Strong Convergence of the Mann Iteration for Demicontractive Mappings

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1 Applied Mathematical Sciences, Vol. 9, 015, no. 4, HIKARI Ltd, Strong Convergence of the Mann Iteration for Demicontractive Mappings Ştefan Măruşter West University of Timişoara, Romania Copyright c 015 Ştefan Măruşter. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The strong convergence of the Mann iteration for a demicontractive mapping T : C C, where C is a subset of a real Hilbert space, is investigated. The main result states that if T is demicontractive and Fréchet differentiable at some fixed point of T, then the Mann iteration with suitable starting point, converges strongly to that fixed point. In particular, it is shown that if T is strongly demicontractive and Fréchet differentiable at its (unique) fixed point, then T satisfies a Kannan contraction inequality. Mathematics Subject Classification: 65J15 Keywords: Fixed point; Mann iteration; Strong convergence; Demicontractive mappings 1. Introduction Let H be a real Hilbert space (scalar product and norm ) and C H a closed convex subset of H. A mapping T : C C is said to be demicontractive if the set of fixed points of T is nonempty, F ix(t ), and T x p x p + k x T x, x H, p F ix(t ), (1.1) where k (0, 1). T is said to be strongly demicontractive if F ix(t ), and T x p α x p + k x T x, x H, p F ix(t ), (1.) were α, k (0, 1). Remark 1.1. In the paper [10] (01) was proposed the name Firmly pseudodemicontractive for a mapping satisfying (1.). We consider that Strongly

2 06 Ştefan Măruşter demicontractive, terminology which was used in [8] (011), is more appropriate. The demicontractivity and some weak smoothness conditions, like demiclosedness at zero, ensure the weak convergence of the Mann iteration to some fixed point [5, 7]. To get strong convergence, some additional conditions are needed (see, for example, [4]). The problem of additional conditions for getting strong convergence of the Mann iteration for demicontractive mappings was discussed in several papers, including the papers in which the concept of demicontractivity was introduced [5, 7]. For example in [5] it is required, in addition, that I T maps closed bounded subsets of C into closed subsets of C; in particular, this is satisfied if T is demicompact. In [7] the existence of a nonzero solution h H, h 0, of the variational inequality x T x, h 0, x C is required as additional condition. It is obvious that the existence of a nonzero solution of this variational inequality occurs only in very particular cases; an example for linear equations is given in [7]. In [] it is required (as the main additional condition) that the mapping T should be demicompact (Corollary 3.3). Note that this result was proved in [3] for a strictly pseudocontractive mapping (such mappings are more restrictive than the demicontractive ones). The same type of additional conditions (T is demicompact or C is a compact subset of H) appear in [6]. In [8] the concept of α-demicontractivity is introduced and it is proved that strong demicontractivity together with α-demicontractivity ensure the strong convergence. A recent additional condition is given in [1], namely, T x, x x λ x T x, x C, where λ is the constant that appears in the (A) condition [7]. In this paper we prove the strong convergence of the Mann iteration for demicontractive mappings without genuine additional conditions, but with stronger smoothness conditions than demiclosedness at zero, more precisely, the mapping T is supposed to be Fréchet differentiable at some fixed point. We prove also that a strongly demicontractive mapping satisfies an inequality of Kannan type.. Preliminaries and lemmas The following lemma is a simple variant of the Mean Value Theorem. Lemma.1. Suppose that T is F réchet differentiable at a point p C. Then T (u) T (p) = (T (p) + R(u))(u p), u C, where R(u) satisfies the following condition: Given c > 0 there exists r such that if u S(p, r) = {x : x p r} then R(u) c. The proof is immediate if we take R(u) = (T (u) T (p) T (p)(u p))(u p) T / u p.

