Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

New Iterative Methods for Finding Common Fixed Points of Two Non-self Mappings in a Real Hilbert Space

New Iterative Methods for Finding Common Fixed Points of Two Non-self Mappings in a Real Hilbert... Topol. Algebra Appl. 2022; 10:36–46 Research Article Open Access Abebe Regassa Tufa* New Iterative Methods for Finding Common Fixed Points of Two Non-self Mappings in a Real Hilbert Space https://doi.org/10.1515/taa-2022-0111 Received 1 November, 2021; accepted 18 March, 2022 Abstract: In this paper, we introduce new iterative methods for approximating common xed points of two non-self mappings in the framework of real Hilbert spaces. We establish weak and strong convergence results for approximating common xed points of two nonexpansive non-self mappings. In addition, we establish strong convergence results for approximating common xed points of two quasi-nonexpansive non-self map- pings under appropriate conditions. Our results improve and generalize many of the results in the literature. Moreover, our ndings will open the way forward for the study of iterative methods for nding common xed points of two non-self mappings in Banach spaces more general than Hilbert spaces. Keywords: Fixed points, Iterative methods, Nonexpansive mappings PACS: 47H09, 47H10, 47J25 1 Introduction Let C be a nonempty subset of a real Hilbert space H and T : C ! H be a mapping. A point x in C is said to be a xed point of T if x = Tx. The mapping T : C ! H is said to be 1) contraction if there exists k 2 (0, 1) such that jjTx − Tyjj ≤ kjjx − yjj, for all x, y 2 C. 2) nonexpansive if jjTx − Tyjj ≤ jjx − yjj, for all x, y 2 C. 3) quasi-nonexpansive if F(T ) := fx 2 C : x = Txg ≠ ; and jjTx − Tpjj ≤ jjx − pjj, for all x 2 C and for all p 2 F(T ). One can easily see that a contraction mapping is nonexpansive and a nonexpansive mapping with nonempty xed point set is quasi-nonexpansive mapping. However, a quasi-nonexpansive mapping need not be non- expansive (see, for instance, [1]). The theory of xed points of nonlinear maps is applicable in diverse elds of sciences such as dieren- tial equations, optimization, control theory, economics, etc. Thus, existences and approximations of xed points of various nonlinear mappings have been studied by numerous researchers (see, for instance, [2–13]). *Corresponding Author: Abebe Regassa Tufa: University of Botswana, Department of Mathematics,Gaborone, Botswana, E-mail: abykabe@yahoo.com Open Access. © 2022 Boris G. Averbukh, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. Iterative Methods for Common Fixed Points of Two Mappings Ë 37 We recall that the Mann iteration method [14] for a mapping T : C ! C is given by x 2 C, x = α x + (1 − α )Tx , (1) 0 n+1 n n n n where the initial guess x 2 C is arbitrary and fα g  [0, 1] such that lim α = 0 and α = ∞. It is n n n n!∞ well known that the Mann iteration method converges weakly for nonexpansive mappings to a xed point of the mapping T in an innite dimensional Hilbert space. Consequently, several researchers have studied and modied the Mann iteration method to obtain strong convergence results (see, for instance, [15–20]). Many researchers have also studied iterative schemes for approximating common xed points of two self mappings (see, for instance, [10, 21–25]). In 2007, Yao and Chen [25] introduced and studied the following iteration process for common xed points of two self mappings: x 2 C, x = α x + β Tx + Sx , n 2 N, n n n n n n 1 n+1 where fα g,fβ g and f g are in [0, 1] and α + β + = 1. We observe that this iterative scheme reduces n n n n n n to Mann iterative scheme if T or S is an identity map. On the other hand, many researchers have dealt with (see, for instance, [26–28]) approximation of xed points of non-self operators using the concept of metric projection or sunny nonexpansive retraction. But this is computationally expensive and not suitable for practical applications. In [29], Colao and Marino constructed a new iterative method for approximating xed points of non-self mapping T : C ! H without using the concept of metric projection or sunny nonexpansive retraction. Their algorithm is read as follows: x 2 C, α = maxf , h(x )g, 0 0 (2) x = α x + (1 − α )Tx , n+1 n n n n α = maxfα , h(x )g, n ≥ 0, n+1 n n+1 where h(x) := inffλ ≥ 0 : λx + (1 − λ)Tx 2 Cg,8x 2 C  H. They established weak and strong convergence results of the algorithm for approximating xed points of nonexpansive non-self mappings. Motivated and inspired by the above results, our objective is to construct new iterative methods for ap- proximating common xed points of two non-self mappings in the framework of real Hilbert spaces. We establish weak and strong convergence results for approximating common xed points of two nonexpansive non-self mappings. In addition, we establish strong convergence results for approximating common xed points of two quasi-nonexpansive non-self mappings under appropriate conditions. Our results improve, complement and generalize many of the results in the literature. Moreover, our ndings open the way for- ward for the study of iterative methods for nding common xed points of two non-self mappings in Banach spaces more general than Hilbert spaces. 2 Preliminaries In this section, we collect some basic denitions and known results which we may use in the proof of our main results. 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 2 C, we have Tx 2 I (x) := fx + λ(u − x) : for some u 2 C and λ ≥ 1g. 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 sequencefx g in C such that x * x and Tx − x ! 0 as n ! ∞, then x = Tx, n n n n 38 Ë Abebe Regassa Tufa where ” * ” denotes weak convergence, while ” ! ” denotes strong convergence. The set C is called strictly convex if it is convex and with the property that x, y 2 ∂C and t 2 (0, 1) im- plies that tx + (1 − t)y 2 C (see [29]). We will use the following lemmas in the sequel. Lemma 1 (Opial’s Condition). [30] If in a Hilbert space H the sequencefx g is weakly convergent to x then for any x ≠ x lim infjjx − xjj > lim infjjx − x jj. n n n!∞ n!