Fixed Point Approximation of Generalized Nonexpansive Mappings in Hyperbolic Spaces

1Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 631-701, Republic of Korea 2Department of Applied Mathematics, National Institute of Technology, Raipur 492001, India 3Department of Applied Mathematics, Shri Shankaracharya Groups of Institution, Junwani, Bhilai 490020, India 4Department of Applied Mathematics, Sant Guru Ghasidas, Government P. G. College, Kurud, Dhamtari 493663, India

We know that there exist many generalizations of nonexpansive and quasi-nonexpansive mappings.Garcia-Falset et al. [1] introduced two generalizations of nonexpansive mappings which in turn include Suzuki generalized nonexpansive mappings (see [2]).
The following example shows that the class of mappings satisfying the conditions () and (  ), for some  ∈ (0, 1), is larger than the class of mappings satisfying the condition ().

International Journal of Mathematics and Mathematical Sciences
The basic properties and details of CAT(0) spaces can be found in the literature [3][4][5].In [6], Lim introduced a concept of convergence in a general metric space which is called "Δ-convergence." In 2008, Kirk and Panyanak [7] specialized Lim's concept to CAT(0) spaces and showed that many Banach space results involving weak convergence have precise analogs in this setting.Since then, the existence problem and the Δ-convergence problem of iterative sequences to a fixed point for various classes of nonexpansive mappings in the frame work of CAT(0) spaces have been rapidly developed (see [1,[8][9][10][11][12]).
In [13], Leustean proved that CAT(0) spaces are uniformly convex hyperbolic spaces with modulus of uniform convexity (, ) =  2 /8 quadratic in .Thus, the class of uniformly convex hyperbolic spaces are a natural generalization of both uniformly convex Banach spaces and CAT(0) spaces.
Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [14].It is noted that they are different from Gromov hyperbolic spaces [15] or from other notions of hyperbolic spaces that can be found in literature (see [16][17][18][19]).
A hyperbolic space (, , ) is a metric space (, ) together with a convexity mapping  :  2 × [0, 1] →  satisfying A metric space is said to be a convex metric space in the sense of Takahashi [20], where a triple (, , ) satisfy only ( 1 ) (see [21][22][23]).We get the notion of the space of hyperbolic type in the sense of Goebel and Kirk [16], where a triple (, , ) satisfies ( 1 )-( 3 ).The ( 4 ) was already considered by Itoh [24] under the name of "condition III" and it is used by Reich and Shafrir [18] and Kirk [17] to define their notions of hyperbolic spaces.
The class of hyperbolic spaces include normed spaces and convex subsets thereof, the Hilbert space ball equipped with the hyperbolic metric [25], Hadrmard manifold, and the CAT(0) spaces in the sense of Gromov (see [15]).
Let  be a convex subset of a linear space  and let  be a mapping from  into itself.Then the iterative sequence {  } generated from  1 ∈  and defined by where {  } and {  } are sequences in (0, 1) satisfying the certain condition.It is observed that rate of convergence of S-iteration process is similar to the Picard iteration process but faster than the Mann iteration process for contraction mapping (see [26,27]).The purpose of this paper is to prove the strong and Δconvergence theorems for generalized nonexpansive mappings in uniformly convex hyperbolic spaces by using Siteration process.Our results can be viewed as extension and generalization of several well-known results in Banach spaces as well as CAT(0) spaces [10-12, 28, 29].

Preliminaries
Let  be a nonempty subset of metric space  and let {  } be any bounded sequence in .Consider a continuous functional   (⋅, {  }) :  → R + defined by Then, the infimum of   (⋅, {  }) over  is said to be the asymptotic radius of {  } with respect to  and is denoted by   (, {  }).
A point  ∈  is said to be an asymptotic center of the sequence {  } with respect to  if the set of all asymptotic centers of {  } with respect to  is denoted by (, {  }).This set may be empty or a singleton or contain infinitely many points.
It is known that every bounded sequence has a unique asymptotic center with respect to each closed convex subset in uniformly convex Banach spaces and even CAT(0) spaces.
The following lemma is due to Leus ¸tean [30] and ensures that this property also holds in a complete uniformly convex hyperbolic space.Lemma 4 (see [30]).Let (, , ) be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity .Then every bounded sequence {  } in  has a unique asymptotic center with respect to any nonempty closed convex subset  of .
Recall that a sequence {  } in  is said to be Δ-convergent to  ∈ , if  is the unique asymptotic center of {  } for every subsequence {  } of {  }.In this case, we write Δ-lim    =  and call  the Δ-limit of {  }.

