Regularization of ill-posed mixed variational inequalities with non-monotone perturbations
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.
Keywordsmonotone mixed variational inequality non-monotone perturbations regularization convergence rate
where A : X → X* is a monotone-bounded hemicontinuous operator with domain D(A) = X, φ : 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 S0 we denote the solution set of the problem (1). It is easy to see that S0 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 ):
then (1) has at least one solution.
where A h is a monotone operator, α is a regularization parameter, U is the duality mapping of X, x * ∈ X and (A h , f δ , φ ε ) are approximations of (A, f, φ), τ = (h, δ, ε). The convergence rates of the regularized solutions defined by (6) are considered by Buong and Thuy .
where μ is positive small enough, U s is the generalized duality mapping of X (see Definition 1.3) and Open image in new window is in X which plays the role of a criterion of selection, g is defined below.
Assume that the solution set S0 of the inequality (1) is non-empty, and its data A, f, φ are given by A h , f δ , φ ε satisfying the conditions:
(1) || f - f δ || ≤ δ, δ → 0;
where C0 is some positive constant, d(t) has the same properties as g(t).
The convergence rate of the regularized solutions Open image in new window to x0 will be established under the condition of inverse-strongly monotonicity for A and the regularization parameter choice based on the generalized discrepancy principle.
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;
monotone if 〈Ax - Ay, x - y〉 ≥ 0, ∀x, y ∈ X;
- (c)inverse-strongly monotone if
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 ). And an inverse-strongly monotone operator is not strongly monotone (see ).
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 ).
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 ).
2 Main result
Lemma 2.1. Let X* be a strictly convex Banach space. Assume that A is a monotone-bounded 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 α > 0 and f δ ∈ X*.
Due to the monotonicity of A and the strict monotonicity of U s , the last inequality occurs only if x1 = x2.
Since μ ≥ h, we can conclude that each Open image in new window is a solution of (7).
Let Open image in new window be a solution of (7). We have the following result.
Since μ/α → 0 as α → 0 (and consequently, h/α → 0), it follows from (19) and the last inequality that the set Open image in new window are bounded. Therefore, there exists a subsequence of which we denote by the same Open image in new window weakly converges to Open image in new window .
This means that Open image in new window .
Using the property of U s , we have that Open image in new window , ∀x ∈ S0. Because of the convexity and the closedness of S0, and the strictly convexity of X, we can conclude that Open image in new window . The proof is complete.
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 Open image in new window is single-valued and continuous for α≥ α0> 0, where Open image in new window is the solution of (7).
Theorem 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)-(3) are satisfied, the operator U s satisfies condition (13). Then
(i) there exists at least a solution Open image in new window of the equation (30),
(ii) let μ, δ, ε → 0. Then
(1) Open image in new window ;
Therefore, limα→+0α q ρ(α) = 0.
Since ρ(α) is continuous, there exists at leat one Open image in new window which satisfies (30).
Therefore, Open image in new window as μ, δ, ε → 0.
therefore, there exists a positive constant C2 such that (32). On the other hand, because c > 0 so there exists a positive constant C1 satisfied (32). This finishes the proof.
Theorem 2.3. Let X be a strictly convex Banach space and A be a monotone-bounded 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);
Remark 2.2 Condition (34) was proposed in  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 .
Remark 2.3 The generalized discrepancy principle for regularization parameter choice is presented in  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 .
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