Assume that V is a reflexive separable Banach space,H is a separable Hilbert space,and V^∗ is the dual space of the Banach space V . We suppose that V is dense in H and compactly embeds into H. Thus,V ⊆ H ⊆ V^∗ is an evolution triple,and all embeddings are continuous and dense. For the Banach space V ,we denote the duality between V and its dual V^∗ by <·,·>,and the norm by ‖ · ‖v . Given a constant T ∈ (0,+∞),we let V = L²(0,T ; V ),H = L²(0,T ; H),and V^∗ = L²(0,T ; V^∗) be the dual space of V . It is obvious that,since V ⊆ H ⊆ V^∗ is an evolution triple (see Ref. ^[1]),V ⊆ H ⊆ V^∗ also forms an evolution triple with dense and continuous embeddings. For the Banach space V ,the duality between V and V^∗,denoted by <<·,·>>,and the norm,denoted by ‖ · ‖v ,can be specified as follows:

and

Without loss of generality,by its reflexivity,the Banach space V and its dual V^∗ can be assumed to be locally uniformly convex. Otherwise,an equivalent norm can be introduced such that both of them are locally uniformly convex.

Let f be an element in V^∗ and θ be the zero element of V . We suppose that A,B : V → V^∗ are two operators on the Banach space V ,and G₂^◦ (t,u,v) stands for Clarke’s generalized directional derivative of G : [0,T] × V → R at u in the direction of v with respect to the second variable,which will be defined in Section 2. Now,we consider a second-order evolution hemivariational inequality (SOEHVI),which is specified as follows:

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Since the early 1980s when the hemivariational inequality (HVI) was introduced by Panagiotopoulos^[2] to formulate variational principles involving nonconvex and nonsmooth energy functions,a variety of HVIs have been studied widely by a lot of researchers worldwide^{[3, 4, 5, 6, 7, 8, 9, 10, 11, 12]}. The notion of HVI is based on Clarke’s generalized directional derivative and Clarke’s generalized gradient. It has been proved that HVIs are very efficient to treat certain unsolved or partially solved problems in mechanics and engineering^[13]. The readers are referred to the monographs on the HVI by Panagiotopoulos^[14],Naniewicz and Panagiotopoulos^[15],Carl et al.^[16],and Mig´ orski et al.^[17] for more details on the HVI.

In terms of the recent literature on the HVI,an important tool used by many researchers for the existence results of various types of HVIs including static hemivariational inequalities (SHVIs) and evolution hemivariational inequalities (EHVIs) is related to surjectivity theorems concerning pseudomonotone and coercive operators. A great many of existence results for HVIs have been obtained when the operators involved are pseudomonotone and satisfy certain coercivity conditions^{[18, 19, 20, 21, 22]}. However,as pointed by Liu^[4],there also are many economic,engineering,and stochastic models leading to HVIs with involved operators being non-coercive. Some existence theorems available for HVIs become inefficient in this situation. Therefore,in recent years,more and more mathematicians and engineers have paid more attention to obtaining reasonable approximations of solutions to HVIs without coercivity conditions. By applying the Browder-Tikhonov regularization method,which was captured by Giannessi and Khan^[23] to deal with quasi-variational inequalities,Liu^{[4, 24]} studied a class of non-coercive EHVIs,and a strongly convergent approximation procedure was designed for the EHVI considered. By the same method,Xiao and Huang^[25] studied a class of SOEHVIs,and based on the solvability of the regularized EHVIs,they constructed an approximate sequence whose weak clusters are the solutions to the equivalent evolution inclusion. In this paper,we continue the study on the SOEHVI. As in Ref.^[25],rather than the exact data (A,B,G,f) for the SOEHVI considered, only the noisy data (A_αn,B_βn,G_γn,f_θn) with {α_n},{β_n},{γ_n},and {θ_n} being sequences of positive reals are assumed to be available.

The paper is structured as follows. In Section 2,we recall briefly some preliminary materials and give some assumptions. In Section 3,we present the regularized formulations for the SOEHVI and its derived first-order evolution hemivariational inequality (FOEHVI). Some equivalent results among solutions to these problems and some inclusion problems are also obtained. In Section 4,based on the Browder-Tikhonov regularization method for the derived FOEHVI,we construct a sequence of regularized solutions to the regularized SOEHVIs and prove that the whole sequence of regularized solutions converges strongly to a solution to the SOEHVI considered.

In this section,we first recall briefly some useful notions and results in nonsmooth analysis and nonlinear analysis. We refer the readers to Refs.^[1],^[17],and ^[26] for more details and the references therein.

