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Fixed point theorems for N-generalized hybrid mappings in uniformly convex metric spaces

Abstract

In this paper, we prove some fixed point theorems for N-generalized hybrid mappings in both uniformly convex metric spaces and CAT(0) spaces. We also introduce a new iteration method for approximating a fixed point of N-generalized hybrid mappings in CAT(0) spaces and obtain Δ-convergence to a fixed point of N-generalized hybrid mappings in such spaces. Our results improve and extend the corresponding results existing in the literature.

MSC:47H09, 47H10.

1 Introduction and preliminaries

Let C be a nonempty closed subset of a metric space (X,d) and let T be a mapping of C into itself. The set of all fixed points of T is denoted by F(T)={xC:x=Tx}. In 1970, Takahashi [1] introduced the concept of convex metric spaces by using the convex structure as follows.

Definition 1.1 Let (X,d) be a metric space. A mapping W:X×X×[0,1]X is said to be a convex structure on X if for each x,yX and λ[0,1],

d ( z , W ( x , y , λ ) ) λd(z,x)+(1λ)d(z,y)

for all zX. A metric space (X,d) together with a convex structure W is called a convex metric space which will be denoted by (X,d,W).

A nonempty subset C of X is said to be convex if W(x,y,λ)C for all x,yC and λ[0,1]. Clearly, a normed space and each of its convex subsets are convex metric spaces, but the converse does not hold. For each x,yX and λ[0,1], it is known that a convex metric space has the following properties [1, 2]:

  1. (i)

    W(x,x,λ)=x, W(x,y,0)=y and W(x,y,1)=x;

  2. (ii)

    d(x,W(x,y,λ))=(1λ)d(x,y) and d(y,W(x,y,λ))=λd(x,y).

In 1996, Shimizu and Takahashi [3] introduced the concept of uniform convexity in convex metric spaces and studied some properties of these spaces. A convex metric space (X,d,W) is said to be uniformly convex if for any ε>0, there exists δ ε >0 such that for all r>0 and x,y,zX with d(z,x)r, d(z,y)r and d(x,y)rε imply that d(z,W(x,y, 1 2 ))(1 δ ε )r. Obviously, uniformly convex Banach spaces are uniformly convex metric spaces.

Let C be a nonempty closed and convex subset of a convex metric space (X,d,W) and let { x n } be a bounded sequence in X. For xX, we define a mapping r(,{ x n }):X[0,) by

r ( x , { x n } ) = lim sup n d(x, x n ).

Clearly, r(,{ x n }) is a continuous and convex function. The asymptotic radius of { x n } relative to C is given by

r ( C , { x n } ) =inf { r ( x , { x n } ) : x C } ,

and the asymptotic center of { x n } relative to C is the set

A ( C , { x n } ) = { x C : r ( x , { x n } ) = r ( C , { x n } ) } .

It is clear that the asymptotic center A(C,{ x n }) is always closed and convex. It may either be empty or consist of one or many points. The asymptotic center A(C,{ x n }) is singleton for uniformly convex Banach spaces [4, 5] or CAT(0) spaces [6]. The following lemma obtained by Phuengrattana and Suantai [7] is useful for our results.

Lemma 1.2 Let C be a nonempty closed and convex subset of a complete uniformly convex metric space (X,d,W) and let { x n } be a bounded sequence in X. Then A(C,{ x n }) is a singleton set.

One of the special spaces of uniformly convex metric spaces is a CAT(0) space; see [8]. It was noted in [9] that any CAT(κ) space (κ>0) is uniformly convex in a certain sense but it is not a CAT(0) space. Fixed point theory in CAT(0) spaces was first studied by Kirk [9, 10]. He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the fixed point theory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared (e.g., see [1127]).

Let (X,d) be a metric space. A geodesic path joining xX to yX (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]R to X such that c(0)=x, c(l)=y and d(c( t 1 ),c( t 2 ))=| t 1 t 2 | for all t 1 , t 2 [0,l]. In particular, c is an isometry and d(x,y)=l. The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by [x,y]. Write c(α0+(1α)l)=αx(1α)y for α(0,1). The space (X,d) is said to be a geodesic metric space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x,yX. A subset Y of X is said to be convex if Y includes every geodesic segment joining any two of its points.

A geodesic triangle ( x 1 , x 2 , x 3 ) in a geodesic metric space (X,d) consists of three points x 1 , x 2 , x 3 in X (the vertices of ) and a geodesic segment between each pair of vertices (the edges of ). A comparison triangle for the geodesic triangle ( x 1 , x 2 , x 3 ) in (X,d) is a triangle ¯ ( x 1 , x 2 , x 3 ):=( x ¯ 1 , x ¯ 2 , x ¯ 3 ) in the Euclidean plane E 2 such that d E 2 ( x ¯ i , x ¯ j )=d( x i , x j ) for i,j{1,2,3}.

