Regularization of ill-posed mixed variational inequalities with non-monotone perturbations

Correspondence: thuychip04@yahoo.com College of Sciences, Thainguyen University, Thainguyen, Vietnam Abstract In this paper, we study a regularization method for ill-posed mixed variational inequalities with non-monotone perturbations in Banach spaces. The convergence and convergence rates of regularized solutions are established by using a priori and a posteriori regularization parameter choice that is based upon the generalized discrepancy principle.


Introduction
Variational inequality problems in finite-dimensional and infinite-dimensional spaces appear in many fields of applied mathematics such as convex programming, nonlinear equations, equilibrium models in economics, and engineering (see [1][2][3]). Therefore, methods for solving variational inequalities and related problems have wide applicability. In this paper, we consider the mixed variational inequality: for a given f X*, find an element x 0 X such that where A : X X* is a monotone-bounded hemicontinuous operator with domain D (A) = X, : X ℝ is a proper convex lower semicontinuous functional and X is a real reflexive Banach space with its dual space X*. For the sake of simplicity, the norms of X and X* are denoted by the same symbol || · ||. We write 〈x*, x〉 instead of x*(x) for x* X* and x X. By S 0 we denote the solution set of the problem (1). It is easy to see that S 0 is closed and convex whenever it is not empty. For the existence of a solution to (1), we have the following well-known result (see [4]): Theorem 1.1. If there exists u dom satisfying the coercive condition then (1) has at least one solution.
Many standard extremal problems can be considered as special cases of (1). Denote by the indicator function of a closed convex set K in X, Then, the problem (1) is equivalent to that of finding x 0 K such that In the case K is the whole space X, the later variational inequality is of the form of the following operator equation: When A is the Gâteaux derivative of a finite-valued convex function F defined on X, the problem (1) becomes the nondifferentiable convex optimization problem (see [4]): Some methods have been proposed for solving problem (1), for example, the proximal point method (see [5]), and the auxiliary subproblem principle (see [6]). However, the problem (1) is in general ill-posed, as its solutions do not depend continuously on the data (A, f, ), we used stable methods for solving it. A widely used and efficient method is the regularization method introduced by Liskovets [7] using the perturbative mixed variational inequality: where A h is a monotone operator, a is a regularization parameter, U is the duality mapping of X, x * X and (A h , f δ , ε ) are approximations of (A, f, ), τ = (h, δ, ε). The convergence rates of the regularized solutions defined by (6) are considered by Buong and Thuy [8].
In this paper, we do not require A h : x * ∈ X to be monotone. In this case, the regularized variational inequality (6) may be unsolvable. In order to avoid this fact, we introduce the regularized problem of finding x τ α ∈ X such that where μ is positive small enough, U s is the generalized duality mapping of X (see Definition 1.3) and x * is in X which plays the role of a criterion of selection, g is defined below.
Assume that the solution set S 0 of the inequality (1) is non-empty, and its data A, f, are given by A h , f δ , ε satisfying the conditions: ℝ is a proper convex lower semicontinuous functional for which there exist positive numbers c ε and r ε such that where C 0 is some positive constant, d(t) has the same properties as g(t).
In the next section we consider the existence and uniqueness of solutions x τ α of (7), for every a >0. Then, we show that the regularized solutions x τ α converge to x 0 S 0 , the x * -minimal norm solution defined by The convergence rate of the regularized solutions x τ α to x 0 will be established under the condition of inverse-strongly monotonicity for A and the regularization parameter choice based on the generalized discrepancy principle.
1. An operator A : D(A) = X X* is said to be (a) hemicontinuous if A(x + t n y) ⇀ Ax as t n 0 + , x, y X, and demicontinuous if x n x implies Ax n ⇀ Ax; where m A is a positive constant. It is well-known that a monotone and hemicontinuous operator is demicontinuous and a convex and lower semicontinuous functional is weakly lower semicontinuous (see [9]). And an inverse-strongly monotone operator is not strongly monotone (see [10]). Definition 1.2. It is said that an operator A : X X* has S-property if the weak convergence x n ⇀ x and 〈Ax n -Ax, x n -x〉 0 imply the strong convergence x n x as n ∞.
When s = 2, we have the duality mapping U. If X and X* are strictly convex spaces, U s is single-valued, strictly monotone, coercive, and demicontinuous (see [9]).
Let X = L p (Ω) with p (1, ∞) and Ω ⊂ ℝ m measurable, we have Assume that the generalized duality mapping U s satisfies the following condition: where m s is a positive constant. It is well-known that when X is a Hilbert space, then U s = I, s = 2 and m s = 1, where I denotes the identity operator in the setting space (see [12]).

