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Fixed point results for generalized mappings

Abstract

In this paper, first we establish fixed point results for weak asymptotic pointwise contraction type mappings in metric spaces. Then we study the existence of fixed points for weak asymptotic pointwise nonexpansive type mappings in CAT(0) spaces. Our results improve and extend some corresponding known results in the literature.

MSC:47H09, 47H10, 54H25.

1 Introduction

The notion of asymptotic pointwise contraction was introduced by Kirk [1] as follows.

Let (M,d) be a metric space. A mapping T:MM is called an asymptotic pointwise contraction if there exists a function α:M[0,1) such that, for each integer n1,

d ( T n x , T n y ) α n (x)d(x,y)for each x,yM,

where α n α pointwise on M.

Moreover, Kirk and Xu [2] proved that if C be a weakly compact convex subset of a Banach space E and T:CC an asymptotic pointwise contraction, then T has a unique fixed point vC and for each xC the sequence of Picard iterates { T n x} converges in norm to v.

Rakotch [3] proved that if M be a complete metric space and f:MM satisfies d(f(x),f(y))α(d(x,y))d(x,y), for all x,yM, where α:[0,)[0,1) is monotonically decreasing, then f has a unique fixed point z and { f n (x)} converges to z, for each xM.

Boyd and Wong [4] proved that if M be a complete metric space and f:MM satisfies d(f(x),f(y))ψ(d(x,y))d(x,y), for all x,yM, where ψ:[0,)[0,) is upper semicontinuous from the right and satisfies 0ψ(t)<t for t>0, then f has a unique fixed point z and { f n (x)} converges to z, for each xM.

Using the diameter of an orbit, Walter [5] obtained a result that may be stated as follows:

Let (M,d) be a complete metric space and let T:MM be a mapping with bounded orbits. If there exists a continuous, increasing function φ: R + R + for which φ(r)<r for every r>0 and

d(Tx,Ty)φ ( diam ( O T ( x , y ) ) ) for every x,yM,

where O T (x,y)={ T n x}{ T n y}, then T has a unique fixed point x 0 . Moreover, { T n x} converges to x 0 , for each xM.

In [6], the authors proved coincidence results for asymptotic pointwise nonexpansive mappings. Kirk [7], proved that if M be a complete metric space and T:MM satisfies d( T n x, T n y) ϕ n (d(x,y)), for all x,yM, where ϕ n :[0,)[0,), ϕ n ϕ uniformly on the range of d and ϕ is continuous with ϕ(s)<s for all s>0, then T has a unique fixed point z and { T n (x)} converges to z, for each xM.

References [3, 4, 611] present simple and elegant proofs of fixed point results for pointwise contraction, asymptotic pointwise contraction, and asymptotic nonexpansive mappings.

Very recently, Saeidi [11] introduced the concept of (weak) asymptotic pointwise contraction type mappings.

Let (M,d) be a metric space. A mapping T:MM is said to be of asymptotic pointwise contraction type (resp. of weak asymptotic pointwise contraction type) if T N is continuous for some integer N1 and there exists a function α:M[0,1) such that, for each x in M,

lim sup n sup y M { d ( T n x , T n y ) α n ( x ) d ( x , y ) } 0
(1.1)
( resp.  lim inf n sup y M { d ( T n x , T n y ) α n ( x ) d ( x , y ) } 0 ) ,
(1.2)

where α n α pointwise on M. Taking

r n (x)= sup y M { d ( T n x , T n y ) α n ( x ) d ( x , y ) } R + {}

it can easily be seen from (1.1) (resp. (1.2)) that

lim n r n (x)=0
(1.3)
( resp.  lim inf n r n ( x ) 0 )
(1.4)

for all xM and

d ( T n x , T n y ) α n (x)d(x,y)+ r n (x).
(1.5)

It is easy to see that the class of asymptotic pointwise contraction type mappings contains the class of an asymptotic pointwise contraction mappings, but the converse is not true [11]. Furthermore, if C is a nonempty weakly compact subset of a Banach space E and T:CC a mapping of weak asymptotic pointwise contraction type, then T has a unique fixed point vC and, for each xC, the sequence of Picard iterates { T n x} converges in norm to v (see [11]). Golkarmanesh and Saeidi [12] obtained some results for this mappings in modular spaces.