∞ Lemma 2. [31] Let C be a closed convex subset of a real Hilbert space H and T : C ! H be quasi-nonexpansive mapping. Then F(T ) is closed and convex. Lemma 3. [32] Let H be a real Hilbert space. Then for all x 2 H and α 2 [0, 1] for i = 1, 2, 3,··· , n such that i i α + α + ··· + α = 1 the following equality holds: 1 2 X X 2 2 2 jjα x + α x + ··· + α x jj = α jjx jj − α α jjx − x jj . n n 1 1 2 2 i i i j i j i=1 1≤i,j≤n The following lemma can be obtained from Proposition 2.1 of [33]. Lemma 4. [33] Let C be a closed convex subset of a real Hilbert space H and T : C ! H be a nonexpansive mapping. Then the following hold. (i) I − T is demiclosed at zero; (ii) F(T ) is closed convex subset of C. Lemma 5. [29] Let C be a nonempty, closed and convex subset of a real Hilbert space H and T : C ! H be a mapping. Dene h : C ! R by h(x) = inffλ ≥ 0 : λx + (1 − λ)Tx 2 Cg. Then for any x 2 C the following hold: 1) h(x) 2 [0, 1] and h(x) = 0 if and only if Tx 2 C; 2) if β 2 [h(x), 1], then βx + (1 − β)Tx 2 C; 3) if T is inward, then h(x) < 1; 4) if Tx 2̸ C, then h(x)x + (1 − h(x))Tx 2 ∂C. 3 Main Results In this section, we construct an algorithm which involves two non-self mappings in the framework of real Hilbert spaces. Then we establish convergence results to a common xed point of the mappings. We rst prove the following lemma. Lemma 6. Let C be a nonempty, closed and convex subset of a real Hilbert space H and T, S : C ! H be two mappings. For each x 2 C and some θ 2 [0, 1], dene f : C ! [0,∞] by f (x) = inffα ≥ 0 : α θx + (1 − θ)Tx + (1 − α)Sx 2 Cg. Then for any x 2 C the following hold: (1) If θx + (1 − θ)Tx 2 C, then f (x) 2 [0, 1] and f (x) = 0 i Sx 2 C. θ θ (2) If θx + (1 − θ)Tx 2 C, then β(θx + (1 − θ)Tx + (1 − β)Sx 2 C for any β 2 [f (x), 1]. (3) If T and S are inward mappings, then f (x) < 1. (4) If θx + (1 − θ)Tx 2 C and Sx 2̸ C, then f (x) θx + (1 − θ)Tx + 1 − f (x) Sx 2 ∂C. θ θ Iterative Methods for Common Fixed Points of Two Mappings Ë 39 Proof. We need to prove only (3) and (4) as the proofs of (1) and (2) are trivial. (3) Let T and S be inward mappings and x 2 C. Then for some u , u 2 C and c , c ≥ 1, we have 1 2 1 2 Tx = x + c (u − x) and Sx = x + c (u − x), 1 1 2 2 which implies that 1 1 1 1 u = Tx + 1 − x and u = Sx + 1 − x. 1 2 c c c c 1 1 2 2 This yields 1 1 1 1 1 1 1 1 u + u = Tx + 1 − x + Sx + 1 − x 1 2 2 2 2c 2 c 2c 2 c 1 1 2 2 1 c c 1 2 2 = 1 − Tx + 1 − x + Sx 2c c (2c − 1) c (2c − 1) 2c 2 1 2 1 2 2 and since C is convex, we obtain that 1 c c 1 2 2 1 − Tx + 1 − x + Sx 2 C. 2c 2c c (2c − 1) c (2c − 1) 2 1 2 1 2 2 It is easy to verify that θ := 2 (0, 1]. Hence, c (2c −1) 1 2 f (x) ≤ 1 − < 1. 2c (4) Let θx + (1 − θ)Tx 2 C and Sx 2̸ C. Then f (x) 2 [0, 1] by (1). Let fw g be a sequence of real numbers in (0, f (x)) such that w ! f (x). By the denition of f , we obtain θ θ θ z := w (θx + (1 − θ)Tx) + (1 − w )Sx 2̸ C. n n n Thus, we have jjz − f (x)(θx + (1 − θ)Tx) − (1 − f (x))Sxjj ≤ jw − f (x)jjjθx + (1 − θ)Tx − Sx)jj. n n θ θ θ Now, since w ! f (x), it follows that z ! f (x)(θx + (1 − θ)Tx) + (1 − f (x))Sx 2 C θ θ and since z = w θx + (1 − θ)Tx + (1 − w )Sx 2̸ C, for all n ≥ 1, we obtain n n n f (x) θx + (1 − θ)Tx + 1 − f (x) Sx 2 ∂C. θ θ This completes the proof. Remark 1. If θ = 1 or T is an identity map, then Lemma 6 reduces to Lemma 5. We are now in a position to construct our algorithm which involves two non-self mappings. Let C be a nonempty, closed and convex subset of a real Hilbert space H and T, S : C ! H be two non-self inward mappings. Given x 2 C, let h(x ) := inffθ ≥ 0 : θx + (1 − θ)Tx 2 Cg. 1 1 1 Take θ = maxf , h(x )g. Then θ x + (1 − θ )Tx 2 C. 1 1 1 1 1 1 Now, let f (x ) := inffα ≥ 0 : α[θ x + (1 − θ )Tx ] + (1 − α)Sx 2 Cg. 1 1 1 1 1 1 Putting α = maxf , f (x )g, we obtain that 1 1 2 1 x := α [θ x + (1 − θ )Tx ] + (1 − α )Sx 2 C. 2 1 1 1 1 1 1 1 40 Ë Abebe Regassa Tufa Inductively, we dene a sequencefx g as follows: > x 2 C, x = α [θ x + (1 − θ )Tx ] + (1 − α )Sx n n n n n n n n+1 (3) > θ = maxfθ , h(x )g, n+1 n+1 α = maxfα , f (x )g, n+1 θ n+1 n+1 where h : C ! R and f : C ! [0,∞] are dened, respectively, by h(x) = inffλ ≥ 0 : λx + (1 − λ)Tx 2 Cg and f (x) = inffα ≥ 0 : α[θx + (1 − θ)Tx] + (1 − α)Sx 2 Cg, n ≥ 1, for some θ 2 [0, 1]. We observe that Algorithm (3) reduces to Mann iterative scheme when T or S is an identity map. Theorem 1. Let C be a nonempty, closed and convex subset of a real Hilbert space H and S, T : C ! H be two nonexpansive inward mappings with F = F(T )\ F(S) ≠ ;. Letfx g be a sequence dened in (3). i) If there is b 2 (0, 1) such that θ , α ≤ b,8n ≥ 1, thenfx g converges weakly to a point in F. n n n ∞ ∞ X X ii) If C is strictly convex, (1 − θ ) < ∞ and (1 − α ) < ∞, thenfx g converges strongly to a point in F. n n n n=1 n=1 Proof. Let p 2 F. Since T and S are nonexpansive, from (3) and the triangle inequality, we have jjx − pjj = jjα [θ x + (1 − θ )Tx ] + (1 − α )Sx − pjj n n n n n n n n+1 ≤ α jjθ x + (1 − θ )Tx − pjj + (1 − α )jjSx − pjj n n n n n n n ≤ α θ jjx − pjj + α (1 − θ )jjTx − pjj + (1 − α )jjSx − pjj n n n n n n n n ≤ α θ jjx − p)jj + α (1 − θ )jjx − pjj + (1 − α )jjx − pjj n n n n n n n n = jjx − pjj. (4) Then fjjx − pjjg is decreasing which implies that lim jjx − pjj exists. In addition, fx g is bounded and so n n n n!∞ are Tx , Sx andfy g. n n n i) Assume that there is b 2 (0, 1) with θ , α ≤ b,8n ≥ 1. Then by Lemma 3, we have: n n 2 2 jjx − pjj = jjα θ x + α (1 − θ )Tx + (1 − α )Sx − pjj n n n n n n n n n+1 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjTx − pjj + (1 − α )jjSx − pjj n n n n n n n n 2 2 −α θ (1 − θ )jjTx − x jj n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjx − pjj + (1 − α )jjx − pjj n n n n n n n n 2 2 −α θ (1 − θ )jjTx − x jj n n n n 2 2 2 = jjx − pjj − α θ (1 − θ )jjTx − x jj . n n n n n This provides 2 2 2 2 α θ (1 − θ )jjTx − x jj ≤ jjx − pjj −jjx − pjj . (5) n n n n n n n+1 Also, since lim jjx − pjj exists, it follows that n!∞ 2 2 2 jjx − pjj −jjx − pjj ≤ jjx − pjj . (6) n+1 1 n=1 Then since ≤ α , θ < b, (5) and (6) yields n n ∞ ∞ X X 2 2 2 (1 − b)jjTx − x jj ≤ α θ (1 − θ )jjTx − x jj < ∞. n n n n n n n n=1 n=1 Hence, we have lim jjx −Tx jj = 0. Similarly, one can easily show that lim jjx −Sx jj = 0. Also, sincefx g n n n n n n!∞ n!∞ is bounded, the set of weak cluster points offx g is nonempty, that is w(x ) := fx 2 H : x * x, for some subsequencefx g offx gg ≠ ;. n n n n i i Iterative Methods for Common Fixed Points of Two Mappings Ë 41 Let x 2 w(x ) and fx g be a subsequence of fx g such that x * x as i ! ∞. Then since I − T and I − S n n n n i i are demiclosed at zero (see Lemma 4), it follows that x 2 F = F(T ) \ F(S). This gives w(x )  F. To show uniqueness, let x, y 2 w(x ) and letfx g andfx g be subsequences offx g such that x * x and x * y n n n n n n i j i j as i, j ! ∞. Suppose x ≠ y. Then since lim jjx −xjj exists for all x 2 F and Hilbert space satises the Opial’s n!∞ property, by Lemma 1, we get lim jjx − xjj = lim jjx − xjj < lim jjx − yjj n n n i i n!∞ i!∞ i!∞ = lim jjx − yjj = lim jjx − yjj n n n!∞ j!∞ < lim jjx − xjj = lim jjx − xjj. n n n!∞ j!∞ However, this is a contradiction and hence x = y. Thus, every subsequence of fx g converges weakly to x and hencefx g converges weakly to x 2 F. ∞ ∞ X X ii) Suppose that C is strictly convex, (1 − θ ) < ∞ and (1 − α ) < ∞. n n n=1 n=1 Then since α (1 − θ ) ≤ (1 − θ ), it follows that α (1 − θ ) < ∞. Moreover, from the fact thatfx g,fTx g n n n n n n n n=1 andfSx g are bounded andjjx − x jj = α (1 − θ )jjx − Tx jj + (1 − α )jjx − Sx jj, we have n n n+1 n n n n n n n jjx − x jj < ∞, n+1 n=1 * * which implies thatfx g is Cauchy sequence and hence there is x 2 C such that x ! x as n ! ∞. n n On the other hand, since lim θ = 1 and θ = maxfθ , h(x )g, we can pick a sub-sequencefx g such that n n n−1 n n n!