ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES

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1 TJMM 6 (2014), No. 1, ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES ADESANMI ALAO MOGBADEMU Abstract. In this present paper, we employed a modified Noor iteration method introduced by Rafiq [9]. Some strong convergence theorems of this iteration scheme are established for three nearly uniformly Lipschitzian mappings if at least one of these maps is uniformly Lipschitzian mapping. Our results extend and improve the recent ones proved by Chang et al., Kim et al., Olaleru and Mogbademu, Ofoedu and many others. 1. Introduction Let denote by J the normalized duality mapping from X into 2 X by J(x) = {f X : x, f = x 2 = f 2 }, where X denotes the dual space of real normed linear space X and.,. denotes the generalized duality pairing between elements of X and X. We first recall and define some concepts as follows (see, [7]): Let K be a nonempty subset of real Banach space X. The mapping T is said to be asymptotically pseudocontractive if there exists a sequence (k n ) [1, ) with lim n k n = 1 and for any x, y K there exists j(x y) J(x y) such that T n x T n y, j(x y) k n x y 2, n 1. The concept of asymptotically pseudocontractive mappings was introduced by Schu [10]. A mapping T : K X is called Lipschitzian if there exists a constant L > 0 such that T x T y L x y, for all x, y K and is called generalized Lipschitzian if there exists a constant L > 0 such that T x T y L( x y + 1), for all x, y K. A mapping T : K K is called uniformly L-Lipschitzian if for each n N, there exists a constant L > 0 such that T n x T n y L x y, for all x, y K. It is obvious that the class of generalized Lipschitzian map includes the class of Lipschitz map. Moreover, every mapping with a bounded range is a generalized Lipschitzian mapping Mathematics Subject Classification. 47H10, 46A03. Key words and phrases. modified Noor iteration method, nearly Lipschitzian mappings, uniformly Lipschitzian maps, Banach space, Common fixed point. 45

2 46 ADESANMI ALAO MOGBADEMU Sahu [11] introduced the following new class of nonlinear mappings which is more general than the class of generalized Lipschitzian mappings and the class of uniformly L- Lipschitzian mappings. Fix a sequence {r n } in [0, ] with r n 0. A mapping T : K K is called nearly Lipschitzian with respect to {r n } if for each n N, there exists a constant k n > 0 such that T n x T n y k n ( x y + r n ) for all x, y K. A nearly Lipschitzian mapping T with sequence {r n } is said to be nearly uniformly L-Lipschitzian if k n = L for all n N. Observe that the class of nearly uniformly L-Lipschitzian mapping is more general than the class of uniformly L-Lipschitzian mappings. In recent years, many authors have given much attention to iterative methods for approximating fixed points of Lipschitz type pseudocontractive type nonlinear mappings (see [1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12]). Ofoedu [7] used the modified Mann iteration process introduced by Schu [10], x n+1 = (1 α n )x n + α n T n x n, n 0, (1) to obtain a strong convergence theorem for uniformly Lipschitzian asymptotically pseudocontractive mapping in real Banach space setting. This result itself is a generalization of many of the previous results (see [7] and the references therein). Recently, Chang et al. [3] proved a strong convergence theorem for a pair of L- Lipschitzian mappings instead of a single map used in [7]. In fact, they proved the following theorem : Theorem 1 ([3]). Let E be a real Banach space, K be a nonempty closed convex subset of E, T i : K K, (i = 1, 