Convergence Results for Elliptic Variational-Hemivariational Inequalities

Problem

𝓟. Find an element u ∈ K such that

A short survey of some results concerning Problem 𝓟 is the following. First, its unique solvability was proved in [19] and, under slightly weaker assumptions, in [27]. The dependence of the solution with respect to the data, including the set K and the element f, was proved in [34, 37], under different assumptions. These results have been completed in [34] and [15] by considering an associate optimal control problem and an evolution inequality problem, respectively. Results on the well-posedness of Problem 𝓟 in the sense of Tykhonov have been obtained in [31]. There, given a sequence of positive numbers {ε_n}, the following perturbation of Problem 𝓟 was considered, for each n ∈ ℕ.

Problem

𝓟_εn. Find an element u_n ∈ K such that

Note that the solution of Problem 𝓟 is solution of the perturbed problem 𝓟_εn. Nevertheless, the solution of Problem 𝓟_εn could fail to be unique. Denote by u_n a solution of Problem 𝓟_εn, for each n ∈ ℕ. Then, under appropriate assumptions, it was proved in [31] that the sequence {u_n}, called approximating sequence, converges to u in X. This property represents the main ingredient for the well-posedness of Problem 𝓟 in the sense of Tykhonov, introduced in the study of variational inequalities in [17, 18] and extended to a particular class of hemivariational inequalities in [8]. References in the field include [1, 13, 14, 16, 29, 30, 35].

Other convergence results concerning Problem 𝓟 are related to the penalty method. Given a sequence of positive numbers {λ_n} and a penalty operator G : X → X^*, the classical penalty method consists to replace Problem 𝓟 by a sequence of problems {𝓟_λn} which, for every n ∈ ℕ, can be formulated as follows.

Problem

𝓟_λn. Find an element u_n ∈ X such that

Note that Problem 𝓟_λn is formally obtained from Problem 𝓟 by removing the constraint u ∈ K and including a penalty term governed by a parameter λ_n > 0 and an operator G : X → X^*. Penalty methods have been used in [6, 7, 26] and [19, 27, 28] as an approximation tool to treat constraints in variational inequalities and variational-hemivariational inequalities, respectively. In particular, the existence of a unique solution to Problem 𝓟_λn together with its convergence to the solution of Problem 𝓟 as λ_n → 0 was proved in [19, 27]. An extension of this convergence result was obtained in [33] where the operator G in (1.3) was replaced by an operator G_n : X → X^*, which depends on n.

Another type of convergence results for the variational-hemivariational inequality (1.1) arise from its numerical analysis and, more precisely, from its numerical approximation. Given a sequence {K_n}, an approximation of Problem 𝓟 is stated as follows.

Problem

𝓟_Kn. Find an element u_n ∈ K_n such that

Note that in various applications K_n = X_n ∩ K where X_n is a finite-dimensional space constructed with the finite element method. We refer the reader to [11, 12] for convergence results related to internal numerical approximations, and [9] for both internal and external numerical approximations of such inequalities. A comprehensive reference on the numerical analysis of Problem 𝓟 can be found in the survey paper [10].

The aim of this paper is threefold. The first one is to construct a sequence of Problems {𝓟_n} and to show that for each n ∈ ℕ, Problem 𝓟_n has at least a solution u_n which convergences to the solution u of Problem 𝓟, as n → ∞. Our main result on this matter is Theorem 2 which states an existence and convergence result. Our second aim is to show that Theorem 2 can be used to recover various convergence results in the study of Problems 𝓟_εn, 𝓟_λn, 𝓟_Kn described above. To this end, we use the theorem with a particular choice of sets, operators and parameters. Finally, our third aim is to illustrate the use of our abstract result in the study of a frictional contact problem and to provide the corresponding mechanical interpretations. The novelty of our paper arises from the generality of our main result which unifies various convergence results in the study of Problem 𝓟 and provides a new and nonstandard mathematical tool in the variational analysis of frictional contact problems with elastic materials.

