International Journal of Analysis and Applications ISSN 2291-8639 Volume 6, Number 2 (2014), 144-153 http://www.etamaths.com APPROXIMATING FIXED POINTS OF GENERALIZED NONEXPANSIVE MAPPINGS IN BANACH SPACES BAPURAO C. DHAGE Abstract. In this paper, we prove a fixed point theorem for the selfmaps of a closed convex and bounded subset of the Banach space satisfying a generalized nonexpansive type condition. Some results concerning the approximations of fixed points with Krasnoselskii and Mann type iterations are also proved under suitable conditions. Our results include the well-known result of Kannan (1968) and Bose and Mukherjee (1981) as the special cases with a different and constructive method. 1. Introduction Let (X,d) be a metric space. Then Banach contraction principle states that if X is complete and f : X → X satisfies the condition (1.1) d(fx,fy) ≤ αd(x,y) for all x,y ∈ X and 0 ≤ α < 1, then f has a unique fixed point. The mapping f satisfying the condition (1.1) is called contraction and when α = 1, f is called nonexpansive. The nonexansive mappings have been studied by Kirk and Goebel [6] for fixed points. Bogin [1] considered a class of generalized nonexpansive mappings characterized by the inequality (1.2) d(fx,fy) ≤ ad(x,y) + b[d(x,fx) + d(y,fy)] + b[d(x,fy) + d(y,fx)] for all x,y ∈ X, where a,b,c are nonnegative real numbers satisfying (1.3) a + 2b + 2c = 1 for the study of fixed points. Recently Ciric [3] generalized the above class of mappings (1.2)-(1.3) to a wider class mappings characterized by the inequality d(fx,fy) ≤ a max { d(x,y),d(x,fx),d(y,fy), 1 2 [d(x,fy) + d(y,fx)] } + b max{d(x,fx),d(y,fy)} + c[d(x,fy) + d(y,fx)](1.4) for all x,y ∈ X, where the real numbers a,b,c ≥ 0 satisfy the condition (1.5) a + b + 2c = 1. 2010 Mathematics Subject Classification. 47H10. Key words and phrases. Banach space; Nonexpansive mappings; fixed points. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 144 GENERALIZED NONEXPANSIVE MAPPINGS IN BANACH SPACES 145 Similarly, the study of nonexansive mappings in Banach spaces has been made extensively by several authors. Bose and Mukherjee [2] studied the class of gen- eralized nonexansive mappings for the study of fixed points characterized by the inequality (1.6) ‖fx−fy‖≤ a‖x−y‖ + b [‖x−fx‖ + ‖y −fy‖] + c [‖x−fx‖ + ‖y −fy‖] for all x,y ∈ X, where a,b,c are nonnegative real numbers, a > 0 satisfying the condition (1.7) 3a + 2b + 4c = 1. The aim of the present note is to generalize the above class of mappings (1.6)- (1.7) and prove a couple of fixed point theorems under a generalized contraction condition with a different method which in turn generalize fixed point theorems of Bose and Mukherjee [2] as the special cases. 