5.2.1. GE-Semigroup and Mild Solution of System (54)

In this part, the existence and uniqueness of the mild solution to system (54) are considered by GE-semigroup theory.

**Definition 21** ([73–77])**.** *Suppose* {*U*(*t*) : *t* ≥ 0} *is one parameter family of bounded linear operators in Banach space H, and A is a bounded linear operator. If*

$$\mathcal{U}(t+s) = \mathcal{U}(t) A \mathcal{U}(s), t, s \ge 0, \frac{1}{2}$$

*then* {*U*(*t*) : *t* ≥ 0} *is called a GE-semigroup induced by A. If the GE-semigroup U*(*t*) *satisfies*

$$\lim\_{t \to 0^+} \|\mathcal{U}(t)x - \mathcal{U}(0)x\|\_H = 0\_\prime$$

*for arbitrary x* ∈ *H*, *then it is called strongly continuous on H.*

**Lemma 1** ([73,74,76,77])**.** *If GE-semigroup U*(*t*) *is strongly continuous on H, then there exist M* ≥ 1 *and ω* > 0*, such that*

$$\|\|\boldsymbol{L}(t)\|\|\_{\boldsymbol{L}(H,H)} \leq \mathcal{M}e^{\omega t}, t \geq 0, \boldsymbol{\zeta}$$

*i.e., U*(*t*) *is exponentially bounded.*

**Definition 22** ([75–77])**.** *Suppose U*(*t*) *is strongly continuous GE-semigroup induced by A. If*

*Bx* = lim *<sup>h</sup>*→0<sup>+</sup> *AU*(*h*)*A* − *AU*(0)*A <sup>h</sup> <sup>x</sup>*,

*for every x* ∈ *D*1*, where*

$$D\_1 = \{ \mathbf{x} : \mathbf{x} \in \text{dom}(B) \subseteq H, \mathcal{U}(0)A\mathbf{x} = \mathbf{x}, \exists \lim\_{h \to 0^+} \frac{\mathcal{A}\mathcal{U}(h)A - \mathcal{A}\mathcal{U}(0)A}{h}\mathbf{x} \},$$

*then B is called a generator of GE-semigroup U*(*t*) *induced by A.*

Now, we consider the initial value problem (54).

**Definition 23.** *If <sup>B</sup> is a generator of GE-semigroup <sup>U</sup>*(*t*) *induced by A, <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*1)*, and <sup>v</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>U</sup>*)*; Cv*(*t*), *Ddw*(*t*) <sup>∈</sup> *<sup>A</sup>*(*L*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>D</sup>*1))*, the mild solution <sup>x</sup>*(*t*, *<sup>x</sup>*0) *to (54) is defined by*

$$\mathbf{x}(t, \mathbf{x}\_0) = \mathcal{U}(t) A \mathbf{x}\_0 + \int\_0^t \mathcal{U}(t - \tau) \mathbb{C} \boldsymbol{v}(\tau) d\tau + \int\_0^t \mathcal{U}(t - \tau) D d\mathbf{w}(\tau). \tag{55}$$

From the above knowledge, we have the following proposition.

**Proposition 1** ([76,77])**.** *If B is the generator of GE-semigroup U*(*t*) *induced by A, v*(*t*) ∈ *<sup>L</sup>*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>U</sup>*)*, <sup>x</sup>*<sup>0</sup> <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>D</sup>*1)*; Cv*(*t*), *Ddw*(*t*) <sup>∈</sup> *<sup>A</sup>*(*L*2([0, *<sup>b</sup>*], <sup>Ω</sup>, *<sup>D</sup>*1))*, and <sup>U</sup>*(0) *is a definite operator, then there exists unique mild solution x*(*t*, *x*0) *to (54), which is given by (55).*

In the following, we suppose that Proposition 1 holds true.