The rest of the manuscript is structured as follows. In Section 2 we introduce some preliminary material, then we recall the existence and uniqueness result obtained in [19, 27]. In Section 3 we state and prove our main result, Theorem 2. Its proof is based on arguments of compactness, monotonicity, pseudomonotonicity, lower semicontinuity, combined with the properties of the Clarke subdifferential. In Section 4 we deduce some consequences of Theorem 2 that we present in a form of relevant particular cases. Finally, in Section 5 we illustrate the use of our abstract results in the analysis of a mathematical model of contact.

Definition 1

A multivalued operator T : X → 2^X* is said to be pseudomonotone if:

1. For every u ∈ X, the set Tu ⊂ X^* is nonempty, closed and convex.

2. T is upper semicontinuous (u.s.c.) from each finite dimensional subspace of X into Xw∗ .

3. For any sequences {u_n} ⊂ X and { un∗ } ⊂ X^* such that u_n → u weakly in X, un∗ ∈ Tu_n for all n ∈ ℕ and lim sup 〈 un∗ , u_n – u〉 ≤ 0, we have that for every v ∈ X there exists u^*(v) ∈ Tu such that

   〈u∗(v),u−v〉≤lim inf〈un∗,un−v〉.

Definition 2

A multivalued operator T : X → 2^X* is said to be generalized pseudomonotone if for any sequences {u_n} ⊂ X and { un∗ } ⊂ X^* such that u_n → u weakly in X, un∗ ∈ Tu_n for all n ∈ ℕ, un∗ → u^* in Xw∗ and lim sup 〈 un∗ , u_n – u 〉 ≤ 0, we have u^* ∈ Tu and

Definition 3

An singlevalued operator A : X → X^* is said to be:

1. monotone, if for all u, v ∈ X, we have 〈Au – Av, u – v〉 ≥ 0;

2. strongly monotone, if there exists m_A > 0 such that

   〈Au−Av,u−v〉≥mA∥u−v∥X2for all u,v∈X;

3. bounded, if A maps bounded sets of X into bounded sets of X^*;

4. pseudomonotone, if it is bounded and u_n → u weakly in X with

   lim sup〈Aun,un−u〉≤0

   imply lim inf 〈Au_n, u_n – v〉 ≥ 〈Au, u – v〉 for all v ∈ X;

5. demicontinuous, if u_n → u in X implies Au_n → Au weakly in X^*.

Proposition 1

1. If the operator A : X → X^* is bounded, demicontinuous and monotone, then A is pseudomonotone.

2. If A, B : X → X^* are pseudomonotone operators, then the sum A + B : X → X^* is pseudomonotone.

Definition 4

A function j : X → ℝ is said to be locally Lipschitz if for every x ∈ X, there exists U_x a neighborhood of x and a constant L_x > 0 such that |j(y) – j(z)| ≤ L_x ∥y – z∥_X for all y, z ∈ U_x. For such functions the generalized (Clarke) directional derivative of j at the point x ∈ X in the direction v ∈ X is defined by

The generalized gradient (Clarke subdifferential) of j at x is a subset of the dual space X^* given by

The function j is said to be regular (in the sense of Clarke) at the point x ∈ X if for all v ∈ X the one-sided directional derivative j′(x; v) exists and j⁰(x; v) = j′(x; v).

Proposition 2

Assume that j : X → ℝ is a locally Lipschitz function. Then the following hold:

1. For every x ∈ X, the function X ∋ v ↦ j⁰(x; v) ∈ ℝ is positively homogeneous and subadditive, i.e., j⁰(x; λv) = λj⁰(x; v) for all λ ≥ 0, v ∈ X and j⁰ (x; v₁ + v₂) ≤ j⁰(x; v₁) + j⁰(x; v₂) for all v₁, v₂ ∈ X, respectively.

2. For every v ∈ X, we have j⁰(x; v) = max{〈ξ, v〉 : ξ ∈ ∂j(x)}.

3. For every x ∈ X, the gradient ∂j(x) is a nonempty, convex, and compact subset of Xw∗ which is bounded by the Lipschitz constant L_x > 0 of j near x.