In this paper, motivated by [1, 911], we combine the above results and obtain fixed point results for classes of mappings that extend the notions of asymptotic contraction and asymptotic pointwise contraction introduced by Kirk [1] and Saeidi [11].

2 Preliminaries

Let M be a metric space and a family of subsets of M. We say that defines a convexity structure of M if it contains the closed ball and is stable by intersection. For instance A(M), the class of the admissable subsets of M, defines a convexity structure on any metric space M. Recall that a subset of M is admissable if it is a nonempty intersection of closed balls.

At this point we introduce some notation which will be used throughout the remainder of this work. For a subset A of a metric space M, set:

r x (A)=sup{d(x,y):yA};

R(A)=inf{ r x (A):xA};

diam(A)=sup{d(x,y):x,yA};

C A (A)={xA: r x (A)=R(A)};

cov(A)={B:B is a closed ball and BA}.

diam(A) is called the diameter of A, R(A) is called the Chebyshev radius of A, C A (A) is called the Chebyshev center of A and cov(A) is called the cover of A.

Definition 1 ([9])

Let be a convexity structure on M.

  1. (i)

    We will say that is compact if any family ( A α ) α Γ of elements of , has a nonempty intersection provided α F A α for any finite subset FΓ.

  2. (ii)

    We will say that is normal if for any AF, not reduced to one point, we have R(A)<diam(A).

  3. (iii)

    We will say that is uniformly normal if there exists c(0,1) such that, for any AF, not reduced to one point, we have R(A)c(diam(A)). It is easy to check that c1/2.

Let M be a metric space and a convexity structure. We will say that a function Φ:MM is -convex if {x:Φ(x)r}F for any r0. Also we define a type to be a function Φ:M[0,] such that

Φ(u)= lim sup n d( x n ,u),

where ( x n ) is a bounded sequence in M. Types are very useful in the study of the geometry of Banach spaces and the existence of fixed point of mappings. We will say that a convexity structure on M is T-stable if types are -convex. We have the following lemma.

Lemma 1 ([9])

Let M be a metric space and a compact convexity structure on M which is T-stable. Then, for any type Φ, there exists x 0 M such that

Φ( x 0 )=inf { Φ ( x ) : x M } .

Hussain and Khamsi [9] and Nicolae [10] proved the following results in metric spaces.

Theorem 1 ([9])

Let M be a bounded metric space. Assume that there exists a convexity structure which is compact and T-stable. T:MM be an asymptotic pointwise contraction. Then T has a unique fixed point x 0 . Moreover, the orbit { T n x} converges to x 0 for each xM.

Theorem 2 ([10])

Let M be a bounded metric space, T:MM and suppose that there exists a convexity structure which is compact and T-stable. Assume that

d ( T n x , T n y ) α n (x) r x ( O T ( y ) ) for every x,yM,

where α n :MR for each nN and the sequence { α n } n N converges pointwise to a function α:M[0,1). Then T has a unique fixed point x 0 . Moreover, the orbit { T n x} converges to x 0 , for each xM.

3 Fixed point results for asymptotic pointwise contractive type mappings in metric spaces

In this section, we generalize the results obtained by Hussain and Khamsi [9] and Nicolae [10] for the wider class of weak asymptotic pointwise contraction type mappings.

Theorem 3 Let (M,d) be a bounded metric space. Assume that there exists a convexity structure which is compact and T-stable. Let T:MM be a weak asymptotic pointwise contraction type mapping. Then T has a unique fixed point x 0 . Moreover, the orbit { T n x} converges to x 0 for each xM.