2. Generalized Nonexpansive Mappings Given a non-empty, closed, convex and bounded subset C of the Banach space X, consider the class of nonexpansive type mappings f : C → C characterized by the inequality ‖fx−fy‖≤ a max { ‖x−y‖,‖x−fx‖,‖y −fy‖, 1 2 [‖x−fy‖ + ‖y −fx‖] } + b [ ‖x−fx‖ + ‖y −fy‖ ] + c max { ‖x−fy‖,‖y −fx‖ } (2.1) for all x,y ∈ X, where the real numbers a,b,c ≥ 0 satisfy the inequality (2.2) a + b + c ≤ 1 2 . The generalized nonexpansive mappings characterized by the inequalities (2.1) and (2.2) have been considered in Dhage [4] in the setting of a metric space for fixed points and are different from the class of Ciric’s mappings characterized by the inequalities (1.6) and (1.7). In this section we prove a couple of results concerning the existence of fixed point for the class of generalized nonexpansive mappings (2.1) and (2.2) in a Banach space via a scheme of Krasnoselskii type iterations. Theorem 2.1. Let C be a non-empty, closed, convex and bounded subset of the normed linear space X and let f : C → C be a mapping satisfying the inequality (2.1) and (2.2) with a > 0. If the sequence {xn} defined by (2.3) xn+1 = (1 − t)xn + tfxn, n = 0, 1, 2, ...; for some t ∈ (0, 1) and for some x = x0 ∈ C converges to u, then u is a unique fixed point of f. 146 DHAGE Proof. By (2.1), one gets ‖xn+1 −fu‖≤ (1 − t)‖xn −fu‖ + t‖fxn −fu‖ ≤ (1 − t)‖xn −fu‖ + a { ‖xn −u‖,‖xn −fx‖,‖u−fu‖, 1 2 [‖xn −fu‖ + ‖u−fxn‖] } + b [‖xn −fxn + ‖u−fu‖ + ‖] + c max{‖xn −fu‖,‖u−fxn‖}.(2.4) Now, xn+1 = (1 − t)xn + tfxn, and so we have (xn+1 −xn) = −t(xn −fxn). This further implies that ‖xn+1 −xn‖ = t‖xn −fxn‖−→ 0 as n →∞. Taking the limit as n →∞ in (2.4), we obtain ‖u−fu‖≤ (1 − t)‖u−fu‖ + ta max { 0, 0,‖u−fu‖, 1 2 ‖u−fu‖ } + tb [0 + ‖u−fu‖] + tc max{‖u−fu‖, 0} ≤ [(1 − t) + ta + tb + tc]‖u−fu‖ ≤ (1 − t + a + b + c)‖u−fu‖. Since a + b + c < 1, we may choose t ∈ (0, 1) such that t > a + b + c. Then from the above inequality, we obtain so u = fu. To prove uniqueness, let v(6= u) be another fixed point of f. Then by (2.1), ‖u−v‖ = ‖fu−fv‖ ≤ a max { ‖u−v‖,‖u−fu‖,‖v −fv‖, 1 2 [‖u−fv‖ + ‖v −fu‖] } + b [‖u−fu‖ + ‖v −fv‖] + c max{‖u−fv‖,‖v −fu‖} = (a + c)‖u−v)‖ which is a contradiction. Hence u = v and the proof of the theorem is complete. � Theorem 2.2. Let C be a non-empty, closed, convex and bounded subset of a Banach space X. If f : C → C satisfies the inequalities (2.1) and (2.2) with a > 0, b > 0, then f has a unique fixed point. Proof. Let x = x0 ∈ C be arbitrary and consider the sequence {xn} defined by (2.3). Then, we have x1 −x2 = (1 − t)(x0 −x1) + t(fx0 −fx2). GENERALIZED NONEXPANSIVE MAPPINGS IN BANACH SPACES 147 Then, by (2.1), we obtain ‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + t‖fx0 −fx1‖ ≤ (1 − t)‖x0 −x1‖ + ta max { ‖x0 −x1‖,‖x0 −fx0‖,‖x1 −fx1‖, 1 2 [‖x0 −x1‖ + ‖x1 −x2‖] } + tb [‖x0 −fx0‖ + ‖x1 −fx1‖] + tc max{‖x0 −fx1‖,‖x1 −fx0‖}.(2.5) Now, x1 = (1 − t)x0 + tfx0, and so we have ⇒ x1 −x0 = −t(x0 −fx0). This further implies that t‖x0 −fx0‖ = ‖x0 −x1‖. Again, x2 = (1 − t)x1 + tfx1, and so we have x2 −x1 = −t(x1 −fx1) which again implies that t‖x1 −fx1‖ = ‖x1 −x2‖. Similarly, (x0 −fx1) = (x0 −x1) + (x1 −fx1), implies t(x0 −fx1) = t(x0 −x1) + t(x1 −x2)), and t‖x0 −fx1‖≤ t‖x0 −x1‖ + t‖x1 −x2‖. Again, x1 −fx0 = x1 −x0 + x0 −fx0 = (x1 −x2) + (x0 −fx0), which gives t(x1 −fx0) = t(x1 −x0) + t(x0 −f(x0) = (1 − t)(x0 −x1), or, t‖x1 −fx0‖ = (1 − t)‖x0 −x1‖. 