History-dependent fractional hemivariational inequality with time-delay system for a class of new frictionless quasistatic contact problems

Abstract

We study a new frictionless quasistatic contact problem for viscoelastic materials, in which contact conditions are described by the fractional Clarke generalized gradient of nonconvex and nonsmooth functions and a time-delay system. In addition, our constitutive relation is modeled using the fractional Kelvin–Voigt law with long memory. The existence of mild solutions for new history-dependent fractional differential hemivariational inequalities with a time-delay system are obtained by the Rothe method, properties of the Clarke generalized gradient, and a fixed-point theorem.

Keywords

Frictionless quasistatic contact problem history-dependent fractional differential hemivariational inequalities time-delay system Rothe method fixed-point theorem

1. Introduction

The mathematical theory of hemivariational and variational–hemivariational inequalities was first studied by Panagiotopoulos [1, 2]. Hemivariational inequalities can be used to describe various physical phenomena, which include nonconvex, nondifferentiable local Lipschitz functions, and have been considered extensively in a variety of mathematical theory analysis and engineering applications during the last 30 years, see [3 –9] and the references therein. Recently, fractional hemivariational inequalities have attracted extensive attention and achieved good results in mathematical, physics and engineering research [10 –14]. In particular, Zeng et al. [10] considered a class of fractional differential hemivariational inequalities with application to contact problems, using the Rothe method combined with the surjectivity result of multivalued pseudomonotone operators. Han et al. [11] investigated the creep and relaxation behavior for the Caputo fractional-order Maxwell model and the fractional Kelvin-Voigt model, and established the solvability, based on the Rothe method. Bartosz and Sofonea [15] proved an existence and uniqueness result using a time-discretization scheme known as the Rothe method for variational–hemivariational inequalities. Migórski and Zeng [16] showed, by applying the Rothe method, that unique solvability and solution regularity are about history-dependent hemivariational inequalities. For the fractional Kelvin–Voigt constitutive law and the fractional Maxwell constitutive law, see [11 –13]. In addition, Jiang et al. [17] considered the existence of a global attractor for fractional differential hemivariational inequalities by applying a measure of noncompactness, a fixed-point theorem of a condensing multivalued map.

As we know, in real life, many systems are influenced by time-delay equations, for example, population dynamics, predator–prey models, and physiological systems [18, 19]. However, there are few studies on hemivariational inequalities with time delay. For hemivariational inequalities with time delay or viscoelastic materials with time delay, see [14, 20 –22].

Motivated by these works, we consider a frictionless quasistatic contact problem for viscoelastic materials, in which contact conditions are described using the fractional Clarke generalized gradient of nonconvex and nonsmooth functions, and time delay.

Next, we briefly introduce the innovations of this paper, which are mainly divided into the following four points.

The motivation to study the inequality of the system comes from a quasistatic contact problem for a viscoelastic body and the fractional Kelvin–Voigt constitutive law, which includes a generalized Clarke subdifferential operator and long memory.

As far as we know, no scholars have considered that the pointwise fractional density of active bonds is affected by a delay system; this is different from the previous case [21, 22].

Notice that the generalized Clarke of $j$ contains ${{}_{0}^{C}D}_{t}^{α} u (t)$ and has a certain physical meaning. Therefore, the form of the hemivariational inequality with a time-delay system is different from that in [14, 20]. In particular, it is necessary to mention that our fractional hemivariational inequality is different from that of Zeng et al. [10].

The fractional hemivariational inequality with a history-dependent operator is not considered. In this paper, we will provide a global unique solvability for a history-dependent fractional hemivariational inequality, which is mainly based on the Rothe method.

The rest of the paper is structured as follows. In Section 2, we recall the physical setting of contact problem, notation, and definitions. In Section 3, we establish hemivariational inequalities for the quasistatic frictionless contact problem. In Section 4, we consider the abstract problem and prove its solvability.

2. Physical setting of contact problem and preliminaries

In this section, we review the physical setting of the contact problem and introduce the basic notation, definitions, and some results.

We study a deformable viscoelastic body, which occupies a domain $Ω \subset R^{d}, d = 2, 3$ . The boundary of $Ω$ , denoted $Γ = \partial Ω$ , is assumed to be Lipschitz continuous. By Figure 1, we know that the boundary $Γ$ is consisted of three disjoint measurable parts $Γ_{D}, Γ_{N}$ , and $Γ_{C}$ , with means $Γ_{D} > 0$ .

Figure 1.

A deformable body in contact with a foundation.

Let $υ$ be a unit outward normal vector and $S^{d}$ be the space of second-order symmetric tensors on $R^{d}$ . The standard inner products and norm on $R^{d}$ and $S^{d}$ are given by

\begin{matrix} u \cdot v = u_{i} v_{i}, ∥ v ∥_{R^{d}} = (v \cdot v)^{1 / 2} for all u = (u_{i}), v = (v_{i}) \in R^{d}, \\ σ \cdot τ = σ_{ij} τ_{ij}, ∥ τ ∥_{S^{d}} = (τ \cdot τ)^{1 / 2} for all σ = (σ_{ij}), τ = (τ_{ij}) \in S^{d}, \end{matrix}

where $u = (u_{i})$ , $σ = (σ_{ij})$ , and $ε (u) = (ε_{ij} (u))$ denote the displacement vector, the stress tensor, and the linearized strain tensor, respectively. We denote by $u_{ν} = u \cdot ν$ and $u_{τ} = u - u_{ν} ν$ the normal and tangential components of the displacement $u$ , respectively. The normal and tangential components of the displacement $u$ on $Γ$ are denoted by $σ_{ν} = (σ ν) \cdot ν$ and $σ_{τ} = σ ν - σ_{ν} ν$ , respectively. The contact problem will be discussed for a finite time interval $[0, T]$ . The classical formulation of the contact problem reads as follows.

Problem 1. Find a displacement field $u : Ω \times [0, T] \to R^{d}$ , a stress field $σ : Ω \times [0, T] \to S^{d}$ , and a bonding field $β : Γ_{C} \times [- τ, T] \to [0, 1]$ , such that

σ (t) \in A ε ({{}_{0}^{C}D}_{t}^{α} (u (t))) + B (ε (u (t))) + \int_{0}^{t} L (t - s) ε (u (s)) d s + \partial Ψ (ε (u (t))) in Ω \times [0, T],

(1)

Div σ (t) + f_{0} (t) = 0 in Ω \times [0, T],

(2)

u (t) = 0 on Γ_{D} \times [0, T],

(3)

σ (t) υ = f_{N} (t) on Γ_{N} \times [0, T],

(4)

- σ_{ν} (t) \in \partial j_{ν} (β (t), {{}_{0}^{C}D}_{t}^{α} u_{ν} (t)) on Γ_{C} \times [0, T],

(5)

β' (t) = A β (t) + B β (t - τ) + F (t, u (t), β (t)) on Γ_{C} \times [0, T],

(6)

β (t) = φ (t) on Γ_{C} \times [- τ, 0],

(7)

- σ_{τ} (t) = 0 on Γ_{C} \times [0, T],

(8)

u (0) = u_{0} in Ω .

(9)

We now give a brief description of the equations and relations in Problem 1. Equation (1) is the fractional Kelvin–Voigt viscoelastic constitutive law with long memory, where $A$ and $B$ stand for the viscosity and elasticity operators, respectively, $L$ denotes the relaxation operator, and $Ψ$ is a generalized Clarke subdifferential operator. For the fractional Kelvin–Voigt viscoelastic constitutive law and the Kelvin–Voigt viscoelastic constitutive law with long memory, we can refer to [11] and [16], respectively. In addition, motivated by Section 5 of [23], we consider the constitutive law that includes a generalized Clarke subdifferential operator.

Equation (2) is the equilibrium equation for the quasistatic process. Equations (3) and (4) reveal the displacement and traction boundary conditions on parts $Γ_{D}$ and $Γ_{N}$ .

Equations (5) to (8) represent the contact and frictionless conditions. Equation (5) is called the multivalued normal compliance condition, which is governed by the subdifferential of a nonconvex potential $j_{ν}$ . Note that equation (5) contains ${{}_{0}^{C}D}_{t}^{α} u_{ν} (t)$ . Therefore, it is necessary to introduce it here. By the definition of fractional integral and the Caputo derivative operators, we know that ${{}_{0}^{C}D}_{t}^{α} u_{ν} (t)$ stands for the memory function on the surface $Γ_{C}$ . Conversely, the function $β$ denotes the pointwise density of active bonds on the contact surface. In [10], Zeng et al. introduce the physical meanings of $β = 0$ , $β = 1$ , and $β \in (0, 1)$ , respectively. We will not introduce them individually here. However, in practical situations, the roughness of the contact surface $Γ_{C}$ is different, and the contact surface $Γ_{C}$ is easily affected by temperature and other factors, see Figure 1. Therefore, it is necessary to consider that the function $β$ is driven by the delay evolution equation, that is,

β' (t) = F_{1} (t, u (t), β (t), β (t - τ)),

where $F_{1} : Γ_{C} \times [0, T] \times R^{d} \times R \to R$ . However, in this paper, we consider a special case, equations (6) and (7).

Moreover, equation (8) represents the frictionless contact problem. Finally, the initial displacement is specified in equation (9).

To study Problem 1, we need to recall some basic notation, definitions, and preliminary results.

Assume that $X$ is a Banach space; $X^{*}$ is the topological dual of $X$ , and $〈 \cdot, \cdot 〉_{X \times X^{*}}$ denotes the duality pairing between $X^{*}$ and $X$ . The norms of these spaces are defined as

∥ x ∥_{X} = sup_{t \in I} ∥ x ∥_{X}

where $I = [0, T]$ . We denote by $L (E, F)$ the space of linear and bounded operators from a Banach space $E$ to a Banach space $F$ with the usual norm ∥·∥.

Next, we recall the properties of the fractional integral and the Caputo derivative operators.

Proposition 1. [24]. Assume that $X$ is a Banach space and $α, δ > 0$ . Then:

For $y \in L^{1} (0, T; X)$ , we have ${{}_{0}I}_{t}^{α} {{}_{0}I}_{t}^{δ} y (t) = {{}_{0}I}_{t}^{α + δ} y (t) .$

For $y \in L^{1} (0, T; X)$ and $α \in (0, 1]$ , we have ${{}_{0}I}_{t}^{α} {{}_{0}^{C}D}_{t}^{α} y (t) = y (t) - y (0) .$

For $y \in L^{1} (0, T; X)$ , we have ${{}_{0}^{C}D}_{t}^{α} {{}_{0}I}_{t}^{α} y (t) = y (t) .$

The fractional integral of order $α$ of $y$ is given by

{{}_{0}I}_{t}^{α} y (t) = \frac{1}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} y (s) d s .

Next, we will review the definition of generalized Clarke for a locally Lipschitz functional $F : X \to R$ on a Banach space $X$ and recall some definitions.

