- Open Access
Regularization of ill-posed mixed variational inequalities with non-monotone perturbations
© Thuy; licensee Springer. 2011
- Received: 10 February 2011
- Accepted: 21 July 2011
- Published: 21 July 2011
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.
- monotone mixed variational inequality
- non-monotone perturbations
- 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 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 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;
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 ).
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 is a solution of (7).
Let be a solution of (7). We have the following result.
Then converges strongly to the -minimal norm solution x0 ∈ S0.
Since μ/α → 0 as α → 0 (and consequently, h/α → 0), it follows from (19) and the last inequality that the set are bounded. Therefore, there exists a subsequence of which we denote by the same weakly converges to .
Finally, the S property of A implies the strong convergence of to .
This means that .
Using the property of U s , we have that , ∀x ∈ S0. Because of the convexity and the closedness of S0, and the strictly convexity of X, we can conclude that . The proof is complete.
with , where is the solution of (7) with , 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 is single-valued and continuous for α≥ α0> 0, where is the solution of (7).
Obviously, as μ → 0 and α1 → α2. It means that the function is continuous on [α0; +∞). Therefore, ρ(α) is also continuous on [α0; +∞).
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 of the equation (30),
(ii) let μ, δ, ε → 0. Then
Therefore, limα→+0α q ρ(α) = 0.
Since ρ(α) is continuous, there exists at leat one which satisfies (30).
Therefore, as μ, δ, ε → 0.
By Theorem 2.1 the sequence converges to x0 ∈ S0 with -minimal norm 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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