systems with (without) delay since it is easier to be verified in applications, see [14,31,34,42] among others. The outstanding feature of this technique is that the nonlinear functions in the considered systems should commonly satisfy the condition of uniform boundedness. Our objective in this work is to study the approximate controllability of the system (1.1). Observe that the term L(y t ) in (1.1) is not uniformly bounded since L is a bounded linear operator on B, thus it can not be regarded as a special case of System (1.4), and its approximate controllability can not be investigated straightly by cosine family and the above resolvent condition method as in [31,40] (where the the nonlinear terms were required to be uniformly bounded). To overcome the difficulties brought by the non-uniform boundedness of L(y t ), as in [25,35], we are going to discuss this problem by constructing the fundamental solution for the associated linear second order evolution equations with infinite delay. In this manner we will represent the mild solutions of (1.1) via the fundamental solution and Laplace transform. As a result, we can still explore the controllability by applying the technique of the resolvent condition (see the condition (H 3 ) in Section 3).
On the other hand, in many practical cases the nonlinear function f in (1.1) frequently involves a term of spatial derivative, like in Example 5.2 discussed in Section 5. It can be seen that, when take X = L 2 (0, π), then the function f in (5.9) is defined on [0, T ] × C g, 1 2 , not on [0, T ] × C g . This means that one can not discuss the existence of mild solutions for (5.9) directly on space X, but on X 1 2 . For this reason, in order to study the approximate controllability of System (1.1) for this situation, we shall also restrict this equation in Section 4.2 in the Banach space X 1 2 (⊂ X) induced by fractional power operators A 1 2 to get the existence result. Namely, we will discuss the existence of mild solutions by applying fractional power operators theory and the 1 2 -norm. We note that, unlike analytic semigroups, there are few similar estimates on the fractional power operators with analytic cosine and sine operators. Fortunately we can take advantage of the assumption (F ) due to Travis and Webb [44] (see (H ′ 4 ) in Section 4.2) to carry on our discussion. We stress that, we do not use Schauder fixed point theorem to prove the existence of solutions in Theorem 4.3 as the compactness of the operator P λ in space X 1 2 is unknown. It is easy to see that our obtained results extend and develop directly the work on approximate controllability of second order evolution systems with infinite delay appearing in literature mainly in two aspects: one is that the considered system (1.1) involves infinite state-dependent delay and a non-uniformly bounded term; the second is that the results can be applied to the control systems in which the nonlinear functions admit derivative terms of spatial variables. To the best of our knowledge, there are seemly few similar results on this kind of second order evolution systems. In addition, it is worth pointing out here that, the fundamental solution theory for second order linear retarded evolution systems with infinite delay founded in this article are theoretically meaningful and can also be applied to discuss other important issues such as qualitative properties and optimal controls of solutions for second order semilinear FDEs with infinite delay.
Subsequently, we first present in Section 2 some notations and properties of strongly continuous cosine operator families. In addition, notations about phase spaces for infinite delay are also introduced in this section. Then in Section 3, we construct the fundamental solution G(t) and discuss some regularity properties for it. After that, we represent the mild solutions of System (1.1) explicitly via the fundamental solution G(t) using Laplace transform techniques. Based on this, in Section 4 we exploit the approximate controllability of (1.1) for two cases respectively by employing the resolvent condition and the uniform boundedness of the function F (·, ·), and some sufficient conditions of approximate controllability for (1.1) are obtained. In Section 5 two examples are provided to show the applications of the main results. Finally, Section 6 is an appendix in which we prove for completeness the existence and uniqueness of mild solutions of the corresponding second order linear system. This is the start point of constructing the fundamental solution.

Preliminaries
In this section, we collect some concepts, notations, and properties on strongly continuous cosine family and the phase spaces for infinite delay to be used in the whole paper. Let X be a Hilbert space with inner product ⟨·, ·⟩ and norm ∥ · ∥, and L (X) be the Banach space of bounded linear operators from X into X.

