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.
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 .
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 ).
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/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 .
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).
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
(ii) let μ, δ, ε → 0. Then
Therefore, limα→+0α q ρ(α) = 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 .
- Badriev IB, Zadvornov OA, Ismagilov LN: On iterative regularization methods for variational inequalities of the second kind with pseudomonotone operators. Comput Meth Appl Math 2003,3(2):223–234.MathSciNetGoogle Scholar
- Konnov IV: Combined Relaxation Methods for Variational Inequalities. Springer, Berlin; 2001.View ArticleGoogle Scholar
- Konnov IV, Volotskaya EO: Mixed variational inequalities and economic equilibrium problems. J Appl Math 2002,2(6):289–314. 10.1155/S1110757X02106012MathSciNetView ArticleGoogle Scholar
- Ekeland I, Temam R: Convex Analysis and Variational Problems. North-Holland Publ. Company, Amsterdam; 1970.Google Scholar
- Noor MA: Proximal methods for mixed variational inequalities. J Opt Theory Appl 2002,115(2):447–452. 10.1023/A:1020848524253View ArticleGoogle Scholar
- Cohen G: Auxiliary problem principle extended to variational inequalities. J Opt Theory Appl 1988,59(2):325–333.Google Scholar
- Liskovets OA: Regularization for ill-posed mixed variational inequalities. Soviet Math Dokl 1991, 43: 384–387. (in Russian)MathSciNetGoogle Scholar
- Buong Ng, Thuy NgTT: On regularization parameter choice and convergence rates in regularization for ill-posed mixed variational inequalities. Int J Contemporary Math Sci 2008,4(3):181–198.Google Scholar
- Alber YaI, Ryazantseva IP: Nonlinear Ill-Posed Problems of Monotone Type. Springer, New York; 2006.Google Scholar
- Liu F, Nashed MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal 1998, 6: 313–344. 10.1023/A:1008643727926MathSciNetView ArticleGoogle Scholar
- Zeidler E: Nonlinear Functional Analysis and Its Applications. Springer, New York; 1985.View ArticleGoogle Scholar
- Alber YaI, Notik AI: Geometric properties of Banach spaces and approximate methods for solving nonlinear operator equations. Soviet Math Dokl 1984, 29: 611–615.Google Scholar
- Hanke M, Neubauer A, Scherzer O: A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer Math 1995, 72: 21–37. 10.1007/s002110050158MathSciNetView ArticleGoogle Scholar
- Buong Ng: Convergence rates in regularization for ill-posed variational inequalities. CUBO, Math J 2005,21(3):87–94.MathSciNetGoogle Scholar
- Engl HW: Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates. J Opt Theory Appl 1987, 52: 209–215. 10.1007/BF00941281MathSciNetView ArticleGoogle Scholar
- Buong Ng: Generalized discrepancy principle and ill-posed equation involving accretive operators. J Nonlinear Funct Anal Appl Korea 2004, 9: 73–78.Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.