Skip to main content

On new generalizations of Smarzewski’s fixed point theorem

Abstract

In this work, we prove some new generalizations of Smarzewski’s fixed point theorem and some new fixed point theorems which are original and quite different from the well-known results in the literature.

MSC:46B20, 47H09, 54H25.

1 Introduction and preliminaries

Let (X,) be a normed space with its zero vector θ. We use B(X) and S(X) to denote respectively the closed unit ball and unit sphere centered at θ with radius 1, that is,

B(X)= { x X : x 1 }

and

S(X)= { x X : x = 1 } .

The notion of uniformly convex (UC, for short) Banach space was introduced by Clarkson [1], and the research of geometric properties of the Banach space started from 1936. The function δ X :[0,2][0,1], defined by

δ X (ε)=inf { 1 x + y 2 : x , y B ( X ) , x y ε } for ε[0,2],

is called the modulus of convexity of X. The normed space X is called uniformly convex if δ X (ε)>0 for every ε(0,2]. It is well known that a uniformly convex Banach space is reflexive and all Hilbert spaces and Banach spaces p and L p (1<p<) all are uniformly convex; see, e.g., [27] for more details. The normed space X is said to be strictly convex if x+y<2 whenever x,yS(X) with xy>0. It is obvious that a Banach space X is strictly convex if and only if δ X (2)=1. It is well known that the strict convexity of a normed space X can be characterized by the properties: for any nonzero vectors x,yX, if x+y=x+y, then y=cx for some real c>0. For each ε>0, the modulus of convexity of X in the direction zS(X) is defined by

δ X (ε,z)=inf { 1 x + y 2 : x , y B ( X ) , x y = λ z , | λ | ε } .

Clearly, δ X (ε)=inf{ δ X (ε,z):zS(X)}. The Banach space X is called uniformly convex in every direction (UCED, for short) if for any zS(X) and ε>0, δ(ε,z)>0. Some characterizations of UCED Banach spaces were proved by Day et al. [7]; see also [4].

Fact 1.1 (see, e.g., [24, 7])

  1. (a)

    Every UC Banach space is UCED.

  2. (b)

    Every UCED Banach space is strictly convex.

Let (X,) be a Banach space and K be a given nonempty closed subset of X. For xX and a bounded sequence { x n }X, define the asymptotic radius of { x n } at x as the number

r ( x , { x n } ) = lim sup n x x n .

The asymptotic radius of { x n } with respect to K is defined by

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

and the set

A ( K , { x n } ) = { x K : r ( x , { x n } ) = r ( K , { x n } ) }

is called the asymptotic center of { x n } with respect to K. For any bounded sequence { x n } in X, r(x,{ x n }) is easily seen to be a nonnegative, continuous and convex functional of xX. Moreover, if K is a nonempty convex subset of X, then A(K,{ x n }) is also convex.

Fact 1.2 [[8], Lemma 2.2]

Every bounded sequence in a UCED Banach space X has a unique asymptotic center with respect to any nonempty weakly compact convex subset of X.

Definition 1.1 A normed space (X,) is said to have the (UAC)-property if every bounded sequence in X has a unique asymptotic center with respect to any nonempty weakly compact convex subset of X.

According to Facts 1.1 and 1.2, it is easy to know that Hilbert spaces, UC Banach spaces and UCED Banach spaces all have the (UAC)-property.

Let C be a nonempty subset of a normed space (X,) and T:CX be a mapping. T is said to be nonexpansive if

TxTyxyfor all x,yC.

The concept of firmly nonexpansive mappings was introduced by Bruck [9]. Let λ(0,1). The mapping T is said to be λ-firmly nonexpansive [9] if

TxTy ( 1 λ ) ( x y ) + λ ( T x T y ) for all x,yC.

It is obvious that every λ-firmly nonexpansive mapping is nonexpansive, but the converse is not true. The following example shows that there exists a nonexpansive mapping which is not a λ-firmly nonexpansive mapping for some λ(0,1).

Example A Let X=R with the absolute-value norm || and C=[2,10]. Let T:CX be defined by Tx=x. Then T is a nonexpansive mapping. For x=6, y=4 and λ= 1 2 , we have

|T(x)T(y)|=2>0=|(1λ)(xy)+λ ( T ( x ) T ( y ) ) |,

which deduces that T is not a 1 2 -firmly nonexpansive mapping. In fact, T is not λ-firmly nonexpansive for all λ(0,1).

