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Halpern–Ishikawa type iterative method for approximating fixed points of non-self pseudocontractive mappings
Halpern–Ishikawa type iterative method for approximating fixed points of non-self...
Zegeye, Habtu; Tufa, Abebe
2018-06-04 00:00:00
Department of Mathematics, In this paper, we deﬁne a Halpern–Ishikawa type iterative method for approximating Botswana International University of Science and Technology, Palapye, a ﬁxed point of a Lipschitz pseudocontractive non-self mapping T in a real Hilbert Botswana space settings and prove strong convergence result of the iterative method to a ﬁxed Full list of author information is point of T under some mild conditions. We give a numerical example to support our available at the end of the article results. Our results improve and generalize most of the results that have been proved for this important class of nonlinear mappings. MSC: 37C25; 47H10; 47J05 Keywords: Fixed points; Monotone mappings; Pseudocontractive mappings 1 Introduction Let H be a real Hilbert space with norm · and C be a nonempty subset of H. A mapping T : C → H is said to be L-Lipschitz if there exists L ≥ 0such that Tx – Ty≤ Lx – y for all x, y ∈ C.(1) T is said to be contraction if L ∈ [0, 1) and is called nonexpansive mapping if L =1. We observe that every contraction mapping is nonexpansive and every nonexpansive mapping is Lipschitz. A mapping T : C → H is said to be k-strictly pseudocontractive if there exists k ∈ [0, 1) such that 2 2 Tx – Ty ≤x – y + k x – y –(Tx – Ty) , ∀x, y ∈ C.(2) We remark that every k-strictly pseudocontractive mapping is Lipschitz and hence the class of k-strictly pseudocontractive mappings includes properly the class of nonexpansive mappings. An important class of mappings more general than the class of k-strictly pseudocontrac- tive mappings is the class of pseudocontractive mappings. T is said to be pseudocontractive if 2 2 Tx – Ty ≤x – y + x – y –(Tx – Ty) , ∀x, y ∈ C.(3) © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 2 of 14 The class of pseudocontractive mappings is related to one of the important classes of operators known as monotone mappings. A mapping A : C → H is said to be monotone if Ax – Ay, x – y ≥ 0, ∀x, y ∈ C. Note that a mapping A : C → H is monotone if and only if T := I – A is pseudocontrac- tive, where I is an identity mapping on C.Thus, thezeros of A are ﬁxed points of T,that is, N(A):= {x ∈ C : Ax =0} = F(T):= {x ∈ C : x = Tx}. Several authors have studied iterative methods for approximating ﬁxed points of non- expansive, k-strictly pseudocontractive and pseudocontractive mappings (see, e.g., [3, 6, 15, 17, 22, 27, 28] and the references contained therein). In 1953, Mann [15]introduced the following scheme, which is refereed to as Mann iteration method: x = α x +(1– α )Tx,(4) n+1 n n n n where the initial guess x ∈ C is arbitrary and {α }⊆ [0, 1] such that lim α =0 and 0 n n→∞ n α = ∞. The Mann iteration method has been extensively investigated for approxi- mating ﬁxed points of nonexpansive mappings (see, e.g., [17]). In an inﬁnite-dimensional Hilbert space, the Mann iteration method can provide only weak convergence (see, e.g., [7]). To obtain strong convergence, numerous authors have modiﬁed the Mann iterative method (see, e.g., [8, 10, 11]) in many ways. In 1967, Halpern [8] studied the following