# Common fixed points and best approximations in locally convex spaces

- Saleh Abdullah Al-Mezel
^{1}Email author

**2011**:99

https://doi.org/10.1186/1687-1812-2011-99

© Al-Mezel; licensee Springer. 2011

**Received: **13 July 2011

**Accepted: **7 December 2011

**Published: **7 December 2011

## Abstract

We extend the main results of Aamri and El Moutawakil and Pant to the weakly compatible or *R*-weakly commuting pair (*T, f*) of maps, where *T* is multivalued. As applications, common fixed point theorems are obtained for new class of maps called *R*-subcommuting maps in the setup of locally convex topological vector spaces. We also study some results on best approximation via common fixed point theorems.

**2000 MSC** 41A65; 46A03; 47H10; 54H25.

## Keywords

*R*-subcommuting maps

*R*-weakly commuting.

## 1. Introduction and preliminaries

The study of common fixed points of compatible mappings has emerged as an area of vigorous research activity ever since Jungck [1] introduced the notion of compatible mappings. The concept of compatible mappings was introduced as a generalization of commuting mappings. In 1994, Pant [2] introduced the concept of *R*-weakly commuting maps which is more general than compatibility of two maps. Several authors discussed various results on coincidence and common fixed point theorem for compatible single-valued and multivalued maps. Among others Kaneko [3] extended well-known result of Nadler [4] to multivalued *f*-contraction maps as follows.

**Theorem 1.1**. *Let* (*X, d*) *be a complete metric space and f* : *X* → *X be a continuous map. Let T be closed bounded valued f-contraction map on X which commutes with f and T*(*X*) ⊆ *f*(*X*). *Then, f and T have a coincidence point in X. Suppose moreover that one of the following holds: either* (*i*) *fx* ≠ *f*^{2}*x implies fx* ∉ *Tx or* (*ii*) *fx* ∈ *Tx implies* lim *f*^{
n
}*x exists. Then, f and T have a common fixed point*.

It is pointed out in [5] that condition (i) in the above result implies condition (ii). A great deal of work has been done on common fixed points for commutative, weakly commutative, *R*-weakly commutative and compatible maps (see [1, 2, 6–11]). The following more general common fixed point theorem for 1-subcommutative maps was proved in [12].

**Theorem 1.2**.

*Let M be a nonempty τ-bounded, τ-sequentially complete and q-starshaped subset of a Hausdorff locally convex space*(

*E, τ*).

*Let T and I be selfmaps of M. Suppose that T is I-nonexpansive, I*(

*M*) =

*M, Iq*=

*q, I is nonexpansive and affine. If T and I are*1

*-subcommutative maps, then T and I have a common fixed point provided one of the following conditions holds:*

- (i)
*M is τ-sequentially compact*. - (ii)
*T is a compact map*. - (iii)
*M is weakly compact in*(*E, τ*),*I is weakly continuous and I - T is demiclosed at*0. - (iv)
*M is weakly compact in an Opial space*(*E, τ*)*and I is weakly continuous*.

In this article, we begin with a common fixed point result for a pair (*T, f*) of weakly compatible as well as *R*-weakly commuting maps in the setting of a Hausdorff locally convex space. This result provides a nonmetrizable analogue of Theorem 1.2 for weakly compatible as well as *R*-weakly commutative pair of maps and improves main results of Davies [13] and Jungck [14]. As applications, we establish some theorems concerning common fixed points of a new class, *R*-subcommuting maps, which in turn generalize and strengthen Theorem 1.2 and the results due to Dotson [15], Jungck and Sessa [16], Lami Dozo [17] and Latif and Tweddle [18]. We also extend and unify well-known results on fixed points and common fixed points of best approximation for *R*-subcommutative maps.

