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1 Introduction

Let be three reflexive and separable Banach spaces, and let be the dual space of. For a prefixed , . This paper focuses on the following fractional impulsive differential hemivariational inequality (FIDHVI): find and such that

where stands for the Caputo derivative of fractional order is given by with and being the left and right limit , respectly, and .

We remark that for appropriate and suitable choices of the spaces and the above defined maps, FIDHVI includes a number of differential variational inequalities as special cases [5,12,13,26, 29].

It is worth mentioning that FIDHVI is a new model, which captures the required char- acteristics of both the hemivariational inequality and the fractional impulsive differential equation within the same framework. In addition, FIDHVI can be used to describe the frictional contact problem with the surface traction driven by the fractional impulsive differential equation (see Section 5).

The study of differential variational inequality (DVI) can ascend to the work of Aubin and Cellina [1]. DVI described by the following generalized abstract system

was then examined by Pang and Stewart [20] in finite dimensional Euclidean spaces. Here K is a nonempty, closed, and convex subset of,, and are three given functions. As pointed out by Pang and Stewart [20], DVI provides a powerful tool of describing many practical problems such as fluid mechanical problems, engineering operation research, dynamic traffic networks, economical dynamics, and frictional contact problems [6,23, 25]. In 2010, Li et al. [11] discussed the solvability for a class of DVI in finite dimensional spaces. Later, Chen and Wang [4] employed the regularized time-stepping method to consider a class of parametric DVI and provided convergence analysis for this method in finite dimensional Euclidean spaces. Liu et al. [14] studied a class of nonlocal semilinear evolution DVI in Banach spaces. By using the theory of topological degree they obtained some existence results for their model under some suitable assumptions. Recently, in order to describe a free boundary problem raising from contact mechanics, Sofonea et al. [21] studied a differential quasivariational inequality and proved the stability of the solutions for such a problem. For more works related to DVIs, we refer the reader to [10,13, 15, 28] and the the references therein.

As is well known, fractional calculus, that is, the noninteger calculus, allows us to define derivatives of arbitrary order and has many applications in practical problems [9]. Recently, by applying the fixed point approach, Ke et al. [8] discussed the solvability of a class of fractional DVI in finite dimensional spaces. Using the Rothe method, Zeng et al. [30] studied a class of parabolic fractional differential hemivariational inequalities in Banach spaces. Xue et al. [29] discussed the existence of the mild solutions of a class of fractional DVIs in Banach spaces under some appropriate hypotheses. Very recently, Weng et al. [27] considered a fractional nonlinear evolutionary delay system driven by a hermivariational inequality in Banach spaces and established an existence theorem for such a system by employing the KKM theorem, fixed point theorem for condensing set-valued operators, and the theory of fractional calculus.

It is worth noting that, in the real world, many systems are often disturbed suddenly, and systems changes suddenly in a short time. These phenomena are called impulsive effects. We note that diverse numerical methods and theoretical results have been widely studied for differential equations with impulsive effects using different assumptions in the literature; for instance, we refer the reader to [2] and the references therein. In [17] and [16], Migórski and Ochal studied the existence of the solutions for two class of nonlinear second-order impulsive evolution inclusions problems. Recently, Li et al. [12] introduced a class of impulsive DVI in finite dimensional spaces and presented some existence and stability results of the solutions under some suitable assumptions. However, in some practical situations applications, it is necessary to consider FIDHVI. To illustrate this point, a fractional contact problem with the surface traction driven by the fractional impulsive differential equation will be considered as an application of FIDHVI in Section 5. The discipline of FIDHVI is still not explored, and very little is known. To fill this gap, in this paper, we seek to make a contribution in this new direction.

The outline of this work is as follows. In the next section, we present some necessary preliminaries and notations. After that, Section 3 establishes an existence and uniqueness result concerning FIDHVI under some mild conditions. In Section 4, we provide a stability result of the solution of FIDHVI with respect to the perturbation of data. Finally, we apply our main results for FIDHVI to the frictional contact problem with the surface traction driven by the fractional impulsive differential equation in Section 5.