2) be two uniformly L i -Lipschitzian mappings with F (T 1 ) F (T 2 ) φ, where F (T i ) is the set of fixed points of T i in K and ρ be a point in F (T 1 ) F (T 2 ). Let k n [1, ) be a sequence with k n 1. Let {α n } n=1 and {β n } n=1 be two sequences in [0, 1] satisfying the following conditions: (i) n=1 α n = (ii) n=1 α2 n < (iii) n=1 β n < (iv) n=1 α n(k n 1) <. For any x 1 K, let {x n } n=1 be the iterative sequence defined by x n+1 = (1 α n )x n + α n T n 1 y n y n = (1 β n )x n + β n T n 2 x n. If there exists a strictly increasing function Φ : [0, ) [0, ) with Φ(0) = 0 such that for all j(x ρ) J(x ρ) and x K, (i = 1, 2), then {x n } n=1 converges strongly to ρ. The result above extends and improves the corresponding results of [7] from one uniformly Lipschitzian asymptotically pseudocontractive mapping to two uniformly Lipschitzian mappings. In fact, if the iteration parameter {β n } n=0 in Theorem 1 above is equal to zero for all n and T 1 = T 2 = T then, we have the main result of Ofoedu [7]. Rafiq [9], introduced a new type of iteration the modified three-step iteration process, to approximate the common fixed point of three nonlinear mappings in real Banach spaces. It is defined as follows: (2)

3 ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD Let T 1, T 2, T 3 : K K be three mappings. For any given x 1 K, the modified Noor iteration {x n } n=1 K is defined by x n+1 = (1 α n )x n + α n T 1 y n y n = (1 β n )x n + β n T 2 z n z n = (1 γ n )x n + γ n T 3 x n, n 1, (3) where {α n } n=1, {β n } n=1 and {γ n } n=1 are three real sequences satisfying some conditions. It is clear that the iteration scheme (3) includes iterations defined in (1) and (2). It is also worth mentioning that, several authors, (for example, see [8]), have recently used the iteration in equation (3) to approximate the common fixed points of some nonlinear operators in Banach spaces. In this present paper, we employed a modified Noor iteration method introduced by Rafiq [9] and prove that it converges to a common fixed point of three nearly uniformly Lipschitzian mappings if at least one of these maps is uniformly Lipschitzian mapping. Thus, our results extend and improve the recent ones proved by Chang et al., Kim et al., Olaleru and Mogbademu, Ofoedu and many others. We need the following Lemmas. Lemma 1 ([1, 7]). Let E be real Banach Space and J : E 2 E be the normalized duality mapping. Then, for any x, y E x + y 2 x y, j(x + y), j(x + y) J(x + y). Lemma 2 ([6]). Let Φ : [0, ) [0, ) be an increasing function with Φ(x) = 0 x = 0 and let {b n } n=0 be a positive real sequence satisfying b n = + and n=0 lim b n = 0. n Suppose that {a n } n=0 is a nonnegative real sequence. If there exists an integer N 0 > 0 satisfying a 2 n+1 < a 2 n + o(b n ) b n Φ(a n+1 ), n N 0 where lim n o(b n) b n = 0, then lim n a n = Main results Theorem 2. Let K be a nonempty closed convex subset of a real Banach space X and T 1, T 2, T 3 : K K be three nearly uniformly L i -Lipschitzian mappings with sequences {r ni },(i = 1, 2, 3) such that F (T 1 ) F (T 2 ) F (T 3 ) φ, where F (T i ) (i = 1, 2, 3) is the set of fixed points of T 1, T 2,T 3 in K and, ρ be a point in F (T 1 ) F (T 2 ) F (T 3 ). Let k n [1, ) be a sequence with k n 1. Let {α n } n=1, {β n } n=1 and {γ n } n=1 be real sequences in [0, 1] satisfying (i) n 1 α n = (ii) lim n α n, β n, γ n = 0. Let T 1 be uniformly Lipschitzian mapping and {x n } n=1 be the iterative sequence defined by (3). If there exists a strictly increasing function Φ : [0, ) [0, ) with Φ(0) = 0 such that for all j(x ρ) J(x ρ) and x K, (i = 1, 2, 3), then {x n } n=1 converges strongly to ρ.