∞ fh(x )g is increasing and lim h(x ) = 1. In particular, for any < 1, n n i i i!∞ t := x + (1 − )Tx 2̸ C, eventually holds. n n n i i i Now choose , 2 (h(x ), 1) such that ≠ and let 1 2 1 2 * * * * v = x + (1 − )Tx and v = x + (1 − )Tx . 1 1 1 2 2 2 Then for any 2 [ , ], we have 1 2 * * v := x + (1 − )Tx 2 C. * * Also, since x ! x and Tx ! Tx as i ! ∞, we obtain that t ! v as i ! ∞ and hence v 2 ∂C. This n n n i i i yields [v , v ]  ∂C since is arbitrary point in [ , ]. Then by strict convexity of C, we obtain that v = v . 1 2 1 2 1 2 * * Thenjjv − Tx jj = jjv − Tx jj which implies that 1 2 * * * * jjx − Tx jj = jjx − Tx jj. 1 2 * * * * Then since ≠ , we can see that x = Tx and hence, x 2 F(T ). It remains to show that x 2 F(S). To this 1 2 end , we observe that y := θ x + (1 − θ )Tx ! x as n ! ∞. n n n n n Then, since lim α = 1 and α = maxfα , f (x )g, repeating the above arguments provides the required n n n−1 n n!∞ result. Indeed, we can pick a sub-sequencefx g such thatff (x )g is increasing and lim f (x ) = 1. n n n j θ j j!∞ θ j n n j j In particular, for any μ < 1, s := μy + (1 − μ)Sx 2̸ C, eventually holds. n n n j j j 42 Ë Abebe Regassa Tufa Now choose μ , μ 2 (f (x ), 1) such that μ ≠ μ and let 1 2 1 2 * * * * u = μ x + (1 − μ )Sx and u = μ x + (1 − μ )Sx . 1 1 1 2 2 2 Thus, for any μ 2 [μ , μ ], we have 1 2 * * u := μx + (1 − μ)Sx 2 C. * * Since y ! x and Sx ! Sx as j ! ∞, it follows that s ! u as j ! ∞ and hence u 2 ∂C. Furthermore, n n n j j j since μ is arbitrary, it follows that [u , u ]  ∂C. Then the strict convexity of C gives u = u and d(u , Sx ) = 1 2 1 2 1 * * * * * * * d(u , Sx ) which in turn gives μ d(x , Sx ) = μ d(x , Sx ). Then since μ ≠ μ , we have that x = Sx and 2 1 2 1 2 * * hence, x 2 F(S). Therefore,fx g converges strongly to x 2 F = F(T )\ F(S). The proof is complete. Next, we prove strong convergence results using the condition (I). For this, we rst modify the denition of condition (I) for a pair of mappings. Recall that a mapping T : C ! H is said to satisfy condition (I) if there exists a nondecreasing function f : [0,∞) ! [0,∞) with f (0) = 0 and f (r) > 0, for all r 2 (0,∞) such that jjx − Txjj ≥ f (d(x, F(T ))), for all x 2 C, where d(x, F(T )) = inffjjx − pjj : p 2 F(T )g. Now, we modify this denition for a pair of two mappings as follows. A pair of mappings S and T denoted by fS, Tg is said to satisfy condition (I) if there exists a nondecreas- ing function f : [0,∞) ! [0,∞) with f (0) = 0 and f (r) > 0, for all r 2 (0,∞) such thatjjx − Txjj ≥ f (d(x, F)) orjjx − Sxjj ≥ f (d(x, F)), for all x 2 C, where d(x, F) = inffjjx − pjj : p 2 F = F(T )\ F(S)g. Example 1. Let H = R with the Euclidean norm and S, T : [0, 1] ! R be dened by Sx = x − x, and x, x 2 [0, 1/2), Tx = : 1 − x, x 2 [1/2, 1]. Then the pair (S, T ) satisfy the condition (I). Proof. Note that F = F(S)\ F(T ) = f0g, d(x, p) = jxj and jxj, x 2 [0, 1/2), jx − Txj = jxj, x 2 [1/2, 1]. Then one can consider f (r) = r to complete the proof. Now, we consider the following Algorithm: > x 2 C, > θ = max , h(x ) , 1 1 > 2 α = maxfθ , f (x )g, 1 1 1 (7) > x = α [θ x + (1 − θ )Tx ] + (1 − α )Sx , n+1 n n n n n n n θ 2 maxfθ , h(x )g, 1 , > n+1 n n+1 α 2 maxfα , θ , f (x )g, 1 , n+1 n n n+1 n+1 where h : C ! R and f : C ! [0,∞] are dened, respectively, by h(x) = inffλ ≥ 0 : λx + (1 − λ)Tx 2 Cg and f (x) = inffα ≥ 0 : α[θx + (1 − θ)Tx] + (1 − α)Sx 2 Cg, n ≥ 1, for some θ 2 [0, 1]. θ Iterative Methods for Common Fixed Points of Two Mappings Ë 43 Theorem 2. Let C be a nonempty, closed and convex subset of a real Hilbert space H and S, T : C ! H be two quasi-nonexpansive inward mappings. Let fx g be a sequence as dened in (7) such that (1 − α ) = ∞. If n n n=1 the pair fS, Tg satises the condition (I) and F = F(T )\ F(S) ≠ ;, then fx g converges strongly to a common xed point of S and T. Proof. The method of the proof of Theorem 1 yields lim jjx − pjj exists for each p 2 F. Moreover, by Lemma n!∞ 3, we have: 2 2 jjx − pjj = jjα θ x + α (1 − θ )Tx + (1 − α )Sx − pjj n+1 n n n n n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjTx − p)jj + (1 − α )jjSx − pjj n n n n n n n n 2 2 −α θ (1 − θ )jjTx − x jj n n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjx − pjj + (1 − α )jjx − pjj n n n n n n n n 2 2 −α θ (1 − θ )jjTx − x jj n n n n n 2 2 2 = jjx − pjj − α θ (1 − θ )jjTx − x jj . n n n n n n This implies that 2 2 2 2 α θ (1 − θ )jjTx − x jj ≤ jjx − pjj −jjx − pjj , n n n n n n n+1 which in turn implies that 2 2 α θ (1 − θ )jjTx − x jj < ∞. (8) n n n n n=1 Also, from Lemma 3, we have 2 2 jjx − pjj = jjα θ x + α (1 − θ )Tx + (1 − α )Sx − pjj n+1 n n n n n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjTx − pjj + (1 − α )jjSx − pjj n n n n n n n n −α θ (1 − α )jjSx − x jj n n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjx − pjj + (1 − α )jjx − pjj n n n n n n n n −α β (1 − α )jjSx − x jj n n n n n 2 2 = jjx − pjj − α θ (1 − α )jjSx − x jj . n n n n n n Hence, we obtain α θ (1 − α )jjSx − x jj < ∞. (9) n n n n n n=1 Since (1 − α ) = ∞ and α ≥ θ ≥ for each n, it follows that n n n n=1 ∞ ∞ X X α θ (1 − θ ) = ∞ = α θ (1 − α ). n n n n n n=1 n=1 Thus, from (8) and (9), we get lim infjjx − Tx jj = 0 = lim infjjx − Sx jj. n n n n n!∞ n!∞ Then, sincefS, Tg satises the Condition (I), lim inf f (d(x , F)) = 0 for some increasing function f : [0,∞) ! n!∞ [0,∞) with f (0) = 0, f (r) > 0 when r > 0. This gives lim inf d(x , F) = 0. Moreover, since jjx − pjj ≤ jjx − pjj, taking inmum over all p 2 F, we n n+1 n n!∞ obtain d(x , F) ≤ d(x , F). Then, the sequencefd(x , F)g is decreasing and hence lim d(x , F) = 0. n+1 n n n n!∞ 44 Ë Abebe Regassa Tufa Now, for arbitrary p 2 F and any n, m ≥ 1, we have jjx − x jj ≤ jjx − pjj +jjx − pjj ≤ 2jjx − pjj, n+m n n+m n n which implies that jjx − x jj ≤ 2d(x , F). n+m n n Thenfx g is Cauchy sequence and hence x ! x 2 C. Thus, we have n n * * d(x , F) ≤ jjx − x jj + d(x , F) ! 0 as n ! ∞. n n Then, it follows from Lemma 2, that x 2 F. If, in Theorem 2, T and S are nonexpansive and F = F(T ) \ F(S) ≠ ;, then T and S are quasi-nonexpansive and hence the following corollary. Corollary 1. Let C be a nonempty, closed convex subset of a real Hilbert space H and S, T : C ! H be two nonexpansive inward mappings. Letfx g be a sequence as dened in (7) such that (1 − α ) = ∞. If the pair n n n=1 fS, Tg satises the condition (I) and F = F(T )\ F(S) ≠ ;, thenfx g converges strongly to a common xed point of S and T. A mapping T : C ! H is called hemicompact if for any sequence fx g in C such that d(x , Tx ) ! 