2 Preliminaries

For a Banach space$\textcolor{red}{}\chi$, we denote the space of all functions $\textcolor{red}{}x$: that is continuous, the space of all $\textcolor{red}{}p$th power Bochner integrable functions on $\textcolor{red}{}Q$ taking values in the space of all functions such that is continuous, and and exist with the set of all closed subsets of the set of all compact (closed and bounded) convex subsets of $\textcolor{red}{}X$ . For a set

In the sequel, let denote the gamma function.

Definition 1. (See [9].) The$\textcolor{red}{}q$th fractional integral of with is defined by

Definition 2. (See [9].) For , the Caputo fractional-order derivative of $\textcolor{red}{}\alpha$ of

Definition 3. (See [3].) The generalized directional derivative of a locally Lipschitz functional in the direction and the generalized gradient of function $\textcolor{red}{}F$ at $\textcolor{red}{}v$, denoted respectively by and , are respectively defined by

and

Definition 4. (See [22].) An operator is said to be

1. (i) monotone if for all

2. (ii) strongly monotone if there exists satisfying

3. (iii) pseudomomotone if $\textcolor{red}{}B$ is bounded and weakly in $z^2$ with lim sup yields that for all y

4. (iv) demicontinuous if in $z^2$ implies that weakly in

5. (v) bounded if is bounded implies is bounded.

Definition 5. (See [18].) A set-valued operator is said to be pseudo- momotone if

1. (i) for every

2. (ii) for any subspace is upper semicontinuous from $\textcolor{red}{}H$ to $z_2^{*}$ endowed with the weak topology;

3. (iii) if weakly in $z_2$ and such that then for every , there exists such that

Lemma 1. (See [7, Prop. 5.6].) Assume that $U_1$ and $U_2$ are two reflexive Banach spaces, is the adjoint operator of is a locally Lipschitz functional satisfying , for all , is pseudomonotone.

Lemma 2. (See [30, Cor. 7].) Assume that $U_0$ is a reflexive Banach spaces, and let the following conditions hold:

1. (i) is pseudomomotone and strong monotone with constant

2. (ii) is pseudomomotone, and there exist two constants satisfyng

3. (iii)

Then is surjective in $U_0^{*}$.

According to [18, Prop. 3.37], we can rewrite FIDHVI as follows.

Problem 1. Find and such that

To study Problem 1, we consider the following fractional impulsive Cauchy problem

Noting the fact that

solves the Cauchy problem

we have the following result immediately.

Lemma 3. Let and . Then the Cauchy problem

is equivalent to the integral equation

Lemma 4. For (0, 1) and , the Cauchy problem

is equivalent to the integral equation

Proof. Assume that (1) holds. If , then for all with . Clearly,

If , then

and so Lemma 3 implies that

If , then using Lemma 3 again, we have

Similarly, if , then we can show that

Conversely, suppose that (2) holds. If , then we know that (1) holds by the fact that is the inverse of . If , since the Caputo fractional derivate for a constant is zero, one has and .

From Lemma 4 we have the following definition.

Definition 6. A pair is said to be a solution of Problem 1 if it satisfies the following system:

Finally, we recall the following nonlinear impulsive Gronwall inequality.

Lemma 5. (See [24, Lemma 3.4].) Let satisfy the following inequality:

where are constants. Then

where , and$\textcolor{red}{}E$γ is the Mittag-Leffler function [9] defined by for all

3 Existence and uniqueness

To study the solvability of Problem 1, we need the following assumptions.

1. ($H_f$ ) is a map such that

   1. (i) for any given is continuous;

      (ii) for any , there exists satisfying

      (iii) there exists satisfying

2. ($H_I$ ) For each is bounded, and there exists satisfying for all

3. ($H_A$) is a map such that

   1. (i) for any given is continuous;

      (ii) for any given is bounded, demicontinuous, and strongly monotone with the constant$m_A$.