A geodesic metric space is said to be a CAT(0) space if all geodesic triangles satisfy the following comparison axiom: Let be a geodesic triangle in X and let ¯ be a comparison triangle for . Then is said to satisfy the CAT(0) inequality if for all x,y and all comparison points x ¯ , y ¯ ¯ ,

d(x,y) d E 2 ( x ¯ , y ¯ ).

It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples include pre-Hilbert spaces [8], -trees [16], the complex Hilbert ball with a hyperbolic metric [5], and many others.

If z, x, y are points in a CAT(0) space and if m[x,y] is the midpoint of the segment [x,y], then the CAT(0) inequality implies

d ( z , m [ x , y ] ) 2 1 2 d ( z , x ) 2 + 1 2 d ( z , y ) 2 1 4 d ( x , y ) 2 .
(CN)

This is the (CN) inequality of Bruhat and Tits [28], which is equivalent to

for any λ[0,1]. The (CN*) inequality has appeared in [29]. Moreover, if X is a CAT(0) space and x,yX, then for any λ[0,1], there exists a unique point λx(1λ)y[x,y] such that

d ( z , λ x ( 1 λ ) y ) λd(z,x)+(1λ)d(z,y)

for any zX. It follows that CAT(0) spaces have a convex structure W(x,y,λ)=λx(1λ)y.

Remark 1.3

  1. (i)

    By using the (CN) inequality, it is easy to see that CAT(0) spaces are uniformly convex.

  2. (ii)

    A geodesic metric space is a CAT(0) space if and only if it satisfies the (CN) inequality; see [8].

In 2012, Dhompongsa et al. [12] introduced the following notation in CAT(0) spaces: Let x 1 ,, x N be points in a CAT(0) space X and λ 1 ,, λ N (0,1) with i = 1 N λ i =1, we write

i = 1 N λ i x i :=(1 λ N ) ( λ 1 1 λ N x 1 λ 2 1 λ N x 2 λ N 1 1 λ N x N 1 ) λ N x N .
(1.1)

The definition of is an ordered one in the sense that it depends on the order of points x 1 ,, x N . Under (1.1) we obtain that

d ( i = 1 N λ i x i , y ) i = 1 N λ i d( x i ,y)for each yX.

In 1976, Lim [30] introduced the concept of Δ-convergence in a general metric space. Later in 2008, Kirk and Panyanak [15] extended the concept of Lim to a CAT(0) space.

Definition 1.4 [15]

A sequence { x n } in a CAT(0) space X is said to Δ-converge to xX if x is the unique asymptotic center of { u n } for every subsequence { u n } of { x n }. In this case, we write Δ- lim n x n =x and call x the Δ-limit of { x n }.

Lemma 1.5 [15]

Every bounded sequence in a complete CAT(0) space has a Δ-convergent subsequence.

For any nonempty subset C of a CAT(0) space X, let π:= π C be the nearest point projection mapping from X to a subset C of X. In [8], it is known that if C is closed and convex, the mapping π is well defined, nonexpansive, and the following inequality holds:

d ( x , y ) 2 d ( x , π x ) 2 +d ( π x , y ) 2

for all xX and yC. By using the same argument as in [[31], Lemma 3.2], we can prove the following result for nearest point projection mappings in CAT(0) spaces.

Lemma 1.6 Let C be a nonempty closed and convex subset of a complete CAT(0) space X, let π:XC be the nearest point projection mapping, and let { x n } be a sequence in X. If d( x n + 1 ,p)d( x n ,p) for all pC and nN, then {π x n } converges strongly to some element in C.

Proof Let m>n. By the (CN) inequality and the property of π, it follows that

d ( π x m , π x n ) 2 2 d ( x m , π x m ) 2 + 2 d ( x m , π x n ) 2 4 d ( x m , π x m π x n 2 ) 2 2 d ( x m , π x m ) 2 + 2 d ( x m , π x n ) 2 4 d ( x m , π x m ) 2 = 2 d ( x m , π x n ) 2 2 d ( x m , π x m ) 2 2 d ( x n , π x n ) 2 2 d ( x m , π x m ) 2 .
(1.2)

This implies that

d ( x m , π x m ) 2 d ( x n , π x n ) 2 for m>n.