According to [6, 7], we denote by $F^{0} (x; v)$ the Clarke generalized directional derivative of $F$ at $x$ in the direction $v$ , that is

F^{0} (x; v) = lim_{y \to x} sup_{λ \to 0^{+}} \frac{F (y + λ v) - F (y)}{λ} .

Recall also that the generalized Clarke subdifferential of $F$ at x, denote $\partial F$ , is a subset of $X^{*}$ , given by

\partial F (x) = {x^{*} \in X^{*} : F^{0} (x; v) \geq 〈 x^{*}, v 〉},

for all $v \in X$ .

Definition 1. An operator $G : X \to X^{*}$ on a reflexive Banach space is called

(i) monotone, if

〈 G x_{1} - G x_{2}, x_{1} - x_{2} 〉 \geq 0 for x_{1}, x_{2} \in X;

(ii) strong monotone, if there exists $m_{G} > 0$ such that

〈 G x_{1} - G x_{2}, x_{1} - x_{2} 〉 \geq m_{G} ∥ x_{1} - x_{2} ∥_{X}^{2} for x_{1}, x_{2} \in X;

(iii) pseudomonotone, if for every sequence ${x_{n}} \subseteq X$ converging weakly $x \in X$ such that $limsup 〈 G x_{n}, x_{n} - x 〉 \leq 0$ we have

\underset{n \to \infty}{liminf} 〈 G x_{n}, x_{n} - y 〉 \geq 〈 Gx, x - y 〉 for all y \in X .

Moreover, we will recall the notion of pseudomonotonicity of a multivalued operator.

Definition 2. A multivalued operator $T : X \to 2^{X^{*}}$ is pseudomonotone if:

for every $v \in X$ , the set $Tv \subset X^{*}$ is nonempty, closed, and convex;

$T$ is upper semicontinuous from each finite-dimensional subspace of $X$ to $X^{*}$ endowed with the weak topology;

for any sequences ${u_{n}} \subset X$ and ${u_{n}^{*}} \subset X^{*}$ , such that $u_{n} \to u$ weakly in $X, u_{n}^{*} \in T u_{n}$ for all $n \geq 1$ and $limsup 〈 u_{n}^{*}, u_{n} - u 〉 \leq 0$ , there exists $u^{*} (v)$ such that

〈 u^{*} (v), u - v 〉 \leq \underset{n \to \infty}{liminf} 〈 u_{n}^{*}, u_{n} - v 〉 for all v \in X .

Lemma 1 Lemma 5 of [10]. Let $X$ and $Y$ be reflexive Banach spaces. The operator $M : Y \to X$ is linear, bounded, and compact. Denote by $M^{*} : X^{*} \to Y^{*}$ the adjoint operator of $M$ . Let $J : X \to R$ be a locally Lipschitz function. There exists $m > 0$ such that

∥ \partial J (y) ∥_{X^{*}} \leq m (1 + ∥ y ∥_{X}),

for all $y \in X$ . Then the multivalued operator $Θ : Y \to 2^{Y^{*}}$ defined by $Θ (y) = M^{*} \partial J (M (y))$ for $y \in Y$ is pseudomonotone.

We recall the following fundamental surjectivity theorem.

Theorem 1. Theorem 1.3.70 of [25]. Let $X$ be a reflexive Banach space and $T : X \to 2^{X^{*}}$ be pseudomonotone and coercive. Then $T$ is surjective.

Lemma 2. [26]. The solution of

{\begin{matrix} β' (t) = A β (t) + B β (t - τ) + g (t, u (t), β (t)) for a . e . t \in I, \\ β (t) = φ (t), t \in [- τ, 0], \end{matrix}

(10)

has the form

β (t) = e^{A (t + τ)} e_{τ}^{B_{2} t} φ (- τ) + \int_{- τ}^{0} e^{A (t - s)} e_{τ}^{B_{2} (t - τ - s)} [φ' (s) - A φ (s)] d s + \int_{0}^{t} e^{A (t - τ - s)} e^{B_{2} (t - τ - s)} g (t, u (s), β (s)) d s,

(11)

where $B_{2} = e^{- A τ} B$ and $e_{τ}^{B_{2} t}$ is defined as

e_{τ}^{B_{2} t} = {\begin{matrix} Θ, & t < - τ, \\ E, & - τ \leq t < 0, \\ E + B_{2} t + \frac{B_{2}^{2} {(t - τ)}^{2}}{2} + \dots + \frac{B_{2}^{k} {(t - (k - 1) τ)}^{k}}{k!}, & (k - 1) τ \leq t \leq k τ, k = 1, 2, \dots . \end{matrix}

Here $Θ$ and $E$ are the $n$ -dimensional zero and identity matrices.

Lemma 3. Lemma 3 of [27]. For all $t \in R$ ,

∥ e_{τ}^{Bt} ∥ \leq e^{∥ B ∥ (t + τ)} .

Lemma 4. Lemma 4 of [26]. For all $t \in R$ ,

\frac{d}{d t} e_{τ}^{Bt} = {Be}_{τ}^{B (t - τ)} .

Finally, we give the generalized discrete Gronwall inequality.

Lemma 5. Lemma 2 of [28]. Assume that ${z_{n}}, {a_{n}}$ , and ${w_{n}}$ are nonnegative sequences and satisfy

z_{n} \leq a_{n} + \sum_{k = 1}^{n - 1} w_{k} z_{k} for n \geq 1 .

Then we have

z_{n} \leq v_{n} + \sum_{k = 1}^{n - 1} w_{k} a_{k} \exp (\sum_{j = k + 1}^{n - 1} w_{j}) for n \geq 1 .

If ${z_{n}}$ and ${w_{n}}$ satisfy

z_{n} \leq ρ + \sum_{k = 1}^{n - 1} w_{k} z_{k} for n \geq 1,

where $ρ > 0$ is a constant, we have

z_{n} \leq ρ \exp (\sum_{k = 1}^{n - 1} w_{k}) .

3. History-dependent fractional differential hemivariational inequality

In this paper, we consider the spaces $X, H, H$ , and $V$ , as follows:

X = {v \in H^{1} (Ω; R^{d}) | v = 0 on Γ_{D}}, H = L^{2} (Ω; R^{d}), H = L^{2} (Ω; S^{d}), and V = L^{2} (Γ_{C}) .

The space $H$ is endowed with the Hilbert structure

(σ, τ)_{H} = \int_{Ω} σ_{ij} τ_{ij} d x for σ, τ \in H,

and the associated norm $∥ \cdot ∥_{H}$ . For space $X$ , we consider the inner product

(u, v)_{X} = (ε (u), ε (v))_{H} for u, v \in X .

and the associated norm ∥·∥; $V$ is a separable and reflexive Banach space and has the associated norm $∥ \cdot ∥_{V}$ . We denote by $C$ the space of fourth-order tensor fields given by

C = {ε = (ε_{ijkl}) | ε_{ijkl} = ε_{jikl} = ε_{klij} \in L^{\infty} (Ω), 1 \leq i, j, k, l \leq d} .

From the Sobolev trace theorem, there exists the smallest constant $r_{0} > 0$ , such that

∥ v ∥_{L^{2} (Γ_{C}; R^{d})} \leq r_{0} ∥ v ∥ for v \in X .

To study Problem 1, we give some hypotheses on the relevant data.

$H (A)$ : the viscosity operator $A : Ω \times S^{d} \to S^{d}$ satisfies the following conditions:

$A (x, ε) = a (x) ε$ for a.e. $x \in Ω$ and $ε \in S^{d}$ .

$a (x) = (a_{ijkl} (x))$ with $a_{ijkl} \in L^{\infty} (Ω)$ .

There exists $m_{a} > 0$ such that

a_{ijkl} (x) ε_{ij} ε_{kl} \geq m_{a} ∥ ε ∥_{S^{d}}^{2}

for a.e. $x \in Ω$ .

$H (B)$ : the elasticity operator $B : Ω \times S^{d} \to S^{d}$ satisfies the following conditions:

$B (x, ε) = b (x) ε$ for a.e $x \in Ω$ and $ε \in S^{d}$ .

$b (x) = (b_{ijkl} (x))$ with $b_{ijkl} \in L^{\infty} (Ω)$ .

$H (L)$ : the relaxation $L : [0, T] \to C$ is Lipschitz continuous with constant $L_{L} > 0$ .

$H (Ψ)$ : $Ψ : Ω \times S^{d} \to R$ satisfies the following conditions:

$Ψ (\cdot, ε)$ is measurable on $Ω$ for all $ε \in S^{d}$ .

$Ψ (x, \cdot)$ is locally Lipschitz continuous on $x \in Ω$ .

$∥ \partial Ψ (x, ε) ∥ \leq c_{Ψ} (1 + ∥ ε ∥_{S^{d}})$ for $x \in Ω, ε \in S^{d}$ with $c_{Ψ} > 0$ .

There exists a constant $c_{Ψ_{1}} > 0$ such that

Ψ^{0} (x, ε_{1}; ε_{2} - ε_{1}) + Ψ^{0} (x, ε_{2}; ε_{1} - ε_{2}) \leq c_{Ψ_{1}} ∥ ε_{1} - ε_{2} ∥_{S^{d}}^{2}

for $x \in Ω, ε_{i} \in S^{d}$ , $i = 1, 2$ .

$H (j_{ν})$ : the function $j_{ν} : Γ_{C} \times R \times R \to R$ is such that

$j_{ν} (\cdot, r, s)$ is measurable on $Γ_{C}$ for all $r, s \in R$ , and $j_{ν} (\cdot, 0, 0) \in L^{1} (Γ_{C})$ .

$j_{ν} (x, r, \cdot)$ is locally Lipschitz on $R$ for all $r \in R$ and $x \in Γ_{C}$ .

There exists $c_{j_{ν}} > 0$ such that $| \partial j_{ν} (x, r, s) | \leq c_{j_{ν}} (1 + | s |)$ for all $r, s \in R$ and a.e. $x \in Γ_{C}$ .

$j_{ν}^{0} (x, r_{1}, s_{1}; s_{2} - s_{1}) + j_{ν}^{0} (x, r_{2}, s_{2}; s_{1} - s_{2}) \leq α_{j_{ν}} (| s_{1} - s_{2} |^{2} + | r_{1} - r_{2} | | s_{1} - s_{2} |)$ for $x \in Γ_{C}$ , all $r_{1}, r_{2} \in R$ with $α_{j_{ν}} > 0$ .

$(r, ε) \to j_{ν}^{0} (x, r, ε; z)$ is upper semicontinuous for a.e. $x \in Γ_{C}$ , all $z \in R$ .

$H (f)$ : the force $f_{0}, f_{N}$ satisfies the following conditions:

f_{0} \in L^{\infty} ([0, T]; L^{2} (Ω, R^{d})), f_{N} \in L^{\infty} ([0, T]; L^{2} (Γ_{N}, R^{d})) .