3 Strong convergence of the Mann iteration 063 In the following the mapping T : C C will be said to satisfy condition (A) if T has a fixed point p C, T is Fréchet differentiable at p and I T (p) is invertible. Lemma.. Suppose that T satisfies the condition (A). Then, for any positive number c satisfying the condition cη < 1, where η := (I T (p)) 1, there exists r such that where β = η 1 ηc. x p β x T (x), x S(p, r), Proof. Let c be a positive number satisfying cη < 1 and r defined in Lemma 1. Since η R(x) ηc < 1, from Perturbation Lemma it results that there exists [I T (p) R(x)] 1 and [I T (p) R(x)] 1 Now, using again Lemma 1, we have and x T (x) Finally we obtain (I T (p)) 1 1 (I T (p)) 1 R(x) = η 1 η R(x) < η 1 ηc. = (x p) (T (x) p) = [I T (p) R(x)](x p) [I T (p) R(x)] 1 1 x p, x p [I T (p) R(x)] 1 x T (x). x p η 1 ηc x T (x), x S(p, r). Remark.3. In the conditions of Lemma the point p is the unique fixed point of T in S(p, r). Indeed, if q is another fixed point of T in S(p, r), q p, then q p β q T (q) = 0, which is a contradiction. 3. Strong convergence We are now able to investigate the strong convergence of the Mann iteration in the case of a mapping that is Fréchet differentiable at a fixed point, for both demicontractive and strongly demicontractive mappings. In the case of demicontractive mappings the strong convergence can be obtained in the usual conditions on the control sequence. Theorem 3.1. Suppose that T is demicontractive on C and satisfies condition (A). Then the sequence {x n } generated by the Mann iteration with control sequence {t k } satisfying the condition 0 < a t n b < 1 k, and for suitable starting point x 0, converges strongly to p.

4 064 Ştefan Măruşter Proof. The inequality (1.1) is equivalent with x T x, x p 1 k x T x, (x, p) C F ix(t ). Thus, if T tn = (1 t n )I + t n T denote the generation function of the Mann iteration, we have T tn (x) p x p t n (1 k t n ) x T (x). As x n+1 = T tn (x n ) and because 1 k t n > 0, it follows that x n+1 p x n p and so x n p l P as n. From 0 < a t n b < 1 k we have further that x n T (x n ) [a(1 k b)] 1 ( x n p x n+1 p ) 0 (n ). Let c be a positive number satisfying the condition c < [I T (p)] 1 1 and r defined in Lemma. If x 0 S(p, r) then {x n } S(p, r), n and we can apply Lemma. It results x n p β x n T (x n ) 0, n. Remark 3.. If T (p) < 1 then also T t n (p) < (1 t n ) + t n T (p) < 1 and the sequence defined by x n+1 = T tn (x n ) converges to p. Note that, generally, the condition T (p) < 1 is more restrictive than the existence of (I T (p)) 1. It is obvious that the same statement concerning the strong convergence of the Mann iteration is valid in this case of strongly demicontractive mappings. Under some appropriate conditions this class of mappings verifies the Kannan contraction inequality and, on this basis, the strong convergence can be obtained. The theorem below highlights this property. For the sake of simplicity, we consider a Mann iteration with constant control sequence, t n = t, n = 0, 1,,..., usually known as Krasnoselski iteration, i.e., the iteration is defined by T t = (1 t) + tt. Recall that a mapping T : C C is a Kannan contraction if d(t x, T y) ϱ[d(x, T x) + d(y, T y)], x, y C, (3.1) where ϱ (0, 1/). There holds The Kannan fixed point theorem or Kannan principle: A mapping T that satisfies (3.1) has a unique fixed point p and the Picard iteration converges to p. Note that (3.1) does not imply the continuity of T, as the well known Banach contraction does, and that the fixed point theorem of Kannan characterizes the metric completeness of the underlying space [9]. In the sequel we will suppose that T is strongly demicontractive with constants α, k. Suppose that there exists a number ϱ satisfying the conditions: 0 < ϱ < 0.5,

