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Results on n-tupled fixed points in metric spaces with uniform normal structure

Abstract

In this paper, we introduce the concept of new notions related to n-tupled fixed point and prove some related results for an asymptotically regular one-parameter semigroup ={F(t):tG,where G is an unbounded subset of [0,)} of Lipschitzian self-mappings on i = 1 n X in the case when (X,d) is a complete bounded metric space with uniform normal structure. Our results extend the results due to Yao and Zeng (J. Nonlinear Convex Anal. 8(1):153-163, 2007) and Soliman (Fixed Point Theory Appl. 2013:346, 2013; J. Adv. Math. Stud. 7(2):2-14, 2014).

1 Introduction

The Banach contraction principle is the most natural and significant result of fixed point theory. In complete metric spaces it continues to be an indispensable and effective tool in theory and applications, which guarantees the existence and uniqueness of fixed points of contraction self-mappings besides offering a constructive procedure to compute the fixed point of the underlying mapping. There already exists an extensive literature on this topic. Keeping in view the relevance of this paper, we merely refer to [15]. In 1987, the idea of coupled fixed point was initiated by Guo and Lakshmikantham [6]; it was also followed by Bhaskar and Lakshmikantham [7] wherein authors proved some interesting coupled fixed point theorems for mappings satisfying the mixed monotone property. Many authors obtained important coupled, tripled and n-tupled fixed point theorems (see [716]). In this continuation, Lakshmikantham and Ćirić [13] introduced coupled common fixed point theorems for nonlinear ϕ-contraction mappings in partially ordered complete metric spaces which indeed generalize the corresponding fixed point theorems contained in Bhaskar and Lakshmikantham [7]. In 2010, Samet and Vetro [17] introduced the concept of fixed point of n-tupled fixed point (where n=2,3,4,) for nonlinear mappings in complete metric spaces. They obtained the existence and uniqueness theorems for contractive type mappings. Their results generalized and extended coupled fixed point theorems established by Bhaskar and Lakshmikantham [7]. Recently, Imdad et al. [18] introduced a generalization of n-tupled fixed point and n-tupled coincidence point by considering n even besides using the idea of mixed g-monotone property on i = 1 n X and proved an n-tupled (where n is even) coincidence point theorem for nonlinear ϕ-contraction mappings satisfying the mixed g-monotone property. For more information about n-tupled fixed points, see [10, 1719].

On the other hand, normal structure is one of the most important aspects of metric fixed point theory. It was introduced by Brodskii and Milman in [20]. They found the first application of normal structure to fixed point theory. In 1965, Kirk [21] introduced the following theorem: Every nonexpansive self-mapping on a weakly compact convex subset of a Banach space with normal structure has a fixed point. In 1969, Kijima and Takahashi [22] established the metric space version of Kirk’s theorem [21]. Subsequently, many authors successfully generalized certain fixed point theorems and structure properties from Banach spaces to metric spaces. For example, Khamsi [23] defined normal and uniform normal structure for metric spaces and proved that if (X,d) is a complete bounded metric space with uniform normal structure, then it has the fixed point property for nonexpansive mappings and a kind of intersection property which extends a result of Maluta [24] to metric spaces. In 1995, Lim and Xu [25] proved a fixed point theorem for uniformly Lipschitzian mappings in metric spaces with both property (P) and uniform normal structure, which extended the result of Khamsi [23]. This is the metric space version of Casini and Maluta’s theorem [2]. In 2007, Yao and Zeng [26] established a fixed point theorem for an asymptotically regular one-parameter semigroup of uniformly k-Lipschitzian mappings with property () in a complete bounded metric space with uniform normal structure, which extended the results of Lim and Xu [25]. Recently, the idea of coupled and tripled fixed point results in a complete bounded metric space X with uniform normal structure was initiated by Soliman [27, 28]. He proved that every asymptotically regular one-parameter semigroup ={F(t):tG,} of Lipschitzian mappings on X×X has a coupled fixed point and on X×X×X has a tripled fixed point.

In the present paper, we prove an n-tupled fixed point theorem for asymptotically regular Lipschitzian one-parameter semigroups ={F(t):tG} on i = 1 n X, where X is a complete bounded metric space with uniform normal structure. Also, some corollaries of our main theorem are presented.

2 Preliminaries

Definition 2.1 [7]

An element (x,y)X×X is called a coupled fixed point of the mapping F:X×XX if

F(x,y)=xandF(y,x)=y.

Definition 2.2 [7]

Let (X,) be a partially ordered set and F:X×XX. We say that F has the mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any x,yX, x 1 , x 2 X, x 1 x 2 F( x 1 ,y)F( x 2 ,y) and y 1 , y 2 X, y 1 y 2 F(x, y 1 )F(x, y 2 ).

Theorem 2.1 [7]

Let (X,) be a partially ordered set and suppose that there is a metric d on X such that (X,d) is a complete metric space. Let F:X×XX be a continuous mapping having the mixed monotone property on X. Assume that there exists a constant k[0,1) with

d ( F ( x , y ) , F ( u , v ) ) k 2 [ d ( x , u ) + d ( y , v ) ] xu,yv.