$H (F)$ the adhesive evolution rate function $F : Γ_{C} \times [0, T] \times R^{d} \times R \to R$ is such that

$F (\cdot, \cdot, ξ, r)$ is measurable on $Γ_{C} \times [0, T]$ for all $(ξ, r) \in R^{d} \times R$ .

$| F (x, t, ξ_{1}, r_{1}) - F (x, t, ξ_{2}, r_{2}) | \leq L_{F} (∥ ξ_{1} - ξ_{2} ∥ + | r_{1} - r_{2} |)$ for a.e. $(x, t) \in Γ_{C} \times [0, T]$ and all $(ξ_{i}, r_{i}) \in R^{d} \times R, i = 1, 2$ with $L_{F} > 0$ .

There exists $ϕ \in L^{p} [0, T]$ satisfying

∥ F (t, x, ξ, r) ∥ \leq ϕ (t) for all (t, ξ, r) \in (0, T) \times R^{d} \times R, x \in Γ_{C} .

$H (AB)$ $d \times d$ matrices $A$ and $B$ satisfy $AB = BA .$

$H (φ)$ $φ \in C_{τ}^{1} = C^{1} ([- τ, 0], R^{d})$ .

It is necessary to emphasize here that because of the form of the solution of equation (10), $φ$ satisfies $H (φ)$ .

Next, employing Green’s formula, it is not difficult to obtain the following variational formulation.

Problem 2. Find a displacement vector $u \in W^{1, 2} (0, T; Ω)$ and $β \in W^{1, 2} (- τ, T; L^{2} (Γ_{C}))$ , such that

{\begin{matrix} {〈 A (ε ({{}_{0}^{C}D}_{t}^{α} (u (t)))) + (B ε (u (t))) + \partial Ψ (ε (u (t))), ε (v) 〉}_{H} + 〈 \int_{0}^{t} L (t - s) ε (u (s)) d s, ε (u (t)) 〉 \\ + \int_{Γ_{C}} j_{ν}^{0} (β (t), {{}_{0}^{C}D}_{t}^{α} u_{ν} (t); v_{ν}) d Γ \geq {〈 f_{0} (t), v 〉}_{H} + {〈 f_{N} (t), v 〉}_{L^{2} (Γ_{N}; R^{d})}, for all v \in V, a . e . t \in (0, T), \\ β' (t) = A β (t) + B β (t - τ) + F (t, u (t), β (t)) on Γ_{C} \times (0, T), \\ β (t) = φ (t), on Γ_{C} \times [- τ, 0], \\ u (0) = u_{0} in Ω . \end{matrix}

To this end, we need to define the following operators:

\begin{matrix} 〈 A_{1} u, v 〉_{X^{*} \times X} = 〈 A (ε (u)), ε (v) 〉_{H}, \\ 〈 B_{1} u, v 〉_{X^{*} \times X} = 〈 B (ε (u)), ε (v) 〉_{H}, \\ 〈 \partial ψ (u (t)), v 〉_{X^{*} \times X} = 〈 \partial Ψ (ε (u (t))), ε (v) 〉_{H}, \\ 〈 f (t), v 〉 = 〈 f_{0} (t), v 〉_{H} + 〈 f_{N} (t), v 〉_{L^{2} (Γ_{N}; R^{d})} \end{matrix}

and

J^{0} (β, {{}_{0}^{C}D}_{t}^{α} u; v) = \int_{Γ_{C}} j_{ν}^{0} (β, {{}_{0}^{C}D}_{t}^{α} u_{ν}; v_{ν}) d Γ .

Meanwhile, we define the operators $R : X \to X^{*}$ by

〈 (R u) (t), v 〉_{X^{*} \times X} = 〈 \int_{0}^{t} L (t - s) ε (u (s)) d s, ε (v) 〉,

for all $u, v \in X$ , a.e. $t \in [0, T]$ . Moreover, let $M = r_{0}$ and $g : [0, T] \times X \times V \to V$ be defined by

g (t, u, β) (x) = F (x, t, β (x), u (x)) for all β \in V, u \in X, x \in Γ_{C} .

Meanwhile, by these definitions, Problem 2 can be reformulated as the following abstract fractional differential hemivariational inequality:

{\begin{matrix} β' (t) = A β (t) + B β (t - τ) + g (t, u (t), β (t)) for a . e . t \in [0, T], \\ 〈 A_{1} ({{}_{0}^{C}D}_{t}^{α} u (t)) + B_{1} (u (t)) + \partial ψ (u (t)), v 〉 + 〈 (R u) (t), v 〉 + J^{0} (β (t), M ({{}_{0}^{C}D}_{t}^{α} u (t)); Mv) \geq 〈 f (t), v 〉, \\ u (0) = u_{0} for v \in X, a . e . t \in I, \\ β (t) = φ (t), t \in [- τ, 0] . \end{matrix}

(12)

Conversely, by conditions $H (A), H (B), H (F), H (Ψ)$ , and $H (j)$ , we come to the following conclusions.

$H (A_{1})$ the operator $A_{1} : X \to X^{*}$ is linear, bounded, coercive, i.e.:

$A_{1} \in L (X, X^{*})$ .

There exists $m_{A_{1}} > 0$ such that

〈 A_{1} v, v 〉 \geq m_{A_{1}} ∥ v ∥^{2} for all v \in X .

$H (B_{1})$ $B_{1} \in L (X, X^{*})$ .

$H (ψ)$ $ψ : X \to R$ satisfies the following conditions:

$ψ$ is locally Lipschitz.

$∥ \partial ψ (x) ∥_{X^{*}} \leq c_{ψ} (1 + ∥ x ∥)$ for all $x \in X$ with $c_{ψ} > 0$ .

There exists a constant $c_{ψ_{1}} > 0$ such that

〈 ξ_{1} - ξ_{2}, θ_{1} - θ_{2} 〉 \geq - c_{ψ_{1}} ∥ θ_{1} - θ_{2} ∥^{2}

for all $ξ_{i} \in \partial ψ (θ_{i}), θ_{i} \in X$ .

$H (J)$ the function $J : V \times X \to R$ satisfies the following hypotheses:

$J (\cdot, z)$ is continuous for all $z \in X$ .

$J (v, \cdot)$ is locally Lipschitz for all $v \in V$ .

$∥ \partial J (v, z) ∥_{X^{*}} \leq c_{J} (1 + ∥ z ∥),$ for all $z \in X, v \in V$ with $c_{J} > 0$ .

$J^{0} (v_{1}, z_{1}; z_{2} - z_{1}) + J^{0} (v_{2}, z_{2}; z_{1} - z_{2}) \leq m_{J} ∥ z_{1} - z_{2} ∥^{2} + m_{J_{1}} ∥ z_{1} - z_{2} ∥ ∥ v_{1} - v_{2} ∥_{V}$

for all $β_{i} \in V, u_{i} \in X, i = 1, 2$ , a.e. $t \in I$ with $m_{J}, m_{J_{1}} > 0$ .

$(v, z) \to J^{0} (v, z; y)$ is upper semicontinuous for all $y \in X$ .

$H (f)$ $f \in L^{\infty} (I; X^{*})$ .

$H (g)$ , where the function $g : I \times V \times X \to V$ , satisfies the following hypotheses:

(i) For all $(v, z) \in V \times X, g (\cdot, v, z)$ is measurable.

(ii) There exists $L_{g} > 0$ satisfying

∥ g (t, v_{1}, z_{1}) - g (t, v_{2}, z_{2}) ∥ \leq L_{g} (∥ v_{1} - v_{2} ∥_{V} + ∥ z_{1} - z_{2} ∥),

for all $(t, v_{i}, z_{i}) \in I \times V \times X, i = 1, 2$ .

(iii) There exists $ϕ \in L^{p} [I]$ and $p \in (1, \infty)$ satisfying

∥ g (t, v, z) ∥_{V} \leq ϕ (t),

for all $(t, v, z) \in I \times V \times X$ .

Remark 1. According to conditions $H (A)$ , $H (B)$ , $H (F)$ , $H (Ψ)$ , and $H (j)$ , it is not difficult to get $H (A_{1})$ , $H (B_{1})$ , $H (g)$ , $H (ψ)$ , and $H (J)$ . Here, we will give the relationship of some parameters. $m_{A_{1}} = m_{a}$ , $c_{ψ} = c_{Ψ}$ , $c_{ψ_{1}} = c_{Ψ_{1}}$ , $c_{J} = c_{j_{ν}} meas (Γ_{C}) ∥ r_{0} ∥$ , $m_{J} = α_{j_{ν}} ∥ r_{0} ∥^{2} meas (Γ_{C})$ , $m_{J_{1}} = α_{j_{ν}} ∥ r_{0} ∥ meas (Γ_{C})$ , and $r_{0} = M$ .

Here are our main results.

Theorem 2. Under the assumptions of $H (AB)$ , $H (A_{1})$ , $H (B_{1})$ , $H (J)$ , $H (f)$ , $H (ψ)$ , and $H (g)$ , for any given $β \in W^{1, 2} (0, T; V)$ , and if the inequalities

m_{A_{1}} > c_{J} ∥ M ∥^{2} + \frac{(c_{ψ} + c_{L} T + ∥ B_{1} ∥) T^{α}}{Γ (α + 1)} and m_{A_{1}} > m_{J} ∥ M ∥^{2} + c_{ψ_{1}} + \frac{c_{L} T^{α + 1}}{Γ (α + 1)} + \frac{∥ B_{1} ∥ T^{α}}{Γ (α + 1)}

hold, the history-dependent fractional differential hemivariational inequality has a unique solution. Here, $c_{L} = max_{(t, s) \in I \times I} L (t - s)$ .

Theorem 3. Under the assumptions of $H (AB)$ , $H (A_{1})$ , $H (B_{1})$ , $H (J)$ , $H (f)$ , $H (ψ)$ , and $H (g)$ , and if the inequalities

m_{A_{1}} > c_{J} ∥ M ∥^{2} + \frac{(c_{ψ} + c_{L} T + ∥ B_{1} ∥) T^{α}}{Γ (α + 1)} and m_{A_{1}} > m_{J} + c_{ψ_{1}} + \frac{c_{L} T^{α + 1}}{Γ (α + 1)} + \frac{∥ B_{1} ∥ T^{α}}{Γ (α + 1)}

hold, then equation (12) has at least one solution.

According to Lemma 2, we give the following definition.