4. ($H_N$ ) is a compact operator.

5. ($H_J$) is a functional satisfying

   1. (i) for any given is continuous;

      (ii) for any given is locally Lipschitz;

      (iii) there exists satisfying for all

      (iv) there exists satisfying for all

6. ($H_g$) is a map such that

   1. (i) for any given is continuous;

      (ii) there exists satisfying for all

7. ($H_0$) , where

   1. (ii)

We first consider nonlinear inclusion (4).

Lemma 6. For any given, nonlinear inclusion (4) has a unique solution y providing that assumptions (HA), (HN ), (HJ ), (Hg), and (H0) hold. Moreover, for any, one has

where are the solutions of (4) with respect to $z_1$ and$z_2$, respectively.

Proof. For given and define two operators

For simplicity, we do not indicate their dependence. Using (HA), (HN ), (HJ ), (H0), Lemma 1, and [22, Lemma 3], we deduce that the operators A and N are pseudomomotone and

By applying Lemma 2 with and we know that inclusion (4) has a solution $y\left(t\right)$ for all Next, we show that the solution $\gamma \left(t\right)$ is unique. Let be solutions to (4). Then there exist satisfying . Subtracting the two equations and taking the result in duality with

By assumptions ($H_A$) and ($H_J$ ) one has

and so assumption ($H_0$) implies that , which is our claim.

In what follows, we start by showing that (5) holds. Let and denote . It follows from (4) that . Subtracting the two equations and taking the result in duality with , we have

By assumptions ($H_J$ ), ($H_A$), and ($H_g$) one has

Thus, assumption ($H_0$) implies that

It follows from (6) that the map is continuous for all. Since , we know that . By (6) we conclude that, for any given, nonlinear inclusion (4) has a unique solution . Moreover, for any given , (5) holds due to (6).

Theorem 1. Problem 1 admits a unique solution providing that assumptions hold.

Proof. For any given, Lemma 6 shows that nonlinear inclusions (4) admits a unique solution$y_z$. Define an operator by setting

Then assumption ($H_f$ ) implies that $\textcolor{red}{}\sum$ is well defined. To prove Theorem 1, we only need to show that $\textcolor{red}{}\sum$ admits a unique fixed point in .

To this end, we first show that for any. In fact, let and be given. When , by the Hölder inequality and assumption ($H_f$ ) we have

as, where and . This shows that . When , using the same argument, one has

which implies that . Similarly, when , we can show that

and so .

Combining all the above, we see that for any .

Next, we prove that $\textcolor{red}{}\sum$ is a contractive map. For given , by assumption ($H_f$ ) it follows from (5) that

and so

Now assumption ($H_f$) implies that $\textcolor{red}{}\sum$ is a contractive map, and so $\textcolor{red}{}\sum$ admits a unique solution by employing the Banach fixed point theorem.

4 A convergence result

We investigate the perturbation problem of Problem 1 to prove a convergence result, which describes the stability of the solution in relation to perturbation data. To this end, let be the perturbed data of $\textcolor{red}{}J$ such that $\textcolor{red}{}J$δ satisfies assumptions ($H_J$) and ($H_0$). More precisely, we examine the following perturbation problem: find a pair of functions such that

We denote the constants involved in assumption ($H_J$) by. Furthermore, we introduce the following assumptions.

1. ($H_{\mathrm{J*}}$) is a functional satisfying

   1. (i) there exists a functionsatisfying, for any and for all

   2. (ii)

2. ($H_{\mathrm{0*}}$) There exists such that

   1. (i)

   2. (ii)

The following example indicates that assumption ($H_{\mathrm{J*}}$) can be satisfied for some functions.

