Fixed point theorem for α -nonexpansive mappings in Banach spaces

Nonlinear Analysis 74 (2011) 4387–4391

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Fixed point theorem for α -nonexpansive mappings in Banach spaces Koji Aoyama a , Fumiaki Kohsaka b,∗ a

Department of Economics, Chiba University, Yayoi-cho, Inage-ku, Chiba-shi, Chiba 263-8522, Japan

b

Department of Computer Science and Intelligent Systems, Oita University, Dannoharu, Oita-shi, Oita 870-1192, Japan

article

info

Article history: Received 24 August 2010 Accepted 29 March 2011 Communicated by Ravi Agarwal MSC: primary 47H06 secondary 47H09 47H10

abstract We introduce the class of α -nonexpansive mappings in Banach spaces. This class contains the class of nonexpansive mappings and is related to the class of firmly nonexpansive mappings in Banach spaces. In addition, we obtain a fixed point theorem for α nonexpansive mappings in uniformly convex Banach spaces. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Firmly nonexpansive mapping Fixed point theorem Nonexpansive mapping Nonspreading mapping Hybrid mapping

1. Introduction Let E be a Banach space and let C be a nonempty subset of E. We denote the fixed point set of a mapping T : C → E by F (T ). Following Bruck [1], we say that a mapping T : C → E is firmly nonexpansive if

‖Tx − Ty‖ ≤ ‖r (x − y) + (1 − r )(Tx − Ty)‖ for all r > 0 and x, y ∈ C ; see also [2]. It is obvious that every firmly nonexpansive mapping is nonexpansive, i.e., ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ C . See, for example, [2–7] for more information on firmly nonexpansive mappings. In 2008, Kohsaka and Takahashi [8] studied the existence and approximation of fixed points of mappings of firmly nonexpansive type in Banach spaces. This is another generalization of firmly nonexpansive mappings in Hilbert spaces; see also [8–11]. Kohsaka and Takahashi [10] also introduced the class of nonspreading mappings in Banach spaces, which is wider than the class of mappings of firmly nonexpansive type, and obtained some fixed point theorems for these mappings. Takahashi [12] introduced the class of hybrid mappings in Hilbert spaces, which contains the class of firmly nonexpansive mappings in Hilbert spaces. Recently, Aoyama et al. [13] introduced the class of λ-hybrid mappings in the Hilbert space setting (see Definition 2.1) and obtained a fixed point theorem and an ergodic theorem for these mappings. This class contains the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings in Hilbert spaces. In this paper, we introduce the class of α -nonexpansive mappings in Banach spaces (see Definition 2.2) and generalize some of the results obtained in [13] from Hilbert spaces to more general Banach spaces. It should be noted that every firmly

∗

Corresponding author. E-mail addresses: [email protected] (K. Aoyama), [email protected] (F. Kohsaka).

0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.03.057

4388

K. Aoyama, F. Kohsaka / Nonlinear Analysis 74 (2011) 4387–4391

nonexpansive mapping is α -nonexpansive for all real number α such that 0 ≤ α ≤ 1/2 and every nonexpansive mapping is 0-nonexpansive. In Section 3, we will prove the following fixed point theorem. Theorem 1.1. Let E be a uniformly convex Banach space, let C be a nonempty, closed, and convex subset of E, and let T : C → C be an α -nonexpansive mapping for some real number α such that α < 1. Then F (T ) is nonempty if and only if there exists x ∈ C such that {T n x} is bounded. 2. Preliminaries Throughout this paper, every linear space is real. For a Banach space E, we denote its conjugate space by E ∗ . The norms of E and E ∗ are denoted by ‖·‖. Strong convergence of a sequence {xn } in E to x ∈ E is denoted by xn → x. The sets of real numbers and positive integers are denoted by R and N, respectively. For a Banach space E, we denote the unit sphere and the closed unit ball centered at the origin of E by SE and BE , respectively. We also denote the closed ball with radius r > 0 centered at the origin of E by rBE . A Banach space E is said to be uniformly convex if for each ε ∈ (0, 2], there exists δ > 0 such that ‖(x + y)/2‖ ≤ 1 − δ whenever x, y ∈ SE and ‖x − y‖ ≥ ε . Every uniformly convex Banach space is reflexive (cf. [14]). We know that E is uniformly convex if and only if ‖·‖2 is uniformly convex on bounded convex sets, i.e., for each r > 0 and ε ∈ (0, 2r ], there exists δ > 0 such that