5 Strong convergence of the Mann iteration 065 ϱ < 1 (1 α)(1 k), (3.) { 1 ϱ } 1 + k ϱ η < min,, 1 α 1 α where = 1 ϱ (1 α)(1 k). Remark 3.3. In the particular case of a strongly demicontractive mapping T with α = 0.5 and k = 0., for any ϱ (0.45, 0.5) we have ϱ < 1 (1 α)(1 k) and 1 ϱ 1 + k ϱ 0.53, α 1 α Thus, if η < 0.53 then (3.) is satisfied. From (3.) it follows that we can find a number c greater than zero so that { } 1 c < min η 1 α 1 ϱ, 1 1 α η. (3.3) 1 + k ϱ Now let r be the radius defined in Lemma and β = η 1 ηc. Theorem 3.4. Suppose that T : C C is a strongly demicontractive mapping and satisfies condition (A). Let c be a positive number such that ηc < 1 and suppose that there exists ϱ satisfying conditions (3.). Then T t x T t y ϱ( x T t x y T t y ), x, y S(p, r), where t (t 1, t ) (0, 1) and t 1, t are the roots of the polynomial P (t) = (1 ϱ )t [1 k + (1 α)β ]t + 1 k. Proof. The inequality (1.) is equivalent with x T x, x p 1 α x p + 1 k x T x, and we obtain T t x p = x p t(x T x) x p t(1 α) x p t(1 k) x T x + t x T x = (1 t + tα) x p + (t t + tk) x T x. Using Lemma, there exists r such that x p β x T (x), x S(p, r) and, taking into account that x T (x) = x T t (x) /t, we have T t (x) p (1 t + tα)β + t t + tk t x T t (x), x S(p, r). (3.4) Consider the polynomial P (y) := (1 α)y 1 ϱ y + 1 k. Because = 1 ϱ (1 α)(1 k) > 0, P has two real roots, y 1, y, and y 1 = ( 1 ϱ )/(1 α). As β < y 1 it follows that P (β) > 0.

6 066 Ştefan Măruşter Consider now the polynomial Q(t) = (1 ϱ )t dt + β, where d := 1 k + (1 α)β. P (β) > 0 implies that d 4(1 ϱ )β > 0 and Q has also two real roots t 1, = d ± d 4(1 ϱ )β. (1 ϱ ) Obviously, t > 0. The lowest root t 1 is less than 1, t 1 < 1. Indeed, from (3.3) we have and 1 η c > 1 α 1 + k ϱ (1 α)β < 1 + k ϱ = (1 ϱ ) 1 + k. Thus d (1 ϱ ) < 0, from which it follows that d (1 ϱ ) < d 4(1 )β and d d 4(1 ϱ )β < (1 ϱ ). Therefore, (t 1, t ) (0, 1). For t (t 1, t ) (0, 1) we have that Q(t) < 0 which means that (1 t + tα)β + t t + tk t < ϱ. From (3.4) we obtain, for t (t 1, t ) (0, 1) Finally we have T t x p ϱ x T t x, x S(p, r). T t x T t y T x p + T t y p ϱ( x T t x + y T t y ), x, y S(p, r). Remark 3.5. If, besides the conditions of Theorem, we suppose that T (S(p, r)) S(p, r), then T t : S(p, r) S(p, r) and T t is a Kannan contraction. We can use the Kannan fixed point theorem to conclude that the Mann iteration with control sequence satisfying t (t 1, t ) (0, 1) converges to p. The conditions required by Theorem concerning the control sequence are quite different from those required in Theorem 1. Corollary 3.6. Suppose that T : C C is a strongly demicontractive mapping and satisfies condition (A). Suppose further that the following two conditions are satisfied 3 4(1 α)(1 k) > 0, (3.5) 1.75 (I T (p) < 1 k. (3.6) Then T satisfies a Kannan contraction condition with ϱ = Proof. It is sufficient to prove inequalities (3.). From (3.5) it follows that 1 < 1 (1 α)(1 k) and if we take ϱ < 1 then 4 ϱ < 1 < 1 (1 α)(1 k). Thus the first two inequalities of (3.) are fulfilled. 4 Let f1, f be two real functions of three variable, each defined by 1 ϱ f1(α, k, ϱ) = 1 k 1 + k ϱ, f(α, k, ϱ) = 1 k 1 α α 1.75.