recursive formula: x = α u +(1– α )Tx , n ≥ 0, (5) n+1 n n n where α is a sequence of numbers in (0, 1). He proved strong convergence of {x } to a ﬁxed n n –a point of T,where α := n ,for a ∈ (0, 1), in the framework of Hilbert spaces. Halpern’s scheme (5) has been studied extensively by many authors (see, e.g., [2, 12, 18, 21]). In particular, Reich [18] proved that the result of Halpern remains true in uniformly smooth Banach spaces (see also [19]). In 1977, Lions [12] improved the result of Halpern, still in Hilbert spaces, by proving strong convergence of {x } to a ﬁxed point of T,where therealsequence {α } satisﬁes the n n following conditions: α – α n n–1 (i) lim α = 0; (ii) α = ∞; (iii) lim =0. n n n→∞ n→∞ n=0 In 2002, Xu [24](seealso[25]) improved the result of Lion in two directions. First, he weakened the condition (iii) by removing the square in the denominator so that we can choose the sequence α = . Second, he proved the strong convergence of Halpern’s n+1 scheme (5) in the framework of real uniformly smooth Banach spaces. For approximating ﬁxed points of a Lipschitz pseudocontractive self-mapping T, Ishikawa [9] introduced the following process known as Ishikawa iteration: x ∈ C, ⎪ 0 (6) y = β x +(1– β )Tx , n n n n n x = α x +(1– α )Ty , n ≥ 0, n+1 n n n n Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 3 of 14 where {α }, {β } are sequences of positive numbers satisfying the conditions: n n (i) 0 ≤ α ≤ β ≤ 1; n n (ii) lim β =0; n→∞ n (iii) α β = ∞. n n He showed that the sequence {x } converges strongly to a ﬁxed point of the mapping T, provided that C is a compact convex subset of a Hilbert space H. Several authors have extended the results of Ishikawa [9] to Banach spaces without compactness assumption on C (see, e.g., [13, 23]). However, we observe that all the above results are valid only for self-mappings. For ap- proximating ﬁxed points of non-self mappings, several iterative schemes have been stud- ied (see, e.g., [16, 20]) with the use of metric projection or sunny nonexpansive retraction mapping which are generally diﬃcult to compute in practical applications. In 2015, Colao and Marino [4] introduced a new searching strategy for the coeﬃcient α which makes the Mann algorithm well-deﬁned for non-self mappings in the setting of arealHilbert space H. In fact, they studied the following scheme: x ∈ C, ⎪ 0 ⎨ 1 α = max{ , h(x )}, 0 0 (7) ⎪ x = α x +(1– α )Tx , n+1 n n n n α = max{α , h(x )}, n ≥ 0, n+1 n n+1 where h(x):= inf{λ ≥ 0: λx +(1– λ)Tx ∈ C}, ∀x ∈ C ⊆ H and T is a non-self mapping of C into H. Indeed, they obtained weak and strong convergence of the algorithm to a ﬁxed point of nonexpansive non-self mappings under appropriate conditions. Recently, Colao et al. [5] extended this result of Colao and Marino [4]toaclassof k- strictly pseudocontractive mappings. We observe that these results (the results obtained in [4]and [5]) provide a way forward to avoid the use of metric projection or sunny non- expansive mapping in constructing algorithms for approximating ﬁxed points of a more general class of non-self mappings. It is our purpose in this paper to construct and study a Halpern–Ishikawa type itera- tive scheme for non-self mappings in the setting of Hilbert spaces. As a result, we obtain strong convergence of the scheme to a ﬁxed point of a Lipschitz pseudocontractive non- self mapping under some mild conditions. Our results extend and generalize many results in the literature. 