148 DHAGE Substituting the above values in (2.5), ‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + a max { ‖x0 −x1‖,‖x0 −x1‖,‖x1 −x2‖, 1 2 [ [ (1 − t)‖x0 −x1‖ + t‖x0 −x1‖ + ‖x1 −x2‖ ]} + b [ ‖x0 −x1‖ + ‖x1 −x2‖ ] + c max { (1 − t)‖x0 −x1‖, t‖x0 −x1‖ + ‖x1 −x2‖ } = (1 − t)‖x0 −x1‖ + a max { ‖x0 −x1‖,‖x1 −x2‖, 1 2 [ [ ‖x0 −x1‖ + ‖x1 −x2‖ ]} + b [ ‖x0 −x1‖ + ‖x1 −x2‖ ] + c max { (1 − t)‖x0 −x1‖, t‖x0 −x1‖ + ‖x1 −x2‖ } .(2.6) Now there are three cases: Case I: Suppose that max { ‖x0 −x1‖,‖x1 −x2‖, 1 2 [ ‖x0 −x1‖ + ‖x1 −x2‖ ]} = ‖x0 −x1‖ and max { (1 − t)‖x0 −x1‖, t‖x0 −x1‖ + ‖x1 −x2‖ } = ‖x0 −x1‖ for t > 1 2 . Then from (2.6), (1 − b)‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + (a + b)‖x0 −x1‖ + ct‖x0 −x1‖ + c‖x1 −x2‖. Therefore, ‖x1 −x2‖≤ ( (1 − t) + a + b + ct 1 − b− c ) ‖x0 −x1‖ ≤ ( (1 − t) + a + b + c 1 − b− c ) ‖x0 −x1‖ = α1‖x0 −x1‖ Case II: Suppose that max { ‖x0 −x1‖,‖x1 −x2‖, 1 2 [ ‖x0 −x1‖ + ‖x1 −x2‖ ]} = ‖x1 −x2‖. Then, ‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + a‖x1 −x2‖ + b‖x0 −x1‖ + b‖x1 −x2‖ + ct‖x0 −x1‖ + c‖x1 −x2‖ ≤ ( 1 − t + b + c 1 −a− b− c ) ‖x0 −x1‖ ≤ α2‖x0 −x1‖ [t > a + 2b + 2c] GENERALIZED NONEXPANSIVE MAPPINGS IN BANACH SPACES 149 Case III: Suppose that max { ‖x0 −x1‖,‖x1 −x2‖, 1 2 [ ‖x0 −x1‖+‖x1 −x2‖ ]} = 1 2 [‖x1 −x2‖+‖x0 −x1‖]. Then, ‖x1 −x2‖≤ (1 − t)‖x0 −x1‖ + a 2 ‖x0 −x1‖ + a 2 ‖x1 −x2‖ + b‖x0 −x1‖ + b‖x1 −x2‖ + c‖x0 −x1‖ + c‖x1 −x2‖ ≤ ( 1 − t + a 2 + b + c 1 − a 2 − b− c ) ‖x0 −x1‖ ≤ α3‖x0 −x1‖ [t > a + 2b + 2c]. Let α = max{α1,α2,α3}, then in all above three cases we obtain ‖x1 −x2‖≤ α‖x0 −x1‖. Therefore, ‖xn −xn+1‖≤ n+p∑ i=n ‖xi −xi+1‖ ≤ αn 1 −α ‖x0 −x1‖ −→ 0 as n →∞. This shows that {xn} is a Cauchy is a sequence in C. Since C is a closed subset of a complete space, it is complete. Hence {xn} is convergent and converse to a point u ∈ C. The rest of the proof is similar to Theorem 2.1 and so we omit the details. � Corollary 2.1. Let C be a non-empty, closed, convex and bounded subset of the normed linear space X and let f : C → C. Suppose that there exists a positive integer r such that f satisfies the contraction condition ‖frx−fry‖≤ a max { ‖x−y‖,‖x−frx‖,‖y −fry‖, 1 2 [‖x−fry‖ + ‖y −frx‖] } + b [ ‖x−frx‖ + ‖y −fry‖ ] + c max { ‖x−fry‖,‖y −frx‖ } (2.7) for all x,y ∈ C, where the real numbers a,b,c ≥ 0, a > 0, satisfy the inequality (2.8) a + b + c ≤ 1 2 . If the sequence {xn} defined by (2.9) xn+1 = (1 − t)xn + tfrxn, n = 0, 1, 2, ...; for some t ∈ (0, 1) and for some x = x0 ∈ C converges to u, then u is a unique fixed point of f. Proof. By Theorem 2.1 above, the mapping fr has a unique fixed point, say p ∈ C. Then we have fr(p) = p. Therefore, fr(fp) = fr+1(p) = f(fr(p)) = fp showing that fp is again a fixed point of fr. By uniqueness of p, we get fp = p. Thus, f has 150 DHAGE a unique fixed point p in C and the sequence of iterations given by (2.9) converges to p. The proof of the theorem is complete. � In the following section we prove that the Mann iterations of the mapping f in a uniformly convex Banach space satisfying (2.1) and (2.2). 