Proof Fix xM and define a function f by

f(u)= lim sup n d ( T n x , u ) ,uM.

Since is compact and T-stable, there exists x 0 M such that

f( x 0 )=inf { f ( x ) ; x M } .

Let us show that f( x 0 )=0. Indeed, for any m1 we have

f ( T m x 0 ) = lim sup n d ( T n x , T m x 0 ) = lim sup n d ( T m + n x , T m x 0 ) = lim sup n d ( T m ( T n x ) , T m x 0 ) lim sup n α m ( x 0 ) d ( T n x , x 0 ) + r m ( x 0 ) = α m ( x 0 ) f ( x 0 ) + r m ( x 0 ) ,

which implies

f( x 0 )=inf { f ( x ) ; x C } f ( T m x 0 ) α m ( x 0 )f( x 0 )+ r m ( x 0 ).
(3.1)

Now, by (1.3) and (3.1), we obtain

f( x 0 ) lim inf m [ α m ( x 0 ) f ( x 0 ) + r m ( x 0 ) ] =α( x 0 )f( x 0 ),

which forces f( x 0 )=0 as α( x 0 )<1. Hence, d( T n x, x 0 )0 as n. From this and the continuity of T N , for some N1, it follows that

T N x 0 = T N ( lim n T n x ) = lim n T n + N x= x 0 ,

namely, x 0 is a fixed point of T N . Now, repeating the above proof for x 0 instead of x, we deduce that { T n x 0 } is convergent to an element of M. But T k N x 0 = x 0 for all k1. Hence, T n x 0 x 0 . We show that T x 0 = x 0 . For this purpose, consider an arbitrary ϵ>0. Then there exists a k 0 >0 such that d( T n x 0 , x 0 )<ϵ for all n> k 0 . So, choosing a natural number k> k 0 /N, we obtain

d(T x 0 , x 0 )=d ( T ( T k N x 0 ) , x 0 ) =d ( T k N + 1 x 0 , x 0 ) <ϵ.

Since the choice of ϵ>0 is arbitrary, we get T x 0 = x 0 .

It is easy to verify that T has only one fixed point. Indeed, if a,bM are two fixed points of T, then we have

d(a,b)=d ( T n a , T n b ) α n (a)d(a,b)+ r n (a).

Taking lim inf in the above inequality, it follows that d(a,b)α(a)d(a,b). Since α(a)<1, we immediately get a=b. □

In the following, we present an example in a bounded metric space which shows that a mapping of asymptotic pointwise contraction type is not necessary an asymptotic pointwise contraction.

Example 1 Let M= i = 1 n I i ( I i =[0,1]), equipped with the Euclidean norm. Then M is a bounded metric space. For each ( x 1 , x 2 ,, x n )M, define

T( x 1 , x 2 ,, x n )= ( f ( x 2 ) , f ( x 3 ) , , f ( x n ) , 0 ) ,

where f:[0,1][0,1] is some discontinuous function with f(0)=0. We deduce that T is discontinuous, and then it would not be an asymptotic pointwise contraction. But we see that T n x=0 for all xM, and so T is of asymptotic pointwise contraction type.

Theorem 4 Let (M,d) be a bounded metric space, T:MM and suppose there exists a convexity structure which is compact and T-stable and T N is continuous for some integer N1. Assume

lim inf n sup y M { d ( T n x , T n y ) α n ( x ) r x ( O T ( y ) ) } 0for every x,yM,

where α n :MR for each nN and the sequence { α n } converges pointwise to a function α:M[0,1). Then T has a unique fixed point z. Moreover, the orbit { T n x} converges to z for each xM.

Proof Taking

γ n (x)= sup y M { d ( T n x , T n y ) α n ( x ) r x ( O T ( y ) ) } R,

it can easily be seen that

lim inf n γ n (x)0,
(3.2)

for all xM, and

d ( T n x , T n y ) α n (x) r x ( O T ( y ) ) + γ n (x).
(3.3)

Fix xM and define a function f by

f(u)= lim sup n d ( T n x , u ) ,uM.