Basic conceptions of cosine family
First, let us recall in this subsection some definitions and properties of cosine families.
is called a strongly continuous cosine family on X.
The corresponding strongly continuous sine family {S(t) : t ∈ R} ⊂ L (X) is defined by (2.1) The generator of a strongly continuous cosine family {C(t) : t ∈ R} is the linear operator A : D(A) ⊂ X → X given by There is a necessary and sufficient condition guaranteeing that an operator (A, D(A)) generates a strongly continuous cosine family, which is analogous to the Hille-Yoside generation theorem of operator semigroup theory.
Next we present some basic properties of cosine families. Let It was proved in Kisyński [27] that E is a Banach space endowed with the norm   (2. 3) The following lemma which can be found in [12] gives a characterization of resolvent of the infinitesimal generator of a strongly continuous cosine family.

Fractional power operators
Throughout this paper, we will always assume that the linear operator (−A, D(−A)) in System (1.1) satisfies that (H 0 ) For any λ > 0, the resolvent R(λ 2 ; −A) exists, and it is strongly infinitely differentiable, satisfying Then from Theorem 2.1, (−A, D(−A)) generates a uniformly bounded (strongly continuous) cosine family (C(t)) t∈R on the Hilbert space X. Moreover, since the spectrum σ(−A) is merely contained in the negative real axis, (−A, D(−A)) also generates an analytic semigroup (T (t)) t≥0 on X, and actually we have the Weierstrass formula (see [12], Remark 5.11) In the next section we shall, as mentioned in Section 1, study the fractional power theory for the fundamental solutions which will be applied to discuss the controllability problem of System (1.1). For this purpose, in the sequel we present some results on the fractional power theory for cosine families. Let 0 ∈ ρ(A), then we can define in the following way the fractional power A α , for 0 < α < 1, as a closed linear operator on its domain D(A α ) (see [38] for more details). Define the bounded linear operator A −α as where Γ(·) is the well known Γ function given by the resolvent operator of −A, is compact for some λ > 0. Hereafter, we denote by In the light of the above statements we can now establish the following result.
Theorem 2.2. Let (C(t)) t∈R be the cosine family generated by (−A, D(−A)), (S(t)) t∈R be the associated sine family. Then C(t) and S(t) commute with the operator A α , that is, for each α ∈ (0, 1), x ∈ D(A α ) and for all t ≥ 0, there hold we have that The proof is completed.

Phase space for infinite delay
In this subsection we turn to introduce some notations on the phase spaces for infinite delay. Throughout this paper, we will employ an axiomatic definition of the phase space B introduced by Hale and Kato [15]. Adopting the terminologies used in [23], B will be a linear space of functions mapping (−∞, 0] into X endowed with a seminorm ∥ · ∥ B and satisfies the following axioms: is locally bounded, and H, K(·), M (·) are independent of x(t).
(B) The space B is complete.
We denote by B α the set of all the elements in B which take values in space X α , that is, [23]), which consists of all classes of Then the seminorm in B α is defined by See also the space C g, 1 2 presented in Section 5. Hence, since X α is still a Banach space, we see that the subspace B α will satisfy the following conditions: x σ ∈ B α , then for every t ∈ [σ, σ + a] the followings hold: Here H, K(·) and M (·) are as in (A)(iii) above.
For any ϕ ∈ B α , the notation ϕ t , t ≤ 0, represents the function ϕ t (θ) = ϕ(t+θ), θ ∈ (−∞, 0]. Then, if the function x(·) in axiom (A ′ ) with x 0 = ϕ, we may extend the mapping t → x t to the whole interval (−∞, T ] by setting x t = ϕ t for t ≤ 0. On the other hand, for the function ρ : and give the following hypothesis on ϕ t : the function t → ϕ t is continuous from Z(ρ − ) into B α and there exists a continuous and bounded function Then we have the following lemma, which plays an important role in our proofs in the next section. where