Main Results
We begin with the definition of Fejér monotone sequences.
Definition 6.Let  be a nonempty subset of hyperbolic space  and let {  } be a sequence in .Then {  } is said to be Fejér monotone with respect to  if for all  ∈  and  ∈ N Example 7. Let  be a nonempty subset of hyperbolic space  and let  :  →  be a quasi-nonexpansive (in particular, nonexpansive) mapping such that () ̸ = 0. Then the sequence {  } of Picard iteration is Fejér monotone with respect to ().
We can easily prove the following proposition.Proposition 8. Let {  } be a sequence in  and let  be a nonempty subset of .Suppose that {  } is Fejér monotone with respect to .Then we have the following: (1) {  } is bounded; (2) the sequence {(  , )} is decreasing and convergent for all  ∈ ().
We now define S-iteration process in hyperbolic spaces: Let  be a nonempty closed convex subset of a hyperbolic space  and let  be a mapping of  into itself.For any  1 ∈  the sequence {  } of S-iteration process is defined as where {  } and {  } are real sequences with 0 <  ≤   ,   ≤  < 1.

Lemma 9.
Let  be a nonempty closed convex subset of a hyperbolic space  and let  :  →  be a mapping which satisfies the condition (  ) for some  ∈ (0, 1).If {  } is a sequence defined by (11), then {  } is Fejér monotone with respect to ().
Using (11), we have Again, using ( 11) and ( 14 for all  ∈ N. Taking the limit supremum on both sides, we get lim sup International Journal of Mathematics and Mathematical Sciences Similarly, we have lim sup Taking the limit supremum on both sides of ( 14), we have lim sup by using ( 19), (20), and Lemma 5, we get lim Next, we know that Hence, from ( 23) and ( 24), we have Now we observe that which yields that From the estimates of ( 21) and ( 27), we have that lim Thus, from (11), we have lim which gives lim Conversely, suppose that {  } is bounded and lim  → ∞ (  ,   ) = 0. Let (, {  }) = {}.Then, by Lemma 4,  ∈ .As  satisfies the condition (  ) on , there exists  > 1 such that which implies that lim sup By using the uniqueness of asymptotic center,  = , so  is a fixed point of .
Theorem 11.Let  be a nonempty closed convex subset of a complete uniformly convex hyperbolic space  with monotone modulus of uniform convexity  and let  :  →  be a mapping which satisfies conditions (  ) and (), for some  ∈ (0, 1) on  with () ̸ = 0. Then the sequence {  } which is defined by (11), is Δ-convergent to a fixed point of .
Similarly, we can prove that V = V.Thus,  and V are fixed points of .Now we show that  = V.If not, then by the uniqueness of asymptotic center, lim sup which is a contradiction.Hence  = V.Theorem 12. Let  be a nonempty closed convex subset of a complete uniformly convex hyperbolic space  with monotone modulus of uniform convexity  and let  :  →  be a mapping which satisfies conditions (  ) and (), for some  ∈ (0, 1) on  with () ̸ = 0.
By taking the limit on both sides, we obtain lim In view of the uniqueness of the limit, we have  = , so that () is closed.Suppose that lim inf Then, from ( 15) where {  } is in ().By Lemma 9, we have which implies that Then, we know that {  } converges to .In fact, since we have lim Since lim  → ∞ (  , ) exists, the sequence {  } is convergent to .
We recall the definition of condition () due to Senter and Doston [32].
Theorem 13.Let  be a nonempty closed convex subset of a complete uniformly convex hyperbolic space  with monotone modulus of uniform convexity  and let  :  →  be a mapping which satisfies conditions (  ) and (), for some  ∈ (0, 1) on .Moreover,  satisfies condition () with () ̸ = 0. Then the sequence {  } which is defined by (11) converges strongly to some fixed point of .Remark 14.Our Theorems 11, 12, and 13 improve and extend the previous well-known results from Banach spaces and CAT(0) spaces to more general class of uniformly convex hyperbolic spaces (see [10,28,29], in particular, Theorems 3.4 and 3.6 of [12]).In our results, we considered the faster iteration process to approximate the fixed point of underlying mapping  in the framework of uniformly convex hyperbolic spaces.