Since is compact and T-stable, there exists zM such that

f(z)=inf { f ( x ) : x M } .

Let us prove that f(z)=0. Indeed, for any m1 we have

f ( T m z ) = lim sup n d ( T n x , T m z ) = lim sup n d ( T m + n x , T m z ) = lim sup n d ( T m ( T n x ) , T m z ) lim sup n α m ( z ) r z ( O T ( T n x ) ) + γ m ( z ) = α m ( z ) lim sup n r z ( O T ( T n x ) ) + γ m ( z ) ,

which implies

f(z)=inf { f ( x ) ; x C } f ( T m z ) α m (z) lim sup n r z ( O T ( T n x ) ) + γ m (z).
(3.4)

By (3.2) we have lim inf n γ m (z)0, thus, for the subsequence { γ m k (z)} of { γ m (z)}, we have

lim k γ m k (z)0.
(3.5)

Now, by (3.4) and (3.5) we obtain

f(z) lim inf k [ α m k ( z ) lim sup n r z ( O T ( T n x ) ) + γ m k ( z ) ] =α(z)f(z),

which forces f(z)=0 as α(z)<1. Hence, d( T n x,z)0 as n. From this and the continuity of T N , for some N1, it follows that

T N z= T N ( lim n T n x ) = lim n T n + N x=z;

namely, z is a fixed point of T N . Now, repeating the above proof for z instead of x, we deduce that T n z is convergent to a member of M. But T k N z=z for all k1. Hence, T n zz. We show that Tz=z; for this purpose, consider an arbitrary ϵ>0. Then there exists a k 0 >0 such that d( T n z,z)<ϵ for all n> k 0 . So, by choosing a natural number k> k 0 /N, we obtain

d(Tz,z)=d ( T ( T k N x 0 ) , z ) =d ( T k N + 1 z , z ) <ϵ.

Since ϵ>0 is arbitrary, we get Tz=z.

Assume that T has two fixed points a,bM, then, for each nN,

d(a,b)=d ( T n a , T n b ) α n (a) r a ( O T ( b ) ) + γ n (a)= α n (a)d(a,b)+ γ n (a).

Taking the lim inf in the above inequality, it follows that

d(a,b)α(a)d(a,b).

Since α(a)<1, we immediately get a=b. □

4 Fixed point results for weak asymptotic pointwise nonexpansive type mappings in metric spaces

In this section we introduce weak asymptotic pointwise nonexpansive type mappings in metric spaces and we extend the results found of [9].

Definition 2 Let (M,d) be a metric space. A mapping T:MM is said to be of asymptotic pointwise nonexpansive type (resp. weak asymptotic pointwise nonexpansive type) if T N is continuous for some integer N1 and there exists a sequence α n :M[0,+) such that

lim sup n sup y M { d ( T n x , T n y ) α n ( x ) d ( x , y ) } 0
(4.1)
( resp.  lim inf n sup y M { d ( T n x , T n y ) α n ( x ) d ( x , y ) } 0 ) .
(4.2)

where lim sup n α n (x)1. Taking

r n (x)= sup y M { d ( T n x , T n y ) α n ( x ) d ( x , y ) } R + {}

it can easily be seen from (4.1) (resp. (4.2)) that

lim n r n (x)=0
(4.3)
( resp.  lim inf n r n ( x ) 0 )
(4.4)

for all xM and

d ( T n x , T n y ) α n (x)d(x,y)+ r n (x).
(4.5)

A metric space (M,d) is said to be a length space if each two points of M are joined by a rectifiable path (that is, a path of finite length) and the distance between any two points is taken to be the infimum of the lengh of all rectifiable paths joining them. In this case, d is said to be a length metric (otherwise, known an inner metric or intrinsic metric). In the case that there is no rectifiable path joins two points of the space, the distance between them is said to be ∞.