0, there n n n exist a subsequence fx g of fx g such that x ! p 2 C. We note that if C is compact, then every mapping n n n j j T : C ! H is hemicompact. Theorem 3. Let C be a nonempty, closed and convex subset of a real Hilbert space H and S, T : C ! H be two nonexpansive inward mappings. Let fx g be a sequence as dened in (7) such that (1 − α ) = ∞. Assume n n n=1 that F = F(T )\ F(S) ≠ ; and T or S is hemicompact. Thenfx g converges strongly to a point in F. Proof. From the proof of Theorem 2, we have lim infjjx − Tx jj = 0 = lim infjjx − Sx jj. n n n n n!∞ n!∞ Then, there exists a subsequence sayfx g offx g such that m n lim jjx − Tx jj = 0. m m m!∞ Without loss of generality, assume that T is hemicompact. Then there is a subsequence fx g of fx g such m m that x ! x 2 C as k ! ∞. Then the continuity of T implies * * lim jjx − Tx jj = jjx − Tx jj = 0. (10) m m k k k!∞ Moreover, we have * * * jjx − Sx jj = lim jjx − Sx jj k!∞ = lim infjjx − Sx jj k!∞ ≤ lim inf[jjx − Sx jj +jjSx − Sx jj] m m m k k k k!∞ ≤ lim inf[jjx − Sx jj +jjx − x jj] = 0. (11) m m m k k k k!∞ * * * * From (10) and (11), we obtain x 2 F. Then lim jjx −x jj exists by (4). Thus, lim jjx −x jj = lim jjx −x jj = n n m n!∞ n!∞ k!∞ 0 and hencefx g converges strongly to x 2 F. n Iterative Methods for Common Fixed Points of Two Mappings Ë 45 If, in Theorem 3, we assume that C is compact, then T and S are hemicompact. Consequently, we have the following corollary. Corollary 2. Let C be a nonempty, compact and convex subset of a real Hilbert space H and S, T : C ! H be two nonexpansive inward mappings with F(T ) \ F(S) ≠ ;. Let fx g be a sequence dened in (7) such that (1 − α ) = ∞. Thenfx g converges strongly to a common xed point T and S. n n n=1 4 Conclusion In this paper, new iterative methods for approximating common xed points of two non-self mappings are studied in the setting of real Hilbert spaces. Weak convergence or strong convergence results (depending on the nature of the iteration parameter) of the scheme to a common xed point of two nonexpansive non- self mappings are obtained. In addition, strong convergence results for a pair of quasi-nonexpansive non- self mappings are established under some appropriate conditions. The results obtained in this paper extend, generalize and complement many of the results in the literature. For instance, our results extends the results of Yao and Chen [25] from self mappings to non-self mappings and generalizes the results of Colao and Marino [29] in the sense that our results are valid for two non-self mappings. Moreover, our ndings will open the way forward for the study of iterative methods for nding common xed points of two non-self mappings in Banach spaces more general than Hilbert spaces. References [1] N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal. Theory Methods Appl., 71 (2009), no. 3, 838–844. [2] H. H. Bauschke, The approximation of xed points of compositions of nonexpansive mappings in Hilbert spaces, J. Math Anal. Appl., 202 (1996), 150–159. [3] C.E. Chidume and C.O. Chidume, Iterative approximation of xed points of nonexpansive mappings, J. Math. Anal. Appl., 318 (2006), no. 1, 288–295. [4] O. A. Daman and H. Zegeye, Strong convergence theorems for a common xed point of a nite family of pseudocontractive mappings, Int. J. Math. Math. Sci., 2012 (2012), Article ID 405315, 18 pages. [5] G. Das and J. P. Debata, Fixed points of quasinonexpansive mappings, Indian J. Pure Appl. Math., 17 (1986), no. 11, 1263– [6] Y. Kimura, W. Takahashi and M. Toyoda, Convergence to common xed points of a nite family of nonexpansive mappings, Arch. Math., 84 (2005), no. 4, 350–363. [7] P. E. Maing´e, Approximation methods for common xed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469–479. [8] M.S. Narayanan and M. Marudai, Fixed points of nonexpansive and quasi-nonexpansive mappings, J. Anal., 27, (2019), 75– [9] J.G. O’Hara, P. Pillay and H.K. Xu, Iterative approaches to nding nearest common xed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal., 54 (2003), 1417–1426. [10] S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and nonexpansive map- ping in a Hilbert space, Nonlinear Anal., 69 (2008), 1025–1033. [11] A. R. Tufa and H. Zegeye, Convergence theorems for Lipschitz pseudocontractive non-self mappings in Banach spaces, J. Nonlinear Anal. Optim., 6 (2015), no. 2, 1–17. [12] R. Wittman, Approximation of xed points of nonexpansive mappings, Arch. Math., 58 (1992), 486–491. [13] Q. B. Zhang and C.Z. Cheng, Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space, Math. Comput. Modelling, 48 (2008), 480–485. [14] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. [15] A. Genel and J. Lindenstrauss, An example concerning xed points, Israel J. Math., 22 (1975), 81–86. [16] B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 73 (1967), 957–961. [17] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. 46 Ë Abebe Regassa Tufa [18] T.H. Kim and H. K. Xu, Strong convergence of modied Mann iterations, Nonlinear Anal., 61 (2005), 51–60. [19] T.H. Kim and H.K. Xu, Strong convergence of modied Mann iterations for asymptotically nonexpansive mappings and semi- groups, Nonlinear Anal., 64 (2006), 1140–1152. [20] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), no. 2, 274–276. [21] S. H. Khan and H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal., 61 (2005), no. 8, 1295–1301. [22] S. H. Khan and J. K. Kim, Common xed points of two nonexpansive mappings by a modied faster iteration scheme, Bull. Korean Math. Soc., 47 (2010), no. 5, 973–985. [23] S. H. Khan and W. Takahashi, Approximating common xed points of two asymptotically nonexpansive mappings, Sci. Math. Jpn., 53 (2001), no. 1, 143–148. [24] W. Takahashi and T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal., 5 (1998), no. 1, 45–56. [25] Y. Yao and R. Chen, Weak and strong convergence of a modied Mann iteration for asymptotically nonexpansive mappings, Nonlinear Funct. Anal. Appl., 12 (2007), no. 2, 307–315. [26] S. Matsushita and W. Takahashi, Strong convergence theorems for nonexpansive non-self mappings without boundary con- ditions, Nonlinear. Anal., 68 (2008), 412–419. [27] N. Shahzad, Approximating xed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal., 61 (2005), no. 6, 1031–1039. [28] Y. Song and R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl., 321 (2006), no. 1, 316–326. [29] V. Colao and G. Marino, Krasnoselskii-Mann method for non-self mappings, Fixed Point Theory and Appl., 39 (2015), 1–7. [30] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597. [31] W. G. Dotson, Fixed points of quasi-nonexpansive mappings, J. Aust. Math. Soc., 13 (1972), 167–170. [32] H. Zegeye and N. Shahzad, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive map- pings, Comput. Math. Appl., 62 (2011), 4007–4014. [33] G. Marino and H.K. Xu, Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007) 336–346. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Topological Algebra and its Applications de Gruyter

New Iterative Methods for Finding Common Fixed Points of Two Non-self Mappings in a Real Hilbert Space

Download PDF
11 pages

Loading next page...