2Preliminaries Let C be a nonempty subset of a Hilbert space H. A mapping T : C → H is said to be inward if, for any x ∈ C,wehave Tx ∈ I (x):= x + λ(w – x): for some w ∈ C and λ ≥ 1 . The set I (x) is called inward set of C at x. A mapping I – T,where I is an identity mapping on C, is called demiclosed at zero if for any sequence {x } in C such that x x and Tx – n n n x → 0as n →∞,then x = Tx. In what follows, we shall make use of the following lemmas. Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 4 of 14 Lemma 2.1 Let H be a real Hilbert space. Then, for any given x, y ∈ H, the following in- equality holds: 2 2 x + y ≤x +2y, x + y . Lemma 2.2 ([1]) Let C be a convex subset of a real Hilbert space H and let x ∈ H. Then x = P xif and only if 0 C z – x , x – x ≤ 0, ∀z ∈ C. 0 0 Lemma 2.3 ([24]) Let {a } be a sequence of nonnegative real numbers satisfying the fol- lowing relation: a ≤ (1 – α )a + α δ , n ≥ 0, n+1 n n n n where {α }⊂ (0, 1) and {δ }⊂ R satisfy the conditions α = ∞ and lim sup δ ≤ 0. n n n n n=0 n→∞ Then lim a =0. n→∞ n Lemma 2.4 ([28]) Let C be a closed convex subset of a real Hilbert space H and T : C → C be a continuous pseudo-contractive mapping. Then (i) F(T) is a closed convex subset of C; (ii) I – T is demiclosed at zero. Lemma 2.5 ([14]) Let {a } be sequence of real numbers such that there exists a subsequence {n } of {n} such that a < a for all i ∈ N. Then there exists a nondecreasing sequence i n n +1 i i {m }⊂ Nsuch that m →∞ and the following properties are satisﬁed by all (suﬃciently k k large) numbers k ∈ N: a ≤ a and a ≤ a . m m +1 k m +1 k k k In fact, m = max{j ≤ k : a < a }. k j j+1 Lemma 2.6 ([26]) Let H be a real Hilbert space. Then, for all x, y ∈ Hand α ∈ [0, 1], the following equality holds: 2 2 2 αx +(1– α)y = αx +(1– α)y – α(1 – α)x – y . Lemma 2.7 ([4]) Let C be a nonempty, closed and convex subset of a real Hilbert space H and T : C → H be a mapping. Deﬁne h : C → R by h(x)= inf λ ≥ 0: λx +(1– λ)Tx ∈ C . Then, for any x ∈ C, the following hold: (1) h(x) ∈ [0, 1] and h(x)=0 if and only if Tx ∈ C; (2) if β ∈ [h(x), 1], then βx +(1– β)Tx ∈ C; (3) if T is inward, then h(x)<1; (4) if Tx ∈/ C, then h(x)x +(1– h(x))Tx ∈ ∂C. Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 5 of 14 3 Results and discussion Now, let C be a nonempty, closed and convex subset of a real Hilbert space H and let T : C → H be an inward L-Lipschitz mapping. Let β ∈ (1 – ,1) and {α }⊆ (0, 1) such 1+ L +1 that lim α =0 and α = ∞. We deﬁne a Halpern–Ishikawa type iterative scheme n→∞ n n as follows. Choose u, x ∈ C.Let h(x ):= inf λ ≥ 0: λx +(1– λ)Tx ∈ C and λ ∈ max β, h(x ) ,1 . 0 0 0 0 0 Then by Lemma 2.7 it follows that y := λ x +(1– λ )Tx ∈ C. 0 0 0 0 0 Let l(y ):= inf{θ ≥ 0: θx +(1– θ)Ty ∈ C} and θ ∈ [max{λ , l(y )},1). Again by 0 0 0 0 0 0 Lemma 2.7, θ x +(1– θ )Ty ∈ C, and hence it follows that 0 0 0 0 x := α u +(1– α ) θ x +(1– θ )Ty ∈ C. 1 0 0 0 0 0 0 Thus, by mathematical induction, we have λ ∈ [max{β, h(x )},1); ⎪ n n y = λ x +(1– λ )Tx ; n n n n n (8) ⎪ θ ∈ [max{λ , l(y )},1); n n n x = α u +(1– α )(θ x +(1– θ )Ty ), n+1 n n n n n n where h(x ):= inf{λ ≥ 0: λx +(1– λ)Tx ∈ C} and l(y ):= inf{θ ≥ 0: θx +(1– θ)Ty ∈ C}. n n n n n n Next, we prove the following theorem. Theorem 3.1 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T : C → H be an L-Lipschitz pseudocontractive inward mapping with F(T) = ∅. Let {x } be a sequence deﬁned by (8). If there exists >0 such that θ ≤ 1– ∀n ≥ 0, then {x } n n n converges strongly to a ﬁxed point of T nearest to u. Proof We make use of some ideas of the paper [27]. Let p ∈ F(T). Then from (8)and Lemma 2.6,wehave x – p = α u +(1– α ) θ x +(1– θ )Ty – p n+1 n n n n n n ≤ α u – p +(1– α ) θ (x – p)+(1 – θ )(Ty – p) n n n n n n 2 2 2 ≤ α u – p +(1– α ) θ x – p +(1– θ )Ty – p n n n n n n –(1– α )θ (1 – θ )Ty – x , n n n n n and hence from (3)weobtain 2 2 2 x – p ≤ α u – p +(1– α )θ x – p +(1– α )(1 – θ ) n+1 n n n n n n 2 2 2 × y – p + y – Ty –(1– α )θ (1 – θ )Ty – x n n n n n n n n 2 2 ≤ α u – p +(1– α )(1 – θ )y – p n n n n Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 6 of 14 +(1– α )(1 – θ )y – Ty n n n n 2 2 +(1– α )θ x – p –(1– θ )Ty – x .(9) n n n n n n Moreover, from (8), Lemma 2.6,and (3), we have y – p = λ (x – p)+(1 – λ )(Tx – p) n n n n n 2 2 = λ x – p +(1– λ )Tx – p n n n n – λ (1 – λ )x – Tx n n n n 2 2 2 ≤ λ x – p +(1– λ ) x – p + x – Tx n n n n n n – λ (1 – λ )x – Tx n n n n 2 2 2 = x – p +(1– λ ) x – Tx . (10) n n n n Furthermore, (8) and Lemma 2.6 imply that y – Ty = λ (x – Ty )+(1 – λ )(Tx – Ty ) n n n n n n n n 2 2 = λ x – Ty +(1– λ )Tx – Ty n n n n n n – λ (1 – λ )x – Tx n n n n 2 2 2 ≤ λ x – Ty +(1– λ )L x – y n n n n n n – λ (1 – λ )x – Tx n n n n 2 3 2 2 = λ x – Ty +(1– λ ) L x – Tx n n n n n n – λ (1 – λ )x – Tx n n n n = λ x – Ty n n n 2 2 2 –(1– λ ) λ – L (1 – λ ) x – Tx . (11) n n n n n Substituting (10)and (11)into(9), we obtain 2 2 2 x – p ≤ α u – p +(1– α )(1 – θ ) x – p n+1 n n n n 2 2 2 +(1– λ ) x – Tx +(1– α )(1 – θ ) λ x – Ty n n n n n n n n 2 2 2 –(1– λ ) λ – L (1 – λ ) x – Tx n n n n n 2 2 +(1– α )θ x – p –(1– α )θ (1 – θ )Ty – x n n n n n n n n 2 2 = α u – p +(1– α )x – p –(1– α )(1 – θ )(1 – λ ) n n n n n n 2 2 2 × 1– L (1 – λ ) +2(1 – λ ) x – Tx n n n n +(1– α )(1 – θ )(λ – θ )Ty – x . (12) n n n n n n Then since, from the hypothesis, we have 2 2 2 2 1–2(1– λ )– L (1 – λ ) ≥ 1–2(1– β)– L (1 – β) > 0, (13) n n Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 7 of 14 and θ ≥ λ , for all n ≥ 0, (14) n n inequality (12)implies that 2 2 2 x – p ≤ α u – p +(1– α )x – p . (15) n+1 n n n Thus, by induction, 2 2 2 x – p ≤ max u – p , x – p , ∀n ≥ 0, n+1 0 which provides that {x } and hence {y } are bounded. n n Now, let x = P (u). Then, using (8), Lemma 2.1, and following the methods used to F(T) get (12), we obtain 2 2 ∗ ∗ x – x = α u +(1– α ) θ x +(1– θ )Ty – x n+1 n n n n n n ∗ ∗ = α u – x +(1– α ) θ x +(1– θ )Ty – x n n n n n n ∗ ∗ ∗ ≤ (1 – α ) θ x +(1– θ )Ty – x +2α u – x , x – x n n n n n n n+1 2 2 ∗ ∗ ≤ (1 – α )θ x – x +(1– α )(1 – θ ) Ty – x n n n n n n 2 ∗ ∗ –(1– α )θ (1 – θ )Ty – x +2α u – x , x – x , n n n n n n n+1 and 2 2 ∗ ∗ x – x ≤ (1 – α )θ x – x n+1 n n n ∗ 2 +(1– α )(1 – θ ) y – x + y – Ty n n n n n 2 ∗ ∗ –(1– α )θ (1 – θ )Ty – x +2α u – x , x – x n n n n n n n+1 ≤ (1 – α )θ x – x +(1– α )(1 – θ ) n n n n n ∗ 2 2 × x – x +(1– λ ) x – Tx +(1– α )(1 – θ ) n n n n n n 2 2 2 2 × λ x – Ty –(1– λ ) λ – L (1 – λ ) x – Tx n n n n n n n n 2 ∗ ∗ –(1– α )θ (1 – θ )Ty – x +2α u – x , x – x , n n n n n n n+1 which implies that 2 2 ∗ ∗ x – x ≤ (1 – α ) x – x –(1– α )(1 – θ )(1 – λ ) n+1 n n n n n 2 2 2 × 1– L (1 – λ ) –2(1 – λ ) x – Tx n n n n +(1– α )(1 – θ )(λ – θ )x – Ty n n n n n n ∗ ∗ +2α u – x , x – x (16) n n+1 ∗ ∗ ∗ ≤ (1 – α ) x – x +2α u – x , x – x n n n n +2α u – x x – x . (17) n n+1 n Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 8 of 14 Now, we consider two cases. Case 1. Suppose that there exists n ∈ N such that {x – x } is decreasing for all n ≥ n . 