‖tx + (1 − t )y‖2 ≤ t ‖x‖2 + (1 − t ) ‖y‖2 − t (1 − t )δ for all t ∈ (0, 1) and for all x, y ∈ rBE with ‖x − y‖ ≥ ε (cf. [15]). A function g of a nonempty subset C of a Banach space E into R is said to be coercive if g (zn ) → ∞ whenever {zn } is a sequence in C such that ‖zn ‖ → ∞. Let ℓ∞ denote the Banach space of bounded real sequences with the supremum norm. It is known that there exists a bounded linear functional µ on ℓ∞ such that the following three conditions hold: (1) If {tn } ∈ ℓ∞ and tn ≥ 0 for every n ∈ N, then µ({tn }) ≥ 0; (2) if tn = 1 for every n ∈ N, then µ({tn }) = 1; (3) µ({tn+1 }) = µ({tn }) for all {tn } ∈ ℓ∞ . Such a functional µ is called a Banach limit and the value of µ at {tn } ∈ ℓ∞ is denoted by µn tn (cf. [14]). We know the following result. Proposition 2.1 (cf. [14]). Let E be a reflexive Banach space, let C be a nonempty, closed, and convex subset of E, and let g: C → R be a convex, continuous, and coercive function. Then there exists u ∈ C such that g (u) = inf g (C ). Recall the definition of λ-hybrid mappings in Hilbert spaces. Definition 2.1 ([13]). Let H be a Hilbert space, let C be a nonempty subset of H, and let λ be a real number. A mapping T : C → H is said to be λ-hybrid if

‖Tx − Ty‖2 ≤ ‖x − y‖2 + 2(1 − λ) ⟨x − Tx, y − Ty⟩ for all x, y ∈ C . We give the definition of α -nonexpansive mappings. Definition 2.2. Let E be a Banach space, let C be a nonempty subset of E, and let α be a real number such that α < 1. A mapping T : C → E is said to be α -nonexpansive if

‖Tx − Ty‖2 ≤ α ‖Tx − y‖2 + α ‖Ty − x‖2 + (1 − 2α) ‖x − y‖2 for all x, y ∈ C . We can show the following propositions. Proposition 2.2. Let H be a Hilbert space, let C be a nonempty subset of H, and let T : C → H be a mapping. Let λ be a real number such that λ < 2 and put α = (1 − λ)/(2 − λ). Then T is λ-hybrid if and only if T is α -nonexpansive. Proof. Let x, y ∈ C be given. Then we have

‖x − y‖2 + 2(1 − λ) ⟨x − Tx, y − Ty⟩ − ‖Tx − Ty‖2   = ‖x − y‖2 + (1 − λ) ‖x − Ty‖2 + ‖Tx − y‖2 − ‖x − y‖2 − ‖Tx − Ty‖2 − ‖Tx − Ty‖2     = (2 − λ) α ‖x − Ty‖2 + ‖Tx − y‖2 + (1 − 2α)‖x − y‖2 − ‖Tx − Ty‖2 . Since 2 − λ > 0, the result follows.



K. Aoyama, F. Kohsaka / Nonlinear Analysis 74 (2011) 4387–4391

4389

Proposition 2.3. Let E be a Banach space, let C be a nonempty subset of E, and let T : C → E be a firmly nonexpansive mapping. Then T is α -nonexpansive for all real number α such that 0 ≤ α ≤ 1/2. Proof. Let α ∈ [0, 1/2] and let x, y ∈ C be given. Note that 1 − α ≥ 1 − 2α ≥ 0. Since T is firmly nonexpansive, we have

‖Tx − Ty‖ ≤ ‖(1 − α)(x − y) + α(Tx − Ty)‖ = ‖α(Tx − y) + α(x − Ty) + (1 − 2α)(x − y)‖ ≤ α ‖Tx − y‖ + α ‖Ty − x‖ + (1 − 2α) ‖x − y‖ . Since the mapping [0, ∞) ∋ t → t 2 is increasing and convex, we know that T is α -nonexpansive.