Fundamental solution
In this section we will first construct the theory of fundamental solution for the linear system corresponding to Eq. (1.1), then we express the mild solutions of Eq. (1.1) via the established fundamental solution based on Laplace transform arguments. Let us make the following hypotheses on the operators appearing in System (1.1).
(H 1 ) B ∈ L (U ; X), i.e., B is a bounded linear operator from U to X.
(H 2 ) The operator L : B → X is a bounded linear operator with ∥L∥ = l for some l > 0.
Consider now the following linear second order functional differential equation where (−A, D(−A)) and L : B → X are operators given above and (ϕ, The mild solutions of System (3.1) can be defined by the cosine family C(t) and sine family S(t) generated by (−A, D(−A)) as (see [44]) Remark 3.1. Note that, if ϕ(0) ∈ E, then the mild solution y(·; ϕ, x 0 ) is continuously differentiable and verifies Obviously, from Lemma 2.1 (iv), one has y ′ (0) = x 0 in this case.
Then, we may establish the following existence and uniqueness result for System (3.1). Its proof is standard but not trivial, and for the sake of completeness we will present the whole proof in details in Appendix, see Section 6.
Making use of the preceding theorem we now set about to construct the theory of fundamental solution. Let y(·; 0, x 0 ) denote the solution of (3.1) through (0; 0, x 0 ). For any x ∈ X, we define the fundamental solution G(t) ∈ L (X) as That is to say, from (3.2), G(t) is the unique solution of the operator equation . This definition is well guaranteed by Theorem 3.1. For the fundamental solution G(t), one has the following properties.
is a strongly continuous one-parameter family of bounded linear operators on X and satisfies that Moreover, x is still continuously differentiable for any t ≥ 0 and there hold It is readily seen that this commuting property is verified for any systems, also see Example 5.2 in Section 5.) In order to prove Assertion (iv), firstly, for x ∈ E, by the definition of S(t) and Lemma 2.1, we see S(t)x ∈ D(A), then G(·)x ∈ D(A) and the function G(·)x is continuously differentiable. In particular, (3.9) follows from (3.7), and by Assertion (i), we have Due to Lemma 2.1, S(t) is uniformly continuous on [0, T ], then from (3.5) again we deduce that G(t) is also uniformly continuous for t ∈ [0, T ] which proves Assertion (v).
Next we further consider the following linear inhomogeneous second-order function differential equation with infinite delay on X: where the function f (t) ∈ L 1 (R + , X). The mild solutions of Eq. (3.10) are represented by cosine and sine families as t ≤ 0. (3.11) For the subsequent discussion, we need to represent the mild solutions (3.11) via the fundamental solution G(t) established previously, that is Theorem 3.3. For (ϕ, x 0 ) ∈ B × X, the mild solutions (3.11) of System (3.10) can be expressed equivalently by where the functionφ(·) is defined as Proof. We prove this theorem by Laplace transform arguments. Thanks to Theorem 3.1 we may calculate the Laplace transformŷ(λ) of y(t) for all λ ∈ C with Reλ > max{0, γ}.
which manifests that the initial condition y ′ (0) = x 0 is also involved in the expression (3.12).
(ii) The value of this new expression of mild solutions is that y(·) does not appear directly in the right hand side of (3.12).
Accordingly, the mild solutions of Eq. (1.1) expressed by the fundamental solution are defined as  Next we turn to present the concept of approximate controllability. As mentioned in Section 1, in the sequel, we shall study the approximate controllability for System (1.1) by using a so-called resolvent operator condition (the condition (H 3 ) below). For this purpose, we introduce the following resolvent operator. Let where B * and G * denote respectively the adjoint operators of B and G, then the resolvent operator R(λ, −Γ T 0 ) ∈ L (X) for λ > 0 is defined as Since the operator Γ T 0 is clearly positive, R(λ, −Γ T 0 ) is well defined. We will always assume that (H 3 ) λR(λ, −Γ T 0 ) → 0 as λ → 0 + in the strong operator topology. The above condition (H 3 ) is equivalent to the approximate controllability of the (3.16) More precisely, we have that

Approximate controllability
With the preceding preparation, in this section we devote to investigating the approximate controllability for System (1.1) to obtain the main results of this article .