4 48 ADESANMI ALAO MOGBADEMU Proof. Since T 1, T 2 and T 3 are nearly uniformly L i -Lipschitzian mappings with {r ni }, we have that for all x, y K T n i x T n i y L i ( x y + r ni ), (i = 1, 2, 3). For convenience, denote L = max{l i } and r n = sup{r ni } : n N. And, since T 1 is a nearly uniformly L-Lipschitzian with sequence r n, then there exists a strictly increasing continuous function Φ : [0, ) [0, ), Φ(0) = 0 such that and for x K, ρ F (T ), that is T n 1 x n T n 1 ρ L( x n ρ + r n ) T n 1 x n T n 1 ρ, j(x n ρ) k n x n ρ 2 Φ( x n ρ ), (4) (k n I T n 1 )x n (k n I T n 1 )ρ, j(x n ρ) Φ( x n ρ ). (5) Step 1. We first show that {x n } n=1 is a bounded sequence. For this, if x n1 = T 1 x n1, n 1 then it clearly holds. So, let if possible, there exists a positive integer x n1 K such that x n1 T 1 x n1, thus denote x n1 = x 1 and a 1 = (k n + L) x 1 ρ 2 + L x 1 ρ. Thus by (5) for any n 1, k n (x 1 ρ) (T n 1 x 1 ρ), j(x 1 ρ) Φ( x 1 ρ ), (6) that is, (k n + L) x 1 ρ 2 + L x 1 ρ Φ( x 1 ρ ). Thus, on simplifying we have x 1 ρ Φ 1 (a 1 ). (7) Now, we claim that x n ρ 2Φ 1 (a 1 ), n 0. Clearly, inview of (7), the claim holds for n = 1. We next assume that x n ρ 2Φ 1 (a 1 ), for some n and we shall prove that x n+1 ρ 2Φ 1 (a 1 ). Suppose this is not true, i.e. x n+1 ρ > 2Φ 1 (a 1 ). Since {r n } [0, ] with r n 0 and set M = sup{r n : n N} and M = sup{ 1 α n : n N}. Denote τ 0 = min 1 3 { 1, Φ(2(Φ 1 (a 1))) 18(Φ 1 (a 1)), 2 Φ(2(Φ 1 (a 1))) 6[(2+3L)Φ 1 (a 1)+ML](Φ 1 (a 1)), 2 Φ(2(Φ 1 (a 1)) 12L[(2+3L)Φ 1 (a 1)+ML+MM ](Φ 1 (a 1)) 2, 3Φ 1 (a 1) (2+3L)Φ 1 (a, 3Φ 1 (a 1) 1)+ML 2(1+L)Φ 1 (a 1)+ML Since lim n α n, β n, γ n = 0, without loss of generality, let 0 α n, β n, γ n, k n 1 τ 0 for any n 1. Then, we have the following estimates from (3) z n ρ = (1 γ n )x n + γ n T3 n x n ρ x n ρ + γ n T3 n x n x n x n ρ + γ n [(1 + L) x n ρ + r n L 2Φ 1 (a 1 ) + τ 0 [(1 + L)2Φ 1 (a 1 ) + ML] 3Φ 1 (a 1 ). y n ρ = (1 β n )x n + β n T2 n z n ρ x n ρ + β n T2 n z n x n x n ρ + β n (L( z n ρ + r n ) + x n ρ ) 2Φ 1 (a 1 ) + β n [L(2Φ 1 (a 1 ) + M) + 2Φ 1 (a 1 )] 2Φ 1 (a 1 ) + β n [(2 + 3L)Φ 1 (a 1 ) + ML] 2Φ 1 (a 1 ) + τ 0 [(2 + 3L)Φ 1 (a 1 ) + ML] 3Φ 1 (a 1 ). }. (8)

5 ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD Also, we have the following estimates: (i) T1 n y n x n x n ρ + T1 n y n ρ x n ρ + L( y n ρ + r n ) 2Φ 1 (a 1 ) + L(3Φ 1 (a 1 ) + M) (2 + 3L)Φ 1 (a 1 ) + ML. (ii) x n+1 ρ 3Φ 1 (a 1 ). (iii) x n+1 x n τ 0 [(2 + 3L)Φ 1 (a 1 ) + ML]. (iv) x n T2 n z n (2 + 3L)Φ 1 (a 1 ) + ML. (v) y n x n+1 y n x n + x n+1 x n β n T2 n z n x n + x n+1 x n β n [(2 + 3L)Φ 1 (a 1 ) + ML] + α n [(2 + 3L)Φ 1 (a 1 ) + ML] 2τ 0 [(2 + 3L)Φ 1 (a 1 ) + ML]. (9) Using Lemma 1 and the above estimates, we have x n+1 ρ 2 = (1 α n )x n + α n T1 n y n ρ 2 = x n ρ + α n (T1 n y n x n ) 2 x n ρ 2 2 x n