If there exist x 0 , y 0 X such that x 0 F( x 0 , y 0 ) and y 0 F( y 0 , x 0 ), then there exist x,yX such that x=F(x,y) and y=F(y,x).

Definition 2.3 [17]

An element (x,y,z)X×X×X is called a tripled fixed point of the mapping F:X×X×XX if

F(x,y,z)=x,F(y,z,x)=yandF(z,x,y)=z.

Definition 2.4 [10]

Let X be a nonempty set. An element ( x 1 , x 2 , x 3 ,, x r ) i = 1 r X is called an r-tupled fixed point of the mapping F: i = 1 r XX if

x 1 = F ( x 1 , x 2 , x 3 , , x r ) , x 2 = F ( x 2 , x 3 , , x r , x 1 ) , x 3 = F ( x 3 , , x r , x 1 , x 2 ) , x r = F ( x r , x 1 , x 2 , , x r 1 ) .

Definition 2.5 [23]

Suppose that (X,d) is a metric space, and let μ denote a nonempty family of subsets of X. Then μ defines a convexity structure on X if it is stable under intersection.

Definition 2.6 [23]

Let μ be a convexity structure on a metric (X,d). Then μ has property (R) if any decreasing sequence { C n } of nonempty bounded closed subsets of X with C n μ has a nonempty intersection.

Definition 2.7 [5]

A subset of X is said to be admissible if it is an intersection of closed balls.

Remark 2.1 Let A(X) be a family of all admissible subsets of X. Then we note that A(X) defines a convexity structure on X.

In this paper any other convexity structure μ on X is always assumed to contain A(X).

Let M be a bounded subset of X. Following Lim and Xu [25], we shall adopt the following notations:

B(x,r) is the closed ball centered at x with radius r,

r(x,M)=sup{d(x,y):yM} for xX,

δ(M)=sup{r(x,M):xM},

R(M)=inf{r(x,M):xM}.

For a bounded subset A of X, we define the admissible hull of A, denoted by ad(A), as the intersection of all those admissible subsets of X which contain A, i.e.,

ad(A)={B:ABX with B admissible}.

Proposition 2.1 [25]

For a point xX and a bounded subset A of X, we have

r ( x , ad ( A ) ) =r(x,A).

Definition 2.8 [23]

A metric space (X,d) is said to have normal (resp. uniform normal) structure if there exists a convexity structure μ on X such that R(A)<δ(A) (resp. R(A)cδ(A) for some constant c(0,1)) for all Aμ which is bounded and consists of more than one point. In this case μ is said to be normal (resp. uniformly normal) in X.

We define the normal structure coefficient N(X) of X (with respect to a given convexity structure μ) as the number

sup { R ( A ) δ ( A ) } ,

where the supremum is taken over all bounded AF with δ(A)>0. X then has uniform normal structure if and only if N(X)<1.

Khamsi proved the following result that will be very useful in the proof of our main theorem.

Proposition 2.2 [23]

Let X be a complete bounded metric space and μ be a convexity structure of X with uniform normal structure. Then μ has property (R).

Definition 2.9 [26]

Let (X,d) be a metric space and ={F(t):tG} be a semigroup on i = 1 r X. Let us write the set

w()= { { t n } : { t n } G  and  t n } .

Lemma 2.1 [26]

If { t n }ω(), then { t n + 1 t n }ω().

Definition 2.10 [25]

A metric space (X,d) is said to have property (P) if given any two bounded sequences { x n } and { z n } in X, one can find some z n = 1 ad{ z j :jn} such that

lim sup n d(z, x n ) lim sup j lim sup n d( z j , x n ).

3 Main results

Let G be a subsemigroup of [0,) with addition ‘+’ such that

tsGt,sG with ts.

This condition is satisfied if G=[0,) or G= Z + , the set of nonnegative integers. Let ={F(t):tG} be a family of self-mappings on i = 1 r X. Then is called a (one-parameter) semigroup on i = 1 r X if the following conditions are satisfied:

  1. (i)

    F(0)( x 1 , x 2 , x 3 ,, x r )= x 1 , F(0)( x 2 , x 3 ,, x r , x 1 )= x 2 , …, F(0)( x r , x 1 , x 2 ,, x r 1 )= x r x 1 , x 2 , x 3 ,, x r X;

  2. (ii)

    F(s)(F(t)( x 1 , x 2 , x 3 ,, x r ),F(t)( x 2 , x 3 ,, x r , x 1 ),,F(t)( x r , x 1 , x 2 ,, x r 1 ))=F(s+t)( x 1 , x 2 , x 3 ,, x r ) s,tG and x 1 , x 2 , x 3 ,, x r X;

  3. (iii)

    x 1 , x 2 , x 3 ,, x r X, the self-mappings tF(t)( x 1 , x 2 , x 3 ,, x r ), tF(t)( x 2 , x 3 ,, x r , x 1 ), …, tF(t)( x r , x 1 , x 2 ,, x r 1 ) from G into X are continuous when G has the relative topology of [0,).