Definition 3. A pair $(u, β) \in W^{1, 2} (0, T; X) \times W^{1, 2} (- τ, T; V)$ is said to be a solution of equation (12) if it satisfies the following system:

{\begin{matrix} β (t) = e^{A (t + τ)} e_{τ}^{B_{2} t} φ (- τ) + \int_{- τ}^{0} e^{A (t - s)} e_{τ}^{B_{2} (t - τ - s)} [φ' (s) - A φ (s)] d s + \int_{0}^{t} e^{A (t - τ - s)} e^{B_{2} (t - τ - s)} g (s, u (s), β (s)) d s, \\ 〈 A_{1} ({{}_{0}^{C}D}_{t}^{α} u (t)) + B_{1} (u (t)), v 〉 + 〈 (R u) (t), v 〉 + 〈 \partial ψ (u (t)), v 〉 + J^{0} (β (t), M ({{}_{0}^{C}D}_{t}^{α} u (t)); Mv) \geq 〈 f (t), v 〉, \\ u (0) = u_{0}, for v \in X, a . e . t \in I, \\ β (t) = φ (t), t \in [- τ, 0] . \end{matrix}

(13)

4. Existence solution to contact problem

To study Theorem 2, we first consider the following history-dependent fractional differential hemivariational inequality.

Problem 3. For any given $β \in W^{1, 2} (0, T; V)$ , find $u \in W^{1, 2} (0, T; X)$ such that

{\begin{matrix} 〈 A_{1} ({{}_{0}^{C}D}_{t}^{α} u (t)) + B_{1} (u (t)) + (R_{1} u) (t), v 〉 + ψ^{0} (u (t); v) + J^{0} (β (t), M ({{}_{0}^{C}D}_{t}^{α} u (t)); Mv) \\ \geq 〈 f (t), v 〉 for all v \in X, and a . e . t \in I, \\ x (0) = x_{0}, \end{matrix}

(14)

where $(R_{1} u) : C (0, T; X) \to C (0, T; X^{*})$ is an operator defined by

(R_{1} u) (t) = E (\int_{0}^{t} q (t, s) u (s) d s + μ),

where $E \in L (X, X^{*}), μ \in X$ , $q : I \times I \to L (X, X)$ satisfies $H (q)$ , and there exists $L_{q} > 0$ such that

∥ q (t_{1}, s) - q (t_{2}, s) ∥ \leq L_{q} | t_{1} - t_{2} |, for all t_{1}, t_{2}, s \in [0, T] .

In this paper, we set $c_{E} = ∥ E ∥$ and $c_{q} = max_{(t, s) \in [0, T] \times [0, T]} ∥ q (t, s) ∥$ .

Let $u \in C (0, T; X)$ be a solution to Problem 3 and $θ = {{}_{0}^{C}D}_{t}^{α} u$ . By Proposition 1, we have

u (t) =_{0} I_{t}^{α} θ (t) + u_{0}, for a . e . t \in I .

Therefore, we can transform Problem 3 into the following problem.

Problem 4. For any given $β \in W^{1, 2} (0, T; V)$ , find $θ \in L^{1} (0, T; X)$ such that

{\begin{matrix} 〈 A_{1} (θ (t)), v 〉 + 〈 B_{1} (_{0} I_{t}^{α} θ (t) + u_{0}) + R_{1} (_{0} I_{t}^{α} θ (t) + u_{0}), v 〉 + ψ^{0} (_{0} I_{t}^{α} θ (t) + u_{0}; v) + J^{0} (β (t), M (θ (t)); Mv) \\ \geq 〈 f (t), v 〉 for all v \in X and a . e . t \in I . \end{matrix}

(15)

Through the definition of the generalized directional derivative, we can easily get the following problem.

Problem 5. For any given $β \in W^{1, 2} (0, T; V)$ , find $θ \in L^{1} (0, T; X)$ such that

{\begin{matrix} A_{1} (θ (t)) + B_{1} (_{0} I_{t}^{α} θ (t) + u_{0}) + R_{1} (_{0} I_{t}^{α} θ (t) + u_{0}) + M^{*} \partial J (β (t), M (θ (t))) + \partial ψ (_{0} I_{t}^{α} θ (t) + u_{0}) ∋ f (t) \\ for a . e . t \in I . \end{matrix}

(16)

Motivated by the methods of [10, 15, 16], we will present a result on existence and uniqueness of solution for Problem 5.

Theorem 4. Under the assumptions of $H (AB)$ , $H (A_{1})$ , $H (B_{1})$ , $H (J)$ , $H (f)$ , $H (ψ)$ , and $H (g)$ , for any given $β \in W^{1, 2} (0, T; V)$ , and if the inequalities

m_{A_{1}} > c_{J} ∥ M ∥^{2} + \frac{(c_{ψ} + c_{E} c_{q} T + ∥ B_{1} ∥) T^{α}}{Γ (α + 1)} and m_{A_{1}} > m_{J} ∥ M ∥^{2} + c_{ψ_{1}} + \frac{c_{E} c_{q} T^{α + 1}}{Γ (α + 1)} + \frac{∥ B_{1} ∥ T^{α}}{Γ (α + 1)}

hold. Then the history-dependent fractional differential hemivariational inequality (equation (16)) has a unique solution.

To reach our conclusion, we discuss the following discretized problem corresponding to Problem 5, called the Rothe problem.

Problem 6. Find ${θ_{τ}^{k}}_{k = 1}^{N} \subset X$ such that

A_{1} θ_{τ}^{k} + B_{1} u_{τ}^{k} + y_{τ}^{k} + λ_{τ}^{k} + M^{*} ζ_{τ}^{k} ∋ f_{τ}^{k},

(17)

and

θ_{τ}^{0} = 0,

for $k = 1, 2, \dots, N$ , where $u_{τ}^{k}$ is defined by

u_{τ}^{k} = u_{0} + \frac{τ^{α}}{Γ (α + 1)} \sum_{j = 1}^{k} θ_{τ}^{j} [(k - j + 1)^{α} - (k - j)^{α}],

(18)

and $y_{τ}^{k}$ is defined by

y_{τ}^{k} = (R_{1} u_{τ}^{k}) (t) = E (Σ_{j = 1}^{k} \int_{t_{j - 1}}^{t_{j}} q (t_{k}, s) u_{τ}^{j} (s) d s + μ),

(19)

and $f_{τ}^{k}$ is defined by

f_{τ}^{k} = \frac{1}{τ} \int_{t_{k - 1}}^{t_{k}} f (s) d s,

and $λ_{τ}^{k} \in \partial ψ (u_{τ}^{k})$ , $ζ_{τ}^{k} \in \partial J (β_{τ}^{k}, M (θ_{τ}^{k}))$ , $τ = T / N$ , $t_{k} = k τ$ , and

β_{τ}^{k} = \frac{1}{τ} \int_{t_{k - 1}}^{t_{k}} β (s) d s .

Next, we will prove the existence of a solution to Problem 6.

Indeed, from equation (18) and condition $H (B_{1})$ , we imply that

\begin{matrix} 〈 B_{1} u_{τ}^{n}, θ_{τ}^{n} 〉 = 〈 B_{1} (u_{0} + \frac{τ^{α}}{Γ (α + 1)} \sum_{j = 1}^{n} θ_{τ}^{j} [(n - j + {1)}^{α} - {(n - j)}^{α}]), θ_{τ}^{n} 〉 \\ \geq - ∥ B_{1} u_{0} ∥_{X^{*}} ∥ θ_{τ}^{n} ∥ - \frac{τ^{α}}{Γ (α + 1)} \sum_{j = 1}^{n - 1} [(n - j + 1)^{α} - (n - j)^{α}] ∥ B_{1} ∥ ∥ θ_{τ}^{j} ∥ ∥ θ_{τ}^{n} ∥ - \frac{τ^{α} ∥ B_{1} ∥}{Γ (α + 1)} ∥ θ_{τ}^{n} ∥^{2} . \end{matrix}

(20)

According to equations (19) and (18), we have

\begin{matrix} 〈 y_{τ}^{n}, θ_{τ}^{n} 〉 = 〈 E (Σ_{j = 1}^{n} \int_{t_{j - 1}}^{t_{j}} q (t_{n}, s) u_{τ}^{j} (s) d s + μ), θ_{τ}^{n} 〉 \\ \leq ∥ θ_{τ}^{n} ∥ ‖ E (Σ_{j = 1}^{n - 1} \int_{t_{j - 1}}^{t_{j}} q (t_{n}, s) u_{τ}^{j} (s) d s + μ) ‖_{X^{*}} + c_{q} c_{E} τ ∥ u_{τ}^{n} ∥ ∥ θ_{τ}^{n} ∥ + c_{E} μ ∥ θ_{τ}^{n} ∥ \\ \leq ∥ θ_{τ}^{n} ∥ ‖ E (Σ_{j = 1}^{n - 1} \int_{t_{j - 1}}^{t_{j}} q (t_{n}, s) u_{τ}^{j} (s) d s + μ) ‖_{X^{*}} + c_{q} c_{E} τ ∥ u_{τ}^{n} ∥ ∥ θ_{τ}^{n} ∥ + c_{E} μ ∥ θ_{τ}^{n} ∥ \\ \leq ∥ θ_{τ}^{n} ∥ ‖ E (Σ_{j = 1}^{n - 1} \int_{t_{j - 1}}^{t_{j}} q (t_{n}, s) u_{τ}^{j} (s) d s + μ) ‖_{X^{*}} \\ + c_{q} c_{E} τ (∥ u_{0} ∥ ∥ θ_{τ}^{n} ∥ + \frac{τ^{α} ∥ θ_{τ}^{n} ∥}{Γ (α + 1)} \sum_{j = 1}^{n - 1} ∥ θ_{τ}^{j} ∥ [(n - j + 1)^{α} - (n - j)^{α}] + \frac{c_{ψ} τ^{α} ∥ θ_{τ}^{n} ∥^{2}}{Γ (α + 1)}) + c_{E} μ ∥ θ_{τ}^{n} ∥ . \end{matrix}

(21)

According to $H (J)$ , we have

\begin{matrix} 〈 \partial J (β_{τ}^{n} (t), M (θ_{τ}^{n} (t))), M θ_{τ}^{n} 〉 \geq - ∥ \partial J (β_{τ}^{n} (t), M (θ_{τ}^{n} (t))) ∥_{X^{*}} ∥ M θ_{τ}^{n} ∥ \\ \geq - c_{J} (1 + ∥ M θ_{τ}^{n} ∥) ∥ M θ_{τ}^{n} ∥ \\ \geq - c_{J} ∥ M ∥ ∥ θ_{τ}^{n} ∥ - c_{J} ∥ M ∥^{2} ∥ θ_{τ}^{n} ∥^{2} . \end{matrix}

(22)

Conversely, according to $H (ψ)$ , we have

\begin{matrix} 〈 λ_{τ}^{n}, θ_{τ}^{n} 〉 \geq - c_{ψ} (1 + ∥ u_{τ}^{n} ∥) ∥ θ_{τ}^{n} ∥ \\ = - c_{ψ} (1 + u_{0} + \frac{τ^{α}}{Γ (α + 1)} \sum_{j = 1}^{n} ∥ θ_{τ}^{j} ∥ [(n - j + {1)}^{α} - {(n - j)}^{α}]) ∥ θ_{τ}^{n} ∥ \\ \geq - c_{ψ} (1 + ∥ u_{0} ∥) ∥ θ_{τ}^{n} ∥ - \frac{c_{ψ} τ^{α} ∥ θ_{τ}^{n} ∥}{Γ (α + 1)} \sum_{j = 1}^{n - 1} ∥ θ_{τ}^{j} ∥ [(n - j + 1)^{α} - (n - j)^{α}] - \frac{c_{ψ} τ^{α} ∥ θ_{τ}^{n} ∥^{2}}{Γ (α + 1)} . \end{matrix}