 
/lp/de-gruyter/new-iterative-methods-for-finding-common-fixed-points-of-two-non-self-2LU6g5Zthi
Publisher
de Gruyter
Copyright
© 2022 Abebe Regassa Tufa, published by De Gruyter
ISSN
2299-3231
eISSN
2299-3231
DOI
10.1515/taa-2022-0111
Publisher site
See Article on Publisher Site

Abstract

Topol. Algebra Appl. 2022; 10:36–46 Research Article Open Access Abebe Regassa Tufa* New Iterative Methods for Finding Common Fixed Points of Two Non-self Mappings in a Real Hilbert Space https://doi.org/10.1515/taa-2022-0111 Received 1 November, 2021; accepted 18 March, 2022 Abstract: In this paper, we introduce new iterative methods for approximating common xed points of two non-self mappings in the framework of real Hilbert spaces. We establish weak and strong convergence results for approximating common xed points of two nonexpansive non-self mappings. In addition, we establish strong convergence results for approximating common xed points of two quasi-nonexpansive non-self map- pings under appropriate conditions. Our results improve and generalize many of the results in the literature. Moreover, our ndings will open the way forward for the study of iterative methods for nding common xed points of two non-self mappings in Banach spaces more general than Hilbert spaces. Keywords: Fixed points, Iterative methods, Nonexpansive mappings PACS: 47H09, 47H10, 47J25 1 Introduction Let C be a nonempty subset of a real Hilbert space H and T : C ! H be a mapping. A point x in C is said to be a xed point of T if x = Tx. The mapping T : C ! H is said to be 1) contraction if there exists k 2 (0, 1) such that jjTx − Tyjj ≤ kjjx − yjj, for all x, y 2 C. 2) nonexpansive if jjTx − Tyjj ≤ jjx − yjj, for all x, y 2 C. 3) quasi-nonexpansive if F(T ) := fx 2 C : x = Txg ≠ ; and jjTx − Tpjj ≤ jjx − pjj, for all x 2 C and for all p 2 F(T ). One can easily see that a contraction mapping is nonexpansive and a nonexpansive mapping with nonempty xed point set is quasi-nonexpansive mapping. However, a quasi-nonexpansive mapping need not be non- expansive (see, for instance, [1]). The theory of xed points of nonlinear maps is applicable in diverse elds of sciences such as dieren- tial equations, optimization, control theory, economics, etc. Thus, existences and approximations of xed points of various nonlinear mappings have been studied by numerous researchers (see, for instance, [2–13]). *Corresponding Author: Abebe Regassa Tufa: University of Botswana, Department of Mathematics,Gaborone, Botswana, E-mail: abykabe@yahoo.com Open Access. © 2022 Boris G. Averbukh, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. Iterative Methods for Common Fixed Points of Two Mappings Ë 37 We recall that the Mann iteration method [14] for a mapping T : C ! C is given by x 2 C, x = α x + (1 − α )Tx , (1) 0 n+1 n n n n where the initial guess x 2 C is arbitrary and fα g  [0, 1] such that lim α = 0 and α = ∞. It is n n n n!∞ well known that the Mann iteration method converges weakly for nonexpansive mappings to a xed point of the mapping T in an innite dimensional Hilbert space. Consequently, several researchers have studied and modied the Mann iteration method to obtain strong convergence results (see, for instance, [15–20]). Many researchers have also studied iterative schemes for approximating common xed points of two self mappings (see, for instance, [10, 21–25]). In 2007, Yao and Chen [25] introduced and studied the following iteration process for common xed points of two self mappings: x 2 C, x = α x + β Tx + Sx , n 2 N, n n n n n n 1 n+1 where fα g,fβ g and f g are in [0, 1] and α + β + = 1. We observe that this iterative scheme reduces n n n n n n to Mann iterative scheme if T or S is an identity map. On the other hand, many researchers have dealt with (see, for instance, [26–28]) approximation of xed points of non-self operators using the concept of metric projection or sunny nonexpansive retraction. But this is computationally expensive and not suitable for practical applications. In [29], Colao and Marino constructed a new iterative method for approximating xed points of non-self mapping T : C ! H without using the concept of metric projection or sunny nonexpansive retraction. Their algorithm is read as follows: x 2 C, α = maxf , h(x )g, 0 0 (2) x = α x + (1 − α )Tx , n+1 n n n n α = maxfα , h(x )g, n ≥ 0, n+1 n n+1 where h(x) := inffλ ≥ 0 : λx + (1 − λ)Tx 2 Cg,8x 2 C  H. They established weak and strong convergence results of the algorithm for approximating xed points of nonexpansive non-self mappings. Motivated and inspired by the above results, our objective is to construct new iterative methods for ap- proximating common xed points of two non-self mappings in the framework of real Hilbert spaces. We establish weak and strong convergence results for approximating common xed points of two nonexpansive non-self mappings. In addition, we establish strong convergence results for approximating common xed points of two quasi-nonexpansive non-self mappings under appropriate conditions. Our results improve, complement and generalize many of the results in the literature. Moreover, our ndings open the way for- ward for the study of iterative methods for nding common xed points of two non-self mappings in Banach spaces more general than Hilbert spaces. 2 Preliminaries In this section, we collect some basic denitions and known results which we may use in the proof of our main results. 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 2 C, we have Tx 2 I (x) := fx + λ(u − x) : for some u 2 C and λ ≥ 1g. 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 sequencefx g in C such that x * x and Tx − x ! 0 as n ! ∞, then x = Tx, n n n n 38 Ë Abebe Regassa Tufa where ” * ” denotes weak convergence, while ” ! ” denotes strong convergence. The set C is called strictly convex if it is convex and with the property that x, y 2 ∂C and t 2 (0, 1) im- plies that tx + (1 − t)y 2 C (see [29]). We will use the following lemmas in the sequel. Lemma 1 (Opial’s Condition). [30] If in a Hilbert space H the sequencefx g is weakly convergent to x then for any x ≠ x lim infjjx − xjj > lim infjjx − x jj. n n n!∞ n!