Definition 3.1 A semigroup ={F(t):tG} on i = 1 r X is said to be asymptotically regular at a point ( x 1 , x 2 , x 3 ,, x r ) i = 1 r X if

lim t d ( F ( t + h ) ( x 1 , x 2 , x 3 , , x r ) , F ( t ) ( x 1 , x 2 , x 3 , , x r ) ) =0hG.
(1)

If is asymptotically regular at each ( x 1 , x 2 , x 3 ,, x r ) i = 1 r X, then is called an asymptotically regular semigroup on i = 1 r X.

A semigroup {F(n):nN(the set of all natural numbers)} on i = 1 r X is called simplest asymptotically regular at a point ( x 1 , x 2 , x 3 ,, x r ) i = 1 r X if

lim n d ( F n + 1 ( x 1 , x 2 , x 3 , , x r ) , F n ( x 1 , x 2 , x 3 , , x r ) ) =0 x 1 , x 2 , x 3 ,, x r X.

Definition 3.2 A semigroup ={F(t):tG} on i = 1 r X is called a uniformly Lipschitzian semigroup if

sup { k ( t ) : t G } =k<,

where

k ( t ) = r sup { d ( F ( t ) ( x 1 , x 2 , x 3 , , x r ) , F ( t ) ( y 1 , y 2 , y 3 , , y r ) ) [ d ( x 1 , y 1 ) + d ( x 2 , y 2 ) + + d ( x r , y r ) ] 0 : x 1 , x 2 , , x r , y 1 , y 2 , , y r X } .
(2)

The simplest uniformly Lipschitzian semigroup is a semigroup of iterates of a mapping F: i = 1 r XX with

sup { k n : n N } = k < , k n = r sup { d ( F n ( x 1 , x 2 , x 3 , , x r ) , F n ( y 1 , y 2 , y 3 , , y r ) ) [ d ( x 1 , y 1 ) + d ( x 2 , y 2 ) + + d ( x r , y r ) ] 0 : x 1 , x 2 , , x r , y 1 , y 2 , , y r X } .

x 1 , x 2 ,, x r , y 1 , y 2 ,, y r X, where F n ( x 1 , x 2 , x 3 ,, x r )= F n 1 (F( x 1 , x 2 , x 3 ,, x r ),F( x 2 , x 3 ,, x r , x 1 ),,F( x r , x 1 , x 2 ,, x r 1 )).

Definition 3.3 A mapping F(t): i = 1 r XX has an r-tupled fixed point ( x 1 , x 2 , x 3 ,, x r ) i = 1 r X if x 1 =F(t)( x 1 , x 2 , x 3 ,, x r ), x 2 =F(t)( x 2 , x 3 ,, x r , x 1 ), x 3 =F(t)( x 3 ,, x r , x 1 , x 2 ), …, x r =F(t)( x r , x 1 , x 2 ,, x r 1 ).

Definition 3.4 Let (X,d) be a complete bounded metric space and ={F(t):tG} be a semigroup on i = 1 r X. Then has property () if for each xX and each { t n }w(), the following conditions are satisfied:

  1. (a)

    the sequences {F( t n )( x 1 , x 2 , x 3 ,, x r )}, …, {F( t n )( x r , x 1 , x 2 ,, x r 1 )} are bounded;

  2. (b)

    for any sequence { s n 1 } in ad{F( t n )( x 1 , x 2 , x 3 ,, x r ):n1}, there exists some s 1 n = 1 ad{ s j 1 :jn} such that

    lim sup n d ( s 1 , F ( t n ) ( x 1 , x 2 , x 3 , , x r ) ) lim sup j lim sup n d ( s j 1 , F ( t n ) ( x 1 , x 2 , x 3 , , x r ) ) ,

for any sequence { s n r } in ad{F( t n )( x r , x 1 , x 2 ,, x r 1 ):n1}, there exists some s r n = 1 ad{ s j r :jn} such that

lim sup n d ( s r , F ( t n ) ( x r , x 1 , x 2 , , x r 1 ) ) lim sup j lim sup n d ( s j r , F ( t n ) ( x r , x 1 , x 2 , , x r 1 ) ) .

Remark 3.1 If X is a complete bounded metric space with property (P), then each semigroup ={F(t):tG} on i = 1 r X has property ().

Lemma 3.1 Let (X,d) be a complete bounded metric space with uniform normal structure, and let ={F(t):tG} be a semigroup on i = 1 r X with property (). Then, for each xX, each { t n }ω() and for any constant N ˜ (X)<c, the normal structure coefficient with respect to the given convexity structure μ, there exist some a 1 n = 1 ad{ a j 1 :jn}, …, a r n = 1 ad{ a j r :jn} satisfying the following properties:

  1. (I)
    lim sup n d ( a 1 , F ( t n ) ( x 1 , x 2 , x 3 , , x r ) ) c A 1 ( { F ( t n ) ( x 1 , x 2 , x 3 , , x r ) } ) , lim sup n d ( a r , F ( t n ) ( x r , x 1 , x 2 , , x r 1 ) ) c A r ( { F ( t n ) ( x r , x 1 , x 2 , , x r 1 ) } ) ,

where

A 1 ( { F ( t n ) ( x 1 , x 2 , x 3 , , x r ) } ) = lim sup n { d ( F ( t i ) ( x 1 , x 2 , x 3 , , x r ) , F ( t j ) ( x 1 , x 2 , x 3 , , x r ) ) : i , j n } , A r ( { F ( t n ) ( x r , x 1 , x 2 , , x r 1 ) } ) = lim sup n { d ( F ( t i ) ( x r , x 1 , x 2 , , x r 1 ) , F ( t j ) ( x r , x 1 , x 2 , , x r 1 ) ) : i , j n } ;

(II)

d ( a 1 , w ) lim sup n d ( F ( t n ) ( x 1 , x 2 , x 3 , , x r ) , w ) , d ( a r , w ) lim sup n d ( F ( t n ) ( x r , x 1 , x 2 , , x r 1 ) , w ) for all  w X .