T1 n y n, j(x n+1 ρ) = x n ρ 2 + 2α n T1 n x n+1 ρ, j(x n+1 ρ) 2α n x n+1 ρ, j(x n+1 ρ) +2α n T1 n y n T1 n x n+1, j(x n+1 ρ) +2α n x n+1 x n, j(x n+1 ρ) x n ρ 2 + 2α n (k n x n+1 ρ 2 Φ( x n+1 ρ )) 2α n x n+1 ρ 2 + 2α n T1 n y n T1 n x n+1 x n+1 ρ +2α n x n+1 x n x n+1 ρ = x n ρ 2 + 2α n (k n 1) x n+1 ρ 2 2α n Φ( x n+1 ρ ) +2α n L( y n x n+1 + r n ) x n+1 ρ +2α n x n+1 x n x n+1 ρ x n ρ 2 2α n Φ(2(Φ 1 (a 1 )) + 2α n (k n 1) x n+1 ρ 2 +2α n L[2τ 0 ((2 + 3L)Φ 1 (a 1 ) + ML) + M] x n+1 ρ +2α n τ 0 [(2 + 3L)Φ 1 (a 1 ) + ML] x n+1 ρ x n ρ 2 2α n Φ(2(Φ 1 (a 0 )) + 18α n τ 0 (Φ 1 (a 1 )) 2 +6α n L[2τ 0 ((2 + 3L)Φ 1 (a 1 ) + ML) + M)](Φ 1 (a 1 )) 2 +6α n τ 0 [(2 + 3L)Φ 1 (a 0 ) + ML](Φ 1 (a 0 )) 2 x n ρ 2 2α n Φ(2(Φ 1 (a 1 )) + 18α n τ 0 (Φ 1 (a 1 )) 2 +12α n τ 0 L[(2 + 3L)Φ 1 (a 1 ) + ML + MM ](Φ 1 (a 1 )) 2 +6α n τ 0 [(2 + 3L)Φ 1 (a 1 ) + ML](Φ 1 (a 1 )) 2 x n ρ 2 2α n Φ(2(Φ 1 (a 1 )) + α n Φ(2(Φ 1 (a 1 )) x n ρ 2 α n Φ(2(Φ 1 (a 1 )) x n ρ 2 (2(Φ 1 (a 1 ))) 2, (10) which is a contradiction. Hence {x n } n=1 is a bounded sequence. So {y n }, {z n }, {T n 1 y n }, {T n 2 z n } are all bounded sequences. Step 2. We want to prove x n ρ 0.

6 50 ADESANMI ALAO MOGBADEMU Since α n, β n, γ n, (k n 1) 0 as n and {x n } n=1 is bounded. From (9), we observed that lim n x n+1 x n = 0, lim n T1 n y n T1 n x n+1 = 0, lim n (k n 1) = 0. So from (10), we have where x n+1 ρ 2 x n ρ 2 2 x n T1 n y n, j(x n+1 ρ) = x n ρ 2 + 2α n T1 n x n+1 ρ, j(x n+1 ρ) 2α n x n+1 ρ, j(x n+1 ρ) +2α n T1 n y n T1 n x n+1, j(x n+1 ρ) +2α n x n+1 x n, j(x n+1 ρ) x n ρ 2 + 2α n (k n 1) x n+1 ρ 2 2α n Φ( x n+1 ρ ) +2α n T1 n y n T1 n x n+1 x n+1 ρ +2α n x n+1 x n x n+1 ρ = x n ρ 2 2α n Φ( x n+1 ρ ) + o(α n ), 2α n (k n 1) x n+1 ρ 2 + 2α n x n+1 x n x n+1 ρ +2α n T n 1 y n T n 1 x n+1 x n+1 ρ = o(α n ). By Lemma 2, we obtain that This completes the proof. lim x n ρ = 0. n Corollary 1. Let K be a nonempty closed convex subset of a real Banach space X and T 1, T 2 : K K be two nearly uniformly L i -Lipschitzian mappings with sequences {r ni } (i = 1, 2) such that F (T 1 ) F (T 2 ) φ, where F (T i ) (i = 1, 2) is the set of fixed points of T 1, T 2 in K and ρ be a point in F (T 1 ) F (T 2 ). Let k n [1, ) be a sequence with k n 1. Let {α n } n=1 and {β n } n=1 be real sequences in [0, 1] satisfying (i) n 0 α n = (ii) lim n α n, β n = 0. Let T 1 be uniformly Lipschitzian mapping and {x n } n=1 be the iterative sequence defined by (2). If there exists a strictly increasing function Φ : [0, ) [0, ) with Φ(0) = 0 such that for all j(x ρ) J(x ρ) and x K (i = 1, 2), then {x n } n=1 converges strongly to ρ. Corollary 2. Let K be a nonempty closed convex subset of a real Banach space X and T 1 : K K be uniformly Lipschitzian and nearly uniformly L-Lipschitzian mapping with sequence {r n } such that F (T 1 ) φ, where F (T 1 ) is the fixed point of T 1 in K and ρ be a point in F (T 1 ). Let k n [1, ) be a sequence with k n 1. Let {α n } n=1 be a real sequence in [0, 1] satisfying (i) n 0 α n = (ii) lim n α n = 0. For any x 1 K, let {x n } n=1 be the iterative sequence defined by (1). If there exists a strictly increasing function Φ : [0, ) [0, ) with Φ(0) = 0 such that for all j(x ρ) J(x ρ) and x K, then {x n } n=1 converges strongly to ρ. (11)

7 ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD Application 1. Let X = R, K = [0, 1] and T 1 : K K be a map defined by T 1 x = x 4. Clearly, T 1 is nearly uniformly Lipschitzian (r n = 1 4 n ) with F (T 1 ) = 0. Define Φ : [0, + ) [0, + ) by Φ(t) = t2 4, then Φ is a strictly increasing function with Φ(0) = 0. For all x K, ρ F (T 1 ), we get T1 n x T1 n ρ, j(x ρ) = xn 4 0, j(x 0) n = xn 4 0, x n = xn+1 4 n Obviously, T 1 satisfied (4) with sequence {k n } = 1. References x 2 x2 4 x 2 Φ(x). [1] Chang, S.S., Some results for asymptotically pseudocontractive mappings and asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 129, , [2] Chang, S.S., Cho, Y.J., Lee, B.S. and Kang, S.H., Iterative approximation of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl. 224, , [3] Chang, S.S., Cho, Y.J., Kim, J.K., Some results for uniformly L-Lipschitzian mappings in Banach spaces, Applied Mathematics Letters 22, , [4] Goebel, K. and Kirk, W.A., A fixed point theorem for asymptotically nonexpansive mappings, Proceedings of American Mathematical Society, vol. 35, , [5] Kim, J.K., Sahu, D.R. and Nam, Y.M., Convergence theorem for fixed points of nearly uniformly L- Lipschitzian asymptotically generalized Φ-hemicontractive mappings, Nonlinear Analysis 71, e2833- e2838, [6] Moore, C. and Nnoli, B.V.C., Iterative solution of nonlinear equations involving set-valued uniformly accretive operators, Comput. Math. Anal. Appl. 42, , [7] Ofoedu, E.U Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudocontractive mapping in real Banach space, J. Math. Anal. Appl. 321, , [8] Olaleru, J.O. and Mogbademu, A.A., Modified Noor iterative procedure for uniformly continuous mappings in Banach spaces, Boletin de la Asociacion Matematica Venezolana, Vol. XVIII, No. 2, , [9] Rafiq, A., On Modified Noor iteration for nonlinear equations in Banach spaces, Appl. Math. Comput. 182, , [10] Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158, , [11] Sahu, D.R., Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces, Comment. Math. Univ. Carolin 46 (4), , [12] Xue, Z. and Lv, G., Strong convergence theorems for uniformly L-Lipschitzian asymptotically pseudocontractive mappings in Banach spaces, Journal of Inequalities and Applications 2013, 2013: 79. University of Lagos Department of Mathematics Akoka, yaba, Lagos, Nigeria address: amogbademu@unilag.edu.ng