A geodesic path joining xM to yM (or, more briefly, a geodesic from x to y) is a map c from a closed interval [0,l]R to M such that c(0)=x, c(l)=y, and d(c(t),c( t ))=|t t | for all t, t [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. (M,d) is said to be a geodesic space if every two points of (M,d) are joined by a geodesic. M is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x,yM, which we will denote by [x,y], called the segment joining x to y.

A geodesic metric space Δ( x 1 , x 2 , x 3 ) in a geodesic metric space (M,d) consists of three points in M (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle Δ( x 1 , x 2 , x 3 ) in (M,d) is a triangle Δ ¯ ( x 1 , x 2 , x 3 ):=Δ( x ¯ 1 , x ¯ 2 , x ¯ 3 ) in M k 2 such that d R 2 ( x ¯ i , x ¯ j )=d( x i , x j ) for i,j{1,2,3}. If k>0 it is further assumed that perimeter of Δ( x 1 , x 2 , x 3 ) is less than 2 D k , where D k denotes the diameter of  M k 2 . Such a triangle always exists.

A geodesic metric space is said to be a CAT(k) space if all geodesic triangles of appropriate size satisfy the following CAT(k) comparison axiom.

CAT(k): Let Δ be a geodesic triangle in M and Δ ¯ M k 2 be a comparison triangle for Δ. Then Δ is said to satisfy the CAT(k) inequality if for all x,yΔ and all comparison points x ¯ , y ¯ Δ ¯ ,

d(x,y)d ( x ¯ , y ¯ ) .

Complete CAT(0) spaces are often called Hadamard spaces. These spaces are of particular relevance to this study.

Finally, we observe that if x, y 1 , y 2 are points of a CAT(0) space and if y 0 is the midpoint of the segment [ y 1 , y 2 ], which we denote by y 1 y 2 2 , then the CAT(0) inequality implies

d ( x , y 1 y 2 2 ) 2 1 2 d ( x , y 1 ) 2 + 1 2 d ( x , y 2 ) 2 1 4 d ( y 1 , y 2 ) 2 .

Let M be a complete CAT(0) space and x,yM, then for any α[0,1] there exists a unique point αx(1α)y[x,y] such that

d ( z , α x ( 1 α ) y ) αd(z,x)+(1α)d(z,y),
(4.6)

for any zM.

Let M be a complete CAT(0) space. A subset CM is convex if for any x,yC we have [x,y]C. Any type function achieves its infimum, i.e., for any bounded sequence { x n } in a CAT(0) space M, there exists ωM such that f(ω)=inf{f(x):xM}, where

f(x)= lim sup n d( x n ,x).

Theorem 5 Let M be a complete CAT(0) metric space. Let C be a bounded closed nonempty convex subset of M. If T:CC is a weak asymptotic pointwise nonexpansive type, then the fixed point set Fix(T) is closed and convex.

Proof Fix xC and define a function f by

f(u)= lim sup n d ( T n x , u ) ,uC.

Let x 0 C be such that f( x 0 )=inf{f(x);xM}= f 0 . According to the proof of Theorem 3 we have f( T n x 0 ) α n ( x 0 ) f 0 + r n ( x 0 ), for any n1. The CAT(0) inequality implies

d ( T n ( x ) , T m x 0 T h x 0 2 ) 2 1 2 d ( T n x , T m x 0 ) 2 + 1 2 d ( T n x , T h x 0 ) 2 1 4 d ( T m x 0 , T h x 0 ) 2 .

If we let n go to infinity, we get

f 0 2 f ( T m x 0 T h x 0 2 ) 1 2 f ( T m x 0 ) 2 + 1 2 f ( T h x 0 ) 2 1 4 d ( T m x 0 , T h x 0 ) 2 ,

which implies

d ( T m x 0 , T h x 0 ) f 0 2 ( 2 α m 2 ( x 0 ) + 2 α h 2 ( x 0 ) 4 ) + 2 r n 2 ( x 0 ) + 2 r h 2 ( x 0 ) + 4 f 0 ( α m ( x 0 ) r m ( x 0 ) + α h ( x 0 ) r h ( x 0 ) ) .

