An implicit method for finding a common fixed point of a representation

Hussain et al. Fixed Point Theory and Applications 2014, 2014:238 http://www.fixedpointtheoryandapplications.com/content/2014/1/238

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An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces Nawab Hussain1 , Mahmood Lashkarizadeh Bami2 and Ebrahim Soori3* * Correspondence: [email protected]; [email protected] 3 Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran Full list of author information is available at the end of the article

Abstract We introduce an implicit method for finding an element of the set of common fixed points of a representation of nonexpansive mappings. Then we prove the strong convergence of the proposed implicit scheme to the common fixed point of a representation of nonexpansive mappings. MSC: 90C33; 47H10 Keywords: fixed point; nonexpansive mapping; representation; semigroup; sunny nonexpansive retraction

1 Introduction Let C be a nonempty closed and convex subset of a Banach space E and E∗ be the dual space of E. Let ·, · denote the pairing between E and E∗ . The normalized duality mapping J : E → E∗ is defined by   J(x) = f ∈ E∗ : x, f  = x = f  for all x ∈ E. In the sequel, we use j to denote the single-valued normalized duality mapping. Let U = {x ∈ E : x = }. E is said to be smooth or to have a Gâteaux differentiable norm if the limit x + ty – x t→ t lim

exists for each x, y ∈ U. E is said to have a uniformly Gâteaux differentiable norm if for each y ∈ U, the limit is attained uniformly for all x ∈ U. E is said to be uniformly smooth or is said to have a uniformly Féchet differentiable norm if the limit is attained uniformly for x, y ∈ U. It is known that if the norm of E is uniformly Gâteaux differentiable, then the duality mapping J is single-valued and uniformly norm to weak∗ continuous on each bounded subset of E. A Banach space E is smooth if the duality mapping J of E is singlevalued. We know that if E is smooth, then J is norm to weak-star continuous; for more details, see []. Let C be a nonempty closed and convex subset of a Banach space E. A mapping T of C into itself is called nonexpansive if Tx – Ty ≤ x – y for all x, y ∈ C, and a mapping f is an α-contraction on E if f (x) – f (y) ≤ αx – y, x, y ∈ E such that  ≤ α < . ©2014 Hussain et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Hussain et al. Fixed Point Theory and Applications 2014, 2014:238 http://www.fixedpointtheoryandapplications.com/content/2014/1/238

In this paper, motivated by Lashkarizadeh Bami and Soori [] and Hussain and Takahashi [], we introduce the following general implicit algorithm for finding a common element of the set of fixed points of a representation S = {Tt : t ∈ S} of a semigroup S as nonexpansive mappings from C into itself, with respect to a left regular sequence of means defined on an appropriate subspace of bounded real-valued functions of the semigroup. On the other hand, our goal is to prove that there exists a sunny nonexpansive retraction P of C onto Fix(S ) and x ∈ C such that the following sequence {zn } converges strongly to Px: zn = n f (zn ) + ( – n )Tμn zn

(n ∈ N).

2 Preliminaries Let S be a semigroup. We denote by B(S) the Banach space of all bounded real-valued functions defined on S with supremum norm. For each s ∈ S and f ∈ B(S), we define ls and rs in B(S) by (ls f )(t) = f (st),

(rs f )(t) = f (ts) (t ∈ S).

Let X be a subspace of B(S) containing , and let X ∗ be its topological dual. An element μ of X ∗ is said to be a mean on X if μ = μ() = . We often write μt (f (t)) instead of μ(f ) for μ ∈ X ∗ and f ∈ X. Let X be left invariant (resp. right invariant), i.e., ls (X) ⊂ X (resp. rs (X) ⊂ X) for each s ∈ S. A mean μ on X is said to be left invariant (resp. right invariant) if μ(ls f ) = μ(f ) (resp. μ(rs f ) = μ(f )) for each s ∈ S and f ∈ X. X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. X is amenable if X is both left and right amenable. As is well known, B(S) is amenable when S is a commutative semigroup (see p. of []). A net {μα } of means on X is said to be left regular if   limls∗ μα – μα  =  α