∞ Lemma 2. [31] Let C be a closed convex subset of a real Hilbert space H and T : C ! H be quasi-nonexpansive mapping. Then F(T ) is closed and convex. Lemma 3. [32] Let H be a real Hilbert space. Then for all x 2 H and α 2 [0, 1] for i = 1, 2, 3,··· , n such that i i α + α + ··· + α = 1 the following equality holds: 1 2 X X 2 2 2 jjα x + α x + ··· + α x jj = α jjx jj − α α jjx − x jj . n n 1 1 2 2 i i i j i j i=1 1≤i,j≤n The following lemma can be obtained from Proposition 2.1 of [33]. Lemma 4. [33] Let C be a closed convex subset of a real Hilbert space H and T : C ! H be a nonexpansive mapping. Then the following hold. (i) I − T is demiclosed at zero; (ii) F(T ) is closed convex subset of C. Lemma 5. [29] Let C be a nonempty, closed and convex subset of a real Hilbert space H and T : C ! H be a mapping. Dene h : C ! R by h(x) = inffλ ≥ 0 : λx + (1 − λ)Tx 2 Cg. Then for any x 2 C the following hold: 1) h(x) 2 [0, 1] and h(x) = 0 if and only if Tx 2 C; 2) if β 2 [h(x), 1], then βx + (1 − β)Tx 2 C; 3) if T is inward, then h(x) < 1; 4) if Tx 2̸ C, then h(x)x + (1 − h(x))Tx 2 ∂C. 3 Main Results In this section, we construct an algorithm which involves two non-self mappings in the framework of real Hilbert spaces. Then we establish convergence results to a common xed point of the mappings. We rst prove the following lemma. Lemma 6. Let C be a nonempty, closed and convex subset of a real Hilbert space H and T, S : C ! H be two mappings. For each x 2 C and some θ 2 [0, 1], dene f : C ! [0,∞] by f (x) = inffα ≥ 0 : α θx + (1 − θ)Tx + (1 − α)Sx 2 Cg. Then for any x 2 C the following hold: (1) If θx + (1 − θ)Tx 2 C, then f (x) 2 [0, 1] and f (x) = 0 i Sx 2 C. θ θ (2) If θx + (1 − θ)Tx 2 C, then β(θx + (1 − θ)Tx + (1 − β)Sx 2 C for any β 2 [f (x), 1]. (3) If T and S are inward mappings, then f (x) < 1. (4) If θx + (1 − θ)Tx 2 C and Sx 2̸ C, then f (x) θx + (1 − θ)Tx + 1 − f (x) Sx 2 ∂C. θ θ Iterative Methods for Common Fixed Points of Two Mappings Ë 39 Proof. We need to prove only (3) and (4) as the proofs of (1) and (2) are trivial. (3) Let T and S be inward mappings and x 2 C. Then for some u , u 2 C and c , c ≥ 1, we have 1 2 1 2 Tx = x + c (u − x) and Sx = x + c (u − x), 1 1 2 2 which implies that 1 1 1 1 u = Tx + 1 − x and u = Sx + 1 − x. 1 2 c c c c 1 1 2 2 This yields 1 1 1 1 1 1 1 1 u + u = Tx + 1 − x + Sx + 1 − x 1 2 2 2 2c 2 c 2c 2 c 1 1 2 2 1 c c 1 2 2 = 1 − Tx + 1 − x + Sx 2c c (2c − 1) c (2c − 1) 2c 2 1 2 1 2 2 and since C is convex, we obtain that 1 c c 1 2 2 1 − Tx + 1 − x + Sx 2 C. 2c 2c c (2c − 1) c (2c − 1) 2 1 2 1 2 2 It is easy to verify that θ := 2 (0, 1]. Hence, c (2c −1) 1 2 f (x) ≤ 1 − < 1. 2c (4) Let θx + (1 − θ)Tx 2 C and Sx 2̸ C. Then f (x) 2 [0, 1] by (1). Let fw g be a sequence of real numbers in (0, f (x)) such that w ! f (x). By the denition of f , we obtain θ θ θ z := w (θx + (1 − θ)Tx) + (1 − w )Sx 2̸ C. n n n Thus, we have jjz − f (x)(θx + (1 − θ)Tx) − (1 − f (x))Sxjj ≤ jw − f (x)jjjθx + (1 − θ)Tx − Sx)jj. n n θ θ θ Now, since w ! f (x), it follows that z ! f (x)(θx + (1 − θ)Tx) + (1 − f (x))Sx 2 C θ θ and since z = w θx + (1 − θ)Tx + (1 − w )Sx 2̸ C, for all n ≥ 1, we obtain n n n f (x) θx + (1 − θ)Tx + 1 − f (x) Sx 2 ∂C. θ θ This completes the proof. Remark 1. If θ = 1 or T is an identity map, then Lemma 6 reduces to Lemma 5. We are now in a position to construct our algorithm which involves two non-self mappings. Let C be a nonempty, closed and convex subset of a real Hilbert space H and T, S : C ! H be two non-self inward mappings. Given x 2 C, let h(x ) := inffθ ≥ 0 : θx + (1 − θ)Tx 2 Cg. 1 1 1 Take θ = maxf , h(x )g. Then θ x + (1 − θ )Tx 2 C. 1 1 1 1 1 1 Now, let f (x ) := inffα ≥ 0 : α[θ x + (1 − θ )Tx ] + (1 − α)Sx 2 Cg. 1 1 1 1 1 1 Putting α = maxf , f (x )g, we obtain that 1 1 2 1 x := α [θ x + (1 − θ )Tx ] + (1 − α )Sx 2 C. 2 1 1 1 1 1 1 1 40 Ë Abebe Regassa Tufa Inductively, we dene a sequencefx g as follows: > x 2 C, x = α [θ x + (1 − θ )Tx ] + (1 − α )Sx n n n n n n n n+1 (3) > θ = maxfθ , h(x )g, n+1 n+1 α = maxfα , f (x )g, n+1 θ n+1 n+1 where h : C ! R and f : C ! [0,∞] are dened, respectively, by h(x) = inffλ ≥ 0 : λx + (1 − λ)Tx 2 Cg and f (x) = inffα ≥ 0 : α[θx + (1 − θ)Tx] + (1 − α)Sx 2 Cg, n ≥ 1, for some θ 2 [0, 1]. We observe that Algorithm (3) reduces to Mann iterative scheme when T or S is an identity map. Theorem 1. Let C be a nonempty, closed and convex subset of a real Hilbert space H and S, T : C ! H be two nonexpansive inward mappings with F = F(T )\ F(S) ≠ ;. Letfx g be a sequence dened in (3). i) If there is b 2 (0, 1) such that θ , α ≤ b,8n ≥ 1, thenfx g converges weakly to a point in F. n n n ∞ ∞ X X ii) If C is strictly convex, (1 − θ ) < ∞ and (1 − α ) < ∞, thenfx g converges strongly to a point in F. n n n n=1 n=1 Proof. Let p 2 F. Since T and S are nonexpansive, from (3) and the triangle inequality, we have jjx − pjj = jjα [θ x + (1 − θ )Tx ] + (1 − α )Sx − pjj n n n n n n n n+1 ≤ α jjθ x + (1 − θ )Tx − pjj + (1 − α )jjSx − pjj n n n n n n n ≤ α θ jjx − pjj + α (1 − θ )jjTx − pjj + (1 − α )jjSx − pjj n n n n n n n n ≤ α θ jjx − p)jj + α (1 − θ )jjx − pjj + (1 − α )jjx − pjj n n n n n n n n = jjx − pjj. (4) Then fjjx − pjjg is decreasing which implies that lim jjx − pjj exists. In addition, fx g is bounded and so n n n n!∞ are Tx , Sx andfy g. n n n i) Assume that there is b 2 (0, 1) with θ , α ≤ b,8n ≥ 1. Then by Lemma 3, we have: n n 2 2 jjx − pjj = jjα θ x + α (1 − θ )Tx + (1 − α )Sx − pjj n n n n n n n n n+1 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjTx − pjj + (1 − α )jjSx − pjj n n n n n n n n 2 2 −α θ (1 − θ )jjTx − x jj n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjx − pjj + (1 − α )jjx − pjj n n n n n n n n 2 2 −α θ (1 − θ )jjTx − x jj n n n n 2 2 2 = jjx − pjj − α θ (1 − θ )jjTx − x jj . n n n n n This provides 2 2 2 2 α θ (1 − θ )jjTx − x jj ≤ jjx − pjj −jjx − pjj . (5) n n n n n n n+1 Also, since lim jjx − pjj exists, it follows that n!∞ 2 2 2 jjx − pjj −jjx − pjj ≤ jjx − pjj . (6) n+1 1 n=1 Then since ≤ α , θ < b, (5) and (6) yields n n ∞ ∞ X X 2 2 2 (1 − b)jjTx − x jj ≤ α θ (1 − θ )jjTx − x jj < ∞. n n n n n n n n=1 n=1 Hence, we have lim jjx −Tx jj = 0. Similarly, one can easily show that lim jjx −Sx jj = 0. Also, sincefx g n n n n n n!∞ n!∞ is bounded, the set of weak cluster points offx g is nonempty, that is w(x ) := fx 2 H : x * x, for some subsequencefx g offx gg ≠ ;. n n n n i i Iterative Methods for Common Fixed Points of Two Mappings Ë 41 Let x 2 w(x ) and fx g be a subsequence of fx g such that x * x as i ! ∞. Then since I − T and I − S n n n n i i are demiclosed at zero (see Lemma 4), it follows that x 2 F = F(T ) \ F(S). This gives w(x )  F. To show uniqueness, let x, y 2 w(x ) and letfx g andfx g be subsequences offx g such that x * x and x * y n n n n n n i j i j as i, j ! ∞. Suppose x ≠ y. Then since lim jjx −xjj exists for all x 2 F and Hilbert space satises the Opial’s n!∞ property, by Lemma 1, we get lim jjx − xjj = lim jjx − xjj < lim jjx − yjj n n n i i n!∞ i!∞ i!∞ = lim jjx − yjj = lim jjx − yjj n n n!∞ j!∞ < lim jjx − xjj = lim jjx − xjj. n n n!∞ j!∞ However, this is a contradiction and hence x = y. Thus, every subsequence of fx g converges weakly to x and hencefx g converges weakly to x 2 F. ∞ ∞ X X ii) Suppose that C is strictly convex, (1 − θ ) < ∞ and (1 − α ) < ∞. n n n=1 n=1 Then since α (1 − θ ) ≤ (1 − θ ), it follows that α (1 − θ ) < ∞. Moreover, from the fact thatfx g,fTx g n n n n n n n n=1 andfSx g are bounded andjjx − x jj = α (1 − θ )jjx − Tx jj + (1 − α )jjx − Sx jj, we have n n n+1 n n n n n n n jjx − x jj < ∞, n+1 n=1 * * which implies thatfx g is Cauchy sequence and hence there is x 2 C such that x ! x as n ! ∞. n n On the other hand, since lim θ = 1 and θ = maxfθ , h(x )g, we can pick a sub-sequencefx g such that n n n−1 n n n!