The general case
We first discuss the general case, i.e., the approximate controllability problem of System (1.1) under the assumption that the function F (·, ·) is defined on [0, T ] × B. To do this, besides the previous assumptions on operators A, L and B, we will also impose the following restrictions on the sine family (S(t)) t≥0 and the nonlinear function F (·, ·).
(H 4 ) The sine family S(t) is compact for any t ≥ 0.
(4.1) Using this control function, we define the operator P λ on C([0, T ]; X) as  We will prove the assertion by applying Schauder fixed point theorem that for each λ ∈ (0, 1), there is a b 0 > 0 such that the operator P λ given by (4.2) has a fixed point on E(b 0 ).
For any y ∈ E(b), it follows from Lemma 2.3 that Hence, using (H 5 )(ii) we further obtain that However, the left side of (4.3) may go to +∞ as long as b → +∞ since γ 1 < 1 by our assumption. This is a contradiction. Therefore, there is an b 0 (λ) > 0 such that To prove that P λ is a compact operator, we first prove that P λ is continuous on  ρ(s,ys) ) →0 as n → +∞. y ρ(s,ys) ) ds where u n and u are the controls corresponding to y n and y respectively. By carrying on the semilinear estimations as above we may apply the Lebesgue dominated convergence theorem to get that sup 0≤t≤T ∥((P λ y n )(t) − (P λ y)(t))∥ → 0, as n → +∞, i.e., P λ is continuous.
Next, we prove that the family V (·) = {(P λ y)(·) : y ∈ E(b 0 )} is an equicontinuous family of functions. To this end, let 0 ≤ t 1 < t 2 ≤ T , then from which and the previous estimates we see that Now, it remains us to show that, for each t ∈ [0, T ], the set V (·) = (P λ y)(·) : y ∈ E(b 0 ) is relatively compact in X. In fact, since S(t) is a compact operator for t ≥ 0, from Theorem 3.2 (ii) G(t) is also compact. As a result, the set V (·) is relatively compact in X for all t ∈ [0, T ].
The above arguments enable us to infer from Arzela-Ascoli theorem that P λ : is compact and, consequently, by Schauder fixed point theorem we conclude that there exists a fixed point y(·) for P λ on E(b 0 ) whose extension by ϕ on (−∞, 0] is (by Definition 3.1) a mild solution for System (1.1).
By the definition of the operator Γ T 0 we have y ρ(s,ys) ) ds .
By assumption we know that F (s, y λ ρ(s,ys) ) : λ ∈ (0, 1) is bounded uniformly in λ ∈ (0, 1) in X, from which it follows that there is a subsequence, still denoted by F (s, y λ ρ(s,ys) ), that converges weakly to, say, f (s) in X as λ → 0 + for each s ∈ [0, T ]. Meanwhile, the compactness of G(t), t > 0, implies that Then from this we infer that

The case involving spatial derivatives
Note that if the function F (·, ·) is defined on [0, T ]×B 1 2 , then Theorem 4.2 becomes invalid since the mild solutions are obtained in space X by Theorem 4.1 which do not verify F (·, ·). Hence, in this part we will discuss the approximate controllability problem of System (1.1) for this situation. More precisely, when F satisfies the assumption (H ′ 5 ) below, we shall show that, for (ϕ, x 0 ) ∈ B 1 2 × X and for any y T ∈ X, by selecting some u λ (t) for given λ ∈ (0, 1), there is a solution y(·, ϕ, x 0 , u λ ) : (−∞, T ] → X 1 2 for (1.1) such that y λ (·, ϕ, x 0 , u λ ) → y T in space X. To this end, we require here that the operator L, the sine family S(t) and the function F (·, ·) satisfy respectively the following conditions (H (H ′ 4 ) There is a constant C > 0 such that (H ′ 5 ) The function F : [0, T ] × B 1 2 → X satisfies the Lipschitz condition with respect to the second variable uniformly in t ∈ [0, T ], that is, there exists a constant k > 0 such that for any t ∈ [0, T ] and ϕ 1 , Moreover, there is a positive constant K such that is actually a standard assumption for seconder order evolution equations since it coincides with the hypothesis (F ) in [44].
As above, we first need to establish the existence result in the subspace X 1 2 . . Proof. Let the operator P λ and the control u(·) be respectively defined in (4.2) and (4.1). As there is no assumption on compactness of S(t), we will prove P λ has a fixed point on C([0, T ]; X 1 2 ) by Banach fixed point principle. At first we show that, for any λ ∈ (0, 1), there is a b 1 (λ) > 0 such that In fact, for any y(·) ∈ E(b 1 ) with b 1 > 0 determined below, from (4.1) we have Therefore, from (4.2) and Theorem 3.2 (i)(iii)(vii), it follows that y ρ(s,ys) ) + Bu(s) ds 1 2 ≤N ∥ϕ(0)∥ 1 where we have used the community between A 1 2 and G ′ (t) (cf. Theorem 3.2 (vi)).
Next we prove that P n λ is a contraction mapping on E(b 1 ). Indeed, for any
Now for any integer n ≥ 1, by iteration, it follows from (4.6) and (4.7) that It is easy to compute that Thus, for sufficiently large n, there must have b n = W n (λ)T n n! < 1.
We now present the following the approximate controllability result for System (1.1). Proof. Since we have just obtained the mild solution y(·, ϕ, x 0 , u λ ) : (−∞, T ] → X 1 2 for (1.1) for each λ ∈ (0, 1), as in the proof of Theorem 4.2, we only need to prove that y λ (·, ϕ, x 0 , u λ ) → y T in space X (not in X 1 2 ). But obviously the proof of this assertion is very similar to that of Theorem 4.2, so we omit it here.