for each s ∈ S, where ls∗ is the adjoint operator of ls . Let f be a function of the semigroup S into a reflexive Banach space E such that the weak closure of {f (t) : t ∈ S} is weakly compact, and let X be a subspace of B(S) containing all the functions t → f (t), x∗  with x∗ ∈ E∗ . We know from [] that for any μ ∈ X ∗ , there exists a unique element fμ in E such that fμ , x∗  = μt f (t), x∗  for all x∗ ∈ E∗ . We denote such fμ   by f (t) dμ(t). Moreover, if μ is a mean on X, then from [], f (t) dμ(t) ∈ co{f (t) : t ∈ S}. Let C be a nonempty closed and convex subset of E. Then a family S = {Ts : s ∈ S} of mappings from C into itself is said to be a representation of S as a nonexpansive mapping on C into itself if S satisfies the following: () Tst x = Ts Tt x for all s, t ∈ S and x ∈ C; () for every s ∈ S, the mapping Ts : C → C is nonexpansive.  We denote by Fix(S ) the set of common fixed points of S , that is, Fix(S ) = s∈S {x ∈ C : Ts x = x}. Theorem . [] Let S be a semigroup, let C be a closed, convex subset of a reflexive Banach space E, S = {Ts : s ∈ S} be a representation of S as a nonexpansive mapping from C into itself such that weak closure of {Tt x : t ∈ S} is weakly compact for each x ∈ C, and let X be a subspace of B(S) such that  ∈ X and the mapping t → T(t)x, x∗  be an element of X for

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 each x ∈ C and x∗ ∈ E, and μ be a mean on X. If we write Tμ x instead of Tt x dμ(t), then the following hold. (i) Tμ is a nonexpansive mapping from C into C. (ii) Tμ x = x for each x ∈ Fix(S ). (iii) Tμ x ∈ co{Tt x : t ∈ S} for each x ∈ C. (iv) If X is rs -invariant for each s ∈ S and μ is right invariant, then Tμ Tt = Tμ for each t ∈ S. Remark From Theorem .. in [], every uniformly convex Banach space is strictly convex and reflexive. Let D be a subset of B, where B is a subset of a Banach space E, and let P be a retraction of B onto D, that is, Px = x for each x ∈ D. Then P is said to be sunny if for each x ∈ B and t ≥  with Px + t(x – Px) ∈ B, P(Px + t(x – Px)) = Px. A subset D of B is said to be a sunny nonexpansive retract of B if there exists a sunny nonexpansive retraction P of B onto D. We know that if E is smooth and P is a retraction of B onto D, then P is sunny and nonexpansive if and only if for each x ∈ B and z ∈ D, x – Px, J(z – Px) ≤ . For more details, see []. Lemma . [] Let S be a semigroup, and let C be a compact convex subset of a real strictly convex and smooth Banach space E. Suppose that S = {Ts : s ∈ S} is a representation of S as a nonexpansive mapping from C into itself. Let X be a left invariant subspace of B(S) such that  ∈ X, and the function t → Tt x, x∗  is an element of X for each x ∈ C and x∗ ∈ E∗ . If μ is a left invariant mean on X, then Fix(Tμ ) = Tμ C = Fix(S ) and there exists a unique sunny nonexpansive retraction from C onto Fix(S ). Throughout the rest of this paper, the open ball of radius r centered at  is denoted by Br . Let C be a nonempty closed convex subset of a Banach space E. For  >  and a mapping T : C → C, we let F (T) be the set of -approximate fixed points of T, i.e., F (T) = {x ∈ C : x – Tx ≤ }.

3 Main result In this section, we deal with a strong convergence approximation scheme for finding a common element of the set of common fixed points of a representation of nonexpansive mappings. Theorem . Let S be a semigroup. Let C be a nonempty compact convex subset of a real strictly convex and reflexive and smooth Banach space E. Suppose that S = {Ts : s ∈ S} is a representation of S as a nonexpansive mapping from C into itself such that Fix(S ) = ∅. Let X be a left invariant subspace of B(S) such that  ∈ X, and the function t → Tt x, x∗  is an element of X for each x ∈ C and x∗ ∈ E∗ . Let {μn } be a left regular sequence of means on X. Suppose that f is an α-contraction on C. Let n be a sequence in (, ) such that limn n = . Then there exists a unique sunny nonexpansive retraction P of C onto Fix(S ) and x ∈ C such that the following sequence {zn } generated by zn = n f (zn ) + ( – n )Tμn zn strongly converges to Px.