∞ fh(x )g is increasing and lim h(x ) = 1. In particular, for any < 1, n n i i i!∞ t := x + (1 − )Tx 2̸ C, eventually holds. n n n i i i Now choose , 2 (h(x ), 1) such that ≠ and let 1 2 1 2 * * * * v = x + (1 − )Tx and v = x + (1 − )Tx . 1 1 1 2 2 2 Then for any 2 [ , ], we have 1 2 * * v := x + (1 − )Tx 2 C. * * Also, since x ! x and Tx ! Tx as i ! ∞, we obtain that t ! v as i ! ∞ and hence v 2 ∂C. This n n n i i i yields [v , v ]  ∂C since is arbitrary point in [ , ]. Then by strict convexity of C, we obtain that v = v . 1 2 1 2 1 2 * * Thenjjv − Tx jj = jjv − Tx jj which implies that 1 2 * * * * jjx − Tx jj = jjx − Tx jj. 1 2 * * * * Then since ≠ , we can see that x = Tx and hence, x 2 F(T ). It remains to show that x 2 F(S). To this 1 2 end , we observe that y := θ x + (1 − θ )Tx ! x as n ! ∞. n n n n n Then, since lim α = 1 and α = maxfα , f (x )g, repeating the above arguments provides the required n n n−1 n n!∞ result. Indeed, we can pick a sub-sequencefx g such thatff (x )g is increasing and lim f (x ) = 1. n n n j θ j j!∞ θ j n n j j In particular, for any μ < 1, s := μy + (1 − μ)Sx 2̸ C, eventually holds. n n n j j j 42 Ë Abebe Regassa Tufa Now choose μ , μ 2 (f (x ), 1) such that μ ≠ μ and let 1 2 1 2 * * * * u = μ x + (1 − μ )Sx and u = μ x + (1 − μ )Sx . 1 1 1 2 2 2 Thus, for any μ 2 [μ , μ ], we have 1 2 * * u := μx + (1 − μ)Sx 2 C. * * Since y ! x and Sx ! Sx as j ! ∞, it follows that s ! u as j ! ∞ and hence u 2 ∂C. Furthermore, n n n j j j since μ is arbitrary, it follows that [u , u ]  ∂C. Then the strict convexity of C gives u = u and d(u , Sx ) = 1 2 1 2 1 * * * * * * * d(u , Sx ) which in turn gives μ d(x , Sx ) = μ d(x , Sx ). Then since μ ≠ μ , we have that x = Sx and 2 1 2 1 2 * * hence, x 2 F(S). Therefore,fx g converges strongly to x 2 F = F(T )\ F(S). The proof is complete. Next, we prove strong convergence results using the condition (I). For this, we rst modify the denition of condition (I) for a pair of mappings. Recall that a mapping T : C ! H is said to satisfy condition (I) if there exists a nondecreasing function f : [0,∞) ! [0,∞) with f (0) = 0 and f (r) > 0, for all r 2 (0,∞) such that jjx − Txjj ≥ f (d(x, F(T ))), for all x 2 C, where d(x, F(T )) = inffjjx − pjj : p 2 F(T )g. Now, we modify this denition for a pair of two mappings as follows. A pair of mappings S and T denoted by fS, Tg is said to satisfy condition (I) if there exists a nondecreas- ing function f : [0,∞) ! [0,∞) with f (0) = 0 and f (r) > 0, for all r 2 (0,∞) such thatjjx − Txjj ≥ f (d(x, F)) orjjx − Sxjj ≥ f (d(x, F)), for all x 2 C, where d(x, F) = inffjjx − pjj : p 2 F = F(T )\ F(S)g. Example 1. Let H = R with the Euclidean norm and S, T : [0, 1] ! R be dened by Sx = x − x, and x, x 2 [0, 1/2), Tx = : 1 − x, x 2 [1/2, 1]. Then the pair (S, T ) satisfy the condition (I). Proof. Note that F = F(S)\ F(T ) = f0g, d(x, p) = jxj and jxj, x 2 [0, 1/2), jx − Txj = jxj, x 2 [1/2, 1]. Then one can consider f (r) = r to complete the proof. Now, we consider the following Algorithm: > x 2 C, > θ = max , h(x ) , 1 1 > 2 α = maxfθ , f (x )g, 1 1 1 (7) > x = α [θ x + (1 − θ )Tx ] + (1 − α )Sx , n+1 n n n n n n n θ 2 maxfθ , h(x )g, 1 , > n+1 n n+1 α 2 maxfα , θ , f (x )g, 1 , n+1 n n n+1 n+1 where h : C ! R and f : C ! [0,∞] are dened, respectively, by h(x) = inffλ ≥ 0 : λx + (1 − λ)Tx 2 Cg and f (x) = inffα ≥ 0 : α[θx + (1 − θ)Tx] + (1 − α)Sx 2 Cg, n ≥ 1, for some θ 2 [0, 1]. θ Iterative Methods for Common Fixed Points of Two Mappings Ë 43 Theorem 2. Let C be a nonempty, closed and convex subset of a real Hilbert space H and S, T : C ! H be two quasi-nonexpansive inward mappings. Let fx g be a sequence as dened in (7) such that (1 − α ) = ∞. If n n n=1 the pair fS, Tg satises the condition (I) and F = F(T )\ F(S) ≠ ;, then fx g converges strongly to a common xed point of S and T. Proof. The method of the proof of Theorem 1 yields lim jjx − pjj exists for each p 2 F. Moreover, by Lemma n!∞ 3, we have: 2 2 jjx − pjj = jjα θ x + α (1 − θ )Tx + (1 − α )Sx − pjj n+1 n n n n n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjTx − p)jj + (1 − α )jjSx − pjj n n n n n n n n 2 2 −α θ (1 − θ )jjTx − x jj n n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjx − pjj + (1 − α )jjx − pjj n n n n n n n n 2 2 −α θ (1 − θ )jjTx − x jj n n n n n 2 2 2 = jjx − pjj − α θ (1 − θ )jjTx − x jj . n n n n n n This implies that 2 2 2 2 α θ (1 − θ )jjTx − x jj ≤ jjx − pjj −jjx − pjj , n n n n n n n+1 which in turn implies that 2 2 α θ (1 − θ )jjTx − x jj < ∞. (8) n n n n n=1 Also, from Lemma 3, we have 2 2 jjx − pjj = jjα θ x + α (1 − θ )Tx + (1 − α )Sx − pjj n+1 n n n n n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjTx − pjj + (1 − α )jjSx − pjj n n n n n n n n −α θ (1 − α )jjSx − x jj n n n n n 2 2 2 ≤ α θ jjx − pjj + α (1 − θ )jjx − pjj + (1 − α )jjx − pjj n n n n n n n n −α β (1 − α )jjSx − x jj n n n n n 2 2 = jjx − pjj − α θ (1 − α )jjSx − x jj . n n n n n n Hence, we obtain α θ (1 − α )jjSx − x jj < ∞. (9) n n n n n n=1 Since (1 − α ) = ∞ and α ≥ θ ≥ for each n, it follows that n n n n=1 ∞ ∞ X X α θ (1 − θ ) = ∞ = α θ (1 − α ). n n n n n n=1 n=1 Thus, from (8) and (9), we get lim infjjx − Tx jj = 0 = lim infjjx − Sx jj. n n n n n!∞ n!∞ Then, sincefS, Tg satises the Condition (I), lim inf f (d(x , F)) = 0 for some increasing function f : [0,∞) ! n!∞ [0,∞) with f (0) = 0, f (r) > 0 when r > 0. This gives lim inf d(x , F) = 0. Moreover, since jjx − pjj ≤ jjx − pjj, taking inmum over all p 2 F, we n n+1 n n!∞ obtain d(x , F) ≤ d(x , F). Then, the sequencefd(x , F)g is decreasing and hence lim d(x , F) = 0. n+1 n n n n!∞ 44 Ë Abebe Regassa Tufa Now, for arbitrary p 2 F and any n, m ≥ 1, we have jjx − x jj ≤ jjx − pjj +jjx − pjj ≤ 2jjx − pjj, n+m n n+m n n which implies that jjx − x jj ≤ 2d(x , F). n+m n n Thenfx g is Cauchy sequence and hence x ! x 2 C. Thus, we have n n * * d(x , F) ≤ jjx − x jj + d(x , F) ! 0 as n ! ∞. n n Then, it follows from Lemma 2, that x 2 F. If, in Theorem 2, T and S are nonexpansive and F = F(T ) \ F(S) ≠ ;, then T and S are quasi-nonexpansive and hence the following corollary. Corollary 1. Let C be a nonempty, closed convex subset of a real Hilbert space H and S, T : C ! H be two nonexpansive inward mappings. Letfx g be a sequence as dened in (7) such that (1 − α ) = ∞. If the pair n n n=1 fS, Tg satises the condition (I) and F = F(T )\ F(S) ≠ ;, thenfx g converges strongly to a common xed point of S and T. A mapping T : C ! H is called hemicompact if for any sequence fx g in C such that d(x , Tx ) ! 0, there n n n exist a subsequence fx g of fx g such that x ! p 2 C. We note that if C is compact, then every mapping n n n j j T : C ! H is hemicompact. Theorem 3. Let C be a nonempty, closed and convex subset of a real Hilbert space H and S, T : C ! H be two nonexpansive inward mappings. Let fx g be a sequence as dened in (7) such that (1 − α ) = ∞. Assume n n n=1 that F = F(T )\ F(S) ≠ ; and T or S is hemicompact. Thenfx g converges strongly to a point in F. Proof. From the proof of Theorem 2, we have lim infjjx − Tx jj = 0 = lim infjjx − Sx jj. n n n n n!