The following example shows that there is a discontinuous α -nonexpansive mapping. Example 2.4. Let E be a Banach space and let S , T : E → E be firmly nonexpansive mappings such that S (E ) and T (√ E ) are contained by rBE for some positive real number r. Let α and δ be real numbers such that 0 < α ≤ 1/2 and δ ≥ (1 + 2/ α)r. Then the mapping U: E → E defined by

 Ux =

Sx Tx

(x ∈ δ BE ); (otherwise)

is α -nonexpansive. Proof. Note that α(δ − r )2 ≥ 4r 2 . Let x, y ∈ E be given. It follows from Proposition 2.3 that S and T are α -nonexpansive. Thus, in the case when either x, y ∈ δ BE or x, y ∈ E \ δ BE , it obviously holds that

‖Ux − Uy‖2 ≤ α ‖Ux − y‖2 + α ‖Uy − x‖2 + (1 − 2α) ‖x − y‖2 . In the case when x ∈ δ BE and y ∈ E \ δ BE , we have

α ‖Ux − y‖2 + α ‖Uy − x‖2 + (1 − 2α) ‖x − y‖2 ≥ α ‖Ux − y‖2 = α ‖Sx − y‖2 ≥ α (‖y‖ − ‖Sx‖)2 ≥ α (δ − r )2 and 4r 2 ≥ (‖Sx‖ + ‖Ty‖)2 ≥ ‖Sx − Ty‖2 = ‖Ux − Uy‖2 . Thus U is α -nonexpansive.



3. The proof of Theorem 1.1 Theorem 1.1 is a direct consequence of the following two lemmas. Lemma 3.1. Let E be a Banach space, let C be a nonempty subset of E, and let T : C → C be an α -nonexpansive mapping for some real number α such that α < 1. Suppose that {T n x} is bounded for some x ∈ C . Then µn ‖T n x − Ty‖2 ≤ µn ‖T n x − y‖2 for all Banach limits µ and for all y ∈ C . Proof. Let µ be a Banach limit and let y ∈ C be given. Since T is α -nonexpansive, we have

 n+1        T x − Ty2 ≤ α T n+1 x − y2 + α Ty − T n x2 + (1 − 2α) T n x − y2 for all n ∈ N. Since µ is a Banach limit, we have

 2  2  2  2 µn T n x − Ty ≤ αµn T n x − y + αµn Ty − T n x + (1 − 2α)µn T n x − y and hence (1 − α)µn ‖T n x − Ty‖2 ≤ (1 − α)µn ‖T n x − y‖2 . Since 1 − α > 0, the result follows.



Lemma 3.2. Let E be a uniformly convex Banach space, let C be a nonempty, closed, and convex subset of E, and let T : C → C be a mapping. Suppose that there exist x ∈ C and a Banach limit µ such that {T n x} is bounded and

 2  2 µn T n x − Ty ≤ µn T n x − y for all y ∈ C . Then T has a fixed point.

4390

K. Aoyama, F. Kohsaka / Nonlinear Analysis 74 (2011) 4387–4391

Proof. Let g: C → R be the function defined by g (y) = µn ‖T n x − y‖2 for all y ∈ C . Now we assert that g is a convex, continuous, and coercive function. In fact, the convexity of g is obvious. We show that g is continuous. Let {ym } be a sequence in C such that ym → y. Then we have

 2  2           n  T x − ym  − T n x − y  = T n x − ym  − T n x − y T n x − ym  + T n x − y      ≤ ‖ym − y‖ sup T n x − ym  + T n x − y : m, n ∈ N for all m, n ∈ N. This shows that h: C → ℓ∞ defined by





2 2 h(z ) = T 1 x − z  , T 2 x − z  , . . .







for all z ∈ C is continuous. Thus g = µ ◦ h is also continuous. We next show that g is coercive. If {zm } is a sequence in C such that ‖zm ‖ → ∞, then we have

      n    T x − zm 2 ≥ ‖zm ‖ − T n x 2 ≥ ‖zm ‖ ‖zm ‖ − 2 sup T n x n∈N

and hence g (zm ) → ∞. It follows from Proposition 2.1 that there exists u ∈ C such that g (u) = inf g (C ). Since E is uniformly convex, such a point u is unique. The proof of the uniqueness of such a point u can be found in [16]. For the sake of completeness, we give the proof. Suppose that there exist u1 , u2 ∈ C such that u1 ̸= u2 and g (u1 ) = g (u2 ) = inf g (C ). Since ‖·‖2 is uniformly convex on bounded convex sets, for ε = ‖u1 − u2 ‖ > 0, we have δ > 0 such that

2       1 n  (T x − u1 ) + 1 (T n x − u2 ) ≤ 1 T n x − u1 2 + 1 T n x − u2 2 − δ  2 2 2 2   for all n ∈ N. This implies that g (u1 + u2 )/2 ≤ inf g (C ) − δ . On the other hand, since (u1 + u2 )/2 ∈ C , we have   inf g (C ) ≤ g (u1 + u2 )/2 . This is a contradiction. Hence there exists a unique u ∈ C such that g (u) = inf g (C ). By the assumptions on T , we also know that Tu ∈ C and g (Tu) ≤ g (u). Therefore u must be a fixed point of T .  4. Consequences of Theorem 1.1 Theorem 1.1 immediately implies the following corollaries. Corollary 4.1 (cf. [3,4,14,17]). Let E be a uniformly convex Banach space, let C be a nonempty, closed, and convex subset of E, and let T : C → C be a nonexpansive mapping. Then F (T ) is nonempty if and only if there exists x ∈ C such that {T n x} is bounded. Proof. Since T is 0-nonexpansive, the result follows from Theorem 1.1.