Examples
In this section, we apply the results established above to study the controllability of semilinear wave equation with state-dependent delay. Specifically, we discuss the approximate controllability problem of the following controlled systems.
Example 5.1. Consider the boundary value problem of semilinear retarded wave system.
As stated in the introduction part, the second term on the right-hand side in the first equation is not uniformly bounded and hence the approximate controllability of (5.1) can not be obtained by directly the same methods (concerning cosine and sine operator families) as in Ref. [40]. However, as what follows, Theorem 4.2 is well applied to this system due to theory of fundamental solution and its approximate controllability can be obtained.
(iii) The operators C(t) defined by cos(nt)⟨x, e n ⟩e n , is the cosine family in X generated by (−A, D(−A)), and the associated sine family is given by It is clear that C(·)x and S(·)x are periodic functions, and ∥C(t)∥ ≤ 1, ∥S(t)∥ ≤ 1, t ∈ R. Moreover, S(t) is a compact and self-adjoint operator for each t ≥ 0.
Now define the operator L : C g → X ,the map F (·, ·) : [0, 2] × C g → X and the state-dependent function ρ(·, ·) : [0, 2] × C g → (−∞, T ), respectively, as for any t ∈ [0, 2] and φ ∈ C g . Then under these notations System (5.1) is rewritten into the form of (1.1). Evidently, the assumption (a 2 ) ensures that the function F satisfies the hypothesis (H 5 ) and (a 1 ) implies that L : C g → X is linear and bounded because we have So (H 2 ) holds with l given by (5.5).
Here, as in [25,36], we take u n e n : It is easy to compute that for According to Theorem 4.2, to obtain the approximate controllability for System (5.1), it remains to verify the condition (H 3 ) since (H 4 ) is guaranteed by Property (iii) of S(t) above. As one can see that, generally speaking, it is difficult for us to obtain the explicit expression of the fundamental solutions G(t) associated to the linear system. Fortunately, however, we are able to express it through the relationship between G(t) and S(t) on the interval [0, 1], and this is enough to ensure the condition (H 3 ) holds in this situation. Indeed, the mild solutions on the interval [0, 1] of the linear equation Since there is a spatial derivative term in the function f as we mentioned previously, the approximate controllability problem should be obtained by applying Theorem 4.4.
Hence we take the phase space C g, 1 2 (⊂ C g ), which is defined by . Clearly, C g, 1 2 satisfies correspondingly the axioms (A ′ ), (A ′ 1 ) and (B ′ ). We proceed the similar discussion as in Example 5.1 with the same notations.

Appendix
In this section we prove Theorem 3.1 for the sake of completeness. Proof of Theorem 3.1. We shall show the existence and uniqueness of solutions of (3.1) by the contractive mapping theorem. For the functions K(·) and M (·) in Axiom (A)(iii) we put, for any b ∈ (0, T ], Thus ∥Q(y)∥ C ≤ ρ, which shows that Q maps E(T 1 , ρ) into itself since clearly (Qy)(0) = ϕ(0). On the other hand, for any y 1 , y 2 ∈ E(T 1 , ρ), by virtue of (6.2) it yields that, for t ∈ [0, T 1 ], Qy 1 − Qy 2 C ≤ k 0 y 1 − y 2 C , which from (6.2) implies that Q is a contractive mapping on E (T 1 , ρ). Therefore, by the well-known Banach fixed point theorem, there exists a unique fixed point y(·) for the operator Q in E(T 1 , ρ). Now we may extend y(·) on (−∞, T 1 ] by y| (−∞,0] = ϕ so that y(·) : (−∞, T 1 ] → X is a mild solution of Eq. (3.1) defined on (−∞, T 1 ]. Then, using the same technique we can obtain a fixed point y(·) on interval [T 1 , 2T 1 ] for operator Q, which may also be extended to a mild solution on (−∞, 2T 1 ] for Eq. (3.1).