Definition 5

Let {K_n} be a sequence of nonempty subsets of V and K͠ a nonempty subset of X. We say that the sequence {K_n} converges to K͠ in the sense of Mosco if the following conditions hold.

1. For every v ∈ K͠, there exists a sequence {v_n} ⊂ X such that v_n ∈ K_n for each n ∈ ℕ and v_n → v in X.

2. For each sequence {v_n} such that v_n ∈ K_n for each n ∈ ℕ and v_n ⇀ v in X, we have v ∈ K͠.

Proposition 3

Let C be a nonempty closed convex subset of X, C^* a nonempty closed convex and bounded subset of Xw∗ , φ : X → ℝ a proper, convex lower semicontinuous function and let y be arbitrary element of C. Assume that, for each x ∈ C, there exists x^*(x) ∈ C^* such that

Then, there exists y^* ∈ C^* such that

Definition 6

An operator P : X → X^* is said to be a penalty operator of the set K ⊂ X if P is bounded, demicontinuous, monotone and K = {x ∈ X | Px = 0_X^*}.

Proposition 4

Let K be a nonempty closed and convex subset of X, J : X → X^* the duality map, I : X → X the identity map on X, and P͠_K : X → K the projection operator on K. Then P_K = J(I – P͠_K) : X → X^* is a penalty operator of K.

Theorem 1

Assume (2.1)–(2.6). Then, inequality (1.1) has a unique solution u ∈ K.

Problem

𝓟_n. Find u_n ∈ K_n such that

In the study of Problem 𝓟_n we assume that for each n ∈ ℕ, the following hold.

Moreover, we assume that the following conditions are satisfied.

Our main result in this section is the following.

Theorem 2

Assume (2.2)–(2.6) and (3.2)–(3.5). Then, the following statements hold.

1. For each n ∈ ℕ, there exists at least one solution u_n ∈ K_n to Problem 𝓟_n. Moreover, the solution is unique if ε_n = 0.

2. If, in addition, (2.1) and (3.6)–(3.12) hold, u is the solution of Problem 𝓟 and {u_n} ⊂ X is a sequence such that u_n is a solution of Problem 𝓟_n, for each n ∈ ℕ, then u_n → u in X.

Proof

a) Let n ∈ ℕ. Assumptions (3.3), (3.4) and Proposition 1 a) imply that the operator 1λn G_n : X → X^* is pseudomonotone. Therefore (2.2) and Proposition 1 b) show that the operator A_n : X → X^* defined by A_n = A + 1λn G_n is pseudomonotone, too. Moreover, since G_n is monotone and λ_n > 0, using assumption (2.2), again, we deduce that A_n is strongly monotone with constant m_A. We conclude from above that the operator A_n satisfies condition (2.2). On the other hand, recall that the set K_n satisfies condition (3.2). It follows from above that we are in a position to use Theorem 1 with K_n and A_n instead of K and A, respectively. In this way we deduce the existence of a unique element u_n ∈ K_n such that

This proves the unique solvability of Problem 𝓟_n in the case when ε_n = 0. Next, for ε_n > 0, it follows that the solution u_n of (3.13) satisfies inequality (3.1), too. This proves the existence of at least one solution to Problem 𝓟_n.

b) Let n ∈ ℕ. We start by considering the auxiliary problem of finding an element u͠_n ∈ K_n such that

Note that the variational-hemivariational inequality (3.14) is similar to the variational-hemivariational inequality (3.13), the difference arising in the fact that in (3.14) the first argument of φ is the solution u of Problem 𝓟. The existence of a unique solution to inequality (3.14) follows from Theorem 1, by using the same arguments as those used in the proof of unique solvability of inequality (3.13). Next, we divide the rest of the proof into four steps.

1. We claim that there is an element u͠ ∈ K͠ and a subsequence of {u͠_n}, still denoted by {u͠_n}, such that u͠_n ⇀ u͠ in X, as n → ∞.