Then lim n d ( x n , π x n ) 2 exists. Letting m,n in (1.2), we have that {π x n } is a Cauchy sequence in a closed subset C of a complete CAT(0) space X, hence it converges to some element in C. □

Let C be a nonempty closed and convex subset of a Hilbert space H. A mapping T:CC is called generalized hybrid if there exist α,βR such that

α T x T y 2 +(1α) x T y 2 β T x y 2 +(1β) x y 2

for all x,yC. We note that the generalized hybrid mappings generalize several well-known mappings. For example, a generalized hybrid mapping is nonexpansive for α=1 and β=0, nonspreading for α=2 and β=1, and hybrid for α= 3 2 and β= 1 2 . In 2010, Kocourek et al. [32] proved the fixed point theorems for generalized hybrid mappings in Hilbert spaces. Later in 2011, Takahashi and Yao [33] extended the results of Kocourek et al. to uniformly convex Banach spaces.

Recently, Maruyama et al. [34] introduced a new nonlinear mapping in a Hilbert space as follows. Let NN. A mapping T:CC is called N-generalized hybrid if there are α 1 ,, α N , β 1 ,, β N R such that

k = 1 N α k T N + 1 k x T y 2 + ( 1 k = 1 N α k ) x T y 2 k = 1 N β k T N + 1 k x y 2 + ( 1 k = 1 N β k ) x y 2

for all x,yC. They obtained the existence and weak convergence theorems for N-generalized hybrid mappings in Hilbert spaces. Hojo et al. [35] also studied the fixed point theorems for N-generalized hybrid mappings in Hilbert spaces and provided an example of N-generalized hybrid mappings which are not generalized hybrid mappings as follows.

Example 1.7 Let H be a Hilbert space, A={xH:x1} and define a mapping T:HH as follows:

Tx={ 0 for all  x A ; x x for all  x A .

We observe that the N-generalized hybrid mappings generalize several well-known mappings, for instance, nonexpansive mappings, nonspreading mappings, hybrid mappings, λ-hybrid mappings, generalized hybrid mappings, and 2-generalized hybrid mappings. Many researchers have studied the fixed point theorems of those mappings in both Hilbert spaces and Banach spaces (e.g., see [32, 33, 3638]). However, no researcher has studied the fixed point theorems for N-generalized hybrid mappings in more general spaces. So, in this paper, we are interested in studying and extending those mappings to both uniformly convex metric spaces and CAT(0) spaces.

2 Fixed point theorems in uniformly convex metric spaces

We first define N-generalized hybrid mappings in convex metric spaces. Let C be a nonempty subset of a convex metric space (X,d,W). Let NN. A mapping T:CC is called N-generalized hybrid if there are α 1 ,, α N , β 1 ,, β N R such that

k = 1 N α k d ( T N + 1 k x , T y ) 2 + ( 1 k = 1 N α k ) d ( x , T y ) 2 k = 1 N β k d ( T N + 1 k x , y ) 2 + ( 1 k = 1 N β k ) d ( x , y ) 2

for all x,yC. Now, we prove a fixed point theorem for N-generalized hybrid mappings in complete uniformly convex metric spaces.

Theorem 2.1 Let C be a nonempty closed and convex subset of a complete uniformly convex metric space (X,d,W) and let T:CC be an N-generalized hybrid mapping with k = 1 N α k (,0][1,) and k = 1 N β k [0,1]. Then T has a fixed point if and only if there exists an xC such that { T n x} is bounded.

Proof The necessity is obvious. Conversely, we assume that there exists an xC such that { T n x} is bounded. We will show that F(T) is nonempty. From Lemma 1.2, A(C,{ T n x}) is a singleton set. Let A(C,{ T n x})={z}. Since T is N-generalized hybrid, there are α 1 ,, α N , β 1 ,, β N R such that

k = 1 N α k d ( T n + N + 1 k x , T z ) 2 + ( 1 k = 1 N α k ) d ( T n x , T z ) 2 k = 1 N β k d ( T n + N + 1 k x , z ) 2 + ( 1 k = 1 N β k ) d ( T n x , z ) 2 .
(2.1)

If k = 1 N α k [1,) and k = 1 N β k [0,1], then (2.1) becomes

k = 1 N α k d ( T n + N + 1 k x , T z ) 2 k = 1 N β k d ( T n + N + 1 k x , z ) 2 + ( 1 k = 1 N β k ) d ( T n x , z ) 2 + ( k = 1 N α k 1 ) d ( T n x , T z ) 2 .

This implies that

lim sup n d ( T n x , T z ) 2 lim sup n d ( T n x , z ) 2 .

If k = 1 N α k (,0] and k = 1 N β k [0,1], then (2.1) becomes

( 1 k = 1 N α k ) d ( T n x , T z ) 2 k = 1 N β k d ( T n + N + 1 k x , z ) 2 + ( 1 k = 1 N β k ) d ( T n x , z ) 2 k = 1 N α k d ( T n + N + 1 k x , T z ) 2 .

This implies again that

lim sup n d ( T n x , T z ) 2 lim sup n d ( T n x , z ) 2 .