We will follow the same notation as Ref.^[25]. Assume that g : V → R is a locally Lipschitz functional on the Banach space V ,u ∈ V is a given point,and V ∈ V represents a vector in V . Clarke’s generalized directional derivative of the functional g at u in the direction of v (see Ref.^[26]),which is denoted by g^◦(u,v),is defined as follows:

where w ∈ V ,and λ > 0. With Clarke’s generalized directional derivative,Clarke’s generalized gradient of g at u (see Ref.^[26]),which is denoted by ∂g(u),is given by

Proposition 1^{[17, 26]} Let V be a Banach space,u,V ∈ V two points in V,and G : V → R a locally Lipschitz functional. Then,

(1) The function V → g^◦(u,v) is positively homogeneous and subadditive.

(2) g^◦(u,v) is upper semicontinuous on V × V as a function of (u,v) and is Lipschitz as a function of V alone.

(3) For all u ∈ V,∂g(u) is a nonempty,bounded,convex,and weak∗-compact subset of V^∗.

(4) For every V ∈ V,

Definition 1^[1] Let V be a Banach space and V^∗ be its dual space. Suppose that T : V → 2V^∗ is a multi-valued operator on V. T is said to be

(1) monotone,if

(2) strongly monotone with the constant m > 0,if

(3) relaxed monotone with the constant m > 0,if

Remark 1 In Definition 1,if the operator T : V → V^∗ is a single-valued operator, then Definition 1 reduces to the definitions of monotonicity,strong monotonicity,and relaxed monotonicity for single-valued operators.

Definition 2^[17] Let V be a Banach space and V^∗ be its dual space. An operator T : V → V^∗ from V to V^∗ is said to be demicontinuous if and only if,for any sequence {un} on V,un → u as n → ∞ which implies that T u_n T u as n → ∞.

Definition 3^[17] Let V be a Banach space and V^∗ be its dual space. An operator T : V → V^∗ from V to V^∗ is said to be hemicontinuous if,for any u,V ∈ V,the function T → <T (u + t(v − u)),V − u> from ^{[0, 1]} into (−∞,+∞) is continuous at 0+.

Remark 2^[17] In general,for an operator T from a Banach space V to its dual V^∗,the demicontinuity implies the hemicontinuity. However,if T is monotone,then the notions of demicontinuity and the hemicontinuity coincide.

Definition 4^[25] Let m > 0 be a positive constant. By A(m),we denote the class of operators A : V → V^∗,which are demicontinuous,bounded,and strongly monotone with constant m. By G(m),we denote the class of functionals G : [0,T] × V → R which satisfy that

(i) G(·,u) is measurable on [0,T] for all u ∈ V.

(ii) G(t,·) is locally Lipschitz on V for a.e. T ∈ [0,T].

(iii) For a.e. T ∈ [0,T],Clarke’s generalized gradient ∂₂G(t,·) satisfies the following growth condition,i.e.,there exists a constant H > 0 such that for a.e. T ∈ [0,T],

(iv) For a.e. T ∈ [0,T],Clarke’s generalized gradient ∂₂G(t,·) satisfies the relaxed monotonicity condition.

Remark 3 If A ∈ A(m) and G ∈ G(m),then we obviously have that,for a.e. T ∈ [0,T], the sum A + ∂₂G(t,·) is monotone on V . However,it may not be coercive in general.

Now,we introduce a space W which is defined by

where the generalized derivative is characterized by

Endowed with the graph norm of the generalized derivative operator

W is a separable and reflexive Banach space since V and V^∗ are separable and reflexive (see Propositions 23.7 and 23.23 in Ref. ^[1]).

We denote by

where its domain of definition D(L) is given by

It can be shown that the operator L : D(L) ⊂ V → V^∗ is densely defined,closed,and linear maximal monotone^[17]. Moreover,for any u ∈ D(L) and V ∈ V ,

Let J : V → V^∗ be the duality mapping which is determined uniquely by

It is well known that the duality mapping J is strictly monotone,bijective bicontinuous,and of class (S+). For more details on the duality mapping,we can refer to Ref.^[1].

Assume that α_n,β_n,γ_n,and θ_n are four sequences of positive reals,and {Σn} with Σn > 0 is a (strictly) decreasing and convergent sequence with limit being zero. In the remainder of this section,we give some hypotheses on the exact data (A,B,G,f) and the noisy data (A_αn,B_βn, G γn,f_θn) of the SOEHVI considered,which are specified as follows:

(h1) For any z ∈ V ,

where c₁ ≥ 0 is a positive constant.

(h2) For any z ∈ V ,

where c₂ ≥ 0 is a positive constant.

(h3) For any z ∈ V and a.e. t ∈ [0,T],

where c₃ ≥ 0 is a positive constant,and H(P,Q) is the Hausdorff distance of P and Q.

