A Mathematical Journal Vol. 7, No 3, (87 - 94). December 2005. Convergence rates in regularization for ill-posed variational inequalities Nguyen Buong 1 Vietnamse Academy of Science and Technology, Institute of Information Technology 18, Hoang Quoc Viet, q. Cau Giay, Ha Noi, Vietnam nbuong@ioit.ncst.ac.vn ABSTRACT In this paper the convergence rates for ill-posed inverse-strongly monotone variational inequalities in Banach spaces are obtained on the base of choosing the regularization parameter by the generalized discrepancy principle. RESUMEN En este art́ıculo se obtienen tasas de convergencia para desigualdades varia- cionales en problemas inversos mal puestos fuertemente monótonos en espacios de Banach, sobre la base de la elección del parámetro de regularización por medio del principio de discrepancia generalizada. Key words and phrases: Monotone operators, hemi-continuous, strictly convex Banach space, Frechet differentiable and Tikhonov regularization. Math. Subj. Class.: 47H17; CR: G1.8. 1The author would like to express his thanks to the referees for their valuable remarks. This work was supported by the National Fundamental Research Program in Natural Sciences. 88 Nguyen Buong 7, 3(2005) 1 Introduction. Let X be a real reflexive Banach space having the E-property and X∗, the dual space of X, be strictly convex. For the sake of simplicity, the norms of X and X∗ will be denoted by the symbol ‖.‖. We write 〈x∗, x〉 instead of x∗(x) for x∗ ∈ X∗ and x ∈ X. Let A be a hemi-continuous and monotone operator from X into X∗, and K be a closed convex subset of X. For a given f ∈ X∗, consider the variational inequality: find an element x0 ∈ K such that 〈A(x0) − f, x − x0〉 ≥ 0, ∀x ∈ K. (1.1) Variational inequalities and their approximations have been extensively studied in the last two decates. Existence and approximations of solutions of variational inequalities for various classes of operators in Hilbert and Banach spaces have been considered in [1]-[5], [7], [8], [10], [11] and [13]. We mention, in particular, the paper [3], [11], where the operator method or iterative method of regularization are considered. Further, in [7] the convergence rates of the operator method of regularization is investigated under the inverse-strongly monotone A in Hilbert space when the parameter of regularization α is chosen a priory. In the Banach space X, the operator method of regularization is the following variational inequality 〈Ah(xτα) + αU (x τ α − x 0) − fδ, x − xτα〉 ≥ 0, x τ α ∈ K, ∀x ∈ K, (1.2) where Ah are also monotone operators from X into X∗ and approximate A in the sense ‖Ah(x) − A(x)‖ ≤ hg(‖x‖) (1.3) with a nonegative continuous and bounded (image of bounded set is bounded) function g(t), U is the normalized duality mapping of X, i.e., U is the mapping from X onto X∗ satisfying the condition (see [14]) 〈U (x), x〉 = ‖x‖2, ‖U (x)‖ = ‖x‖, fδ are the approximations of f : ‖fδ − f‖ ≤ δ, τ = (h, δ), and x0 is some element in X playing the role of a criterion selection. By the choice of x0, we can influence which solution we want to approximate. In [11], it is showed the existence and uniqueness of the solution xτα for every α > 0 and for arbitrary Ah, fδ. And, the regularized solution xτα converges to x0 ∈ S0, the set of solutions of (1.1) which is assumed to be nonempty, with ‖x0 − x0‖ = min x∈S0 ‖x − x0‖, if (h+δ)/α, α → 0. Moreover, for each fixed τ = (δ, h) the papameter of regularization α can be chosen by the discrepancy principle ρ(α) = (k − 1)(δ + h)p + δp + g(‖xτα‖)h p, 0 < p < 1, k > 1, 7, 3(2005) Convergence rates in regularization for ill-posed variational inequalities 89 where ρ(α) = α‖xτα − x0‖, under the conditions: x0 ∈ int K and ‖Ah(x0) − fδ‖ > (k − 1)(δ + h)p + δp + g(‖x0‖)hp for 0 < δ < δ < 1, 0 < h < h < 1. The case x0 ∈ ∂K also is considered when xτα ∈ int K. In this paper, under the condition x0 ∈ K\S0 without the restriction xτα ∈ int K we shall show that the parameter of regularization α = α(δ, h) can be chosen by the generalized discrepancy principle ρ(α) = (δ + h)pα−q, p, q > 0, (1.4) for arbitrary monotone operator A, and on the base of the result we can estimate the convergence rates when A is an inverse-strongly monotone operator, i.e., A possesses the property 〈A(x) − A(y), x − y〉 ≥ 1 β ‖A(x) − A(y)‖2, ∀x, y ∈ X, (1.5) where β is some positive constant. In facts, variational inequalities with inverse- strongly monotone operator belong to a class of nonlinear ill-posed problems (see [7]). Note that the generalized discrepancy principle for parameter choice is presented first in [6] for the ill-posed operator equation A(x) = f (1.6) when A is a linear and bounded operator in Hilbert space. Recently, it is considered and applied in estimating convergence rates of the regularized solution for equation (1.6) involving an m-accretive (in general nonlinear) operator (see [9]). Later, the symbols ⇀ and → denote weak convergence and convergence in norm, respectively, and the notation a ∼ b is meant that a = O(b) and b = O(a). 2. Main result To obtain the result on the convergence rate for {xτ α(δ,h) } as in [6] we need the following lemmas. Lemma 1. For each p, q, δ, h > 0, there exists at least a value α such that (1.4) holds. Proof. It follows from [11] that ρ(α) is a continuous and nondecreasing function on [α0, +∞), α0 > 0. Moreover, ρ(α) > 0 ∀ α 6= 0. Indeed, if α1 6= 0 with ρ(ατ1 ) = 0, then xτα1 = x 0 and from (1.2) it follows 〈Ah(x0) − fδ, x − x0〉 ≥ 0, ∀x ∈ K. After passing δ and h to zero in this inequality we see x0 ∈ S0. This contradicts the assumption x0 ∈ K\S0. Therefore, αqρ(α) → +∞, as α → +∞. On the other hand, since 0 ≤ ρ(α) = α‖xτα − x 0‖ ≤ δ + hg(‖x0‖‖) + 2α‖x0 − x0‖ 90 Nguyen Buong 7, 3(2005) (see also [11]), we have αqρ(α) → 0, as α → +0. Hence, there exists a value α such that (1.4) holds. Lemma 2. limδ,h→0 α(δ, h) = 0. Proof. Let δn, hn → 0, and αn = α(δn, hn) → ∞ as n → ∞. From (1.3), 〈Ahn (x τn αn ) + αnU (x τn αn − x0) − fδn , x − x τn αn 〉 ≥ 0, ∀x ∈ K, (2.1) the monotone property of Ahn and x 0 ∈ K it follows ‖xτnαn − x 0‖ ≤ ‖Ahn (x 0) − fδn‖/αn → 0, as n → ∞. Therefore, xτnαn → x 0, as n → ∞. On the other hand, by using the monotone property of Ahn and the property of U we can write (2.1) in the form 〈Ahn (x) − fδn , x − x τn αn 〉 ≥ −αn〈U (xτnαn − x 0), x − xτnαn〉 ≥ −αn‖xτnαn − x 0‖‖x − xτnαn‖ ≥ −ρ(αn)‖x − xτnαn‖ ≥ −(δn + hn)pα−qn ‖x − x τn αn ‖ → 0, as n → ∞. It means that 〈A(x0) − f, x − x0〉 ≥ 0, ∀x ∈ K, i.e., x0 is a solution of (1.1). It contradicts x0 /∈ S0. Thus, α(δ, h) remains bounded as δ, h → 0. Let δn, hn → 0 as n → ∞, and meantime αn → c > 0. Since α1+qn ‖xτnαn − x 0‖ = (δn + hn)p, we have ‖xτnαn − x 0‖ → 0, as n → ∞. Again, x0 ∈ S0. Hence, limδ,h→0 α(δ, h) = 0. Lemma 3. If 0 < p < q, then limδ,h→0(δ + h)/α(δ, h) = 0. Proof. It is easy to see that[ δ + h α(δ, h) ]p [(δ + h)pα(δ, h)−q]α(δ, h)q−p = ρ(α(δ, h))α(δ, h)q−p = α(δ, h)‖xτα(δ,h) − x 0‖α(δ, h)q−p ≤ [ δ + hg(‖x0‖) + 2α(δ, h)‖x0 − x0‖ ] α(δ, h)q−p → 0 as δ, h → 0. Therefore, lim δ,h→0 [ δ + h α(δ, h) ]p = 0. The lemma is proved. 