Therefore, we have

r ( T z , { T n x } ) r ( z , { T n x } ) .

Since TzC and r(z,{ T n x})=inf{r(y,{ T n x}):yC}, it implies that Tz=z. Hence, F(T) is nonempty. □

As a direct consequence of Theorem 2.1, we obtain a fixed point theorem for N-generalized hybrid mappings in uniformly convex metric spaces as follows.

Theorem 2.2 Let C be a nonempty bounded closed and convex subset of a complete uniformly convex metric space (X,d,W) and let T:CC be an N-generalized hybrid mapping with k = 1 N α k (,0][1,) and k = 1 N β k [0,1]. Then T has a fixed point.

We can show that if T is an N-generalized hybrid mapping and x=Tx, then for any yC, we get

k = 1 N α k d ( x , T y ) 2 + ( 1 k = 1 N α k ) d ( x , T y ) 2 k = 1 N β k d ( x , y ) 2 + ( 1 k = 1 N β k ) d ( x , y ) 2

and hence d(x,Ty)d(x,y). This means that an N-generalized hybrid mapping with a fixed point is quasi-nonexpansive. Then, using the methods of the proof of Theorem 1.3 in [13], we can prove the following.

Corollary 2.3 Let C be a nonempty convex subset of a complete uniformly convex metric space (X,d,W). Suppose that T:CC is an N-generalized hybrid mapping and has a fixed point. Then F(T) is closed and convex.

Remark 2.4

  1. (i)

    Theorems 2.1 and 2.2 extend and generalize the corresponding results in [17, 3234, 3638] to N-generalized hybrid mappings on uniformly convex metric spaces.

  2. (ii)

    In CAT(0) spaces, if we set W(x,y,λ):=λx(1λ)y, then Theorems 2.1 and 2.2 can be applied to these spaces under the assumption that k = 1 N α k (,0][1,) and k = 1 N β k [0,1].

3 Fixed point theorems in CAT(0) spaces

In this section, we study the existence and Δ-convergence theorems for N-generalized hybrid mappings in complete CAT(0) spaces.

We first recall the definition of a Banach limit. Let μ be a continuous linear functional on l , the Banach space of bounded real sequences, and ( a 1 , a 2 ,) l . We write μ n ( a n ) instead of μ(( a 1 , a 2 ,)). We call μ a Banach limit if μ satisfies μ=μ(1,1,)=1 and μ n ( a n )= μ n ( a n + 1 ) for each ( a 1 , a 2 ,) l . For a Banach limit μ, we know that lim inf n a n μ n ( a n ) lim sup n a n for all ( a 1 , a 2 ,) l . So if ( a 1 , a 2 ,) l with lim n a n =c, then μ n ( a n )=c; see [39] for more details.

Now, we obtain the following lemma in CAT(0) spaces.

Lemma 3.1 Let C be a nonempty closed and convex subset of a complete CAT(0) space X, let { x n } be a bounded sequence in X, and let μ be a Banach limit. If a function f:CR is defined by

f(z)= μ n d ( x n , z ) 2 for all zC,

then there exists a unique z 0 C such that

f( z 0 )=min { f ( z ) : z C } .

Proof It is easy to show that f is continuous. By (CN) inequality, we obtain that

f ( x y 2 ) 1 2 f(x)+ 1 2 f(y) 1 4 d ( x , y ) 2 for all x,yC.

This implies by Proposition 1.7 in [40] that there exists a unique z 0 C such that f( z 0 )=min{f(z):zC}. □

By using Lemma 3.1, we can prove the following fixed point theorem for N-generalized hybrid mappings in CAT(0) spaces without the assumptions that k = 1 N α k (,0][1,) and k = 1 N β k [0,1].

Theorem 3.2 Let C be a nonempty closed and convex subset of a complete CAT(0) space X and let T:CC be an N-generalized hybrid mapping. Then T has a fixed point if and only if there exists an xC such that { T n x} is bounded.

Proof The necessity is obvious. Conversely, we assume that there exists an xC such that { T n x} is bounded. Let μ be a Banach limit. Since T is N-generalized hybrid, there are α 1 ,, α N , β 1 ,, β N R such that

k = 1 N α k d ( T n + N + 1 k x , T z ) 2 + ( 1 k = 1 N α k ) d ( T n x , T z ) 2 k = 1 N β k d ( T n + N + 1 k x , z ) 2 + ( 1 k = 1 N β k ) d ( T n x , z ) 2

for any zC and nN{0}. Since { T n x} is bounded, we have

k = 1 N α k μ n d ( T n + N + 1 k x , T z ) 2 + ( 1 k = 1 N α k ) μ n d ( T n x , T z ) 2 k = 1 N β k μ n d ( T n + N + 1 k x , z ) 2 + ( 1 k = 1 N β k ) μ n d ( T n x , z ) 2 .