Proof For each integer n1, let A n 1 ={F( t j )( x 1 , x 2 , x 3 ,, x r ):jn}, A n 2 ={F( t j )( x 2 , x 3 , x 4 ,, x r , x 1 ):jn}, …, A n r ={F( t j )( x r , x 1 , x 2 ,, x r 1 ):jn}. Then { A n 1 },{ A n 2 },,{ A n r } are decreasing sequences of admissible subsets of X hence A 1 := n = 1 A n 1 ϕ, A 2 := n = 1 A n 2 ϕ, …, A r := n = 1 A n r ϕ by Proposition 2.2. From Proposition 2.1, it is not difficult to see that δ( A n 1 )=δ({F( t i )( x 1 , x 2 , x 3 ,, x r ):in}), δ( A n 2 )=δ({F( t i )( x 2 , x 3 , x 4 ,, x r , x 1 ):in}), …, δ( A n r )=δ({F( t i )( x r , x 1 , x 2 ,, x r 1 ):in}). Indeed, observe that

δ ( A n 1 ) = sup { r ( w , A n 1 ) : w A n 1 } = sup w A n 1 sup j n d ( w , F ( t j ) ( x 1 , x 2 , x 3 , , x r ) ) = sup j n sup w A n 1 d ( w , F ( t j ) ( x 1 , x 2 , x 3 , , x r ) ) = sup j n r ( F ( t j ) ( x 1 , x 2 , x 3 , , x r ) , A n 1 ) = sup j n sup i n d ( F ( t j ) ( x 1 , x 2 , x 3 , , x r ) , F ( t i ) ( x 1 , x 2 , x 3 , , x r ) ) = δ ( { F ( t i ) ( x 1 , x 2 , x 3 , , x r ) : i n } ) .

Similarly, one can obtain

δ ( A n 2 ) = δ ( { F ( t i ) ( x 2 , x 3 , x 4 , , x r , x 1 ) : i n } ) , δ ( A n 3 ) = δ ( { F ( t i ) ( x 3 , x 4 , , x r , x 1 , x 2 ) : i n } ) , δ ( A n r ) = δ ( { F ( t i ) ( x r , x 1 , x 2 , , x r 1 ) : i n } ) .

On the other hand, for any a 1 A 1 and any wX, we have

sup j n d ( w , F ( t j ) ( x 1 , x 2 , x 3 , , x r ) ) =r ( w , A n 1 ) r ( w , A 1 ) d ( w , a 1 ) .

Therefore,

d ( w , a 1 ) lim sup n d ( w , F ( t n ) ( x 1 , x 2 , x 3 , , x r ) ) .

Also, one can deduce that for any a 2 A 2 , …, a r A r and any wX, we have

d ( w , a 2 ) lim sup n d ( w , F ( t n ) ( x 2 , x 3 , x 4 , , x r , x 1 ) ) , d ( w , a r ) lim sup n d ( w , F ( t n ) ( x r , x 1 , x 2 , , x r 1 ) ) ,

from which (II) follows.

We now suppose that for each n1, there exist a n 1 A n 1 , a n 2 A n 2 , …, a n r A n r such that

r ( a n 1 , A n 1 ) cδ ( { F ( t j ) ( x 1 , x 2 , x 3 , , x r ) : j n } ) ,
(3)
r ( a n 2 , A n 2 ) cδ ( { F ( t j ) ( x 2 , x 3 , x 4 , , x r , x 1 ) : j n } ) ,
(4)
r ( a n 2 , A n r ) c δ ( { F ( t j ) ( x r , x 1 , x 2 , , x r 1 ) : j n } ) .
(5)

Indeed, if δ({F( t j )( x 1 , x 2 , x 3 ,, x r ):jn})=0, then δ( A n 1 )=δ({F( t j )( x 1 , x 2 , x 3 ,, x r ):jn}), we conclude that (3) holds. Without loss of generality, we may assume that δ({F( t j )( x 1 , x 2 , x 3 ,, x r ):j0})>0. Then, for N(X)<c, we choose ϵ>0 so small that it satisfies the following:

N ( X ) δ ( { F ( t j ) ( x 1 , x 2 , x 3 , , x r ) : j n } ) + ϵ c δ ( { F ( t j ) ( x 1 , x 2 , x 3 , , x r z ) : j n } ) .
(6)