In 1965, Browder [10], Kirk [11] and Göhde [12] proved respectively that every nonexpansive mapping T from a nonempty weakly compact convex subset K of a uniformly convex Banach space X into itself has a fixed point. It is know that the convexity of sets and mappings plays an important role in fixed point theory and the union of convex sets does not ensure that it is convex. In 1991, Smarzewski [13] proved the following interesting theorem.

Theorem 1.1 (Smarzewski [13])

Let X be a uniformly convex Banach space and C= k = 1 n C k be a finite union of nonempty weakly compact convex subsets C k of X. If T:CC is a λ-firmly nonexpansive mapping for some λ(0,1), then T has a fixed point in C.

Smarzewski’s fixed point theorem (i.e., Theorem 1.1) is not always true if T is merely nonexpansive, even in X=R.

Example B [13]

Let X=R with the absolute-value norm || and C=[2,1][2,1]. Then the mapping T:CC defined by Tx=x is nonexpansive and fixed point free.

In this paper, in order to promote Smarzewski’s fixed point theorem, we first introduce the concept of reactive firmly nonexpansive mappings.

Definition 1.2 Let C be a nonempty subset of a normed space (X,) and φ:C×C[0,1) be a function. A mapping T:CX is said to be reactive firmly nonexpansive with respect to φ if

TxTy ( 1 φ ( x , y ) ) ( x y ) + φ ( x , y ) ( T x T y ) for all x,yC.

Remark 1.1

  1. (a)

    Every reactive firmly nonexpansive mapping is nonexpansive.

  2. (b)

    It is obvious that any λ-firmly nonexpansive mapping is reactive firmly nonexpansive with respect to the function φ defined by φ(s,t)=λ for all (s,t)C×C.

Example C Let X=R with the absolute-value norm || and C=[2,5][10,20]. Let T:CX be defined by

Tx:= { 3 5 x , if  x [ 2 , 5 ] , 1 2 x , if  x [ 10 , 20 ] .

Then the following statements hold.

  1. (a)

    T is a nonexpansive mapping.

  2. (b)

    T is not 1 3 -firmly nonexpansive.

  3. (c)

    Define φ:C×C[0,1) by

    φ(s,t):= { 1 10 , if  s , t [ 2 , 5 ] , 1 3 , otherwise.

Then T is reactive firmly nonexpansive with respect to φ.

Proof Obviously, statement (a) holds. To see (b), let x=4 and y=3. Since

|T(x)T(y)|= 3 5 > 7 15 =| ( 1 1 3 ) (xy)+ 1 3 (TxTy)|,

we show that T is not 1 3 -firmly nonexpansive. Finally, we prove (c). We consider the following four possible cases to verify

|TxTy|| ( 1 φ ( x , y ) ) (xy)+φ(x,y)(TxTy)|
(1.1)

for all x,yC.

Case 1. If x,y[2,5], then

|T(x)T(y)|= 3 5 |xy|

and

| ( 1 φ ( x , y ) ) ( x y ) + φ ( x , y ) ( T x T y ) | = | ( 1 1 10 ) ( x y ) + 1 10 ( T x T y ) | = 21 25 | x y | .

So (1.1) holds for all x,y[2,5].

Case 2. If x[2,5] and y[10,20], then

|T(x)T(y)|= 1 2 y 3 5 x

and

| ( 1 φ ( x , y ) ) ( x y ) + φ ( x , y ) ( T x T y ) | = | ( 1 1 3 ) ( x y ) + 1 3 ( T x T y ) | = 1 2 y 7 15 x .

Since

( 1 2 y 7 15 x ) ( 1 2 y 3 5 x ) = 2 15 x>0,

we prove that (1.1) holds for all x[2,5] and y[10,20].

Case 3. If x[10,20] and y[2,5], then

|T(x)T(y)|= 1 2 x 3 5 y

and

| ( 1 φ ( x , y ) ) (xy)+φ(x,y)(TxTy)|= 1 2 x 7 15 y.

Since

( 1 2 x 7 15 y ) ( 1 2 x 3 5 y ) = 2 15 y>0,

we prove that (1.1) holds for all x[10,20] and y[2,5].