This implies that

μ n d ( T n x , T z ) 2 μ n d ( T n x , z ) 2

for all zC. It follows by Lemma 3.1 that Tz=z. Hence, F(T) is nonempty. □

As a direct consequence of Theorem 3.2, we obtain a fixed point theorem for N-generalized hybrid mappings in CAT(0) spaces as follows.

Theorem 3.3 Let C be a nonempty bounded closed and convex subset of a complete CAT(0) space X and let T:CC be an N-generalized hybrid mapping. Then T has a fixed point.

Remark 3.4 Theorems 3.2 and 3.3 extend and generalize the corresponding results in [17, 3234, 3638] to N-generalized hybrid mappings on CAT(0) spaces.

Next, we study the Δ-convergence theorem for N-generalized hybrid mappings in CAT(0) spaces. Before proving the theorem, we need the following lemma.

Lemma 3.5 Let C be a nonempty closed and convex subset of a complete CAT(0) space X and let T:CC be an N-generalized hybrid mapping with k = 1 N α k (,0][1,) and k = 1 N β k [0,). If { x n } is a bounded sequence in C with Δ- lim n x n =x and lim n d( x n , T i x n )=0 for all i=1,2,,N, then xF(T).

Proof Since T is an N-generalized hybrid mapping, there are α 1 ,, α N , β 1 ,, β N R such that

k = 1 N α k d ( T N + 1 k x n , T x ) 2 + ( 1 k = 1 N α k ) d ( x n , T x ) 2 k = 1 N β k d ( T N + 1 k x n , x ) 2 + ( 1 k = 1 N β k ) d ( x n , x ) 2 .
(3.1)

Case 1: k = 1 N α k [1,) and k = 1 N β k [0,). It follows by (3.1) that

k = 1 N α k d ( T N + 1 k x n , T x ) 2 k = 1 N β k d ( T N + 1 k x n , x ) 2 + ( 1 k = 1 N β k ) d ( x n , x ) 2 + ( k = 1 N α k 1 ) d ( x n , T x ) 2 k = 1 N β k ( d ( T N + 1 k x n , x n ) 2 + 2 d ( T N + 1 k x n , x n ) d ( x n , x ) + d ( x n , x ) 2 ) + ( 1 k = 1 N β k ) d ( x n , x ) 2 + ( k = 1 N α k 1 ) ( d ( x n , T N + 1 k x n ) 2 + 2 d ( x n , T N + 1 k x n ) d ( T N + 1 k x n , T x ) + d ( T N + 1 k x n , T x ) 2 ) = d ( x n , x ) 2 + ( k = 1 N β k + k = 1 N α k 1 ) d ( T N + 1 k x n , x n ) 2 + 2 k = 1 N β k d ( T N + 1 k x n , x n ) d ( x n , x ) + 2 ( k = 1 N α k 1 ) d ( x n , T N + 1 k x n ) d ( T N + 1 k x n , T x ) + ( k = 1 N α k 1 ) d ( T N + 1 k x n , T x ) 2 .

This implies that

d ( T N + 1 k x n , T x ) 2 d ( x n , x ) 2 + ( k = 1 N β k + k = 1 N α k 1 ) d ( T N + 1 k x n , x n ) 2 + 2 k = 1 N β k d ( T N + 1 k x n , x n ) d ( x n , x ) + 2 ( k = 1 N α k 1 ) d ( x n , T N + 1 k x n ) d ( T N + 1 k x n , T x ) .

Since { x n } is bounded and lim n d( x n , T i x n )=0 for all i=1,2,,N, we have that {T x n },{ T 2 x n },,{ T N x n } are bounded. So, we have

d ( T N + 1 k x n , T x ) 2 d ( x n , x ) 2 + ( k = 1 N β k + k = 1 N α k 1 ) d ( T N + 1 k x n , x n ) 2 + 2 k = 1 N β k d ( T N + 1 k x n , x n ) M + 2 ( k = 1 N α k 1 ) d ( x n , T N + 1 k x n ) M = d ( x n , x ) 2 + ( k = 1 N β k + k = 1 N α k 1 ) d ( T N + 1 k x n , x n ) ( d ( T N + 1 k x n , x n ) + 2 M ) ,

where M= max 1 k N sup{d( x n ,x),d( T N + 1 k x n ,Tx):nN}.

Case 2: k = 1 N α k (,0] and k = 1 N β k [0,). In the same way as Case 1, we can show that

d ( T N + 1 k x n , T x ) 2 d ( x n , x ) 2 + ( k = 1 N β k k = 1 N α k ) d ( T N + 1 k x n , x n ) ( d ( T N + 1 k x n , x n ) + 2 M ) .