From the definition of R( A n 1 ), one can find a n 1 A n 1 such that

r ( a n 1 , A n 1 ) < R ( A n 1 ) + ϵ N ( X ) δ ( A n 1 ) + ϵ = N ( X ) δ ( { F ( t j ) ( x 1 , x 2 , x 3 , , x r ) : j n } ) + ϵ c δ ( { F ( t j ) ( x 1 , x 2 , x 3 , , x r ) : j n } ) ,

which implies that (3) holds. Obviously, it follows from (3) that for each n1,

lim sup j r ( a n 1 , x j ) cδ ( { F ( t j ) ( x 1 , x 2 , x 3 , , x r ) : j n } ) ,

which implies

lim sup n lim sup j r ( a n 1 , F ( t j ( x 1 , x 2 , x 3 , , x r ) ) ) c A 1 ( { F ( t n ) ( x 1 , x 2 , x 3 , , x r ) } ) ,
(7)

where A 1 ({F( t n )( x 1 , x 2 , x 3 ,, x r )})={d(F( t j )( x 1 , x 2 , x 3 ,, x r ),F( t i )( x 1 , x 2 , x 3 ,, x r )):i,jn}. Noticing

a n 1 A n 1 ad { F ( t j ) ( x 1 , x 2 , x 3 , , x r ) : j n } for each n1,

we know that property () yields a point a 1 n = 1 ad{ a j 1 :jn} such that

lim sup j d ( a 1 , F ( t j ) ( x 1 , x 2 , x 3 , , x r ) ) lim sup n lim sup j r ( a n 1 , F ( t j ) ( x 1 , x 2 , x 3 , , x r ) ) .
(8)

Since { a j 1 :jn} A n 1 , a 1 A 1 = n = 1 ad{F( t j )( x 1 , x 2 , x 3 ,, x r ):jn} and satisfies

lim sup j d ( a 1 , F ( t j ) ( x 1 , x 2 , x 3 , , x r ) ) c A 1 ( { F ( t j ) ( x 1 , x 2 , x 3 , , x r ) } ) ,by (7),

similarly one can obtain that

lim sup j d ( a 2 , F ( t j ) ( x 2 , x 3 , , x r , x 1 ) ) c A 2 ( { F ( t j ) ( x 2 , x 3 , , x r , x 1 ) } ) , lim sup j d ( a r , F ( t j ) ( x r , x 1 , x 2 , , x r 1 ) ) c A r ( { F ( t j ) ( x r , x 1 , x 2 , , x r 1 ) } ) ,

where a 2 A 2 = n = 1 ad{F( t j )( x 2 , x 3 ,, x r , x 1 ):jn}, …, a r A r = n = 1 ad{F( t j )( x r , x 1 , x 2 ,, x r 1 ):jn}.

Therefore (I) holds. □

We are ready to prove our main theorem for this paper.

Theorem 3.1 Let (X,d) be a complete bounded metric space with uniform normal structure, and let ={F(t):tG} be an asymptotically regular uniformly Lipschitzian semigroup of self-mappings on i = 1 r X with property () and satisfying

k k ˜ < 1 N ( X ) ,

where k= lim inf t k(t) and k ˜ = lim sup t k(t).

Then there exist some x 1 , x 2 , x 3 ,, x r X such that F(t)( x 1 , x 2 , x 3 ,, x r )= x 1 , …, F(t)( x 2 , x 3 ,, x r , x 1 )= x 2 and F(t)( x r , x 1 , x 2 ,, x r 1 )= x r for all tG.

Proof First, we choose a constant c such that N(X)< c ˆ <1 and k k ˜ < 1 c . We can select a sequence { t n }w(), from Lemma 2.1 we find that { t n + 1 t n }w() and lim n k( t n )=k.

Now fix x 0 1 , x 0 2 , x 0 3 ,, x 0 r X. Then, by Lemma 3.1, we can inductively construct sequences { x l 1 } l = 1 , { x l 2 } l = 1 , …, { x l r } l = 1 X such that x l + 1 1 n = 1 ad{F( t i )( x l 1 , x l 2 , x l 3 ,, x l r ):in}, x l + 1 2 n = 1 ad{F( t i )( x l 2 , x l 3 ,, x l r , x l 1 ):in}, …, x l + 1 r n = 1 ad{F( t i )( x l r , x l 1 , x l 2 ,, x l r 1 ):in} for each integer l0,

  1. (III)
    lim sup n d ( F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) , x l + 1 1 ) c A 1 ( { F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) } ) , lim sup n d ( F ( t n ) ( x l 2 , x l 3 , , x l r , x l 1 ) , x l + 1 2 ) c A 2 ( { F ( t n ) ( x l 2 , x l 3 , , x l r , x l 1 ) } ) , lim sup n d ( F ( t n ) ( x l r , x l 1 , x l 2 , , x l r 1 ) , x l + 1 r ) c . A r ( { F ( t n ) ( x l r , x l 1 , x l 2 , , x l r 1 ) } ) ,