   To prove the claim, we establish the boundedness of the sequence {u͠_n} in X. Let n ∈ ℕ and let u₀ be a given element in K. We use assumption (3.6) to deduce that

   〈Au~n,u~n−u0〉≤1λn〈Gnu~n,u0−u~n〉+φ(u,u0)−φ(u,u~n)+j0(u~n;u0−u~n)+〈f,u~n−u0〉.

   Then, by the strong monotonicity of the operator A we obtain

   mA∥u~n−u0∥X2≤〈Au0,u0−u~n〉+1λn〈Gnu~n,u0−u~n〉+φ(u,u0)−φ(u,u~n)+j0(u~n;u0−u~n)+〈f,u~n−u0〉. (3.15)

   Next, assumptions (3.4) and (3.7) imply that

   1λn〈Gnu~n,u0−u~n〉≤0 (3.16)

   and assumption (3.10) yields

   φ(u,u0)−φ(u,u~n)≤cφ(u)∥u~n−u0∥X. (3.17)

   On the other hand, we write

   j0(u~n;u0−u~n)  =  j0(u~n;u0−u~n)+j0(u0;u~n−u0)−j0(u0;u~n−u0)≤  j0(u~n;u0−u~n)+j0(u0;u~n−u0)+∣j0(u0;u~n−u0)∣,

   then we use assumption (2.4)(b) and Proposition 2 b) to see that

   j0(u~n;u0−u~n)≤αj∥u~n−u0∥X2+(c¯0+c¯1∥u0∥X)∥u~n−u0∥X. (3.18)

   And, obviously,

   〈Au0,u0−u~n〉+〈f,u~n−u0〉≤∥f−Au0∥X∗∥u~n−u0∥X. (3.19)

   Next, we combine inequalities (3.15)–(3.19) to find that

   (mA−αj)∥u~n−u0∥X≤cφ(u)+c¯0+c¯1∥u0∥X+∥f−Au0∥X∗.

   We now use condition (2.5) and the above inequality to deduce that {u͠_n} is a bounded sequence in X. Therefore, by the reflexivity of X, there exists an element u͠ ∈ X and a subsequence of {u͠_n}, still denoted by {u͠_n}, such that u͠_n ⇀ u͠ in X. Recall that u͠_n ∈ K_n for each n ∈ ℕ. Then, assumption (3.8) and Definition 5 imply that u͠ ∈ K͠.

2. Next, we claim that u͠ is the solution to Problem 𝓟, i.e., u͠ = u.

   To prove this claim we use assumption (3.8) and consider an element v ∈ K͠ together with a sequence {v_n} ⊂ X such that v_n ∈ K_n for every n ∈ ℕ and v_n → v in X as n → ∞. We now use inequality (3.14) with v = v_n and assumptions (2.2), (3.10), (2.4)(b) to see that

   1λn〈Gnu~n,u~n−vn〉≤〈Au~n−Avn,vn−u~n〉+φ(u,vn)−φ(u,u~n)+j0(u~n;vn−u~n)+〈f,u~n−vn〉+〈Avn,vn−u~n〉≤φ(u,vn)−φ(u,u~n)+j0(u~n;vn−u~n)+〈f−Avn,u~n−vn〉≤cφ(u)∥u~n−vn∥X+(c¯0+c¯1∥u~n∥X)∥u~n−vn∥X+∥f−Avn∥X∗∥u~n−vn∥X≤(cφ(u)+c¯0+c¯1∥u~n∥X+∥f−Avn∥X∗)∥u~n−vn∥X.

   Then, due to the convergence v_n → v in X, the boundedness of sequence {u͠_n} and the boundedness of the operator A, we deduce that there exists a constant D > 0 which does not depend on n such that

   〈Gnu~n,u~n−vn〉≤λnD.