(h4)

(h5)

In this section,by introducing an integral operator,we give an FOEHVI which is derived from the SOEHVI considered. Then,with the duality mapping on the Banach space V ,we present the regularized formulations for the SOEHVI and the derived FOEHVI. At last,some equivalent results among solutions to the derived FOEHVI,the regularized formulations,and some inclusion problems are obtained.

First of all,we define an operator K : V → C(0,T; V ) as follows:

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It is obvious that K is a linear,bounded,and Lipschitz continuous operator on V ^[17]. With the Banach space W introduced and the operator K,we consider the following FOEHVI,which is derived from the SOEHVI (1).

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We have the following lemma on the relationship between solutions to the derived FOEHVI (3) and the SOEHVI (1) (see Ref.^[25]).

Lemma 1 z ∈ W is a solution to the derived FOEHVI (3) if and only if u := Kz is a solution to the SOEHVI (1).

Now,by mean of the duality mapping J on the Banach space V ,we give the regularized formulations of the SOEHVI (1) and the derived FOEHVI (3) as follows:

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and

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where the duality mapping J : V → V^∗ is called the regularized operator,Σ_n is the regularization parameter,and the solutions u_n and z_n to the regularized problems (4) and (5) are called the regularized solutions.

By the definition of the operator K in (2),we have the following lemma on the relationship between the regularized SOEHVIs (4) and the regularized FOEHVIs (5),which is similar to Lemma 1.

Lemma 2 For n ∈ N,z_n ∈ W is a solution to the regularized FOEHVI (5) if and only if u n := Kz_n is a solution to the regularized SOEHVI (4).

By the operators in the SOEHVI and its regularized problems,we define some new operators on the Banach space V as follows:

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For the properties of the operators defined above,we can refer to Lemmas 3.2 and 3.3 in Ref. ^[25] by Xiao and Huang. With the operators defined in (6) on V ,we have the following results for the derived FOEHVI (3) and its regularized problems (5) (see Lemma 3.4 in Ref. ^[25]).

Lemma 3 For n ∈ N,z_n ∈ D(L) is a solution to the regularized FOEHVI (5) if and only if z_n solves the following regularized evolution inclusion (REI): Find z_n ∈ D(L) such that

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At the same time,z ∈ D(L) is a solution to the derived FOEHVI (3) if and only if z solves the following evolution inclusion (EI): Find z ∈ D(L) such that

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By imposing some much stronger hypotheses on A,B,and G,we can also obtain the following lemma on the EI (8),which is crucial to our main results in the next section.

Lemma 4 Let A ∈ A(m),G ∈ G(m),and B be linear bounded,monotone,and symmetric on V . Then,z ∈ D(L) is a solution to the EI (8) if and only if z solves the following inequality: Find z ∈ D(L) such that

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Proof Necessity: Let z ∈ D(L) be a solution to the EI (8). Then,there exists a ξ ∈ N _z satisfying

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Since A ∈ A(m),G ∈ G(m),and B is linear bounded,monotone,and symmetric on V ,it follows from Lemma 3.2 in Ref.^[25] that A + N and B are monotone on V . Therefore,for any y ∈ W and w ∈ N y,we can get by the monotonicity of L and (10) that

which together with the arbitrariness of y ∈ W and w ∈ N y implies that z solves the inequality (9).

Sufficiency: Suppose that z ∈ D(L) solves the inequality (9). Then,

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For any V ∈ W and λ ∈ [0, 1],letting y = y_λ = z + λv ∈ W in the above inequality (11) yields

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Clearly,y_λ → z in W as λ → 0. Thus,{y_λ} is a bounded sequence on V . By Lemma 3.2 in Ref.^[25],the operator A is bounded and demicontinuous on V ,and N is a weakly closed and bounded multivalued operator from V into V^∗. Then,there exists λ_n → 0 as n → ∞ and w_n ∈ N y_λn such that A y_λn A z and w_n w ∈ N z as n → ∞. Therefore,since both the operator L and the operator B are linear,it follows from (12) that

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Taking limit n → ∞ at both sides of the above inequality (13),we can get that

which together with the arbitrariness of v and w ∈ N z implies that

Thus,z ∈ D(L) is a solution to the EI (8).

Corollary 1 Assume that all assumptions in Lemma 4 hold. Then,the set of solutions to the EI (8) and the derived FOEHVI (3),which is denoted by S in the following of the paper, is closed and convex if it is nonempty.

Proof Since it is obvious that the set of solutions to the inequality (9) is nonempty,closed, and convex,the proof follows immediately from Lemmas 3 and 4.

In this section,based on the Browder-Tikhonov regularization method for the derived FOEHVI,we construct a sequence of regularized solutions to the regularized SOEHVI and prove that the whole sequence of regularized solutions converges strongly to a solution to the SOEHVI considered.