Example 1.. Consider the functions definoted by

Then it is easy to check that are locally Lipschitz and nonconvex for all. Moreover, their Clarke subgradients are given by

Thus, we can see that condition ($H_{\mathrm{J*}}$) holds with. Next, we show the stability result for FDQHVI as follows.

Theorem 2. Suppose that assumptions (HA), (Hf ), (HI ), (HN ), (HJ ), (Hg), (H0), (H0∗ ), and (Hj*) hold. Then

1. (i) for each, the perturbation problem (7) has a unique solution

2. (ii) converges to, the solution of Problem 1.

Proof. (i) In view of Theorem 1, the proof is obvious.

(ii) By Definition 6 we consider the problem

Subtracting (9) from (4) and multiplying the result by , we have

Since

one has

Note that assumption ($H_A$) implies

Using assumptions ($H_J$) and ($H_{\mathrm{J*}}$), for any

and

we have

and

We conclude from assumption ($H_g$) that, for any

Combining (10)–(12), one has

Thus, assumption ($H_0$) yields that

Subtracting (8) from (3), by assumptions ($H_f$), ($H_I$) and estimation (13) one has

By Lemma 5 with

there exists such that, where H∗ is independent of . By assumption ($H_{\mathrm{J*}}$) we assert that. It follows from (13) and ($H_{\mathrm{J*}}$) that

5 An application

In this section, we show that the results obtained in Sections 3 and 4 can be applied to study the frictional contact problem (Problem 2) between an elastic body and a foundation over time interval$\textcolor{red}{}Q$. We suppose that the surface traction may change suddenly in a short time, such as shocks, and consequently, which can be described by a fractional impulsive differential equations. We show that the weak form of Problem 2 leads to Problem 1 analyzed in Sections 3 and 4. Then Theorems 1 and 2 are applied to obtain the unique solvability of the frictional contact problem mentioned above as well as the convergence result of the perturbation problem.

We shortly review the basic notations and its mechanical interpretations. A deformabke elastic body occupies a regular Liposchitz domain. The boundary ∂V consists of three measurable disjoint parts with meas. The body is clamped on and subjected to the action of volume force with density. An unknown surface traction (for convenience, we denote by $f_2$ its density) with impulsive effect is applied on . On , the body may contact with an obstacle. We do not show expressly the relation of various functions and $\textcolor{red}{}y$.

Let $\textcolor{red}{}\nu$ be unit outward normal vector, be the space of symmetric matrix of order two onare equipped with, respectively, the following inner products and norms: for all with for all . Here the summation convention is adopted. For any , we denote by the normal components of η, the tangential components of , the normal components of the tangential components of σ. We also denote by ,, respectively, the displacement vector, the stress tensor, and the linearized (small) strain tensor, where

For more details, we refer the reader to [17,18]. We now turn to present a new contact problem with the surface traction governed by a fractional impulsive differential equation.

Problem 2. Find a stress , a surface traction density , and a displacement field such that

.Here relation (14) presents an elastic constitutive law with A being the elasticity operator. Equation (15) is the equilibrium equation, and equation (16) implies that the body is clamped on . Equalities (17)–(18) show that the traction is acted on , and the density of the surface traction is governed by a fractional impulsive differential equation, where F is a function to be specified later. The set-valued relations in (19) denote the friction and contact conditions, respectively, where $j_T$ and $j_{\nu }$ are locally Lipschitz functionals.

To deduce the weak formulation of Problem 2, we consider spaces and equipped with the inner products

and corresponding norms , respectively. We denote by $\textcolor{red}{}\mathrm{V*}$ the dual space of the duality pairing between $\textcolor{red}{}\mathrm{V*}$ and$\textcolor{red}{}V$. The trace theorem states

where $\textcolor{red}{}\gamma$ is the trace operator defined by . In order to study Problem 2, we impose some hypotheses on the relevant data.