   Passing to the upper limit in above inequality and using assumption (3.11) we have

   limsup〈Gnu~n,u~n−vn〉≤0. (3.20)

   On the other hand, we write

   〈Gu~n,u~n−v〉=〈Gu~n,u~n−vn〉+〈Gu~n,vn−v〉=〈Gu~n−Gnu~n,u~n−vn〉+〈Gnu~n,u~n−vn〉+〈Gu~n,vn−v〉≤∥Gu~n−Gnu~n∥X∗∥u~n−vn∥X+〈Gnu~n,u~n−vn〉+〈Gu~n,vn−v〉

   and, using asumption (3.9)(a) we deduce that

   〈Gu~n,u~n−v〉≤cn(1+∥u~n∥X)∥u~n−vn∥X+〈Gnu~n,u~n−vn〉+〈Gu~n,vn−v〉. (3.21)

   We now use hypotheses (3.9)(b), (c), the boundedness of sequence {u͠_n} and the convergence v_n → v in X to see that

   lim[cn(1+∥u~n∥X)∥u~n−vn∥X]=0, (3.22)

   lim〈Gu~n,vn−v〉=0. (3.23)

   Next, we pass to upper limit in inequality (3.21) and use (3.20), (3.22) and (3.23) to find that

   lim sup〈Gu~n,u~n−v〉≤0. (3.24)

   Taking now v = u͠ ∈ K͠ in (3.24) we deduce that lim sup 〈Gu͠_n, u͠_n – u͠〉 ≤ 0. Recall assumption (3.9)(c) which guarantees that the operator G : X → X^* is pseudomonotone. Hence, using the pseudomonotonicity of G we deduce that

   〈Gu~,u~−v〉≤lim inf〈Gu~n,u~n−v〉. (3.25)

   We now combine inequalities (3.24) and (3.25) to see that

   〈Gu~,u~−v〉≤0. (3.26)

   Recall that this inequality is valid for any v ∈ K͠.

   Assume that condition (3.9)(e)(i) is satisfied. Then, inequality (3.26) implies that 〈Gu͠, u͠ – v〉 ≤ 0 for all v ∈ X, which yields Gu͠ = 0_X^* and, therefore, u͠ ∈ K. Assume now that condition (3.9)(e)(ii) is satisfied. Then, by assumptions (3.6) and (3.8) it is easy to see that K ⊂ K͠ and, therefore, using (3.26) we obtain that

   〈Gu~,u~−v〉≤0∀v∈K.

   On the other hand, from the assumption (3.9)(d) we have

   〈Gu~,v−u~〉≤0∀v∈K.

   The last two inequalities imply that 〈Gu͠, v – u͠〉 = 0 for all v ∈ K and, using (3.9) (e)(ii), we infer that u͠ ∈ K. We conclude from above that, either (3.9)(e)(i) or (3.9)(e)(ii) holds, we have

   u~∈K. (3.27)

   Let n ∈ ℕ. Then, using (3.6) and inequality (3.14), we find that

   〈Au~n,v−u~n〉+1λn〈Gnu~n,v−u~n〉+j0(u~n;v−u~n)−〈f,v−u~n〉≥φ(u,u~n)−φ(u,v)∀v∈K. (3.28)

   Next, using Proposition 2 b) we deduce that for each v ∈ K there exists an element ω_n(u͠_n, v) ∈ ∂j(u͠_n) such that j⁰(u͠_n; v – u͠_n) = 〈ω_n(u͠_n, v), v – u͠_n〉, and, therefore, inequality ((3.28) yields

   〈Au~n+1λnGnu~n+ωn(u~n,v)−f,v−u~n〉≥φ(u,u~n)−φ(u,v) (3.29)

   for all v ∈ K. Recall that Proposition 2 c) guarantees the set

   C∗={Au~n+1λnGnu~n+ξn−f:ξn∈∂j(u~n)} (3.30)

   is nonempty closed convex and bounded in Xw∗ . Then, assumption (2.3)(a) allows us to use Proposition 3 with C = K and C^* defined by (3.30), x = v and y = u͠_n. In this way we find that there exists an element ω_n(u͠_n) ∈ ∂j(u͠_n) which does not depend on v such that

   〈Au~n+1λnGnu~n+ωn(u~n)−f,v−u~n〉≥φ(u,u~n)−φ(u,v)∀v∈K.