(23)

It follows from equations (20), (21) (22), and (23) and condition $H (A_{1})$ that

\begin{matrix} 〈 f_{τ}^{n}, θ_{τ}^{n} 〉 = 〈 A_{1} θ_{τ}^{n}, θ_{τ}^{n} 〉 + 〈 B_{1} u_{τ}^{n}, θ_{τ}^{n} 〉 + 〈 \partial J (β_{τ}^{n} (t), M (θ_{τ}^{n} (t))), M (θ_{τ}^{n}) 〉 + 〈 y_{τ}^{n}, θ_{τ}^{n} 〉 + 〈 λ_{τ}^{n}, θ_{τ}^{n} 〉 \\ \geq (m_{A_{1}} - c_{J} ∥ M ∥^{2} - \frac{(c_{ψ} + c_{E} c_{q} τ + ∥ B_{1} ∥) τ^{α}}{Γ (α + 1)}) ∥ θ_{τ}^{n} ∥^{2} - (c_{J} ∥ M ∥ \\ + ∥ B_{1} u_{0} ∥_{X^{*}} + c_{ψ} (1 + ∥ u_{0} ∥) + c_{E} c_{q} τ ∥ u_{0} ∥ + c_{E} μ) ∥ θ_{τ}^{n} ∥ \\ - ‖ E (Σ_{j = 1}^{n - 1} \int_{t_{j - 1}}^{t_{j}} q (t_{n}, s) u_{τ}^{j} (s) d s + μ) ‖_{X^{*}} ∥ θ_{τ}^{n} ∥ \\ - \frac{(c_{ψ} + c_{E} c_{q} τ + ∥ B_{1} ∥) τ^{α} ∥ θ_{τ}^{n} ∥}{Γ (α + 1)} \sum_{j = 1}^{n - 1} ∥ θ_{τ}^{j} ∥ [(n - j + 1)^{α} - (n - j)^{α}] . \end{matrix}

For the convenience of writing, let $M_{k} = \sum_{j = 1}^{k - 1} ∥ θ_{τ}^{j} ∥ [(k - j + 1)^{α} - (k - j)^{α}]$ and

$M_{k}^{1} = E (Σ_{j = 1}^{k - 1} \int_{t_{k - 1}}^{t_{k}} q (t_{k}, s) u_{τ}^{k} (s) d s + μ)$ . Further, we have

\begin{matrix} ∥ f_{τ}^{n} ∥ + c_{J} ∥ M ∥ + ∥ B_{1} u_{0} ∥_{X^{*}} + c_{ψ} (1 + ∥ u_{0} ∥) + c_{E} c_{q} τ ∥ u_{0} ∥ + c_{E} μ + \frac{(c_{ψ} + c_{E} c_{q} τ + ∥ B_{1} ∥) τ^{α}}{Γ (α + 1)} M_{n} + M_{n}^{1} \\ \geq (m_{A_{1}} - c_{J} ∥ M ∥^{2} - \frac{(c_{ψ} + c_{E} c_{q} τ + ∥ B_{1} ∥) τ^{α}}{Γ (α + 1)}) ∥ θ_{τ}^{n} ∥ . \end{matrix}

Choose

τ_{0} = {(\frac{Γ (α + 1) (m_{A_{1}} / 2 - c_{J} ∥ M ∥^{2})}{c_{ψ} + c_{E} c_{q} τ + ∥ B_{1} ∥})}^{1 / α};

for $τ \in (0, τ_{0})$ , we have

m_{A_{1}} - c_{J} ∥ M ∥^{2} - \frac{(c_{ψ} + c_{E} c_{q} T + ∥ B_{1} ∥) τ^{α}}{Γ (α + 1)} > \frac{m_{A_{1}}}{2} .

(24)

Then we have

\begin{matrix} 2 (∥ f_{τ}^{n} ∥ + c_{J} ∥ M ∥ + ∥ B_{1} u_{0} ∥_{X^{*}} + c_{ψ} (1 + ∥ u_{0} ∥) + c_{E} c_{q} τ ∥ u_{0} ∥ + c_{E} μ + \frac{(c_{ψ} + c_{E} c_{q} τ + ∥ B_{1} ∥) τ^{α}}{Γ (α + 1)} M_{n} + M_{n}^{1}) \\ \geq ∥ θ_{τ}^{n} ∥ . \end{matrix}

(25)

Lemma 6. Assume that conditions $H (A_{1})$ , $H (B_{1})$ , $H (J)$ , $H (f)$ , $H (E)$ , $H (q)$ , and $H (ψ)$ hold. There exists $τ_{0} > 0$ such that, for all $τ \in (0, τ_{0})$ , equation (17) has at least one solution.

Proof. According to equation (18), we get $u_{τ}^{0}, u_{τ}^{1}, u_{τ}^{2}, \dots, u_{τ}^{n - 1}$ . Conversely, to get our conclusions, we need to prove that the multivalued operator

X ∋ v \mapsto A_{1} v + B_{1} (v_{0} + c_{0} v) + R_{1} (v_{0} + c_{0} v) + M^{*} \partial J (β, Mv) + \partial ψ (v_{0} + c_{0} v)

is surjective, where

v_{0} = u_{0} + \frac{τ^{α}}{Γ (α + 1)} \sum_{j = 1}^{n - 1} θ_{τ}^{j} [(n - j + 1)^{α} - (n - j)^{α}] and c_{0} = \frac{τ^{α}}{Γ (α + 1)} .

We introduce two multivalued operators $T_{1}, T_{2} : X \to 2^{X^{*}}$ :

T_{1} v = A_{1} v + B_{1} (v_{0} + c_{0} v) + R_{1} (v_{0} + c_{0} v) + \partial ψ (v_{0} + c_{0} v) and T_{2} v = M^{*} \partial J (β, Mv)

We claim that $T_{1}$ is pseudomonotonic. Indeed, by $H (A_{1}), H (B_{1}), H (q)$ , and Lemma 20 of [29], we can easily obtain it. Here we briefly prove it. First, for all $v \in X$ , the set $A_{1} v + B_{1} (v_{0} + c_{0} v) + R_{1} (v_{0} + c_{0} v) + \partial ψ (v_{0} + c_{0} v)$ is nonempty, closed, and convex in $X^{*}$ . By Proposition 5 of [29], we imply that $T_{1}$ is generalized pseudomonotonic. Next, we show that the operator $T_{1}$ is strongly monotonic. By conditions $H (A_{1})$ , $H (B_{1})$ , $H (q)$ , and $H (ψ)$ , we have

〈 T_{1} v_{1} - T_{1} v_{2}, v_{1} - v_{2} 〉 \geq (m_{A_{1}} - c_{0} ∥ B_{1} ∥ - c_{E} c_{q} T - c_{Ψ_{1}}) ∥ v - u ∥^{2} for v, u \in X .

Let $u_{n} \in X, u_{n} \to u$ weakly in $X$ , $u_{n}^{*} \in T_{1} u_{n}, u_{n}^{*} \to u^{*}$ weakly in $X^{*}$ , and $limsup 〈 u_{n}^{*}, u_{n} - u 〉 \leq 0$ . By the strong monotonicity of $T_{1}$ , we have

(m_{A_{1}} - c_{0} ∥ B_{1} ∥ - c_{E} c_{q} T - c_{Ψ_{1}}) ∥ u_{n}^{*} - u ∥_{X}^{2} \leq 〈 u_{n}^{*}, u_{n} - u 〉 - 〈 T_{1} u, u_{n} - u 〉,

we imply that $u_{n} \to u$ in $X$ . By conditions $H (A_{1})$ , $H (B_{1})$ , $H (q)$ , and $H (ψ)$ and Lemma 20 of [29], we have $u^{*} \in T_{1} u$ . By $u_{n}^{*} \to u^{*}$ weakly in $X^{*}$ and $u_{n} \to u$ in $X$ , we have $〈 u_{n}^{*}, u_{n} 〉 \to 〈 u^{*}, u 〉$ . Therefore, $T_{1}$ is generalized pseudomonotonic and pseudomonotonic, which completes the pro $V ∋ v \to M^{*} \partial J (β (t), Mv) \subset X^{*}$ is pseudomonotonic. Indeed, according to $H (M)$ and $H (J)$ , we have

∥ M^{*} \partial J (β (t), Mv) ∥ \leq c_{J} (∥ M ∥ + ∥ M ∥^{2} ∥ v ∥),

for all $v \in X$ . Moreover, by Lemma 1, we complete the proof of the assertion. Conversely, according to $H (ψ)$ , we have

\partial ψ (v_{0} + c_{0} v) \leq c_{ψ} (1 + ∥ v_{0} + c_{0} v ∥) \leq c_{ψ} + c_{ψ} ∥ v_{0} ∥ + c_{ψ} c_{0} ∥ v ∥,

for all $v \in X$ . Let $z_{n} \to z$ in $X$ and $η_{n} \to η$ weakly in $X^{*}$ with $η_{n} (t) \in \partial ψ (v_{0} + c_{0} z_{n})$ , by Proposition 2.4 of [30], we have

η (t) \in \partial ψ (v_{0} + c_{0} z (t)) .

According to the definition of pseudomonotonicity and $H (ψ)$ , $\partial ψ (v_{0} + c_{0} v)$ is pseudomonotonic. Therefore,

T_{1} v + T_{2} v is pseudomonotonic .