∞ n!∞ Then, there exists a subsequence sayfx g offx g such that m n lim jjx − Tx jj = 0. m m m!∞ Without loss of generality, assume that T is hemicompact. Then there is a subsequence fx g of fx g such m m that x ! x 2 C as k ! ∞. Then the continuity of T implies * * lim jjx − Tx jj = jjx − Tx jj = 0. (10) m m k k k!∞ Moreover, we have * * * jjx − Sx jj = lim jjx − Sx jj k!∞ = lim infjjx − Sx jj k!∞ ≤ lim inf[jjx − Sx jj +jjSx − Sx jj] m m m k k k k!∞ ≤ lim inf[jjx − Sx jj +jjx − x jj] = 0. (11) m m m k k k k!∞ * * * * From (10) and (11), we obtain x 2 F. Then lim jjx −x jj exists by (4). Thus, lim jjx −x jj = lim jjx −x jj = n n m n!∞ n!∞ k!∞ 0 and hencefx g converges strongly to x 2 F. n Iterative Methods for Common Fixed Points of Two Mappings Ë 45 If, in Theorem 3, we assume that C is compact, then T and S are hemicompact. Consequently, we have the following corollary. Corollary 2. Let C be a nonempty, compact and convex subset of a real Hilbert space H and S, T : C ! H be two nonexpansive inward mappings with F(T ) \ F(S) ≠ ;. Let fx g be a sequence dened in (7) such that (1 − α ) = ∞. Thenfx g converges strongly to a common xed point T and S. n n n=1 4 Conclusion In this paper, new iterative methods for approximating common xed points of two non-self mappings are studied in the setting of real Hilbert spaces. Weak convergence or strong convergence results (depending on the nature of the iteration parameter) of the scheme to a common xed point of two nonexpansive non- self mappings are obtained. In addition, strong convergence results for a pair of quasi-nonexpansive non- self mappings are established under some appropriate conditions. The results obtained in this paper extend, generalize and complement many of the results in the literature. For instance, our results extends the results of Yao and Chen [25] from self mappings to non-self mappings and generalizes the results of Colao and Marino [29] in the sense that our results are valid for two non-self mappings. Moreover, our ndings will open the way forward for the study of iterative methods for nding common xed points of two non-self mappings in Banach spaces more general than Hilbert spaces. References [1] N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal. Theory Methods Appl., 71 (2009), no. 3, 838–844. [2] H. H. Bauschke, The approximation of xed points of compositions of nonexpansive mappings in Hilbert spaces, J. Math Anal. Appl., 202 (1996), 150–159. [3] C.E. Chidume and C.O. Chidume, Iterative approximation of xed points of nonexpansive mappings, J. Math. Anal. Appl., 318 (2006), no. 1, 288–295. [4] O. A. Daman and H. Zegeye, Strong convergence theorems for a common xed point of a nite family of pseudocontractive mappings, Int. J. Math. Math. Sci., 2012 (2012), Article ID 405315, 18 pages. [5] G. Das and J. P. Debata, Fixed points of quasinonexpansive mappings, Indian J. Pure Appl. Math., 17 (1986), no. 11, 1263– [6] Y. Kimura, W. Takahashi and M. Toyoda, Convergence to common xed points of a nite family of nonexpansive mappings, Arch. Math., 84 (2005), no. 4, 350–363. [7] P. E. Maing´e, Approximation methods for common xed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469–479. [8] M.S. Narayanan and M. Marudai, Fixed points of nonexpansive and quasi-nonexpansive mappings, J. Anal., 27, (2019), 75– [9] J.G. O’Hara, P. Pillay and H.K. Xu, Iterative approaches to nding nearest common xed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal., 54 (2003), 1417–1426. [10] S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem and nonexpansive map- ping in a Hilbert space, Nonlinear Anal., 69 (2008), 1025–1033. [11] A. R. Tufa and H. Zegeye, Convergence theorems for Lipschitz pseudocontractive non-self mappings in Banach spaces, J. Nonlinear Anal. Optim., 6 (2015), no. 2, 1–17. [12] R. Wittman, Approximation of xed points of nonexpansive mappings, Arch. Math., 58 (1992), 486–491. [13] Q. B. Zhang and C.Z. Cheng, Strong convergence theorem for a family of Lipschitz pseudocontractive mappings in a Hilbert space, Math. Comput. Modelling, 48 (2008), 480–485. [14] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. [15] A. Genel and J. Lindenstrauss, An example concerning xed points, Israel J. Math., 22 (1975), 81–86. [16] B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 73 (1967), 957–961. [17] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147–150. 46 Ë Abebe Regassa Tufa [18] T.H. Kim and H. K. Xu, Strong convergence of modied Mann iterations, Nonlinear Anal., 61 (2005), 51–60. [19] T.H. Kim and H.K. Xu, Strong convergence of modied Mann iterations for asymptotically nonexpansive mappings and semi- groups, Nonlinear Anal., 64 (2006), 1140–1152. [20] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), no. 2, 274–276. [21] S. H. Khan and H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal., 61 (2005), no. 8, 1295–1301. [22] S. H. Khan and J. K. Kim, Common xed points of two nonexpansive mappings by a modied faster iteration scheme, Bull. Korean Math. Soc., 47 (2010), no. 5, 973–985. [23] S. H. Khan and W. Takahashi, Approximating common xed points of two asymptotically nonexpansive mappings, Sci. Math. Jpn., 53 (2001), no. 1, 143–148. [24] W. Takahashi and T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal., 5 (1998), no. 1, 45–56. [25] Y. Yao and R. Chen, Weak and strong convergence of a modied Mann iteration for asymptotically nonexpansive mappings, Nonlinear Funct. Anal. Appl., 12 (2007), no. 2, 307–315. [26] S. Matsushita and W. Takahashi, Strong convergence theorems for nonexpansive non-self mappings without boundary con- ditions, Nonlinear. Anal., 68 (2008), 412–419. [27] N. Shahzad, Approximating xed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal., 61 (2005), no. 6, 1031–1039. [28] Y. Song and R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl., 321 (2006), no. 1, 316–326. [29] V. Colao and G. Marino, Krasnoselskii-Mann method for non-self mappings, Fixed Point Theory and Appl., 39 (2015), 1–7. [30] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597. [31] W. G. Dotson, Fixed points of quasi-nonexpansive mappings, J. Aust. Math. Soc., 13 (1972), 167–170. [32] H. Zegeye and N. Shahzad, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive map- pings, Comput. Math. Appl., 62 (2011), 4007–4014. [33] G. Marino and H.K. Xu, Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces, J. Math. Anal. Appl., 329 (2007) 336–346.

Journal

Topological Algebra and its Applications – de Gruyter

Published: Jan 1, 2022

Keywords: Fixed points; Iterative methods; Nonexpansive mappings

There are no references for this article.