Case 4. If x,y[10,20], then

|T(x)T(y)|= 1 2 |xy|

and

| ( 1 φ ( x , y ) ) ( x y ) + φ ( x , y ) ( T x T y ) | = | ( 1 1 3 ) ( x y ) + 1 3 ( T x T y ) | = 1 2 | x y | .

So (1.1) holds for all x,y[10,20].

By Cases 1-4, we verify that inequality (1.1) holds for all x,yC. Hence T is reactive firmly nonexpansive with respect to φ and (c) is proved. □

In this paper, we establish some generalizations of Smarzewski’s fixed point theorem for reactive firmly nonexpansive mappings and some new fixed point theorems which are original and quite different from the well-known results in the literature.

2 New generalizations of Smarzewski’s fixed point theorem and applications to fixed point theory

In this section, we first establish a new fixed point theorem for reactive firmly nonexpansive mappings which is generalized Smarzewski’s fixed point theorem. We assume 0<φ(s,t)<1 for all (s,t)C×C in the following main theorem.

Theorem 2.1 Let X be a strictly convex Banach space with its zero vector θ and C= k = 1 n C k be a finite union of nonempty weakly compact convex subsets C k of X. Let φ:C×C(0,1) be a function and T:CC be a mapping. Suppose that

  1. (a)

    X has the (UAC)-property,

  2. (b)

    T is reactive firmly nonexpansive with respect to φ.

Then T has a fixed point in C.

Proof Let zC be given. Since C is T-invariant and C is bounded, the sequence { T j z } j = 1 C is bounded. Define the functional r:X[0,), the asymptotic radius of { T j z} at xX, by

r(x):=r ( x , { T j z } ) = lim sup j x T j z .

For each 1kn, let the number

r( C k ):=r ( C k , { T j z } ) =inf { r ( x ) : x C k }

and the set

A( C k ):=A ( C k , { T j z } ) = { x C k : r ( x ) = r ( C k ) }

be respectively the asymptotic radius and the asymptotic center of the sequence { T j z} with respect to C k . Since X has the (UAC)-property, let x k C k be the unique asymptotic center of { T j z} with respect to C k for 1kn. So

r( x k )=r( C k )=inf { r ( x ) : x C k } for each 1kn.

For any k and j, since T is nonexpansive, we have

T x k T j z x k T j 1 z ,

which implies

r(T x k )= lim sup j T x k T j z lim sup j x k T j 1 z =r( x k )for all k.
(2.1)

Let

m=min { r ( x k ) : 1 k n } .

Clearly, m<. For arbitrary xC= k = 1 n C k , x C k x for some k x , 1 k x n. Thus we have

r(x)r( C k )=r( x k )m.
(2.2)

Taking the infimum for x over C yields r(C,{ T j z})m. Conversely, suppose r( x k 0 )=m for some k 0 , 1 k 0 n. Then

m=r( x k 0 ) inf x C r(x)=r ( C , { T j z } ) .
(2.3)

By (2.2) and (2.3), we prove

r ( C , { T j z } ) =m.
(2.4)

Put

L= { x k : r ( x k ) = m } .

It is obvious that (L)n, where (L) is the cardinal number of L. We claim that for uC with r(u)=m if and only if uL. Indeed, it suffices to show that if uC with r(u)=m, then uL. Let uC= k = 1 n C k . Then there is some k u , 1 k u n, such that u C k u . So we obtain

r(u)=mr( x k u )r(u),

and hence

r(u)=m=r( x k u ).

By the uniqueness of asymptotic center x k u , we have u= x k u , which means that uL.

Next, we will prove TLL (i.e., L is T-invariant). For any uL, since r(u)=m, from (2.1) and (2.4) it follows that

m=r(u)r(Tu) inf x C r(x)=r ( C , { T j z } ) =m,

and hence r(Tu)=m. According to our claim, we get TuL, which completes the assertion. Now, we show that T has a fixed point in L. Suppose to the contrary that T has no fixed point in L, that is, Tuu for all uL. Then T x k C k for all x k L. Indeed, if T x k C k for some x k L, since

m=r( x k )r(T x k ) inf x C k r(x)=r( x k )=m,

and by the uniqueness of asymptotic center x k , we obtain T x k = x k . This means that x k is a fixed point for T, contradicting our assumption, hence T x k C k for all x k L. For TLL, there exists wL such that T α w=w for some 1α(L). Since