Corollary 4.2. Let E be a uniformly convex Banach space, let C be a nonempty, closed, and convex subset of E, and let T : C → C be a mapping such that 2 ‖Tx − Ty‖2 ≤ ‖Tx − y‖2 + ‖Ty − x‖2 for all x, y ∈ C . Then F (T ) is nonempty if and only if there exists x ∈ C such that {T n x} is bounded. Proof. Since T is 1/2-nonexpansive, the result follows from Theorem 1.1.



Corollary 4.3. Let E be a uniformly convex Banach space, let C a nonempty, closed, and convex subset of E, and let T : C → C be a mapping such that 3 ‖Tx − Ty‖2 ≤ ‖Tx − y‖2 + ‖Ty − x‖2 + ‖x − y‖2 for all x, y ∈ C . Then F (T ) is nonempty if and only if there exists x ∈ C such that {T n x} is bounded. Proof. Since T is 1/3-nonexpansive, the result follows from Theorem 1.1.



Corollary 4.4 ([13]). Let H be a Hilbert space, let C be a nonempty, closed, and convex subset of H, and let T : C → C be a λ-hybrid mapping for some real number λ such that λ < 2. Then F (T ) is nonempty if and only if there exists x ∈ C such that {T n x} is bounded. Proof. It follows from Proposition 2.2 that T is α -nonexpansive, where α = (1 − λ)/(2 − λ). Note that α < 1. Thus the result follows from Theorem 1.1.  We end the present paper with the following remarks.

K. Aoyama, F. Kohsaka / Nonlinear Analysis 74 (2011) 4387–4391

4391

Remark 4.1. (1) Corollary 4.2 in the Hilbert space setting is also a corollary of the fixed point theorem obtained in [10] for nonspreading mappings in Banach spaces. (2) Corollary 4.3 is a generalization of the fixed point theorem obtained in [12] for hybrid mappings in Hilbert spaces. (3) The assumption on λ in Corollary 4.4 can be replaced with λ ∈ R; see [13] for more details. Acknowledgment The authors would like to express their sincere appreciation to the anonymous referee for several helpful comments on the original version of the manuscript. References [1] R.E. Bruck Jr., Nonexpansive projections on subsets of Banach spaces, Pacific J. Math. 47 (1973) 341–355. [2] R.E. Bruck, S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math. 3 (1977) 459–470. [3] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, in: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. [4] K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, in: Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker Inc., New York, 1984. [5] S. Reich, Extension problems for accretive sets in Banach spaces, J. Funct. Anal. 26 (1977) 378–395. [6] S. Reich, I. Shafrir, The asymptotic behavior of firmly nonexpansive mappings, Proc. Amer. Math. Soc. 101 (1987) 246–250. [7] R. Smarzewski, On firmly nonexpansive mappings, Proc. Amer. Math. Soc. 113 (1991) 723–725. [8] F. Kohsaka, W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19 (2008) 824–835. [9] K. Aoyama, F. Kohsaka, W. Takahashi, Three generalizations of firmly nonexpansive mappings: Their relations and continuity properties, J. Nonlinear Convex Anal. 10 (2009) 131–147. [10] F. Kohsaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel) 91 (2008) 166–177. [11] F. Kohsaka, W. Takahashi, Strongly convergent net given by a fixed point theorem for firmly nonexpansive type mappings, Appl. Math. Comput. 202 (2008) 760–765. [12] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010) 79–88. [13] K. Aoyama, S. Iemoto, F. Kohsaka, W. Takahashi, Fixed point and ergodic theorems for λ-hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 11 (2010) 335–343. [14] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000. [15] C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co. Inc., River Edge, NJ, 2002. [16] S. Reich, Nonlinear semigroups, holomorphic mappings, and integral equations, in: Nonlinear Functional Analysis and its Applications, Part 2, Berkeley, Calif., 1983, in: Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc, Providence, RI, 1986, pp. 307–324. [17] F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (1965) 1041–1044.