Meanwhile, take

τ_{0} = {(\frac{Γ (α + 1) (m_{A_{1}} / 2 - c_{J} ∥ M ∥^{2})}{c_{ψ} + c_{E} c_{q} T + ∥ B_{1} ∥})}^{1 / α};

if $τ \in (0, τ_{0})$ and by equation (24), we obtain $c_{J} ∥ M ∥^{2} + c_{0} ∥ B_{1} ∥ + c_{ψ} c_{0} + c_{E} c_{q} T < m_{A_{1}}$ . It follows from equation (25) and Theorem 1 that the operator $A_{1} v + B_{1} (v_{0} + c_{0} v) + R_{1} (v_{0} + c_{0} v) + M^{*} \partial J (β, Mv) + \partial ψ (v_{0} + c_{0} v)$ is surjective. According to Lemma 1, there exist $u_{τ}^{n} \in X$ , $η_{τ}^{n} \in \partial (β_{τ}^{n}, M (θ_{τ}^{n}))$ and $λ_{τ}^{n} \in \partial ψ (u_{τ}^{n})$ , such that equation (17) holds. We complete the proof of the lemma. □

Lemma 7. Assume that conditions $H (A_{1})$ , $H (B_{1})$ , $H (J)$ , $H (ψ)$ , $H (E)$ , $H (q)$ , and $H (f)$ hold. For all $τ \in (0, τ_{0})$ , the solutions to equation (17) satisfy the following inequalities:

max_{k = 1, 2, 3, \dots, N} ∥ θ_{τ}^{k} ∥ \leq C,

(26)

max_{k = 1, 2, 3, \dots, N} ∥ u_{τ}^{k} ∥ \leq C,

(27)

max_{k = 1, 2, 3, \dots, N} ∥ ζ_{τ}^{k} ∥ \leq C,

(28)

max_{k = 1, 2, 3, \dots, N} ∥ y_{τ}^{k} ∥_{X^{*}} \leq C,

(29)

max_{k = 1, 2, 3, \dots, N} ∥ λ_{τ}^{k} ∥_{X^{*}} \leq C,

(30)

and

max_{k = 1, 2, 3, \dots, N} ∥ f_{τ}^{k} ∥_{X^{*}} \leq C,

(31)

where $ζ_{τ}^{k} \in \partial J (β_{τ}^{k}, M (θ_{τ}^{k}))$ and $λ_{τ}^{k} \in \partial ψ (u_{τ}^{k})$ satisfy

A (θ_{τ}^{k}) + B (u_{τ}^{k}) + y_{τ}^{k} + M^{*} ζ_{τ}^{k} + λ_{τ}^{k} = f_{τ}^{k},

for $k = 1, 2, \dots, N .$

Proof. According to condition $H (f)$ , we can easily infer that there exists $m_{f} > 0$ , such that

max_{k = 1, 2, 3, \dots, N} ∥ f_{τ}^{k} ∥_{X^{*}} \leq m_{f},

for all $τ > 0$ . Conversely, taking $k = n$ , it follows from equation (17) that

〈 A_{1} θ_{τ}^{n}, θ_{τ}^{n} 〉 + 〈 B_{1} u_{τ}^{n}, θ_{τ}^{n} 〉 + 〈 y_{τ}^{n}, θ_{τ}^{n} 〉 + 〈 λ_{τ}^{n}, θ_{τ}^{n} 〉 + 〈 ζ_{τ}^{n}, M θ_{τ}^{n} 〉 = 〈 f_{τ}^{n}, θ_{τ}^{n} 〉 .

According to conditions $H (E)$ and $H (q)$ , we have

\begin{matrix} ‖ E (Σ_{j = 1}^{n - 1} \int_{t_{j - 1}}^{t_{j}} q (t_{n}, s) u_{τ}^{j} (s) d s + μ) ‖_{X^{*}} \\ \leq c_{E} c_{q} T Σ_{j = 1}^{n - 1} ∥ u_{τ}^{j} ∥ + μ ∥ E ∥ \\ \leq c_{E} c_{q} T Σ_{j = 1}^{n - 1} ∥ u_{0} + \frac{τ^{α}}{Γ (α + 1)} \sum_{i = 1}^{j} θ_{τ}^{j} [(j - i + 1)^{α} - (j - i)^{α}] ∥ + μ ∥ E ∥ \end{matrix}

\begin{matrix} \leq c_{E} c_{q} T (n - 1) ∥ u_{0} ∥ + μ c_{E} + c_{E} c_{q} T (n - 1) \frac{τ^{α}}{Γ (α + 1)} \sum_{j = 1}^{n - 1} ∥ θ_{τ}^{j} | | [(j - i + 1)^{α} - (j - i)^{α}] ∥ . \end{matrix}

By Lemma 5 and equations (30) and (25), we take

ρ = \frac{2 (m_{f} + (c_{J} ∥ M ∥ + ∥ B_{1} u_{0} ∥_{X^{*}} + c_{ψ} (1 + ∥ u_{0} ∥) + n c_{E} c_{q} T ∥ u_{0} ∥ + 2 c_{E} μ) ∥)}{m_{A_{1}}},

and

ω_{j} = \frac{(c_{ψ} + n c_{E} c_{q} T + ∥ B_{1} ∥) 2 τ^{α}}{m_{A_{1}} Γ (α + 1)} [(n - j + 1)^{α} - (n - j)^{α}] .

Then, we have

\begin{matrix} ∥ θ_{τ}^{n} ∥ \leq ρ \exp (\frac{(c_{ψ} + n c_{E} c_{q} T + ∥ B_{1} ∥) 2 τ^{α}}{m_{A_{1}} Γ (α + 1)} \sum_{j = 1}^{n - 1} [(n - j + {1)}^{α} - {(n - j)}^{α}]) \\ = ρ \exp (\frac{(c_{ψ} + n c_{E} c_{q} T + ∥ B_{1} ∥) 2 τ^{α}}{m_{A_{1}} Γ (α + 1)} t_{n}^{α}) \\ \leq ρ \exp (\frac{(c_{ψ} + n c_{E} c_{q} T + ∥ B_{1} ∥) 2 τ_{0}^{α}}{m_{A_{1}} Γ (α + 1)} T^{α}) . \end{matrix}

In this case, we deduce equation (26). According to equations (18) and (26), we can easily deduce equation (27). Next, we will prove equation (28). Indeed, according to condition $H (J)$ and equation (26), we have

∥ ζ_{τ}^{n} ∥ \leq c_{J} (1 + ∥ M ∥ ∥ θ_{τ}^{n} ∥) \leq C .

Conversely, according to $H (ψ)$ and equation (27), we have

∥ λ_{τ}^{n} ∥_{X^{*}} \leq c_{ψ} (1 + ∥ u_{τ}^{n} ∥) \leq C .

We will show that equation (30) gives

\begin{matrix} y_{τ}^{n} = E (Σ_{j = 1}^{k} \int_{t_{n - 1}}^{t_{n}} q (t_{n}, s) u_{τ}^{n} (s) d s + μ) \\ \leq T c_{E} c_{q} ∥ u_{τ}^{n} ∥ + c_{E} μ . \end{matrix}

In addition, by equation (27), we infer equation (30). □

Lemma 8. Assume that conditions $H (A_{1})$ , $H (B_{1})$ , $H (J)$ , $H (Ψ)$ , and $H (f)$ hold. Let ${τ_{n}}$ be a sequence such that $τ_{n} \to 0$ , as $n \to \infty$ . For a subsequence, still denoted $τ$ , we have

\begin{matrix} θ_{τ}^{n} \to θ weakly in L^{q} (0, T; X), \\ ζ_{τ}^{n} \to ζ weakly in X^{*}, \\ y_{τ}^{n} \to y weakly in X^{*}, \\ λ_{τ}^{n} \to λ weakly in X^{*}, \end{matrix}

and

{{}_{0}I}_{t}^{α} θ_{τ}^{n} (t) \to_{0} I_{t}^{α} θ (t) weakly in X,

where $q > 1$ and $θ \in L^{q} (0, T; X), ζ, λ, y \in X^{*} .$

Proof. Indeed, according to equation (26), we have

∥ θ_{τ}^{n} ∥^{q} = \int_{0}^{T} ∥ θ_{τ}^{n} ∥^{q} = \sum_{i = 1}^{n} \int_{t_{i - 1}}^{t_{i}} ∥ θ_{τ}^{i} ∥^{q} = τ_{n} \sum_{i = 1}^{n} ∥ θ_{τ}^{i} ∥^{q} \leq C .

Therefore, as $τ \to 0$ , there exists $θ \in L^{q} (0, T; X)$ such that

θ_{τ}^{n} \to θ weakly in L^{q} (0, T; X) .

According to equation (16) of [10], we obtain

{{}_{0}I}_{t}^{α} θ_{τ}^{n} (t) \to_{0} I_{t}^{α} θ (t) weakly in X .

Conversely, by equation (28), the sequence ${ζ_{τ}^{n}}$ is bounded in $X^{*}$ . Without loss of generality, as $τ \to 0$ , there exists $ζ$ , such that

ζ_{τ}^{n} \to ζ weakly in X^{*} .

According to $H (M)$ and equation (26), we have

M θ_{τ}^{n} \to M θ strongly X .

By Lemma 12 of [10], we know that the mapping $(y, x) \to \partial (y, x)$ is upper semicontinuous. And by $M^{*} ζ_{τ}^{n} \in M^{*} \partial J (β_{τ}^{n}, M (u_{τ}^{n}))$ and $β_{τ}^{n} \to β$ in C(0,T;V), we imply that

M^{*} ζ (t) \in M^{*} \partial J (β (t), M (θ (t))) .

(32)

Similar to the proof of equation (31), we can imply that

λ_{τ}^{n} \to λ weakly in X^{*}, and λ \in \partial ψ (_{0} I_{t}^{α} θ (t) + u_{0}) .

By equation (29), it is easy to infer that

y_{τ}^{n} \to y weakly in X^{*}, and y \in X^{*} .

□

Lemma 9. Assume that condition $H (A_{1})$ , $H (B_{1})$ , $H (J)$ , $H (ψ)$ , $H (E)$ , $H (q)$ , and $H (f)$ hold, for all $τ \in (0, τ_{0})$ . Then, $θ \in L^{1} (0, T; X)$ is a solution to Problem 4.

Proof. By Lemma 8, as $τ \to 0$ , we have

\begin{matrix} 〈 A_{1} θ_{τ}^{n}, v 〉_{X^{*} \times X} \to 〈 A_{1} θ, v 〉_{X^{*} \times X}, \\ 〈 B_{1} (_{0} I_{t}^{α} θ_{τ}^{n} (t) + u_{0}), v 〉_{X^{*} \times X} \to 〈 B_{1} (_{0} I_{t}^{α} θ (t) + u_{0}), v 〉_{X^{*} \times X}, \\ 〈 y_{τ}^{n}, v 〉_{X^{*} \times X} \to 〈 y, v 〉_{X^{*} \times X}, \\ 〈 M^{*} ζ_{τ}^{n}, v 〉_{X^{*} \times X} \to 〈 M^{*} ζ, v 〉_{X^{*} \times X}, \\ 〈 λ_{τ}^{n}, v 〉_{X^{*} \times X} \to 〈 λ, v 〉_{X^{*} \times X} \end{matrix}

and

\begin{matrix} 〈 f_{τ}^{n}, v 〉_{X^{*} \times X} \to 〈 f, v 〉_{X^{*} \times X}, \end{matrix}

for all $v \in X$ .