Since T is of weak asymptotic pointwise nonexpansive type, we get

lim sup m , h d ( T m x 0 , T h x 0 ) 0,

which implies { T n x 0 } is a Cauchy sequence. Let v= lim n T n x 0 . By the proof of Theorem 3 Tv=v and Fix(T) is closed. In order to prove that Fix(T) is convex, it is enough to prove x y 2 Fix(T), whenever x,yFix(T). Let z= x y 2 . The CAT(0) inequality implies

d ( T n z , z ) 2 1 2 d ( x , T n z ) 2 + 1 2 d ( y , T n z ) 2 1 4 d ( x , y ) 2

for any n1. Since

d ( x , T n z ) 2 =d ( T n x , T n z ) 2 ( α n ( z ) d ( z , x ) + r n ( z ) ) 2 = ( α n ( z ) d ( x , y ) 2 + r n ( z ) ) 2

and

d ( y , T n z ) 2 =d ( T n y , T n z ) 2 ( α n ( z ) d ( z , y ) + r n ( z ) ) 2 = ( α n ( z ) d ( x , y ) 2 + r n ( z ) ) 2

we get

d ( T n z , z ) 2 1 4 ( α n 2 ( z ) 1 ) d ( x , y ) 2 + r n 2 (z)+2 α n (z) r n (z)d(x,y)

for any n1. Since T is of weak asymptotic pointwise nonexpansive type, we get lim n T n z=z, which implies that T(z)=z, i.e., zFix(T). □

Before we state the next and final result of this work, we need the following notation:

{ x n }zif and only iff(z)= inf x C f(x),

where C is a closed convex subset which contains the bounded sequence { x n } and f(x)= lim sup n d( x n ,x).

Theorem 6 Let M be a complete CAT(0) metric space. Let C be a bounded closed nonempty convex subset of M. Let T:CC be a weak asymptotic pointwise nonexpansive type. Let { x n }C be an approximate fixed point sequence, i.e., lim n d( x n ,T x n )=0 and { x n }z. Then we have Tz=z.

Proof Since { x n } is an approximate fixed point sequence, we have

f(x)= lim sup n d ( T m x n , x ) ,

for any m1. Hence f( T m x) α m (x)f(x)+ r m (x) (see the proof of Theorem 3). In particular, lim n f( T m z)=f(z). The CAT(0) inequality implies

d ( x n , z T n z 2 ) 2 1 2 d ( x n , z ) 2 + 1 2 d ( x n , T m z ) 2 1 4 d ( z , T m z ) 2 ,

for any m,n1. If n, we will get

f ( z T n z 2 ) 2 1 2 f ( z ) 2 + 1 2 f ( T m z ) 2 1 4 d ( z , T m z ) 2 ,

for any m1. The definition of z implies

f ( z ) 2 1 2 f ( z ) 2 + 1 2 f ( T m z ) 2 1 4 d ( z , T m z ) 2

for any m1, or

d ( z , T m z ) 2 2f ( T m z ) 2 2f ( z ) 2 2 ( α m ( z ) f ( z ) + r m ( z ) ) 2 2f ( z ) 2 .

Letting m, we will get lim n d(z, T m z)=0. The rest of the proof is similar to the one used for Theorem 3. □

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Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. The authors, therefore, acknowledge with thanks the DSR for technical and financial support.

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Golkarmanesh, F., Al-Mazrooei, A.E., Parvaneh, V. et al. Fixed point results for generalized mappings. Fixed Point Theory Appl 2014, 217 (2014). https://doi.org/10.1186/1687-1812-2014-217

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Keywords

  • asymptotic center
  • asymptotic pointwise contraction type
  • convexity structure
  • T-stable
  • weak asymptotic pointwise nonexpansive type
  • CAT(0) space