0 n 0 Then it follows that {x – x } is convergent. Thus, from (16), (13), and (14), we have x – Tx →0as n →∞. (18) n n Moreover, from (8)and (18), we obtain y – x =(1 – λ )x – Tx →0as n →∞, (19) n n n n n and hence the Lipschitz continuity of T,(19), and (18)imply that Ty – x ≤Ty – Tx + Tx – x n n n n n n ≤ Ly – x + Tx – x →0as n →∞. (20) n n n n In addition, from (3.1)and (18), we obtain x – x ≤ α u – x +(1– α )(1 – θ )Ty – x → 0. (21) n+1 n n n n n n n Furthermore, since {x } is a bounded subset of H which is reﬂexive, we can choose a subsequence {x } of {x } such that n n ∗ ∗ ∗ ∗ x w and lim sup u – x , x – x = lim u – x , x – x . n n n i i i→∞ n→∞ Then from (18) and Lemma 2.4,wehave w ∈ F(T). Therefore, by Lemma 2.2, we immedi- ately obtain ∗ ∗ ∗ ∗ lim sup u – x , x – x = lim u – x , x – x n n i→∞ n→∞ ∗ ∗ = u – x , w – x ≤ 0. (22) Then it follows from (17), (22), and Lemma 2.3 that x – x → 0as n →∞.Conse- quently, x → x = P (u). n F(T) Case 2. Suppose that there exists a subsequence {n } of {n} such that ∗ ∗ x – x < x – x , ∀i ∈ N. n n +1 i i Then, by Lemma 2.5,there exists anondecreasing sequence {m }⊂ N such that m →∞ k k and ∗ ∗ ∗ ∗ x – x ≤ x – x and x – x ≤ x – x , (23) m m +1 k m +1 k k k for all k ∈ N.Now,from(16), (13), and (14), it follows that x – Tx → 0as k →∞. m m k k Thus,likeinCase1,weobtain ∗ ∗ lim sup u – x , x – x ≤ 0. (24) k→∞ Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 9 of 14 Now, from (17), we have 2 2 ∗ ∗ ∗ ∗ x – x ≤ (1 – α ) x – x +2α u – x , x – x m +1 m m m m k k k k k +2α u – x x – x , (25) m m +1 m k k k and hence (23)and (25)imply that 2 2 2 ∗ ∗ ∗ ∗ ∗ α x – x ≤ x – x – x – x +2α u – x , x – x m m m m +1 m m k k k k k k +2α u – x x – x m m +1 m k k k ∗ ∗ ∗ ≤ 2α u – x , x – x +2α u – x x – x . m m m m +1 m k k k k k Thus, using (21), (24), and the fact that α >0, we obtain ∗ ∗ x – x ≤ 0 and hence x – x →0as k →∞. m m k k ∗ ∗ This together with (25)implies that x – x → 0as k →∞.But,since x – x ≤ m +1 k ∗ ∗ x – x , for all k ∈ N, it follows that x → x = P (u). Therefore, from the above m +1 k F(T) two cases, we can conclude that {x } converges strongly to the ﬁxed point of T nearest to u. If, in Theorem 3.1, we assume that T is k-strictly pseudocontractive, then T is Lipschitz 1+k pseudocontractive with L = , and hence we get the following corollary. Corollary 3.2 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let T : C → H be a k-strictly pseudocontractive inward mapping with F(T) = ∅. Let β ∈ (1 – ,1) and {α }⊆ (0, 1) such that lim α =0 and α = ∞. Let a sequence n n→∞ n n 2 2 k+ (k+1) +k {x } be generated from arbitrary x , u ∈ Cby n 0 λ ∈ [max{β, h(x )},1); ⎪ n n y = λ x +(1– λ )Tx ; n n n n n (26) ⎪ θ ∈ [max{λ , l(y )},1); n n n x = α u +(1– α )(θ x +(1– θ )Ty ), n+1 n n n n n n where h(x ):= inf{λ ≥ 0: λx +(1– λ)Tx ∈ C} and l(y ):= inf{θ ≥ 0: θx +(1– θ)Ty ∈ C}. n n n n n n If there exists >0 such that θ ≤ 1– ∀n ≥ 0, then {x } converges strongly to a ﬁxed n n pointof T nearestto u. If, in Theorem 3.1,weassumethat T is nonexpansive, then we have that T is