*X*will denote a complete Hausdorff locally convex topological vector space unless stated otherwise,

*P*the family of continuous seminorms generating the topology of

*X*and

*K*(

*X*) the family of nonempty compact subsets of

*X*. For each

*p*∈

*P*and

*A, B*∈

*K*(

*X*), we define

*p*is only a seminorm,

*D*

_{ p }is a Hausdorff metric on

*K*(

*X*) (cf. [19]). For any

*u*∈

*X, M*⊂

*X*and

*p*∈

*P*, let

*P*

_{ M }(

*u*) = {

*y*∈

*M*:

*p*(

*y*-

*u*) =

*d*

_{ p }(

*u*,

*M*), for all

*p*∈

*P*} be the set of best

*M*-approximations to

*u*∈

*X*. For any mapping

*f*:

*M*→

*X*, we define (cf. [6])

*M*be a nonempty subset of

*X*. A mapping

*T*:

*M*→

*K*(

*M*) is called multivalued contraction if for each

*p*∈

*P*, there exists a constant

*k*

_{ p }, 0

*< k*

_{ p }

*<*1 such that for each

*x, y*∈

*M*, we have

*T*is called nonexpansive if for each

*x, y*∈

*M*and

*p*∈

*P*,

*f*:

*M*→

*M*be a single-valued map. Then,

*T*:

*M*→

*K*(

*M*) is called an

*f*-contraction if there exists

*k*

_{ p }, 0 <

*k*

_{ p }< 1 such that for each

*x*,

*y*∈

*M*and for each

*p*∈

*P*, we have

*k*

_{ p }= 1 for all

*p*∈

*P*, then

*T*is called an

*f*-nonexpansive mapping. The pair (

*T, f*) is said to be compatible if, whenever there is a sequence {

*x*

_{ n }} in

*M*satisfying $\underset{n\to \infty}{lim}f{x}_{n}\in \underset{n\to \infty}{lim}T{x}_{n}$ (provided $\underset{n\to \infty}{lim}\phantom{\rule{2.77695pt}{0ex}}f{x}_{n}$ exists in

*M*and $\underset{n\to \infty}{lim}\phantom{\rule{2.77695pt}{0ex}}T{x}_{n}$ exists in

*K*(

*M*)), then $\underset{n\to \infty}{lim}{D}_{p}\left(fT{x}_{n},\phantom{\rule{2.77695pt}{0ex}}Tf{x}_{n}\right)=0$, for all

*p*∈

*P*. The pair (

*T, f*) is called

*R*-weakly commuting, if for each

*x*∈

*M, fTx*∈

*K*(

*M*) and

for some positive real *R* and for each *p* ∈ *P*. If *R* = 1, then the pair (*T*, *f*) is called weakly commuting [10]. For *M* = *X* and *T* a single-valued, the definitions of compatibility and *R*-weak commutativity reduce to those given by Jungck [1] and Pant [2], respectively.

A point *x* in *M* is said to be a common fixed point (coincidence point) of *f* and *T* if *x* = *fx* ∈ *Tx*. (*fx* ∈ *Tx*). We denote by *F*(*f*) and *F*(*T*) the set of fixed points of *f* and *T*, respectively. A subset *M* of *X* is said to be *q*-starshaped if there exists a *q* ∈ *M*, called the starcenter of *M*, such that for any *x* ∈ *M* and 0 ≤ *α* ≤ 1, *αq* + (1 - *α*) *x* ∈ *M*.

Shahzad [20] introduced the notion of *R*-subcommuting maps and proved that this class of maps contains properly the class of commuting maps.

*T, f*) of maps when

*T*is not necessarily single-valued. Suppose

*q*∈

*F*(

*I*),

*M*is

*q*-starshaped with

*T*(

*M*) ⊂

*M*and

*f*(

*M*) ⊂

*M*. Then,

*f*and

*T*are

*R*-subcommutative if for each

*x*∈

*M, fTx*∈

*K*(

*M*) and there exists some positive real number

*R*such that

for each *p* ∈ *P*, *h* ∈ (0, 1) and *x* ∈ *M*.

Obviously, commutativity implies *R*-subcommutativity (which in turn implies R-weak commutativity) but the converse does not hold as the following example shows.

**Example 1.1**.

*Consider M*= [1, ∞)

*with the usual metric of reals. Define*

*Further* |*Tfx - ftx*| ≤ (*R/h*)|(*hTx* + (1 - *h*)*q*) - *fx*| *for all x in M, h* ∈ (0, 1) *with R* = 12 *and q* = 1 ∈ *F*(*f*). *Thus, f and T are R-subcommuting but not commuting*.