First of all,let us mainly focus on the EI (8) and its regularized problems (7),which by Lemma 3 are equivalent to the derived FOEHVI (3) and its regularized problems (5). A sequence of regularized solutions to the REI (7) {z_n} was constructed by Xiao and Huang^[25]. In the paper,Xiao and Huang proved the existence and uniqueness of the sequence of regularized solutions to the REI (7) in Theorem 3.1,and the uniform boundedness of the sequence was also obtained in Theorem 3.2.

Now,in terms of the properties of the solution set for the EI (8) in Corollary 1,we are in a position to give the strong convergence result of the sequence of regularized solutions to the REI (7).

Theorem 1 Let A,A_αn ∈ A(m),G,G_γn ∈ G(m),and B,B_βn be symmetric operators which are bounded,linear,and monotone. Suppose that the hypotheses (h1)-(h5) hold. If the EI (8) is solvable,then the whole sequence {z_n} of regularized solutions to the REI (7) converges strongly in V to the minimal norm solution to the EI (8).

Proof Since the REI (7) possesses a unique solution for all n ∈ N by Theorem 3.1 in Ref. ^[25],we let {z_n} be a sequence of solutions to the REI (7). Thus,z_n ∈ D(L) satisfies

which implies that there exists w_n ∈ N_γnz_n such that

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According to the assumptions of Theorem 1,the EI (8) is solvable. Let S ⊆ D(L) be its solution set. Thus,for any z ∈ S,we have

which means that there exists w ∈ N z such that

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Subtracting (15) from (14) and multiplying the obtained result by z_n − z,we can get that

Consequently,for any w_n ∈ N_γnz,we have

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With the monotonicity of the operators L,J,A_αn + N_γn,and B_βn,we can further get from (h4) and (16) that

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Since the above inequality (17) holds for any w_n ∈ N_γnz,by a similar proof as Lemma 3.3 in Ref. ^[25],we can choose w_n ∈ N_γnz such that

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As a result,we can get by the inequalities (17) and (18) that

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Now,in terms of Theorem 3.2 in Ref. ^[25],{z_n} is uniformly bounded in W . Thus,it follows from the reflexivity of W that there exists a weakly convergent subsequence of {z_n}. Without confusion,we still denote it by {z_n}. Then,z_n z^∗ in W for some point z^∗,i.e.,

Moreover,again by Theorem 3.2 in Ref. ^[25],the weak limit z^∗ is also a solution to the EI (8), i.e.,z^∗ ∈ S. Therefore,taking limit as n → ∞ at both sides of the inequality (19),we can get by the hypothesis (h5) that

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By Corollary 1,the solution set S of the EI (8) is convex and closed. For any u ∈ S and λ ∈ ^{[0, 1]},letting

in the above inequality (20) yields

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Taking limit as λ → 0 at both sides of the inequality (21),we can get by the demicontinuity of J that

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which implies that z^∗ is the unique minimal norm element of the solution set S of the EI (8). Therefore,the whole sequence z_n z^∗ in V .

By Lemma 3.2 in Ref. ^[25],J is the duality mapping on the Banach space V . Then,we have the inequality as follows:

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which implies that

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By similar arguments for the inequality (19),we can also obtain that

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which implies by the hypothesis (h5) that

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Taking limit as n → ∞ at both sides of the inequality (22),we get by the weak convergence of {z_n} that

Since V is locally uniformly convex,z_n converges strongly to z^∗ in V from Proposition 21.23 in Ref. ^[1].

Remark 4 By the closedness and convexity of the solution set of the derived FOEHVI (3),Theorem 1 gives a strong convergence result for the constructed sequence of regularized solutions to the regularized FOEHVI (5),which is a generalization of the results in Ref. ^[4] to the SOEHVI and also improves the main result Theorem 3.3 in Ref. ^[25].

With the sequence {z_n} of regularized solutions to the REI (7),we can easily construct the sequence {un} with u_n := Kz_n of the regularized solutions to the regularized SOEHVI (4) and prove the following strong convergence result for the SOEHVI considered.

Theorem 2 Suppose that all assumptions in Theorem 1 hold,and the sequence {un} with u n := Kz_n and {z_n} being the sequence of regularized solutions to the REI (7) is a sequence of regularized solutions to the regularized SOEHVI (4). Moreover,the whole sequence {un} of regularized solutions converges strongly to a solution to the SOEHVI (1).

Proof Obviously,the conclusion that {un} are the regularized solutions to the regularized SOEHVI (4) follows immediately from Lemmas 2 and 3. Moreover,by Theorem 1,the sequence {z_n} of regularized solutions to the REI (7) converges strongly to the minimal norm solution z^∗ to the EI (8). Thus,the Lipschitz continuity of the operator K defined in (2) implies that

which is a solution to the SOEHVI (1) by Lemmas 2 and 3.

Acknowledgements The authors are grateful to the referees for their valuable comments which lead to the improvement of this paper.