By Case 1, Case 2, and the assumption lim n d( x n , T i x n )=0 for all i=1,2,,N, we obtain

lim sup n d( x n ,Tx) lim sup n d( x n ,x).

Since Δ- lim n x n =x, it follows by the uniqueness of asymptotic centers that Tx=x. Hence, xF(T). □

Fixed point iteration methods are very useful for approximating a fixed point of various nonlinear mappings such as Mann iteration, Ishikawa iteration, Noor iteration and so on. We now introduce a new iteration method for approximating a fixed point of mappings in a CAT(0) space X as follows: Let C be a nonempty closed and convex subset of X, let T:CC be a mapping and NN. For x 1 C, the sequence { x n } generated by

x n + 1 = i = 0 N λ n ( i ) T i x n for all nN,
(3.2)

where { λ n ( i ) } is a sequence in [0,1] for all i=0,1,,N with i = 0 N λ n ( i ) =1.

Remark 3.6 If we put

W n ( N ) = i = 0 N λ n ( i ) j = 0 N λ n ( j ) T i x n ,

then by (1.1) we get

W n ( N ) = j = 0 N 1 λ n ( j ) j = 0 N λ n ( j ) W n ( N 1 ) λ n ( N ) j = 0 N λ n ( j ) T N x n .
(3.3)

Indeed, we put δ n ( i , N ) = λ n ( i ) j = 0 N λ n ( j ) for i=0,1,,N. Thus

W n ( N ) = i = 0 N λ n ( i ) j = 0 N λ n ( j ) T i x n = i = 0 N δ n ( i , N ) T i x n = ( 1 δ n ( N , N ) ) ( δ n ( 0 , N ) 1 δ n ( N , N ) x n δ n ( 1 , N ) 1 δ n ( N , N ) T x n δ n ( N 1 , N ) 1 δ n ( N , N ) T N 1 x n ) δ n ( N , N ) T N x n = ( 1 δ n ( N , N ) ) ( δ n ( 0 , N 1 ) x n δ n ( 1 , N 1 ) T x n δ n ( N 1 , N 1 ) T N 1 x n ) δ n ( N , N ) T N x n = ( 1 δ n ( N , N ) ) ( λ n ( 0 ) j = 0 N 1 λ n ( j ) x n λ n ( 1 ) j = 0 N 1 λ n ( j ) T x n λ n ( N 1 ) j = 0 N 1 λ n ( j ) T N 1 x n ) δ n ( N , N ) T N x n = ( 1 δ n ( N , N ) ) W n ( N 1 ) δ n ( N , N ) T N x n = j = 0 N 1 λ n ( j ) j = 0 N λ n ( j ) W n ( N 1 ) λ n ( N ) j = 0 N λ n ( j ) T N x n .

Therefore, (3.3) is justified.

Using Lemma 3.5, we can prove the Δ-convergence theorem for N-generalized hybrid mappings in complete CAT(0) spaces as follows.

Theorem 3.7 Let C be a nonempty closed and convex subset of a complete CAT(0) space X and let T:CC be an N-generalized hybrid mapping with F(T) and k = 1 N α k (,0][1,) and k = 1 N β k [0,). Let π:CF(T) be the nearest point projection mapping. Suppose that { x n } is a sequence in C defined by (3.2) with 0<a λ n ( i ) b<1 for all i=0,1,,N. Then { x n } Δ-converges to a fixed point u of T, where u= lim n π x n .

Proof Since T is N-generalized hybrid and F(T), we get T is quasi-nonexpansive. Then, for pF(T), we have

d ( x n + 1 , p ) = d ( i = 0 N λ n ( i ) T i x n , p ) i = 0 N λ n ( i ) d ( T i x n , p ) i = 0 N λ n ( i ) d ( x n , p ) = d ( x n , p ) .

Therefore, lim n d( x n ,p) exists and hence { x n } is bounded.