(n ∈ N)

()

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Proof By Proposition .. and Theorem .. in [], any compact subset C of a reflexive Banach space E is weakly compact, and from Proposition .. in [], any closed convex subset of a weakly compact subset C of a Banach space E is itself weakly compact, and by Proposition .. in [], any convex subset C of a normed space E is weakly closed if and only if C is closed. Therefore, weak closure of {Tt x : t ∈ S} is weakly compact for each x ∈ C. We divide the proof into five steps. Step . The existence of zn which satisfies (). This follows immediately from the fact that for every n ∈ N, the mapping Nn given by Nn x := n f (x) + ( – n )Tμn x (x ∈ C) is a contraction. To see this, put βn = ( + n (α – )), then  ≤ βn <  (n ∈ N). Then we have   Nn x – Nn y ≤ n f (x) – f (y) + ( – n )Tμn x – Tμn y ≤ n αx – y + ( – n )x – y   =  + n (α – ) x – y = βn x – y. Therefore, by the Banach contraction principle [], there exists a unique point zn ∈ C such that Nn zn = zn . Step . limn→∞ zn – Tt zn  =  for all t ∈ S. Consider t ∈ S and let  > . By Lemma  in [], there exists δ >  such that co Fδ (Tt ) + Bδ ⊆ F (Tt ). By Corollary . in [], there also exists a natural number N such that 

 N N       Tti s y – Tt Tti s y  ≤ δ  N +   N + i=

()

i=

for all s ∈ S and y ∈ C. Let p ∈ Fix(S ) and M be a positive number such that supy∈C y ≤ M . Let t ∈ S, since {μn } is strongly left regular, there exists N ∈ N such that μn –lt∗i μn  ≤ δ for n ≥ N and i = , , . . . , N . Then we have (M )   N      Tti s y dμn (s) supTμn y –   N +  y∈C i=

  N



 

∗ ∗ = sup sup Tμn y, x – Tti s y dμn (s), x

N +  i= y∈C x∗ =

N N

     

∗ ∗ (μn )s Ts y, x – (μn )s Tti s y, x = sup sup

N +  i= y∈C x∗ = N +  i= N

      sup sup (μn )s Ts y, x∗ – lt∗i μn s Ts y, x∗ ∗ N +  i= y∈C x =    ≤ max μn – lt∗i μn  M + p

≤

i=,,...,N

  ≤ max μn – lt∗i μn (M ) i=,,...,N

≤δ

(n ≥ N ).

()

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By Theorem . we have

  N N   T i y dμn (s) ∈ co T i (Ts y) : s ∈ S . N +  i= t s N +  i= t

()

It follows from ()-() that 

 N  T i y : s ∈ S + Bδ Tμn y ∈ co N +  i= t s ⊂ co Fδ (Tt ) + Bδ ⊂ F (Tt ) for all y ∈ C and n ≥ N . Therefore, lim supn→∞ supy∈C Tt (Tμn y) – Tμn y ≤ . Since  >  is arbitrary, we have   lim sup supTt (Tμn y) – Tμn y = . n→∞

()

y∈C

Let t ∈ S and  > , then there exists δ >  which satisfies (). Take L = ( + α)M + f (p) – p. Now, from the condition limn n =  and from (), there exists a natural number N such that Tμn y ∈ Fδ (Tt ) for all y ∈ C and n < Lδ  for all n ≥ N . Since p ∈ Fix(S ), we have        n f (zn ) – Tμn zn  ≤ n f (zn ) – f (p) + f (p) – p + Tμn p – Tμn zn      ≤ n αzn – p + f (p) – p + Azn – p     ≤ n αzn – p + f (p) – p + zn – p    ≤ n ( + α)zn – p + f (p) – p    ≤ n ( + α)M + f (p) – p = n L ≤

δ 

for all n ≥ N . Observe that zn = n f (zn ) + ( – n )Tμn zn   = Tμn zn + n f (zn ) – Tμn zn ∈ Fδ (Tt ) + B δ



⊆ Fδ (Tt ) + Bδ ⊆ F (Tt ) for all n ≥ N . This shows that zn – Tt zn  ≤ 

(n ≥ N ).

Since  >  is arbitrary, we get limn→∞ zn – Tt zn  = .