Main result
Lemma 2.1. Let X* be a strictly convex Banach space. Assume that A is a monotonebounded hemicontinuous operator with D(A) = X and conditions (2) and (3) are satisfied. Then, the inequality (7) has a non-empty solution set S ε for each a >0 and f δ X*.
Proof. Let x ε dom ε . The monotonicity of A and assumption (3) imply the following inequality: for ||x|| > r ε . Consequently, (2) is fulfilled for the pair (A + aU s , ε ). Thus, for each a >0 and f δ X*, there exists a solution of the following inequality: Observe that the unique solvability of this inequality follows from the monotonicity of A and the strict monotonicity of U s . Indeed, let x 1 and x 2 be two different solutions of (14). Then, and Putting z = x 2 in (15) and z = x 1 in (16) and add the obtained inequalities, we obtain Due to the monotonicity of A and the strict monotonicity of U s , the last inequality occurs only if x 1 = x 2 .
Let x δ,ε α be a solution of (14), that is, For all h >0, making use of (8), from (17) one gets Since μ ≥ h, we can conclude that each x δ,ε α is a solution of (7). □ Let x τ α be a solution of (7). We have the following result. Theorem 2.1. Let X and X* be strictly convex Banach spaces and A be a monotonebounded hemicontinuous operator with D(A) = X. Assume that conditions (1)-(3) are satisfied, the operator U s satisfies condition (13) and, in addition, the operator A has the S-property. Let Then {x τ α }converges strongly to the x * -minimal norm solution x 0 S 0 . Proof. By (1) and (7), we obtain This inequality is equivalent to the following The monotonicity of A, assumption (1), and the inequalities (8), (9), (13) and (20) yield the relation Since μ/a 0 as a 0 (and consequently, h/a 0), it follows from (19) and the last inequality that the set x τ α are bounded. Therefore, there exists a subsequence of which we denote by the same x τ α weakly converges tox ∈ X. We now prove the strong convergence of {x τ α } tox. The monotonicity of A and U s implies that In view of the weak convergence of {x τ α } tox, we have By virtue of (8), Using further (7), we deduce By (22)-(24) and (26), it results that Finally, the S property of A implies the strong convergence of {x τ α } tox ∈ X. We show thatx ∈ S 0 . By (8) and take into account (7) we obtain Since the functional is weakly lower semicontinuous, Since {x τ α } is bounded, by (9), there exists a positive constant c 2 such that By letting a 0 in the inequality (7), provided that A is demicontinuous, from (8), (9), (28), (29) and condition (1) imply that This means thatx ∈ S 0 . We show thatx = x 0 . Applying the monotonicity of U s and the inequalities (8), (9) and (13), we can rewrite (17) as Since a 0, ε/a, δ/a, μ/a 0 (and h/a 0), the last inequality becomes Replacing x by tx + (1 − t)x, t (0, 1) in the last inequality, dividing by (1t) and then letting t to 1, we get Using the property of U s , we have that ||x − x * || ≤ ||x − x * ||, ∀x S 0 . Because of the convexity and the closedness of S 0 , and the strictly convexity of X, we can conclude thatx = x 0 . The proof is complete. □ Now, we consider the problem of choosing posteriori regularization parameter α = α(μ, δ, ε) such that lim μ,δ,ε→0 α(μ, δ, ε) = 0 and lim μ,δ,ε→0 To solve this problem, we use the function for selectingα = α(μ, δ, ε) by generalized discrepancy principle, i.e. the relationα = α(μ, δ, ε) is constructed on the basis of the following equation: with ρ(α) =α c + ||x τα − x * || s−1 , where x τα is the solution of (7) with α =α, c is some positive constant. Lemma 2.2. Let X and X* be strictly convex Banach spaces and A : X X* be a monotone-bounded hemicontinuous operator with D(A) = X. Assume that conditions (1), (2) are satisfied, the operator U s satisfies condition (13). Then, the function ρ(α) = α c + ||x τ α − x * || s−1 is single-valued and continuous for a ≥ a 0 >0, where x τ α is the solution of (7).
Proof. Single-valued solvability of the inequality (7) implies the continuity property of the function r(a). Let a 1 , a 2 ≥ a 0 be arbitrary (a 0 >0). It follows from (7) that where x τ α 1 and x τ α 2 are solutions of (7) with a = a 1 and a = a 2 . Using the condition (2) and the monotonicity of A, we have It follows from (13) and the last inequality that Obviously, x τ α 1 → x τ α 2 as μ 0 and a 1 a 2 . It means that the function ||x τ α − x * || s−1 is continuous on [a 0 ; +∞). Therefore, r(a) is also continuous on [a 0 ; +∞).
(i) For 0 < a <1, it follows from (7) that Hence, We invoke the condition (1), the monotonicity of A, (8), (10), (12), and the last inequality to deduce that It follows from (33) and the form of r(a) that Therefore, lim a +0 a q r(a) = 0. On the other hand, Since r(a) is continuous, there exists at leat oneα which satisfies (30).
(ii) It follows from (30) and the form of ρ(α) that Therefore,α → 0 as μ, δ, ε 0. If 0 < p < q, it follows from (30) and (32) that So, By Theorem 2.1 the sequence x τα converges to x 0 S 0 with x * -minimal norm as μ, δ, ε 0. Clearly, therefore, there exists a positive constant C 2 such that (32). On the other hand, because c >0 so there exists a positive constant C 1 satisfied (32). This finishes the proof. □ Theorem 2.3. Let X be a strictly convex Banach space and A be a monotonebounded hemicontinuous operator with D(A) = X. Suppose that (i) for each h, δ, ε >0 conditions (1)-(3) are satisfied; (ii) U s satisfies condition (13); (iii) A is an inverse-strongly monotone operator from X into X*, Fréchet differentiable at some neighborhood of x 0 S 0 and satisfies (iv) there exists z X such that then, if the parameter a = a (μ, δ, ε) is chosen by (30) with 0 < p < q, we have Proof. By an argument analogous to that used for the proof of the first part of Theorem 2.1, we have (21). The boundedness of the sequence {x τ α } follows from (21) and the properties of g(t), d(t) and a. On the other hand, based on (20), the property of U s and the inverse-strongly monotone property of A we get that Hence, Further, by virtue of conditions (iii), (iv) and the last estimate, we obtain
Remark 2.2 Condition (34) was proposed in [13] for studying convergence analysis of the Landweber iteration method for a class of nonlinear operators. This condition is used to estimate convergence rates of regularized solutions of ill-posed variational inequalities in [14].
Remark 2.3 The generalized discrepancy principle for regularization parameter choice is presented in [15] for the ill-posed operator equation (4) when A is a linear and bounded operator in Hilbert space. It is considered and applied to estimating convergence rates of the regularized solution for equation (4) involving an accretive operator in [16].