0wTw= T α w T α + 1 w T α 1 w T α w T w T 2 w wTw,

it follows that

0<ξ:=wTw= T w T 2 w == T α 1 w T α w = T α w T α + 1 w .
(2.5)

Since T is reactive firmly nonexpansive with respect to φ, we have

ξ = T i w T i + 1 w ( 1 φ ( T i 1 w , T i w ) ) ( T i 1 w T i w ) + φ ( T i 1 w , T i w ) ( T i w T i + 1 w ) ( 1 φ ( T i 1 w , T i w ) ) T i 1 w T i w + φ ( T i 1 w , T i w ) T i w T i + 1 w = ξ

for 1iα, where T 0 =I (the identity mapping). Therefore, in view of strict convexity of the norm, there is t i >0 such that

T i 1 w T i w= t i ( T i w T i + 1 w ) for 1iα.
(2.6)

From (2.5) and (2.6), we have

ξ= T i 1 w T i w = t i T i w T i + 1 w = t i ξ,

which implies that t i =1 for all i. So, by (2.6) again, we get

θv:=wTw=Tw T 2 w== T α 1 w T α w.

Since

T α w= T α 1 wv= ( T α 2 w v ) v= T α 2 w2v==wαv,

we obtain

w= T α w=wαv,

which implies v=θ, contradicting the fact that vθ. Therefore T must have a fixed point in LC and this completes the proof. □

The following results are immediate consequences of Theorem 2.1.

Corollary 2.1 Let X be a strictly convex Banach space with its zero vector θ and C= k = 1 n C k be a finite union of nonempty weakly compact convex subsets C k of X. Let T:CC be a mapping. Suppose that

  1. (a)

    X has the (UAC)-property,

  2. (b)

    T is λ-firmly nonexpansive for some λ(0,1).

Then T has a fixed point in C.

Corollary 2.2 Let C= k = 1 n C k be a finite union of nonempty weakly compact convex subsets C k of a UCED Banach space X and φ:C×C(0,1) be a function. If T:CC is T is reactive firmly nonexpansive with respect to φ, then T has a fixed point in C.

Corollary 2.3 [[8], Theorem 2.8]

Let C= k = 1 n C k be a finite union of nonempty weakly compact convex subsets C k of a UCED Banach space X. If T:CC is λ-firmly nonexpansive for some λ(0,1), then T has a fixed point in C.

Remark 2.1 Theorem 2.1 and Corollaries 2.1 and 2.2 all generalize and improve Smarzewski’s fixed point theorem, [[8], Theorem 2.8] and [[14], Theorems 2.8, 2.9].

In Theorem 2.1, if C 1 = C 2 == C n :=C, then we obtain the following new fixed point theorem.

Theorem 2.2 Let X be a strictly convex Banach space with its zero vector θ and C be a nonempty weakly compact convex subset of X. Let φ:C×C(0,1) be a function and T:CC be a mapping. Suppose that

  1. (a)

    X has the (UAC)-property,

  2. (b)

    T is reactive firmly nonexpansive with respect to φ.

Then T has a fixed point in C.

The following results are immediate from Theorem 2.2.

Corollary 2.4 Let X be a strictly convex Banach space with its zero vector θ and C be a nonempty weakly compact convex subset of X. Let φ:C×C(0,1) be a function and T:CC be a mapping. Suppose that

  1. (a)

    X has the (UAC)-property,

  2. (b)

    T is λ-firmly nonexpansive for some λ(0,1).

Then T has a fixed point in C.

Corollary 2.5 Let C be a nonempty weakly compact convex subset of a UCED Banach space X. Let φ:C×C(0,1) be a function. If T:CC is reactive firmly nonexpansive with respect to φ, then T has a fixed point in C.

Corollary 2.6 Let C be a nonempty weakly compact convex subset of a UCED Banach space X. If T:CC is λ-firmly nonexpansive for some λ(0,1), then T has a fixed point in C.

Definition 2.1 Let C be a nonempty subset of a normed space (X,) and α,β(0,1). A mapping T:CX is said to be (α,β)-firmly nonexpansive if

TxTy ( 1 β ) [ ( 1 α ) ( x y ) + α ( T x T y ) ] + β ( T x T y ) for all x,yC.

Finally, by applying Theorem 2.1, we give some new fixed point theorems for (α,β)-firmly nonexpansive mappings.