where

A 1 ( { F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) } ) = lim sup n { d ( F ( t i ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) : i , j n } , A 2 ( { F ( t n ) ( x l 2 , x l 3 , , x l r , x l 1 ) } ) = lim sup n { d ( F ( t i ) ( x l 2 , x l 3 , , x l r , x l 1 ) , F ( t j ) ( x l 2 , x l 3 , , x l r , x l 1 ) ) : i , j n } , A r ( { F ( t n ) ( x l r , x l 1 , x l 2 , , x l r 1 ) } ) = lim sup n { d ( F ( t i ) ( x l r , x l 1 , x l 2 , , x l r 1 ) , F ( t j ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) : i , j n } ;
  1. (IV)
    d ( x l + 1 1 , w ) lim sup n d ( F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) , w ) , d ( x l + 1 2 , w ) lim sup n d ( F ( t n ) ( x l 2 , x l 3 , , x l r , x l 1 ) , w ) , d ( x l + 1 r , w ) lim sup n d ( F ( t n ) ( x l r , x l 1 , x l 2 , , x l r 1 ) , w ) w X .

Let

D l = lim sup n [ d ( x l + 1 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + + d ( x l + 1 r , F ( t n ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) ] , and h = c k k ˜ < 1 .

Observe that for each i>j1, using (IV) we have

d ( F ( t i ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) = d ( F ( t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t j ) F ( t i t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) = d ( F ( t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t j ) ( F ( t i t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t i t j ) ( x l 2 , x l 3 , , x l r , x l 1 ) , , F ( t i t j ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) ) k ( t j ) r [ d ( x l 1 , F ( t i t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + + d ( x l r , F ( t i t j ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) ] k ( t j ) r lim sup n [ d ( F ( t n ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , F ( t i t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + d ( F ( t n ) ( x l 1 2 , x l 1 3 , , x l 1 r , x l 1 1 ) , F ( t i t j ) ( x l 2 , x l 3 , , x l r , x l 1 ) ) + + d ( F ( t n ) ( x l 1 r , x l 1 1 , x l 1 2 , , x l 1 r 1 ) , F ( t i t j ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) ] .
(9)

By the asymptotic regularity of ={F(t):tG} on i = 1 r X, we see that

lim sup n [ d ( F ( t n ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , F ( t n + t i t j ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) ) ] = 0 , lim sup n [ d ( F ( t n ) ( x l 1 2 , x l 1 3 , , x l 1 r , x l 1 1 ) , F ( t n + t i t j ) ( x l 1 2 , x l 1 3 , , x l 1 r , x l 1 1 ) ) ] = 0 , lim sup n [ d ( F ( t n ) ( x l 1 r , x l 1 1 , x l 1 2 , , x l 1 r 1 ) , F ( t n + t i t j ) ( x l 1 r , x l 1 1 , x l 1 2 , , x l 1 r 1 ) ) ] = 0 ,

which implies

lim sup n d ( F ( t n ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , F ( t i t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) lim sup n d ( F ( t n ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , F ( t n + t i t j ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) ) + lim sup n d ( F ( t n + t i t j ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , F ( t i t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) lim sup n d ( F ( t i t j ) ( F ( t n ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , F ( t n ) ( x l 1 2 , x l 1 3 , , x l 1 r , x l 1 1 ) , , F ( t n ) ( x l 1 r , x l 1 1 , x l 1 2 , , x l 1 r 1 ) ) , F ( t i t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) k ( t i t j ) r [ d ( F ( t n ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , x l 1 ) + d ( F ( t n ) ( x l 1 2 , x l 1 3 , , x l 1 r , x l 1 1 ) , x l 2 ) + + d ( F ( t n ) ( x l 1 r , x l 1 1 , x l 1 2 , , x l 1 r 1 ) , x l r ) ] k ( t i t j ) r D l 1 .
(10)

Similarly, one can show that

lim sup n d ( F ( t n ) ( x l 1 2 , x l 1 3 , , x l 1 r , x l 1 1 ) , F ( t i t j ) ( x l 2 , x l 3 , , x l r , x l 1 ) ) k ( t i t j ) r D l 1 ,
(11)
lim sup n d ( F ( t n ) ( x l 1 r , x l 1 1 , x l 1 2 , , x l 1 r 1 ) , F ( t i t j ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) k ( t i t j ) r D l 1 .
(12)

Then it follows from (9), (10), (11) and (12) that for each i>j1,

d ( F ( t i ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) k ( t j ) r k( t i t j ) D l 1 ,

which implies that for each n1,

sup { d ( F ( t i ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) : i , j n } = sup { d ( F ( t i ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) : i > j n } sup { k ( t j ) r k ( t i t j ) D l 1 : i > j n } D l 1 r sup { k ( t j ) : j n } sup { k ( t i t j ) : i > j n } D l 1 r sup { k ( t j ) : j n } sup { k ( t ) : G t t n + 1 t n } .
(13)