1. ($H_A$) The elasticity operator satisfies the conditions:

   1. (i) is symmetric and linear for

   2. (ii) there exists such that for all

   3. (iii) there exists such that

2. ($H_F$) The function is such that

   1. (i) is continuous for all

   2. (ii) there exists such that

   3. (iii) there exists satisfying

3. ($H_I$) is bounded, and there exist for all

4. (Hjν ) The function is such that

   1. (i) For a.e. is locally Lipschitz on$\textcolor{red}{}R$;

   2. (ii) For all is measurable on ;

   3. iii) For all , there exist such that

   4. (iv) For all , there exist such that

5. ($H_{\mathrm{JT}}$) The function is such that

   1. (i) is locally Lipschitz on

   2. (ii) is measurable on for all

   3. (iii) there exist such that ,

   4. (iv) there exist such that

6. ($H_f$) The densities of body force satisfies

7. ($H_0$) (i)

   (ii)

Utilizing the Green formula, we get the variational form of Problem 2.

Problem 3. Find a displacement field and a surface traction density $f_2$ : such that

5.1 Existence and uniqueness for the contact problem

We define the maps and by setting

for all

Then Problem 3 is equivalent to the problem:

Problem 4. Find a displacement vector and a surface traction density $f_2$: such that

Clearly, Problem 4 is the form of Problem 1 with

Theorem 3. Problem 4 has a unique solution providing that hypotheses hold.

Proof. To prove Theorem 3, we only need to check the validity of assumptions ,

Firstly, conditions , and indicate that assumptions, and are fulfilled with Since the trace operator is compact and surjective, we see that assumption holds. Clearly, (21) implies that assumption holds with . By hypotheses and Lemma 14 in [19] it follows from Lemma 14 in [19] that the functional $\textcolor{red}{}J$ in (20) is locally Lipschitz on $\textcolor{red}{}\nu$ and

is the generalized directional derivative of $\textcolor{red}{}J$ at $\textcolor{red}{}u$ in the directional $\textcolor{red}{}w$. Moreover, assumption holds with . Combining Theorem 1 with hypothesis, we see that Theorem 3 holds.

5.2 A convergence result for the contact problem

The above analysis reveals that the solution of Problem 4 relies on the data $j_{\nu }$ and $j_r$ In what follows, we present a continuous dependence result of the solution in relation to these data. We consider the perturbation data jνδ and jτδ of jν and jτ , respectively, which satisfy hypotheses and . For each , define a function by setting

The perturbation problem of Problem 4 can be formulated as follows.

Problem 5. Find a displacement vector and a surface traction density such that

Denote the constants involved in hypotheses and and , respectively. In addition, we impose the following hypotheses on the data.

1. ($H_{\mathrm{J*}}$) There exists a function satisfying

   1. (i)

   2. (ii)

   3. (iii)

2. ($H_{\mathrm{0*}}$) There exists such that

   1. (i)

   2. (ii)

Remark 1. Assumption means that the perturbations of$j_{\upsilon }$ and$j_r$ must satisfy the locally Lipschitz conditions. Moreover, it is easy to see that the functions given in Example 1 satisfy condition .

Theorem 4. Assume that hypotheses hold. Then

1. (i) Problem 5 has a unique solution for each

2. (ii) converges to , the solution of Problem 4.

Proof. (i) In view of Theorem 3, the proof is obvious.

1. (ii) We employ Theorem 2 to prove the conclusion. to this end, we only need to check the validity of assumptions ($H_{\mathrm{0*}}$) and ($H_{\mathrm{J*}}$). Clearly, hypothesis ($H_{\mathrm{0*}}$) implies that assumption ($H_{\mathrm{0*}}$) holds. By Proposition 3.35 of [18], Corollary 4.15 in [18], and hypoth- esis ($H_{\mathrm{J*}}$), for any and we have

which shows that assumption ($H_{\mathrm{J*}}$) holds with . The convergence result now follows from Theorem 2.

Acknowledgments

The authors are grateful to the editor and the referees for their valu- able comments and suggestions.

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