Theorem 2.3 Let X be a strictly convex Banach space with its zero vector θ and C= k = 1 n C k be a finite union of nonempty weakly compact convex subsets C k of X. Let T:CC be a mapping and α, β be positive real numbers satisfying 0<(1α)(1β)<1. Suppose that

  1. (a)

    X has the (UAC)-property,

  2. (b)

    T is (α,β)-firmly nonexpansive.

Then T has a fixed point in C.

Proof Since α,β>0 and 0<(1α)(1β)<1, we have α+βαβ(0,1). Define φ:C×C(0,1) by

φ(s,t):=α+βαβfor (s,t)C×C.

Due to T is (α,β)-firmly nonexpansive, we obtain

T x T y ( 1 β ) [ ( 1 α ) ( x y ) + α ( T x T y ) ] + β ( T x T y ) = ( 1 φ ( x , y ) ) ( x y ) + φ ( x , y ) ( T x T y )

for all x,yC. So, T is reactive firmly nonexpansive with respect to φ. Therefore the conclusion follows from Theorem 2.1. □

Corollary 2.7 Let C be a nonempty weakly compact convex subset of a UCED Banach space X. Let α, β be positive real numbers satisfying 0<(1α)(1β)<1. If T:CC is (α,β)-firmly nonexpansive, then T has a fixed point in C.

Corollary 2.8 Let C be a nonempty weakly compact convex subset of a uniformly convex Banach space X. Let α, β be positive real numbers satisfying 0<(1α)(1β)<1. If T:CC is (α,β)-firmly nonexpansive, then T has a fixed point in C.

References

  1. Clarkson JA: Uniformly convex spaces. Trans. Am. Math. Soc. 1936, 40: 296–414.

    Article  MathSciNet  MATH  Google Scholar 

  2. Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.

    MATH  Google Scholar 

  3. Goebel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.

    Book  MATH  Google Scholar 

  4. Megginson RE: An Introduction to Banach Space Theory. Springer, New York; 1998.

    Book  MATH  Google Scholar 

  5. Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.

    MATH  Google Scholar 

  6. Khamsi MA, Kirk WA: An Introduction to Metric Spaces and Fixed Point Theory. Wiley, New York; 2001.

    Book  MATH  Google Scholar 

  7. Day MM, James RC, Swaminathan S: Normed linear spaces that are uniformly convex in every direction. Can. J. Math. 1971, 23: 1051–1059. 10.4153/CJM-1971-109-5

    Article  MathSciNet  MATH  Google Scholar 

  8. Du W-S, Huang Y-Y, Yen C-L:Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces. Int. J. Math. Math. Sci. 2002, 31: 251–257. 10.1155/S0161171202107113

    Article  MathSciNet  MATH  Google Scholar 

  9. Bruck RE: Nonexpansive projections on subsets of Banach spaces. Pac. J. Math. 1973, 47: 341–355. 10.2140/pjm.1973.47.341

    Article  MathSciNet  MATH  Google Scholar 

  10. Browder FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 1965, 54: 1041–1044. 10.1073/pnas.54.4.1041

    Article  MathSciNet  MATH  Google Scholar 

  11. Kirk WA: A fixed point theorem for mappings which do not increase distances. Am. Math. Mon. 1965, 72: 1004–1006. 10.2307/2313345

    Article  MathSciNet  MATH  Google Scholar 

  12. Göhde D: Zum Prinzip der kontraktiven Abbildung. Math. Nachr. 1965, 30: 251–258. (in German) 10.1002/mana.19650300312

    Article  MathSciNet  MATH  Google Scholar 

  13. Smarzewski R: On firmly nonexpansive mappings. Proc. Am. Math. Soc. 1991, 113: 723–725. 10.1090/S0002-9939-1991-1050023-1

    Article  MathSciNet  MATH  Google Scholar 

  14. Kaczor W: Fixed points of asymptotically regular nonexpansive mappings on nonconvex sets. Abstr. Appl. Anal. 2003,2003(2):83–91. 10.1155/S1085337503205054

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author was supported by Grant No. MOST 103-2115-M-017-001 of the Ministry of Science and Technology of the Republic of China.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wei-Shih Du.

Additional information

Competing interests

The author declares that he has no competing interests.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, WS. On new generalizations of Smarzewski’s fixed point theorem. J Inequal Appl 2014, 493 (2014). https://doi.org/10.1186/1029-242X-2014-493

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-493

Keywords