Hence, by using (III) and (9), we have

D l = lim sup n [ d ( x l + 1 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + d ( x l + 1 2 , F ( t n ) ( x l 2 , x l 3 , , x l r , x l 1 ) ) + + d ( x l + 1 r , F ( t n ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) ] c [ A 1 ( { F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) } ) + A 2 ( { F ( t n ) ( x l 2 , x l 3 , , x l r , x l 1 ) } ) + + A r ( { F ( t n ) ( x l r , x l 1 , x l 2 , , x l r 1 ) } ) ] c lim sup n { d ( F ( t i ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t j ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + d ( F ( t i ) ( x l 2 , x l 3 , , x l r , x l 1 ) , F ( t j ) ( x l 2 , x l 3 , , x l r , x l 1 ) ) + + d ( F ( t i ) ( x l r , x l 1 , x l 2 , , x l r 1 ) , F ( t j ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) : i , j n } c D l 1 lim sup n k ( t n ) lim sup n { k ( t ) : G t t n + 1 t n } c k k ˜ D l 1 h D l 1 h 2 D l 2 = h l D 0 .
(14)

Hence, by the asymptotic regularity of on i = 1 r X, we have, for each integer n1,

d ( x l + 1 1 , x l 1 ) d ( x l + 1 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + d ( x l 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) d ( x l + 1 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + lim sup m d ( F ( t m ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) d ( x l + 1 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + lim sup m d ( F ( t m ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , F ( t m + t n ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) ) + lim sup m d ( F ( t m + t n ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) d ( x l + 1 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + k ( t n ) r lim sup m [ d ( x l 1 , F ( t m ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) ) + d ( x l 2 , F ( t m ) ( x l 1 2 , x l 1 3 , , x l 1 r , x l 1 1 ) ) + + d ( x l r , F ( t m ) ( x l 1 r , x l 1 1 , x l 1 2 , , x l 1 r 1 ) ) ] d ( x l + 1 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + k ( t n ) r D l 1 ,
(15)

which implies

d ( x l + 1 1 , x l 1 ) lim sup n [ d ( x l + 1 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + d ( x l + 1 2 , F ( t n ) ( x l 2 , x l 3 , , x l r , x l 1 ) ) + + d ( x l + 1 r , F ( t n ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) ] + 1 r D l 1 lim sup n k ( t n ) D l + 1 r k D l 1 .
(16)

It follows from (14) that

d ( x l + 1 1 , x l 1 ) D l + 1 r k D l 1 ( h l + 1 r k h l 1 ) D 0 h l 1 2 D 0 max { h , k r } .

Similarly, one can deduce that

d ( x l + 1 2 , x l 2 ) h l 1 2 D 0 max { h , k r } ,
(17)
d ( x l + 1 r , x l r ) h l 1 2 D 0 max { h , k r } .
(18)

Thus, we have l = 0 d( x l + 1 1 , x l 1 )2 D 0 max{h, k r } l = 0 h l 1 <, …, l = 0 d( x l + 1 r , x l r )<. Consequently, { x l 1 },,{ x l r } are Cauchy and hence convergent as X is complete. Let x 1 = lim l x l 1 , …, x r = lim l x l r , then, for each sG, by the continuity of F(s) we have

lim l d ( F ( s ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( s ) ( x 1 , x 2 , x 3 , , x r ) ) = 0 , lim l d ( F ( s ) ( x l 2 , x l 3 , , x l r , x l 1 ) , F ( s ) ( x l 2 , x l 3 , , x l r , x l 1 ) ) = 0 , lim l d ( F ( s ) ( x l r , x l 1 , x l 2 , , x l r 1 ) , F ( s ) ( x l r , x l 1 , x l 2 , , x l r 1 ) ) = 0 .

On the other hand, from (15) we have actually proven the following inequalities:

lim sup n d ( F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) , x l 1 ) k ( t n ) r D l 1 1 r k ( t n ) h l 1 D 0 , lim sup n d ( F ( t n ) ( x l 2 , x l 3 , , x l r , x l 1 ) , x l 2 ) 1 r k ( t n ) h l 1 D 0 , lim sup n d ( F ( t n ) ( x l r , x l 1 , x l 2 , , x l r 1 ) , x l r ) 1 r k ( t n ) h l 1 D 0 .

Since lim n k( t n )=k, it follows that

lim sup n d ( x 1 , F ( t n ) ( x 1 , x 2 , x 3 , , x r ) ) = d ( x 1 , x l 1 ) + lim sup n d ( x l 1 , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) + lim sup n d ( F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) , F ( t n ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) d ( x 1 , x l 1 ) + 1 r lim sup n k ( t n ) h l 1 D 0 d ( x 1 , x l 1 ) + 1 r k h l 1 D 0 0 , l .

Similarly, one can obtain that

lim sup n d ( x 2 , F ( t n ) ( x 2 , x 3 , , x r , x 1 ) ) d ( x 2 , x l 2 ) + 1 r k h l 1 D 0 0 , l , lim sup n d ( x r , F ( t n ) ( x r , x 1 , x 2 , , x r 1 ) ) d ( x r , x l r ) + 1 r k h l 1 D 0 0 , l ,

i.e., lim n d( x 1 ,F( t n )( x 1 , x 2 , x 3 ,, x r ))=0, …, lim n d( x r ,F( t n )( x r , x 1 , x 2 ,, x r 1 ))=0. Hence, for each sG, by the continuity of F(s), we deduce

d ( x 1 , F ( s ) ( x 1 , x 2 , x 3 , , x r ) ) = lim l d ( x l 1 , F ( s ) ( x l 1 , x l 2 , x l 3 , , x l r ) ) lim l lim sup n d ( x l 1 , F ( t n + s ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) ) lim l lim sup n d ( x l 1 , F ( t n ) ( x l 1 1 , x l 1 2 , x l 1 3 , , x l 1 r ) ) + lim l lim sup n d ( F ( t n ) ( x l 1 1 , x l 1 2 , , x l 1 r ) , F ( t n + s ) ( x l 1 1 , x l 1 2 , , x l 1 r ) ) lim l D l 1 lim l h l 1 D 0 = 0 .