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Step . S{zn } ⊂ Fix(S ), where S{zn } denotes the set of strongly limit points of {zn }. Let z ∈ S{zn }, and let {znj } be a subsequence of {zn } such that znj → z, Tt z – z ≤ Tt z – Tt znj  + Tt znj – znj  + znj – z ≤ znj – z + Tt znj – znj , then by Step , Tt z – z ≤  lim znj – z + lim Tt znj – znj  = , j

j

therefore z ∈ Fix(S ). Step . There exists a unique sunny nonexpansive retraction P of C onto Fix(S ) and x ∈ C such that    := lim sup x – Px, J(zn – Px) ≤ .

()

n

By Lemma . there exists a unique sunny nonexpansive retraction P of C onto Fix(S ). The Banach contraction mapping principle guarantees that fP has a unique fixed point x ∈ C. We show that    := lim sup x – Px, J(zn – Px) ≤ . n

Note that from the definition of  and the fact that C is a compact subset of E, we can select a subsequence {znj } of {zn } with the following properties: (i) limj x – Px, J(znj – Px) = ; (ii) {znj } converges strongly to a point z. By Step , we have z ∈ Fix(S ). Since E is smooth, we have      = lim x – Px, J(znj – Px) = x – Px, J(z – Px) ≤ . j

Since fPx = x, we have (f – I)Px = x – Px. From Theorem ..(v) in [], for x, y ∈ E and f ∈ J(y), x – y ≥ (x – y, f ). Therefore, for each n ∈ N, we have n (α – )zn – Px   ≥ n αzn – Px + ( – n )zn – Px – zn – Px     ≥ n f (zn ) – f (Px) + ( – n )Tμn zn – Px – zn – Px     ≥  n f (zn ) – f (Px) + ( – n )(Tμn zn – Px) – (zn – Px), J(zn – Px)   = –n (f – I)Px, J(zn – Px)   = –n x – Px, J(zn – Px) . Hence zn – Px ≤

   x – Px, J(zn – Px) . –α

()

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Step . {zn } strongly converges to Px. Indeed, from (), () and Px ∈ Fix(S ), we conclude lim sup zn – Px ≤ n

   lim sup x – Px, J(zn – Px) ≤ . –α n

That is, zn → Px.



Remark . It would be an interesting problem to prove Theorem . for continuous representations instead of nonexpansive.

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 2 Department of Mathematics, University of Isfahan, Isfahan, Iran. 3 Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran. Acknowledgements This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first author acknowledges with thanks DSR, KAU for financial support. The authors would like to thank the referee of the paper for his helpful comments and invaluable suggestions. This research was supported by the Center of Excellence for Mathematics and the Office of Graduate Studies of the Lorestan University and the University of Isfahan. Received: 31 August 2014 Accepted: 12 November 2014 Published: 04 Dec 2014 References 1. Takahashi, W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000) 2. Lashkarizadeh Bami, M, Soori, E: Strong convergence of a general implicit algorithm for variational inequality problems and equilibrium problems and a continuous representation of nonexpansive mappings. Bull. Iran. Math. Soc. 40, 977-1001 (2014) 3. Hussain, N, Takahashi, W: Weak and strong convergence theorems for semigroups of mappings without continuity in Hilbert spaces. J. Nonlinear Convex Anal. 14(4), 769-783 (2013) 4. Hirano, N, Kido, K, Takahashi, W: Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces. Nonlinear Anal. 12, 1269-1281 (1988) 5. Kido, K, Takahashi, W: Mean ergodic theorems for semigroups of linear operators. J. Math. Anal. Appl. 103, 387-394 (1984) 6. Saeidi, S: Existence of ergodic retractions for semigroups in Banach spaces. Nonlinear Anal. 69, 3417-3422 (2008) 7. Saeidi, S: A note on ‘Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces’. Nonlinear Anal. 72, 546-548 (2010) 8. Agarwal, RP, Oregan, D, Sahu, DR: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Topological Fixed Point Theory and Its Applications, vol. 6. Springer, New York (2009) 9. Shioji, N, Takahashi, W: Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces. J. Approx. Theory 97, 53-64 (1999) 10. Atsushiba, S, Takahashi, W: A nonlinear ergodic theorem for nonexpansive mappings with compact domain. Math. Jpn. 52, 183-195 (2000)

10.1186/1687-1812-2014-238 Cite this article as: Hussain et al.: An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces. Fixed Point Theory and Applications 2014, 2014:238