7 Strong convergence of the Mann iteration 067 Consider now the following two constrained optimization problems: { min f1, min f, 0 < α < 1, 0 < k < 1, ϱ < 0.5. It is easy to show that min f1 = 0 and min f = Note that is the lowest value of ϱ for which min f1 = 0. From (3.6) we obtain the third inequality (3.). Therefore, the conditions (3.) are fulfilled. If α, k (0.134, 1) then 3 4(1 α)(1 k) > 0. Also, the minimum value of 1 (1 α)(1 k) for α, k (0.134, 1) is 0.5. Therefore, if the demicontractive constants α, k belong to this interval and ϱ ( , 0.5) then (3.) are satisfied. We obtain Corollary 3.7. Suppose that T : C C is a strongly demicontractive mapping with α, k (0.134, 1) and satisfies condition (A). Suppose further that (3.6) is satisfied. Then T satisfies a Kannan contraction condition with ϱ = The real function in the example below satisfies the conditions of Corollary. Example 3.8. Let T be the real function T := f : [ 0.5, 0.5] [ 0.5, 0.5] given by f(x) = x 3 1.x. The function f is strongly demicontractive with p = 0 and α = 0.5, k = 0.. For this function I T (p) is invertible and η = (1 f (0)) 1 = The condition (3.6) is satisfied, 1.75 (1 f (0)) 1 = and 1 k = 0.8. For ϱ = and c = 0.1 the polynomial P has the roots t 1 = 0.35 and t = Note that f(s(0, r)) S(0, r) and therefore f t is a Kannan contraction for t (t 1, t ). Note also that (t 1, t ) approximates quite well the interval of good values of t (for which the Mann iteration converges), in our example this interval is (0, 0.9). References [1] B. G. Akuchu, Strong convergence of the Mann sequence for demicontractive maps in Hilbert spaces, Adv. Fixed Point Theory, 4 No. 3 (014) [] D. Boonchari, S. Saejung, Construction of common fixed points of a countable family of λ-demicontractive mappings in arbitrary Banach spaces, Appl. Math. Comput., 16 (010) [3] C. E. Chidume, M. Abbas, B. Ali, Convergence of the Mann iteration algorithm for a class of pseudocontractive mappings, Appl. Math. Comput. 194 (1) (007) [4] C. E. Chidume, S. Maruster, Iterative methods for the computation of fixed points of demicontractive mappings, J. Comput. Appl. Math., 34(3) (010) [5] T. L. Hicks, J. D. Kubicek, On the Mann iteration process in a Hilbert spaces, J. Math. Anal. Appl., 59 (1977) [6] S. M. Kang, A. Rafiq, N. Hussain, Weak and strong convergence of fixed points of demicontractive mappings in smooth Banach spaces, Int. J. Pure Appl. Math., 84 (3) (013)

8 068 Ştefan Măruşter [7] St. Maruster, The solution by iteration of nonlinear equations in Hilbert spaces, Proc. Amer. Math.Soc., 63 (1) (1977) [8] L. Maruster, St. Maruster, Strong convergence of the Mann iteration for α- demiccontractive mappings, Mathematical and Computer Modeling, 54 (9-10) (011) [9] P. V. Subrahmanyam, Completenes and fixed points, Monasth. Math., 80 (1975) [10] Yanrong Yu, Delei Sheng, On the strong convergence of an algorithm about Firmly Pseudo-Demicontractive mappings for the split common fixed point problem, Journal of Applied Mathematics, Volume 01, Hindawi, Article ID Received: February 4, 015; Published: March 14, 015