3. Convergence of Mann Iterations The following definitions is well-known in the literature. Definition 3.1. A self mapping f of a convex subset C of a Banach space X is said to be quasi-nonexpansive provided f has a fixed point and if p is a fixed point of f, then ‖fx−p‖≤‖x−p‖ for all x ∈ C. In a uniformly Banach space, Senter and Dotson, Jr., have conditions under which the sequence of Mann types of iterates of a quasi-nonexpansive mapping converges to a fixed point of the mapping in question. We denote by F(f) the set of all fixed points of f in C. Condition I: Let C be a convex subset of a uniformly convex Banach space X. A mapping f : C → C is said to satisfy Condition I if there is a nondecreasing function β : [0,∞) → [0,∞) with β(0) = 0, f(r) > 0 for r ∈ (0,∞) satisfying ‖x− fx‖ > β(d(x,F(f))) for all x ∈ C, where β(d(x,F(f))) = inf{ ‖x−p‖ : p ∈ F(f)}. Condition I: Let C be a convex subset of a uniformly convex Banach space X. A mapping f : C → C is said to satisfy Condition I if there is a real number α > 0 such that ‖x−fx‖≥ αd(x,F(f)) for all x ∈ C. It can be easily shown that a mapping which satisfies Condition II also satisfies Condition I. Now, we state a key theorem of Senter and Dotson [9] which is used in what follows. Before going to the theorem we define the Mann iterations of the mapping f on a subset C of the Banach space X. Let x1 ∈ C be arbitrary and let M(x1, tn,f) be a sequence {xn} defined by xn+1 = (1 − tn)xn + tnf(xn), where tn ∈ [β,γ], 0 < β < γ < 1 and n ∈ N. Theorem 3.1 (Senter and Dotson [9]). Let X be a uniformly convex Banach space, C a closed, convex and bounded subset of X and let f be a nonexpansive mapping of C into itself. If f satisfies Condition I, then for arbitrary x1 ∈ C, the sequence M(x1, tn,f) converges to a member of F(f). Below we prove a result concerning the convergence of the sequence of Mann iterations to the fixed point of generalized nonexpansive mappings in a uniformly Banach space. Theorem 3.2. let C be a closed, convex and bounded subset of a uniformly Banach space X and let f : C → C be a generalized nonexpansive mapping satisfying the inequalities (2.1) and (2.2). Then f has a unique fixed point p and for arbitrary x1 ∈ C, the sequence M(x1, tn,f) of Mann iterations converges to p. Proof. By Theorem 2.1, f has a unique fixed point p in C. We show that the sequence M(x1, tn,f) of Mann iterations converges to p for arbitrary x1 ∈ C. This will be achieved in the following two steps: GENERALIZED NONEXPANSIVE MAPPINGS IN BANACH SPACES 151 Step I: f is quasi-nonexpansive on C. We first show that f is a quasi-nonexpansive mapping on C into itself. Assume the contrary, that is, ‖fx−p‖ > ‖x−p‖ for some x ∈ C. Then by (2.1), we have ‖fx−p‖ = ‖fx−fp‖ ≤ a max{‖x−p‖,‖x−fx‖,‖p−fp‖, 1 2 [ ‖x−fp‖ + ‖p−fx‖]} + b [ ‖x−fx‖ + ‖p−fp‖ ] + c max{‖x−fp‖,‖p−fx‖} = a max{‖x−p‖,‖x−fx‖, 1 2 [ ‖x−p‖ + ‖fx−p‖ ] } + b‖x−fx‖ + c max{‖x−p‖,‖fx−p‖} ≤ a max{‖x−p‖,‖x−fx‖,‖fx−p‖} + b‖x−fx‖ + c‖fx−p‖ ≤ a max{‖x−fx‖,‖fx−p‖} + b‖x−fx‖ + c‖fx−p‖.(3.1) Now there are two cases: Case I: Suppose that max{‖x−fx‖,‖fx−p‖} = ‖x−p‖. Then from (3.1), we obtain ‖fx−p‖≤ (a + b + c)‖fx−p‖ which is a contradiction, since a + b = c ≤ 1 2 . Case II: Suppose that max{‖x−fx‖,‖fx−p‖} = ‖x−p‖. Then from (3.1), we obtain ‖fx−p‖≤ (a + b + c)‖x−fx‖ ≤ (a + b + c)[‖x−p‖ + ‖fx−p‖] = (a + b + c)‖x−p‖ + (a + b + c)‖fx−p‖ which further implies that ‖fx−p‖≤ [ a + b + c 1 − (a + b + c) ] ‖x−p‖ which is a contradiction, since a + b + c 1 − (a + b + c) ≤ 1. Thus, in both the cases, we obtain a contradiction. Therefore, we conclude that ‖fx−p‖≤‖x−p‖ for all x ∈ C and consequently f is quasi-nonexpansive on C. Step I: f satisfies Condition II on C. let x ∈ C be arbitrary. Then, (3.2) ‖x−p‖≤‖x−fx‖ + ‖fx−p‖. 