Then, we imply that

\begin{matrix} 0 \leq \underset{τ \to 0}{limsup} 〈 A_{1} (θ_{τ}^{n} (t)), v 〉_{X^{*} \times X} + \underset{τ \to 0}{limsup} 〈 B_{1} (_{0} I_{t}^{α} θ_{τ}^{n} (t) + u_{0}), v 〉_{X^{*} \times X} \\ + \underset{τ \to 0}{limsup} 〈 R_{1} (_{0} I_{t}^{α} θ_{τ}^{n} (t) + u_{0}), v 〉_{X^{*} \times X} + \underset{τ \to 0}{limsup} 〈 M^{*} ζ_{τ}^{n}, v 〉_{X^{*} \times X} + \underset{τ \to 0}{limsup} 〈 λ_{τ}^{n}, v 〉_{X^{*} \times X} \\ - \underset{τ \to 0}{liminf} 〈 f_{τ}^{n} (t), v 〉_{X^{*} \times X} \\ \leq 〈 A_{1} (θ (t)), v 〉_{X^{*} \times X} + 〈 B_{1} (_{0} I_{t}^{α} θ (t) + u_{0}), v 〉_{X^{*} \times X} + 〈 R_{1} (_{0} I_{t}^{α} θ (t) + u_{0}), v 〉_{X^{*} \times X} \\ + 〈 M^{*} ζ, v 〉_{X^{*} \times X} + 〈 λ, v 〉_{X^{*} \times X} - 〈 f (t), v 〉_{X^{*} \times X} . \end{matrix}

Then, we have

〈 A_{1} (θ (t)) + B_{1} (_{0} I_{t}^{α} θ (t) + u_{0}) + R_{1} (_{0} I_{t}^{α} θ (t) + u_{0}) + M^{*} ζ + λ, v 〉_{X^{*} \times X} \geq 〈 f (t), v 〉_{X^{*} \times X},

where $ζ (t) \in \partial J (β (t), M (θ (t))), λ (t) \in \partial ψ (_{0} I_{t}^{α} θ (t) + u_{0})$ . Thus, we imply that $θ \in L^{1} (0, T; X)$ is a solution of Problem 5. □

According to Proposition 2(d), we complete the proof of the Problem 3.

Lemma 10 For given $β \in W^{1, 2} (0, T; V)$ , Problem 3 has a unique solution $u \in W^{1, 2} (0, T; X)$ . Moreover, for any $β_{1}, β_{2} \in W^{1, 2} (0, T; V)$ , we have

∥ u_{1} (t) - u_{2} (t) ∥ \leq c_{β} ∥ β_{1} (t) - β_{2} (t) ∥_{V},

where $u_{1} (t), u_{2} (t)$ are the solutions of equation (14) for $β_{1}, β_{2}$ and $θ_{1}, θ_{2}$ .

Proof. According to Lemma 9, we know that equation (16) has at least one solution. Next, for any given $β \in W^{1, 2} (0, T; V)$ , we will show that equation (16) has a unique solution. Indeed, let $θ_{1}, θ_{2} \in L^{1} (0, T; X)$ be the solution to equation (16). Then, there exist $ζ_{i}, λ_{i}$ such that

A_{1} (θ_{i} (t)) + B_{1} (_{0} I_{t}^{α} θ_{i} (t) + u_{0}) + R_{1} (_{0} I_{t}^{α} θ_{i} (t) + u_{0}) + λ_{i} + M^{*} ζ_{i} = f (t),

where $ζ_{i} \in \partial J (β (t), M (θ_{i} (t)))$ and $λ_{i} \in \partial ψ (_{0} I_{t}^{α} θ_{i} (t) + u_{0})$ . Then

\begin{matrix} 〈 A_{1} (θ_{1} (t)) - A_{1} (θ_{2} (t)), θ_{1} - θ_{2} 〉 + 〈 B_{1} (_{0} I_{t}^{α} θ_{1} (t) + u_{0}) - B_{1} (_{0} I_{t}^{α} θ_{2} (t) + u_{0}), θ_{1} - θ_{2} 〉 \\ + 〈 R_{1} (_{0} I_{t}^{α} θ_{1} (t) + u_{0}) - R_{1} (_{0} I_{t}^{α} θ_{2} (t) + u_{0}), θ_{1} - θ_{2} 〉 = 〈 M^{*} ζ_{2} - M^{*} ζ_{1}, θ_{1} - θ_{2} 〉 + 〈 λ_{2} - λ_{1}, θ_{1} - θ_{2} 〉 . \end{matrix}

(33)

It follows from $H (B_{1})$ that

\begin{matrix} 〈 B_{1} (_{0} I_{t}^{α} θ_{1} (t) + u_{0}) - B_{1} (_{0} I_{t}^{α} θ_{2} (t) + u_{0}), θ_{1} - θ_{2} 〉 \leq \frac{∥ θ_{1} - θ_{2} ∥^{2} ∥ B_{1} ∥}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} d s \\ \leq \frac{∥ θ_{1} - θ_{2} ∥^{2} ∥ B_{1} ∥ T^{α}}{α Γ (α)} . \end{matrix}

By $H (E), H (q)$ , we have

\begin{matrix} 〈 R_{1} (_{0} I_{t}^{α} θ_{1} (t) + u_{0}) - R_{1} (_{0} I_{t}^{α} θ_{2} (t) + u_{0}), θ_{1} - θ_{2} 〉 \leq \frac{∥ θ_{1} - θ_{2} ∥^{2} ∥ B_{1} ∥ c_{E} c_{q} T}{Γ (α)} \int_{0}^{t} (t - s)^{α - 1} d s \\ \leq \frac{∥ θ_{1} - θ_{2} ∥^{2} c_{E} c_{q} T^{α + 1}}{α Γ (α)} . \end{matrix}

By conditions $H (A_{1})$ , $H (B_{1})$ , $H (J)$ , $H (ψ)$ , $H (E)$ , and $H (q)$ and equation (32), we have

(m_{A_{1}} - m_{J} ∥ M ∥^{2} - c_{ψ_{1}} - \frac{c_{E} c_{q} T^{α + 1}}{α Γ (α)} - \frac{∥ B_{1} ∥ T^{α}}{α Γ (α)}) ∥ θ_{1} - θ_{2} ∥^{2} \leq 0 .

According to the condition of Theorem 4, we imply that

θ_{1} = θ_{2} .

Conversely, by Proposition 1(d), we have

∥ u_{1} - u_{2} ∥ \leq \frac{T^{α}}{Γ (α + 1)} ∥ θ_{1} - θ_{2} ∥ .

(34)

Therefore, for a given $β \in W^{1, 2} (0, T; V)$ , equation (14) has a unique solution $θ \in L^{1} (0, T; X)$ . Namely, Problem 3 has a unique solution $u \in W^{1, 2} (0, T; X)$ , which completes the proof of Theorem 4.

For a given $β_{i} \in W^{1, 2} (0, T; V)$ , there exists a unique $θ_{i} \in L^{1} (0, T; X)$ such that

〈 A_{1} (θ_{i} (t)), v 〉 + 〈 B_{1} (_{0} I_{t}^{α} θ_{i} (t) + u_{0}), v 〉 + J^{0} (β_{i} (t), M (θ_{i} (t)); Mv) + Ψ^{0} (_{0} I_{t}^{α} θ_{i} (t) + u_{0}; v) \geq 〈 f (t), v 〉 .

Then, by conditions $H (A_{1})$ , $H (B_{1})$ , $H (ψ)$ , $H (E)$ , $H (q)$ , and $H (J)$ , we have

(m_{A_{1}} - m_{J} ∥ M ∥^{2} - c_{ψ_{1}} - \frac{c_{E} c_{q} T^{α + 1}}{Γ (α + 1)} - \frac{∥ B_{1} ∥ T^{α}}{Γ (α + 1)}) ∥ θ_{1} - θ_{2} ∥ \leq m_{J_{1}} ∥ M ∥^{2} ∥ β_{1} - β_{2} ∥_{V} .

Therefore, according to equation (33), we have

∥ u_{1} - u_{2} ∥ \leq c_{β} ∥ θ_{1} - θ_{2} ∥_{V},

where

c_{β} = (m_{J_{1}} T^{α} ∥ M ∥^{2}) / ((m_{A_{1}} - m_{J} ∥ M ∥^{2} - c_{ψ_{1}} - \frac{c_{E} c_{q} T^{α + 1}}{α Γ (α)} - \frac{∥ B_{1} ∥ T^{α}}{α Γ (α)}) Γ (α + 1)) .

□

According to Lemma 10, we complete the proof of Theorem 4. By the definition of $R$ , we take $E = I$ and $L = q$ . Conversely, by Lemma 10, the condition of Theorem 4, and the condition of Theorem 2, we complete the proof of Theorem 2.

Next, we will consider Theorem 3. For equation (10), we will use Schauder’s fixed-point theorem to prove an existence theorem.

Theorem 5. Assume that $H (g)$ and $H (AB)$ hold. Then equation (10) has at least one mild solution.

Proof. For any given $u \in C (0, T; X)$ , we define an operator ∑ as

\begin{matrix} \sum (β (t)) = e^{A (t + τ)} e_{τ}^{B_{2} t} φ (- τ) + \int_{- τ}^{0} e^{A (t - s)} e_{τ}^{B_{2} (t - τ - s)} [φ' (s) - A φ (s)] d s \\ + \int_{0}^{t} e^{A (t - τ - s)} e^{B_{2} (t - τ - s)} g (s, u (s), β (s)) d s . \end{matrix}

According to condition $H (g)$ , we deduce that ∑ is well defined.

Step 1. There exists a bounded closed ball $B_{r} \subset C (- τ, T; V)$ , such that $Σ (B_{r}) \subset B_{r}$ . By Lemmas 3 and 4, we have

\begin{matrix} ‖ \sum β (t) ‖_{V} \leq e^{(∥ A ∥ + ∥ B_{2} ∥) (t + τ)} ∥ φ (- τ) ∥_{V} + \int_{- τ}^{0} e^{∥ A ∥ (t - s)} e^{∥ B_{2} ∥ (t - s)} ∥ φ' (s) - A φ (s) ∥_{V} d s \\ + \int_{0}^{t} e^{∥ A ∥ (t - τ - s)} e^{∥ B_{2} ∥ (t - τ - s)} ϕ (s) d s \\ \leq e^{(∥ A ∥ + ∥ B_{2} ∥) (T + τ)} (∥ φ (- τ) ∥_{V} + \int_{- τ}^{0} ∥ φ' (s) - A φ (s) ∥_{V} d s) \\ + e^{(T + τ) (∥ A ∥ + ∥ B_{2} ∥)} T^{(p - 1) / p} ∥ ϕ ∥_{L^{p} [0, T]} . \end{matrix}

Therefore, we complete the proof of step 1.

Step 2. the operator $Σ$ is continuous. Assume that ${β_{n}} \subset B_{r}$ and $β_{n} \to β$ in $B_{r}$ ; for any $t \in I$ , we have

\begin{matrix} ‖ \sum β_{n} (t) - \sum β (t) ‖_{V} = ‖ \int_{0}^{t} e^{A (t - τ - s)} e^{B_{2} (t - τ - s)} (g (s, u (s), β_{n} (s)) - g (s, u (s), β (s))) d s ‖ \\ \leq e^{(2 T - τ) (∥ A ∥ + ∥ B_{2} ∥)} \int_{0}^{t} ∥ g (s, u (s), β_{n} (s)) - g (s, u (s), β (s)) ∥ d s \\ \leq e^{(2 T - τ) (∥ A ∥ + ∥ B_{2} ∥)} L_{g} T ∥ β_{n} - β ∥_{V} \end{matrix}

In addition, by $H (g) (iv)$ , it is easy to get step 2.