   Therefore, assumptions (3.4) and (3.7) yield

   〈Au~n+ωn(u~n),u~n−v〉≤φ(u,v)−φ(u,u~n)−〈f,v−u~n〉∀v∈K. (3.31)

   We now use the regularity (3.27) to take v = u͠ in (3.31). Then we pass to the upper limit in the resulting inequality, use the convergence u͠_n ⇀ u͠ in X and the lower semicontinuity of φ with respect to its second argument to infer that

   lim sup〈Au~n+ωn(u~n),u~n−u~〉≤0. (3.32)

   Due to the assumption (2.4)(b), the boundedness of the sequence {u͠_n} and the boundedness of the operator A, guaranteed by assumption (2.2), it follows that the sequence {Au͠_n + ω_n(u͠_n)} is bounded in X^*. This implies that there exists a subsequence of the sequence {Au͠_n + ω_n(u͠_n)}, still denoted by {Au͠_n + ω_n(u͠_n)}, and an element θ ∈ X^* such that

   Au~n+ωn(u~n)⇀θinXw∗. (3.33)

   Moreover, as proved in [19, Lemma 20], we know that the multivalued operator A + ∂j : X → 2^X* is generalized pseudomonotone. Exploiting now Definition 2 and the ingredients {u͠_n} ⊂ X, {Au͠_n + ξ_n(u͠_n)} ⊂ X^*, u͠_n ⇀ u͠ in X, Au͠_n + ω_n(u͠_n) ∈ Au͠_n + ∂j(u͠_n), (3.33) and (3.32), we deduce that θ ∈ Au͠ + ∂j(u͠) and

   〈Au~n+ωn(u~n),u~n〉→〈θ,u~〉. (3.34)

   On the other hand, (3.33) implies that

   〈Au~n+ωn(u~n),u~〉→〈θ,u~〉. (3.35)

   We now combine the convergences (3.34) and (3.35) to find that

   〈Au~n+ωn(u~n),u~n−u~〉→0. (3.36)

   Note that the inclusion θ ∈ Au͠ + ∂j(u͠) implies that there exists ω(u͠) ∈ ∂j(u͠) such that

   θ=Au~+ω(u~). (3.37)

   Consider now an element v ∈ K. We write

   〈Au~n+ωn(u~n),u~n−v〉=〈Au~n+ωn(u~n),u~n−u~〉+〈Au~n+ωn(u~n),u~−v〉,

   then we use (3.36), (3.34) and (3.37) to see that

   lim〈Au~n+ωn(u~n),u~n−v〉=〈Au~+ω(u~),u~−v〉.

   Then, by passing to upper limit in (3.31) and using assumption (2.3)(a) we have

   〈Au~+ω(u~),u~−v〉≤φ(u,v)−φ(u,u~)−〈f,v−u~〉

   or, equivalently,

   〈f,v−u~〉≤〈Au~,v−u~〉+φ(u,v)−φ(u,u~)+〈ω(u~),v−u~〉. (3.38)

   On the other hand, the definition of the Clarke subdifferential yields

   〈ω(u~),v−u~〉≤j0(u~;v−u~). (3.39)

   Then, combining (3.38) and (3.39) we deduce that

   〈f,v−u~〉≤〈Au~,v−u~〉+φ(u,v)−φ(u,u~)+j0(u~;v−u~). (3.40)

   Finally, we use (3.27) and (3.40) to see that u͠ is a solution to Problem 𝓟 and, by the uniqueness of the solution we have that u͠ = u, as claimed.