Lipschitz pseudocontractive with L = 1, and hence we get the following corollary. Corollary 3.3 Let C be a nonempty, closed and convex subset of a real Hilbert space H, and let T : C → H be a nonexpansive inward mapping with F(T) = ∅. Let β ∈ (2 – 2, 1) and {α }⊆ (0, 1) such that lim α =0 and α = ∞. Let a sequence {x } be generated n n→∞ n n n Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 10 of 14 from arbitrary x , u ∈ Cby λ ∈ [max{β, h(x )},1); ⎪ n n y = λ x +(1– λ )Tx ; n n n n n (27) ⎪ θ ∈ [max{λ , l(y )},1); n n n x = α u +(1– α )(θ x +(1– θ )Ty ), n+1 n n n n n n where h(x ):= inf{λ ≥ 0: λx +(1– λ)Tx ∈ C} and l(y ):= inf{θ ≥ 0: θx +(1– θ)Ty ∈ C}. n n n n n n If there exists >0 such that θ ≤ 1– ∀n ≥ 0, then {x } converges strongly to a ﬁxed n n pointof T nearestto u. We now state and prove a convergence result for a monotone mapping. Corollary 3.4 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let A : C → H be an L-Lipschitz monotone inward mapping with N(A) = ∅. Let β ∈ (1 – ,1) and {α }⊂ (0, 1) such that lim α =0 and α = ∞. Let a sequence n n→∞ n n 1+ 1+(1+L) {x } be generated from arbitrary x , u ∈ Cby n 0 λ ∈ [max{β, h(x )},1); ⎪ n n y = x –(1– λ )Ax ; n n n n (28) θ ∈ [max{λ , l(y )},1); n n n x = α u +(1– α )(θ x +(1– θ )(I – A)y ), n+1 n n n n n n where h(x ):= inf{λ ≥ 0: x –(1– λ)Ax ∈ C} and l(y ):= inf{θ ≥ 0: θx +(1– θ)(I –A)y ∈ n n n n n n C}. If there exists >0 such that θ ≤ 1– ∀n ≥ 0, then {x } converges strongly to the zero n n pointof A nearestto u. Proof Let Tx := (I – A)x.Then T is a Lipschitz pseudocontractive mapping with Lipschitz constant L := (1 + L)and F(T)= N(A) = ∅.Moreover, if A is replaced with (I – T), then scheme (28)reduces to scheme (8), and hence the conclusion follows from Theorem 3.1. We observe that the method of proof of Theorem 3.1 provides the following result for approximating the minimum-norm point of ﬁxed points of Lipschitz pseudocontractive non-self mappings. Theorem 3.5 Let C be a nonempty, closed and convex subset of a real Hilbert space H containing 0, and let T : C → H be an L-Lipschitz pseudocontractive inward mapping with F(T) = ∅. Let {x } be a sequence deﬁned by (8) with u =0. If there exists >0 such that θ ≤ 1– ∀n ≥ 0, then {x } converges strongly to the minimum-norm point x of F(T). n n Remark 3.6 Note that, in the above results, the coeﬃcients λ and θ canbechosensimply n n as follows: λ = max{β, h(x )} and θ = max{λ , l(y )}. n n n n n Remark 3.7 If, in all the above theorems and corollaries, the set F(T) is a subset of interior of C, then the assumption that there exists >0 such that θ ≤ 1– ∀n ≥ 0may not be required. Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 11 of 14 4 Numerical example Now, we give an example of a nonlinear mapping which satisﬁes the conditions of Theo- rem 3.1. Example 4.1 Let H = R with Euclidean norm. Let C =[–1,1] and T : C → R be deﬁned by –3x, x ∈ [–1, 0], Tx = (29) x, x ∈ (0, 1]. Then we observe that T satisﬁes the inward condition and F(T) = [0, 1]. One can also easily verify that x – Tx –(y – Ty), x – y ≥ 0, ∀x, y ∈ C. Thus, I – T is monotone and hence T is a pseudocontractive mapping. To show that T is a Lipschitz mapping, we consider the following cases. Case 1: Let x, y ∈ [–1, 0]. Then we have |Tx – Ty| = |–3x +3y| =3|x – y|. Case 2: Let x, y ∈ (0, 1]. Then we have |Tx – Ty| = |x – y|. Case 3: Let x ∈ [–1, 0] and y ∈ (0, 1]. Then we have |Tx – Ty| = |–3x – y| = |3x + y| = |x – y +2x +2y| ≤|x – y| +2|x + y| ≤|x – y| +2|x – y| =3|x – y|. From the above cases, it follows that T is L-Lipschitz with L =3. 