7, 3(2005) Convergence rates in regularization for ill-posed variational inequalities 91 Lemma 4. Let 0 < p < q. Then, there exist constants C1, C2 > 0 such that, for sufficiently small δ, h > 0, the relation C1 ≤ (δ + h)pα(δ, h)−1−q ≤ C2 holds. Proof. From (δ + h)pα(δ, h)−1−q = α(δ, h)−1ρ(α(δ, h)) = ‖xτα(δ,h) − x 0‖ ≤ δ α(δ, h) + h α(δ, h) g(‖x0‖) + 2‖x0 − x0‖ and lemma 3, it implies the existence of a positive constant C2 in the lemma. On the other hand, as X is reflexive and {xτ α(δ,h) } is bounded, there exists a subsequence of the sequence {xτ α(δ,h) } that converges weakly to some element x̃0 in K such that ‖x̃0 − x0‖ ≤ lim inf ‖xτα(δ,h) − x 0‖. We can conclude that x̃0 6= x0. Indeed, if x̃0 = x0, then from the monotone hemi- continuous property of Ah and (1.2) it follows 〈Ah(x) + αU (x − x0) − fδ, x − xτα〉 ≥ 0, ∀x ∈ K. After passing δ and h in the last inequality to zero we obtain 〈A(x) − f, x − x̃0〉 ≥ 0, ∀x ∈ K which is equivalent to (1.1). It is meant that x̃0 ∈ S0. It contradicts x0 /∈ S0. Therefore, there exists a constant C1 in the lemma. To estimate the convergence rates for {xτ α(δ,h) } we assume that 〈U (x) − U (y), x − y〉 ≥ mU‖x − y‖s, mU > 0, s ≥ 2, ∀x, y ∈ X. (2.2) It is well-known that when X ≡ H, the Hilbert space, mU = 1, s = 2, and when X = Lp or Wp, mU = p − 1, s = 2 for the case 1 < p < 2. In the case p > 2 we have to use the duality mapping U s satisfying the condition 〈U s(x), x〉 = ‖x‖s, ‖U s(x)‖ = ‖x‖s−1, s ≥ 2 instead of U . Then, mU s = 22−p/p and s = p in (2.2) (see [12]). Theorem 1. Assume that the following conditions hold: (i) A is an inverse-strongly-monotone operator in X with ‖A(x) − A(x0) − A′(x0)(x − x0)‖ ≤ τ̃‖A(x) − A(x0)‖, ∀x ∈ X, where τ̃ is some positive constant; 92 Nguyen Buong 7, 3(2005) (ii) There exists an element z ∈ X such that A′(x0)∗z = U (x0 − x0); (iii) The parameter α is chosen by (1.4) with p < q. Then, we have ‖xτα(δ,h) − x0‖ = O((δ + h) θ), θ = p (1 + q)(2s − 1) . Proof. From (1.1) - (1.3) it follows 〈A(xτα(δ,h)) − A(x0), x τ α(δ,h) − x0〉 + α(δ, h) × 〈U (xτα(δ,h) − x 0) − U (x0 − x0), xτα(δ,h) − x0〉 ≤ (δ + hg(‖xα(δ,h)‖))‖xα(δ,h) − x0‖ +α(δ, h)〈U (x0 − x0), x0 − xτα(δ,h)〉. (2.3) Thus, by using (1.5) and the monotone property of U we obtain ‖A(xτα(δ,h)) − A(x0)‖ ≤ O( √ δ + h + α(δ, h))‖xτα(δ,h) − x0‖ 1/2. On the other hand, from (2.2), (2.3) and the monotone property of A which is followed from (1.5) we have mU‖xτα(δ,h) − x0‖ s ≤ 〈U (xτα(δ,h) − x 0) − U (x0 − x0), xτα(δ,h) − x0〉 ≤ δ + C̃0h α(δ, h) ‖xτα(δ,h) − x 0)‖ + 〈z, A′(x0)(x0 − xτα(δ,h))〉 where C̃0 is some positive constant, and∣∣〈z, A′(x0)(x0−xτα(δ,h))〉∣∣ ≤ ‖z‖(τ̃ + 1)‖A(xτα(δ,h)) − A(x0)‖ ≤ ‖z‖(τ̃ + 1)O( √ δ + h + α(δ, h))‖xτα(δ,h)) − x0‖ 1/2. Now, from lemma 4 it implies that α(δ, h) ≤ C−1/(1+q)1 (δ + h) p/(1+q). and δ + h α(δ, h) ≤ C2(δ + h)1−pα(δ, h)q ≤ C2C −q/(1+q) 1 (δ + h) 1−p(δ + h)pq/(1+q) ≤ C2C −q/(1+q) 1 (δ + h) 1−p/(1+q). 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