Step 3. $Σ$ is equicontinuous. Let us divide $Σ$ into two parts, as

Σ = Σ_{1} + Σ_{2},

where

Σ_{1} = e^{A (t + τ)} e_{τ}^{B_{2} t} φ (- τ) + \int_{- τ}^{0} e^{A (t - s)} e_{τ}^{B_{2} (t - τ - s)} [φ' (s) - A φ (s)] d s

and

Σ_{2} = \int_{0}^{t} e^{A (t - τ - s)} e^{B_{2} (t - τ - s)} g (s, u (s), β (s)) d s .

To prove this part, we will divide it into two cases.

Case 1. For any $β \in B_{r}$ and $ε > 0$ , if $t_{1} = 0, 0 < t_{2} \leq δ$ , we have

\begin{matrix} ∥ Σ_{1} β (t_{2}) - Σ_{1} β (t_{1}) ∥_{V} = ∥ e^{A (t_{2} + τ)} e_{τ}^{B_{2} t_{2}} φ (- τ) + \int_{- τ}^{0} e^{A (t_{2} - s)} e_{τ}^{B_{2} (t_{2} - τ - s)} [φ' (s) - A φ (s)] d s \\ - e^{A (τ)} e_{τ}^{B_{2} 0} φ (- τ) + \int_{- τ}^{0} e^{A (- s)} e_{τ}^{B_{2} (- τ - s)} [φ' (s) - A φ (s)] d s ∥ . \end{matrix}

Then, as $t_{2} \to 0$ , we have

∥ Σ_{1} β (t_{2}) - Σ_{1} β (t_{1}) ∥_{V} \to 0 .

Conversely, according to condition $H (g)$ and the Hölder inequality, we have

\begin{matrix} ∥ Σ_{2} β (t_{2}) - Σ_{2} β (0) ∥_{V} = ‖ \int_{0}^{t_{2}} e^{A (t_{2} - τ - s)} e^{B_{2} (t_{2} - τ - s)} e^{A τ} g (s, u (s), β (s)) d s ‖ \\ \leq ‖ \int_{0}^{t_{2}} e^{A (t_{2} - τ - s)} e^{B_{2} (t_{2} - τ - s)} ϕ (s) d s ‖ \\ \leq e^{(∥ A ∥ + ∥ B_{2} ∥) (T + τ)} ∥ ϕ ∥_{L^{P} [0, T]} t_{2}^{(p - 1) / p} \to 0, as t_{2} \to 0 . \end{matrix}

Case 2. For any $x \in B_{r}$ , if $δ_{0} / 2 \leq t_{1} \leq t_{2} \leq T$ ,

\begin{matrix} ∥ Σ_{1} β (t_{2}) - Σ_{1} β (t_{1}) ∥_{V} = ∥ e^{A (t_{2} + τ)} e_{τ}^{B_{2} t_{2}} φ (- τ) + \int_{- τ}^{0} e^{A (t_{2} - s)} e_{τ}^{B_{2} (t_{2} - τ - s)} [φ' (s) - A φ (s)] d s \\ - e^{A (t_{1} + τ)} e_{τ}^{B_{2} t_{1}} φ (- τ) - \int_{- τ}^{0} e^{A (t_{1} - s)} e_{τ}^{B_{2} (t_{1} - τ - s)} [φ' (s) - A φ (s)] d s ∥ \\ \leq ∥ e^{A (t_{2} + τ)} e_{τ}^{B_{2} t_{2}} φ (- τ) - e^{A (t_{1} + τ)} e_{τ}^{B_{2} t_{2}} φ (- τ) ∥ \\ + ∥ e^{A (t_{1} + τ)} e_{τ}^{B_{2} t_{2}} φ (- τ) - e^{A (t_{1} + τ)} e_{τ}^{B_{1} t_{1}} φ (- τ) ∥ \\ + ‖ \int_{- τ}^{0} e_{τ}^{B_{2} (t_{2} - τ - s)} [φ' (s) - A φ (s)] (e^{A (t_{2} - s)} - e^{A (t_{1} - s)}) d s ‖ \\ + ‖ \int_{- τ}^{0} e^{A (t_{1} - s)} [φ' (s) - A φ (s)] (e_{τ}^{B_{2} (t_{2} - τ - s)} - e_{τ}^{B_{2} (t_{1} - τ - s)}) d s ‖ . \end{matrix}

According to Lemmas 3 and 4, we have

∥ e^{A (t_{2} + τ)} e_{τ}^{B_{2} t_{2}} φ (- τ) - e^{A (t_{1} + τ)} e_{τ}^{B_{2} t_{2}} φ (- τ) ∥ \leq ∥ A ∥ ∥ e^{A ξ} ∥ ∥ φ (- τ) ∥_{V} e^{∥ B_{2} ∥ (T + τ) + ∥ A ∥ τ} (t_{2} - t_{1}),

where $ξ \in (t_{1}, t_{2})$ .

By Lemma 4, we have

\begin{matrix} ∥ e^{A (t_{1} + τ)} e_{τ}^{B_{2} t_{2}} φ (- τ) - e^{A (t_{1} + τ)} e_{τ}^{B_{2} t_{1}} φ (- τ) ∥ \leq e^{∥ A ∥ (T + τ)} ∥ φ (- τ) ∥_{V} ∥ B ∥ e_{τ}^{B_{2} (ε - τ)} (t_{2} - t_{1}) \\ \leq ∥ B ∥ e^{∥ A ∥ (T + τ) + ∥ B_{2} ∥ T} ∥ φ (- τ) ∥_{V} (t_{2} - t_{1}), \end{matrix}

and

\begin{matrix} ‖ \int_{- τ}^{0} e_{τ}^{B_{2} (t_{2} - τ - s)} [φ' (s) - A φ (s)] (e^{A (t_{2} - s)} - e^{A (t_{1} - s)}) d s ‖ \\ \leq τ ∥ A ∥ e^{∥ B_{2} ∥ T + ∥ A ∥ (τ + ε)} \int_{- τ}^{0} ∥ φ' (s) - A φ (s) ∥ d s (t_{2} - t_{1}), \end{matrix}

where $ε \in (t_{1}, t_{2})$ .

Similarly, by Lemma 4, we obtain

\begin{matrix} ‖ \int_{- τ}^{0} e^{A (t_{1} - s)} [φ' (s) - A φ (s)] (e_{τ}^{B_{2} (t_{2} - τ - s)} - e_{τ}^{B_{2} (t_{1} - τ - s)}) d s ‖ \\ \leq e^{∥ A ∥ T} \int_{- τ}^{0} ∥ φ' (s) - A φ (s) ∥ ∥ B_{2} ∥ e_{τ}^{∥ B_{2} ∥ (ε - 2 τ - s)} (t_{2} - t_{1}) d s \\ \leq ∥ B_{2} ∥ e^{∥ A ∥ T + ∥ B_{2} ∥ T} \int_{- τ}^{0} ∥ φ' (s) - A φ (s) ∥ d s (t_{2} - t_{1}) . \end{matrix}

Conversely, we have

\begin{matrix} ∥ Σ_{2} β (t_{2}) - Σ_{2} β (t_{1}) ∥ \leq ‖ \int_{t_{1}}^{t_{2}} e^{A (t_{2} - τ - s)} e^{B_{2} (t_{2} - τ - s)} g (s, u (s), β (s)) d s ‖ \\ + ‖ \int_{0}^{t_{1}} (e^{A (t_{2} - τ - s)} e_{τ}^{B_{2} (t_{2} - τ - s)} - e^{A (t_{1} - τ - s)} e^{B_{2} (t_{2} - τ - s)}) g (s, u (s), β (s)) d s ‖ \\ + ‖ \int_{0}^{t_{1}} (e^{A (t_{1} - τ - s)} e^{B_{2} (t_{2} - τ - s)} - e^{A (t_{1} - τ - s)} e^{B_{2} (t_{1} - τ - s)}) g (s, u (s), β (s)) d s ‖ . \end{matrix}

By $H (g)$ , we have

\begin{matrix} ‖ \int_{t_{1}}^{t_{2}} e^{A (t_{2} - τ - s)} e^{B_{2} (t_{2} - τ - s)} g (s, u (s), β (s)) d s ‖ \leq \int_{t_{1}}^{t_{2}} e^{A (t_{2} - τ - s)} e^{B_{2} (t_{2} - τ - s)} ϕ (s) d s \\ \leq C e^{(∥ A ∥ + ∥ B ∥) (T + τ)} ∥ ϕ ∥_{L^{P} [0, T]} (t_{2} - t_{1})^{\frac{p - 1}{p}} . \end{matrix}

and

\begin{matrix} ‖ \int_{0}^{t_{1}} e^{B_{2} (t_{2} - τ - s)} g (s, u (s), β (s)) (e^{A (t_{2} - τ - s)} - e^{A (t_{1} - τ - s)}) d s ‖ \\ \leq ∥ ϕ ∥_{L^{p} [0, T]} t_{1}^{\frac{p - 1}{p}} e^{(∥ A ∥ + ∥ B_{2} ∥) (T + τ)} ∥ A ∥ e^{∥ A ∥ ε} (t_{2} - t_{1}), \end{matrix}

where $ε \in (t_{1}, t_{2})$ .

Similarly, we have

\begin{matrix} ‖ \int_{0}^{t_{1}} e^{A (t_{1} - τ - s)} g (s, u (s), β (s)) (e^{B_{2} (t_{2} - τ - s)} - e^{B_{2} (t_{1} - τ - s)}) d s ‖ \\ \leq ∥ ϕ ∥_{L^{p} [0, T]} t_{1}^{\frac{p - 1}{p}} e^{(∥ A ∥ + ∥ B_{2} ∥) (T - τ)} ∥ B_{2} ∥ e^{∥ B_{2} ∥ μ} (t_{2} - t_{1}), \end{matrix}

where $μ \in (t_{1}, t_{2})$ .

Thus, by cases 1 and 2, we complete the proof of step 3.

In addition, according to step 1 and the Ascoli–Arzela theorem, we imply that $Σ$ is compact. Therefore, by the Schauder fixed-point theorem, we imply that equation (10) has a fixed point. The proof is complete. □

By Theorems 2 and 5, there exists $(u, β) \in W^{1, 2} (0, T; X) \times W^{1, 2} (- τ, T; V)$ satisfying equation (12). Thus, we complete the proof of Theorem 3.

Footnotes

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant numbers 12161015 and 12001120), Training Object of High Level and Innovative Talents of Guizhou Province (grant number (2016)4006) and Guizhou Data Driven Modeling Learning and Optimization Innovation Team (grant number (2020)5016).

ORCID iD

JinRong Wang

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