3. We now prove the convergence of the whole sequence {u͠_n} to u.

   A careful analysis of the proof in step ii) reveals that every subsequence of {u͠_n} which converges weakly in X has the same weak limit u. Moreover, we recall that the sequence {u͠_n} is bounded in X. Therefore, using a standard argument we deduce that the whole sequence {u͠_n} converges weakly in X to u, as n → ∞. This shows that all the statements in step ii) are valid for the whole sequence {u͠_n}. In particular, (3.36) combined with equality u͠ = u shows that

   〈Au~n+ωn(u~n),u~n−u〉→0. (3.41)

   Let n ∈ ℕ and let ω(u) ∈ ∂j(u). We have

   〈ω(u),u~n−u〉≤j0(u;u~n−u),〈ωn(u~n),u−u~n〉≤j0(u~n;u−u~n),

   which imply that

   〈ω(u),u~n−u〉+〈ωn(u~n),u−u~n〉≤j0(u;u~n−u)+j0(u~n;u−u~n).

   We now use assumption (2.4)(c) to see that

   −αj∥u~n−u∥X2≤〈ω(u),u−u~n〉+〈ωn(u~n),u~n−u〉. (3.42)

   On the other hand, assumption (2.2) yields

   mA∥u~n−u∥X2≤〈Au~n−Au,u~n−u〉. (3.43)

   We now add the inequalities (3.42) and (3.43) to deduce that

   (mA−αj)∥u~n−u∥X2≤〈Au~n+ωn(u~n),u~n−u〉+〈Au+ω(u),u−u~n〉.

   Next, we use the convergences (3.41), u͠_n ⇀ u in X as well as the smallness assumption (2.5) to find that

   ∥u~n−u∥X→0, (3.44)

   which show that u͠_n → u in X as n → ∞, as claimed.

4. In the final step of the proof we prove that u_n → u in X, as n → ∞.

   Let n ∈ ℕ. We test with v = u͠_n in (3.1) and v = u_n in (3.14), then we add the resulting inequalities to see that

   〈Aun−Au~n,un−u~n〉≤1λn〈Gnu~n−Gnun,un−u~n〉+φ(un,u~n)−φ(un,un)+φ(u,un)−φ(u,u~n)+j0(un;u~n−un)+j0(u~n;un−u~n)+εn∥u~n−un∥X.

   Next, using assumptions (3.3), (2.3)(b), (2.4)(c), we deduce that

   〈Aun−Au~n,un−u~n〉≤αφ∥un−u∥X∥u~n−un∥X+αj∥u~n−un∥X2+εn∥u~n−un∥X

   and, therefore, the strong monotonicity of the opeator A yields

   (mA−αj)∥u~n−un∥X≤αφ∥un−u∥X+εn. (3.45)

   We now write

   αφ∥un−u∥X≤αφ∥un−u~n∥X+αφ∥u~n−u∥X

   and substitute this inequality in (3.45) to deduce that

   (mA−αφ−αj)∥u~n−un∥X≤αφ∥u~n−u∥X+εn.

   Then, using the smallness assumption (2.5) we obtain that

   ∥u~n−un∥X≤αφmA−αφ−αj∥u~n−u∥X+εnmA−αφ−αj.

   This inequality, the convergence (3.44) and assumption (3.12) imply that

   ∥u~n−un∥X→0. (3.46)

   Finally, writing ∥u_n – u∥_X ≤ ∥u_n – u͠_n∥_X + ∥u͠_n – u∥_X and using the convergences (3.44), (3.46) we deduce that u_n → u in X which concludes the proof.□

Corollary 1

Assume (2.1)–(2.6), (3.4), (3.5), (3.10)–(3.12) and (4.6). Then, the following statements hold.

a) For each n ∈ ℕ, there exists u_n ∈ X such that

Moreover, the solution is unique if ε_n = 0.

b) If {u_n} ⊂ X is a sequence such that u_n is a solution of (4.9), for each n ∈ ℕ, then u_n → u in X.

Proof

Since K_n = X it follows that conditions (3.2), (3.6), (3.8) are satisfied with K͠ = X. Moreover, since G_n = G and (4.6) holds, it follows that conditions (3.3), (3.7), (3.9) hold, too, with K͠ = X and c_n = 0. Corollary 1 is now a direct consequence of Theorem 2.□

Corollary 2

Assume (2.1)–(2.6), (3.4), (3.5), (3.9)(e)(ii), (3.10)–(3.12), (4.1), (4.2) and (4.6). Then, the following statements hold.