For each pF(T), we obtain, by (3.2), (3.3), and the (CN*) inequality, that

d ( x n + 1 , p ) 2 = d ( i = 0 N λ n ( i ) j = 0 N λ n ( j ) T i x n , p ) 2 = d ( W n ( N ) , p ) 2 = d ( j = 0 N 1 λ n ( j ) j = 0 N λ n ( j ) W n ( N 1 ) λ n ( N ) j = 0 N λ n ( j ) T N x n , p ) 2 j = 0 N 1 λ n ( j ) j = 0 N λ n ( j ) d ( W n ( N 1 ) , p ) 2 + λ n ( N ) j = 0 N λ n ( j ) d ( T N x n , p ) 2 λ n ( N ) j = 0 N λ n ( j ) j = 0 N 1 λ n ( j ) j = 0 N λ n ( j ) d ( W n ( N 1 ) , T N x n ) 2 = j = 0 N 1 λ n ( j ) d ( W n ( N 1 ) , p ) 2 + λ n ( N ) d ( T N x n , p ) 2 λ n ( N ) j = 0 N 1 λ n ( j ) d ( W n ( N 1 ) , T N x n ) 2 = j = 0 N 1 λ n ( j ) d ( j = 0 N 2 λ n ( j ) j = 0 N 1 λ n ( j ) W n ( N 2 ) λ n ( N 1 ) j = 0 N 1 λ n ( j ) T N 1 x n , p ) 2 + λ n ( N ) d ( T N x n , p ) 2 λ n ( N ) j = 0 N 1 λ n ( j ) d ( W n ( N 1 ) , T N x n ) 2 j = 0 N 1 λ n ( j ) ( j = 0 N 2 λ n ( j ) j = 0 N 1 λ n ( j ) d ( W n ( N 2 ) , p ) 2 + λ n ( N 1 ) j = 0 N 1 λ n ( j ) d ( T N 1 x n , p ) 2 j = 0 N 2 λ n ( j ) j = 0 N 1 λ n ( j ) λ n ( N 1 ) j = 0 N 1 λ n ( j ) d ( W n ( N 2 ) , T N 1 x n ) 2 ) + λ n ( N ) d ( T N x n , p ) 2 λ n ( N ) j = 0 N 1 λ n ( j ) d ( W n ( N 1 ) , T N x n ) 2 = j = 0 N 2 λ n ( j ) d ( W n ( N 2 ) , p ) 2 + λ n ( N 1 ) d ( T N 1 x n , p ) 2 + λ n ( N ) d ( T N x n , p ) 2 λ n ( N 1 ) j = 0 N 2 λ n ( j ) j = 0 N 1 λ n ( j ) d ( W n ( N 2 ) , T N 1 x n ) 2 λ n ( N ) j = 0 N 1 λ n ( j ) d ( W n ( N 1 ) , T N x n ) 2 j = 0 N 3 λ n ( j ) d ( W n ( N 3 ) , p ) 2 + λ n ( N 2 ) d ( T N 2 x n , p ) 2 + λ n ( N 1 ) d ( T N 1 x n , p ) 2 + λ n ( N ) d ( T N x n , p ) 2 λ n ( N 2 ) j = 0 N 3 λ n ( j ) j = 0 N 2 λ n ( j ) d ( W n ( N 3 ) , T N 2 x n ) 2 λ n ( N 1 ) j = 0 N 2 λ n ( j ) j = 0 N 1 λ n ( j ) d ( W n ( N 2 ) , T N 1 x n ) 2 λ n ( N ) j = 0 N 1 λ n ( j ) d ( W n ( N 1 ) , T N x n ) 2 λ n ( 0 ) d ( W n ( 0 ) , p ) 2 + k = 1 N λ n ( k ) d ( T k x n , p ) 2 k = 1 N λ n ( k ) j = 0 k 1 λ n ( j ) j = 0 k λ n ( j ) d ( W n ( k 1 ) , T k x n ) 2 k = 0 N λ n ( k ) d ( x n , p ) 2 k = 1 N λ n ( k ) j = 0 k 1 λ n ( j ) j = 0 k λ n ( j ) d ( W n ( k 1 ) , T k x n ) 2 = d ( x n , p ) 2 k = 1 N λ n ( k ) j = 0 k 1 λ n ( j ) j = 0 k λ n ( j ) d ( W n ( k 1 ) , T k x n ) 2 .

This implies that

k = 1 N λ n ( k ) j = 0 k 1 λ n ( j ) j = 0 k λ n ( j ) d ( W n ( k 1 ) , T k x n ) 2 d ( x n , p ) 2 d ( x n + 1 , p ) 2 .

Since lim n d( x n ,p) exists and 0<a λ n ( i ) b<1 for all i=0,1,,N, we get that

lim n d( x n ,T x n )=0and lim n d ( W n ( k 1 ) , T k x n ) =0for all k=2,3,,N.
(3.4)

For k=2,3,,N, we have

d ( x n , T k x n ) d ( x n , W n ( k 1 ) ) + d ( W n ( k 1 ) , T k x n ) = d ( x n , i = 0 k 1 λ n ( i ) j = 0 k 1 λ n ( j ) T i x n ) + d ( W n ( k 1 ) , T k x n ) i = 0 k 1 λ n ( i ) j = 0 k 1 λ n ( j ) d ( x n , T i x n ) + d ( W n ( k 1 ) , T k x n ) = i = 1 k 1 λ n ( i ) j = 0 k 1 λ n ( j ) d ( x n , T i x n ) + d ( W n ( k 1 ) , T k x n ) .