152 DHAGE Now, by (2.1), ‖fx−p‖ = ‖fx−fp‖ ≤ a max{‖x−p‖,‖x−fx‖,‖p−fp‖, 1 2 [ ‖x−fp‖ + ‖p−fx‖]} + b [ ‖x−fx‖ + ‖p−fp‖ ] + c max{‖x−fp‖,‖p−fx‖} = a max{‖x−p‖,‖x−fx‖} + b‖x−fx‖ + c‖x−p‖.(3.3) Now there are two cases: Case I: Suppose that max{‖x−fx‖,‖fx−p‖} = ‖x−p‖. Then from (3.1), we obtain ‖fx−p‖≤ (a + c)‖x−p‖ + b‖x−fx‖. Substituting above value in (3.2), we obtain ‖x−fx‖≥ 1 3 ‖x−p‖ = 1 3 d(x,F(f)). Case II: Suppose that max{‖x−fx‖,‖x−p‖} = ‖x−fx‖. Then from (3.1), we obtain ‖fx−p‖≤ (a + b)‖x−fx‖ + c‖x−p‖. Substituting above value in (3.2), we obtain ‖x−fx‖≥ 1 3 ‖x−p‖ = 1 3 d(x,F(f)). Thus, f satisfies Condition II with α = 1 3 . Consequently f satisfies Condition I and by an application of Theorem 3.1, for arbitrary x1 ∈ C, the sequence M(x1, tn,f) of Mann iterations of f converges to p. This completes the proof. � Corollary 3.1. let C be a closed, convex and bounded subset of a uniformly Banach space X and let f : C → C. Suppose that there exists a positive integer r such that f satisfies the generalized contraction condition (2.7) and (2.8). Then f has a unique fixed point p and for arbitrary x1 ∈ C, the sequence M(x1, tn,fr) of Mann iterations converges to p. Proof. By Theorem 2.1, the mapping f has a unique fixed point p in C which is also a unique fixed point of fr. Now the desired conclusion follows by a direct application of Theorem 3.2. � As a consequence of Theorem 3.2, we obtain the following fixed point theorem of Bose and Mukherjee [2] as a corollary. Corollary 3.2. let C be a closed, convex and bounded subset of a uniformly Banach space X and let f : C → C be a generalized nonexpansive mapping satisfying the inequalities (1.6) and (1.7). Then f has a unique fixed point p and for arbitrary x1 ∈ C, the sequence M(x1, tn,f) of Mann iterations converges to p. GENERALIZED NONEXPANSIVE MAPPINGS IN BANACH SPACES 153 References [1] J. Bogin, A generalization of a fixed point theorem of Goebel, Kirk and Shimi, Canad Math. Bull. 19 (1976),7-12. [2] R. K. Bose, R. N. Mukherjee, Approximating fixed points if some mappings, Proc. Amer. Math. Soc. 82 (1981), 603-606. [3] Lj. B. Ciric, On some nonexpansive type mapppings and fixed points, Indian J. Pure Appl. Math., 24 (1993), 145-149. [4] B. C. Dhage, Generalized nonexpansive mappings and fixed points in metric spaces, (Sub- mitted) [5] W. C. Dotson Jr., Fixed points of quasi-nonexpansive mappings, J. Austral. Math. Soc. 13 (1972), 167-170. [6] K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press 1990. [7] R. Kannan, Some reswults on fixed points, Bull. Cal. Math. Soc. 60 (1968), 71-76. [8] W. R. Mann, Mean value methods in iterations, Proc. Amer. Math. Soc. 4 (1953), 506-510. [9] H. F. Senter, W. G. Dotson, Approximating fixed points if nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-379. Kasubai, Gurukul Colony, Ahmedpur-413 515, Dist: Latur, Maharashtra, India