5 1 2 Now, let β = , u = , x =–1, and α = .Then Tx =3 and 0 n 0 6 2 n+5 h(x )= inf λ ≥ 0: λx +(1– λ)Tx ∈ C 0 0 0 = inf λ ≥ 0:–λ +3(1 – λ) ∈ C = . 5 1 Now, let λ = .Then y = λ x +(1– λ )Tx =– and Ty =1, which gives 0 0 0 0 0 0 0 6 3 l(x )= inf θ ≥ 0: θx +(1– θ)Ty ∈ C =0. 0 0 0 Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 12 of 14 Figure 1 Convergence of x with diﬀerent values of x and u n 0 If we choose θ = ,thenwehave x = α u +(1– α ) θ x +(1– θ )Ty =– . 1 0 0 0 0 0 0 3 5 Thus, Tx = , which implies that h(x )=0. Now, if we choose λ = ,thenweobtain 1 1 1 5 6 1 1 y = λ x +(1– λ )Tx =– , Ty = and l(y )=0. 1 1 1 1 1 1 1 15 5 Again, we can choose θ = ,which yields x = 0.0778. In general, we observe that for 1 2 2 5 u =0.5, x =–1 and α = ,wecan choose λ = θ = . Thus, all the conditions of Theo- 0 n n n n+5 6 rem 3.1 are satisﬁed and x converges to 0.5 = P u (see Fig. 1). n F(T) On the other hand, for u = –0.8, x =1, and α = ,weobtainthat x converges to 0 n n n+5 0.0 = P u.Figure 1 is obtained using MATLAB version 7.5.0.342(R2007b). F(T) 5Conclusion In this paper, we have constructed and studied a Halpern–Ishikawa type iterative scheme for non-self mappings in the setting of Hilbert spaces. As a result, we obtained strong con- vergence of the scheme to a ﬁxed point of a Lipschitz pseudocontractive non-self mapping under some mild conditions. In addition, we provided a numerical example to support our results. Our study can open the door for further research activity in the ﬁeld for a more general class of mappings in Hilbert and/or Banach spaces more general than Hilbert spaces. Our results extend and generalize many results in the literature. More particularly, Theorem 3.1 extends Theorem 8 of Colao et al. [5] in the sense that it provides a convergent scheme for approximating ﬁxed points of Lipschitz pseudocontractive non-self mappings more general than that of k-strictly pseudocontractive non-self mappings. Zegeye and Tufa Fixed Point Theory and Applications (2018) 2018:15 Page 13 of 14 Acknowledgements The authors appreciate the support of their institutes. Funding The second author is supported by the International Mathematical Union (IMU) Breakout Graduate Fellowship Program through The World Academy of Sciences (TWAS). Abbreviations Not applicable. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors contributed equally and signiﬁcantly in writing the article. Both authors read and approved the ﬁnal manuscript. Author details Department of Mathematics, Botswana International University of Science and Technology, Palapye, Botswana. Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Received: 26 October 2017 Accepted: 12 April 2018 References 1. Alber, Y.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Appl. Math., vol. 178, pp. 15–50. Dekker, New York (1996) 2. Chidume, C.E., Chidume, C.O.: Iterative approximation of ﬁxed points of nonexpansive mappings. J. Math. Anal. Appl. 318(1), 288–295 (2006) 3. Chidume, C.E., Zegeye, H.: Approximate ﬁxed point sequences and convergence theorems for Lipschitz pseudocontractive maps. Proc. Am. Math. Soc. 132, 831–840 (2004) 4. 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