This implies by (3.4) that

lim n d ( x n , T k x n ) =0for all k=1,2,,N.
(3.5)

We now let ω Δ ( x n ):=A(C,{ u n }), where the union is taken over all subsequences { u n } of { x n }. We claim that ω Δ ( x n )F(T). Let u ω Δ ( x n ). Then there exists a subsequence { u n } of { x n } such that A(C,{ u n })={u}. By Lemma 1.5, there exists a subsequence { u n k } of { u n } such that Δ- lim k u n k =yC. It implies by (3.5) and Lemma 3.5 that yF(T). Then, lim n d( x n ,y) exists. Suppose that uy. By the uniqueness of asymptotic centers, we get

lim sup k d ( u n k , y ) < lim sup k d ( u n k , u ) lim sup n d ( u n , u ) < lim sup n d ( u n , y ) = lim sup n d ( x n , y ) = lim sup k d ( u n k , y ) .

This is a contradiction, hence u=yF(T). This shows that ω Δ ( x n )F(T).

Next, we show that ω Δ ( x n ) consists of exactly one point. Let { u n } be a subsequence of { x n } with A(C,{ u n })={u} and let A(C,{ x n })={z}. Since u ω Δ ( x n )F(T), it follows that lim n d( x n ,u) exists. We will show that z=u. To show this, suppose not. By the uniqueness of asymptotic centers, we get

lim sup n d ( u n , u ) < lim sup n d ( u n , z ) lim sup n d ( x n , z ) < lim sup n d ( x n , u ) = lim sup n d ( u n , u ) ,

which is a contradiction, and so z=u. Hence, { x n } Δ-converges to a fixed point u of T. Since F(T) is a closed convex subset of X and d( x n + 1 ,p)d( x n ,p) for all pF(T) and nN, we obtain by Lemma 1.6 that {π x n } converges strongly to some element in F(T), say q. Thus, by the property of π, we obtain that

lim sup n d ( x n , q ) lim sup n ( d ( x n , π x n ) + d ( π x n , q ) ) = lim sup n d ( x n , π x n ) lim sup n d ( x n , u ) .

This implies, by the uniqueness of asymptotic centers, that q=u. This means u= lim n π x n . □

Taking N=2 in Theorem 3.7, we obtain the following Δ-convergence theorem of a 2-generalized hybrid mapping in CAT(0) spaces.

Theorem 3.8 Let C be a nonempty closed and convex subset of a complete CAT(0) space X. Let T:CC be a 2-generalized hybrid mapping, i.e., there are α 1 , α 2 , β 1 , β 2 R such that

α 1 d ( T 2 x , T y ) 2 + α 2 d ( T x , T y ) 2 + ( 1 α 1 α 2 ) d ( x , T y ) 2 β 1 d ( T 2 x , y ) 2 + β 2 d ( T x , y ) 2 + ( 1 β 1 β 2 ) d ( x , y ) 2

for all x,yC. Assume that F(T) and α 1 + α 2 (,0][1,) and β 1 + β 2 [0,). Let π:CF(T) be the nearest point projection mapping. For x 1 C, let { x n } be a sequence defined by

x n + 1 = i = 0 2 λ n ( i ) T i x n for all nN,

where { λ n ( i ) } is a sequence in [0,1] with 0<a λ n ( i ) b<1 for all i=0,1,2 and i = 0 2 λ n ( i ) =1. Then { x n } Δ-converges to a fixed point u of T, where u= lim n π x n .

Taking N=1 in Theorem 3.7, we obtain the following Δ-convergence theorem of a generalized hybrid mapping in CAT(0) spaces.

Theorem 3.9 Let C be a nonempty closed and convex subset of a complete CAT(0) space X. Let T:CC be a generalized hybrid mapping, i.e., there are α,βR such that

αd ( T x , T y ) 2 +(1α)d ( x , T y ) 2 βd ( T x , y ) 2 +(1β)d ( x , y ) 2

for all x,yC. Assume that F(T) and α(,0][1,) and β[0,). Let π:CF(T) be the nearest point projection mapping. For x 1 C, let { x n } be a sequence defined by

x n + 1 = λ n ( 0 ) x n λ n ( 1 ) T x n for all nN,

where { λ n ( 0 ) } and { λ n ( 1 ) } are sequences in [0,1] with 0<a λ n ( 0 ) , λ n ( 1 ) b<1 and λ n ( 0 ) + λ n ( 1 ) =1. Then { x n } Δ-converges to a fixed point u of T, where u= lim n π x n .

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Phuengrattana, W. Fixed point theorems for N-generalized hybrid mappings in uniformly convex metric spaces. Fixed Point Theory Appl 2013, 188 (2013). https://doi.org/10.1186/1687-1812-2013-188

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