Similarly, we get that

d ( x 2 , F ( s ) ( x 2 , x 3 , , x r , x 1 ) ) = 0 , d ( x 3 , F ( s ) ( x 3 , x 4 , , x r , x 1 , x 2 ) ) = 0 , d ( x r , F ( s ) ( x r , x 1 , x 2 , , x r 1 ) ) = 0 .

Then we have d( x 1 ,F(s)( x 1 , x 2 , x 3 ,, x r ))=0, …, d( x 2 ,F(s)( x 2 , x 3 ,, x r , x 1 ))=0, i.e., F(s)( x 1 , x 2 , x 3 ,, x r )= x 1 , F(s)( x 2 , x 3 ,, x r , x 1 )= x 2 , …, F(s)( x r , x 1 , x 2 ,, x r 1 )= x r for each sG. □

The following corollary is related to the simplest uniformly Lipschitzian semigroup defined in Definition 3.2.

Corollary 3.1 Let (X,d) be a complete bounded metric space with uniform normal structure, and let ={ F n :nN} be the simplest asymptotically regular uniformly Lipschitzian semigroup of self-mappings on i = 1 r X with property (P) and satisfying

k< 1 N ( X ) .

Then there exist some x 1 , x 2 , x 3 ,, x r X such that F( x 1 , x 2 , x 3 ,, x r )= x 1 , …, F( x 2 , x 3 ,, x r , x 1 )= x 2 and F( x r , x 1 , x 2 ,, x r 1 )= x r for all tG.

From Remark 2.1 and Theorem 3.1, we immediately obtain the following corollary.

Corollary 3.2 Let (X,d) be a complete bounded metric space with property (P) and uniform normal structure, and let ={F(t):tG} be an asymptotically regular semigroup on i = 1 r X satisfying

k k ˜ < 1 N ( X ) .

Then there exist some x 1 , x 2 , x 3 ,, x r X such that F(s)( x 1 , x 2 , x 3 ,, x r )= x 1 , F(s)( x 2 , x 3 ,, x r , x 1 )= x 2 , …, F(s)( x r , x 1 , x 2 ,, x r 1 )= x r for all tG.

For r=1,2,3 in Theorem 3.1, we get the following two corollaries which are due to Soliman [27].

Corollary 3.3 [27]

Let (X,d) be a complete bounded metric space with property () and uniform normal structure, and let ={F(t):tG} be an asymptotically regular semigroup on X×X×X satisfying

k k ˜ < 1 N ( X ) .

Then there exist some x,y,zX such that F(s)(x,y,z)=x, F(s)(y,z,x)=y and F(s)(z,x,y)=z for all tG.

Corollary 3.4 [27]

Let (X,d) be a complete bounded metric space with property (P) and uniform normal structure, and let ={F(t):tG} be an asymptotically regular semigroup on X×X×X satisfying

k k ˜ < 1 N ¯ ( X ) .

Then there exist some x,y,zX such that F(s)(x,y,z)=x, F(s)(y,z,x)=y and F(s)(z,x,y)=z for all tG.

Remark 3.2 It is well known that the Lipschitzian mapping is uniformly continuous. It is natural to ask if there is a contractive mapping definition which does not force it to be continuous. It was answered affirmatively by Kannan. It is clear that Lipschitzian mappings are always continuous and Kannan type mappings are not necessarily continuous. It will be interesting to establish Theorem 3.1 for representative ψ={F(t):tG} on i = 1 r X satisfying the following condition:

d ( F ( t ) ( x 1 , x 2 , , x r ) , F ( t ) ( y 1 , y 2 , , y r ) ) β r [ d ( x 1 , F ( t ) ( x 1 , x 2 , , x r ) ) + d ( y 1 , F ( t ) ( y 1 , y 2 , , y r ) ) + d ( x 2 , F ( t ) ( x 2 , x 3 , , x r , x 1 ) ) + d ( y 2 , F ( t ) ( y 2 , y 3 , , y r , y 1 ) ) + d ( x r , F ( t ) ( x r , x 1 , , x r 1 ) ) + d ( y r , F ( t ) ( y r , y 1 , , y r 1 ) ) ]

for all x 1 , x 2 ,, x r , y 1 , y 2 ,, y r X and 0<β.

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Soliman, A.H. Results on n-tupled fixed points in metric spaces with uniform normal structure. Fixed Point Theory Appl 2014, 168 (2014). https://doi.org/10.1186/1687-1812-2014-168

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Keywords

  • coupled fixed point
  • tripled fixed point
  • n-fixed point
  • asymptotically regular semigroup
  • uniform normal structure
  • convexity structure