*T*from

*M*into 2

^{ X }(the family of all nonempty subsets of

*X*) is said to be demiclosed if for every net {

*x*

_{ α }} in

*M*and any

*y*

_{ α }∈

*Tx*

_{ α }such that

*x*

_{ α }converges strongly to

*x*and

*y*

_{ α }converges weakly to

*y*, we have

*x*∈

*M*and

*y*∈

*Tx*. We say

*X*satisfies Opial's condition if for each

*x*∈

*X*and every net {

*x*

_{ α }} converging weakly to

*x*, we have

The Hilbert spaces and Banach spaces having a weakly continuous duality mapping satisfy Opial's condition [17].

## 2. Main results

We use a technique due to Latif and Tweddle [18], based on the images of the composition of a pair of maps, to obtain common fixed point results for a new class of maps in the context of a metric space.

**Theorem 2.1**.

*Let X be a metric space and f*:

*X*→

*X be a map. Suppose that T*:

*X*→

*CB*(

*X*)

*is an f-contraction such that the pair*(

*T, f*)

*is weakly compatible (or R-weakly commuting) and TX*⊂

*fX such that fX is complete. Then, f and T have a common fixed point provided one of the following conditions holds for all ×*∈

*X:*

- (i)
*fx*≠*f*^{2}*x implies fx*∉*Tx* - (ii)
*fx*∈*Tx implies*$d\left(fx,{f}^{2}x\right)<max\left\{d\left(fx,Tfx\right),d\left({f}^{2}x,Tfx\right)\right\}$

*whenever right-hand side is nonzero*.

- (iii)
*fx*∈*Tx implies*$d\left(fx,{f}^{2}x\right)<max\left\{d\left(Tx,Tfx\right),d\left(fx,Tfx\right),d\left({f}^{2}x,Tfx\right),d\left(Tx,f2x\right)\right\}$

*whenever right-hand side is nonzero*.

- (iv)
*fx*∈*Tx implies*$d\left(x,fx\right)<max\left\{d\left(x,Tx\right),d\left(fx,Tx\right)\right\}$

*whenever the right-hand side is nonzero*.

- (v)
*fx*∈*Tx implies*$\begin{array}{c}d\left(fx,{f}^{2}x\right)<max\{d\left(Tx,Tfx\right),\left[d\left(Tx,fx\right)+d\left({f}^{2}x,Tfx\right)\right]\u22152,\\ \left[d\left(fx,Tfx\right)+d\left({f}^{2}x,Tx\right)\right]\u22152\}\end{array}$

*whenever the right-hand side is nonzero*.

*Proof*. Define

*Jz*=

*Tf*

^{-1}

*z*for all

*z*∈

*fX*=

*G*. Note that for each

*z*∈

*G*and

*x*,

*yf*

^{-1}

*z*, the

*f*-contractiveness of

*T*implies that

*Jz*=

*Ta*for all

*a*∈

*f*

^{-1}

*z*and

*J*is multivalued map from

*G*into

*CB*(

*G*). For any

*w, z*∈

*G*, we have

*x*∈

*f*

^{-1}

*w*and

*yf*

^{-1}

*z*. But

*T*is an

*f-*contraction so there is

*k*∈ (0, 1) such that

which implies that *J* is a contraction. It follows from Nadler's fixed point theorem [4] that there exists *z*_{0} ∈ *G* such that *z*_{0} ∈ *Jz*_{0}. Since *Jz*_{0} *= Tx*_{0} for any *x*_{0} ∈ *f*^{-1}*z*_{0}, so *fx*_{0} = *z*_{0} ∈ *Jz*_{0} = *Tx*_{0}.

*f*and

*T*,

*T*,

*f*) is

*R*-weakly commuting, then

- (i)As
*fx*_{0}∈*Tx*_{0}so we get by (2.1)$f{x}_{0}={f}^{2}{x}_{0}\in fT{x}_{0}=Tf{x}_{0}.$

*fx*

_{0}is the required common fixed point of

*f*and

*T*.

- (ii)Suppose that
*fx*_{0}≠*f*^{2}*x*_{0}. Then,$\begin{array}{cc}\hfill d\left(f{x}_{0},{f}^{2}{x}_{0}\right)& <max\left\{d\left(f{x}_{0},Tf{x}_{0}\right),d\left({f}^{2}{x}_{0},Tf{x}_{0}\right)\right\}\hfill \\ =d\left(f{x}_{0},Tf{x}_{0}\right)\le d\left(f{x}_{0},{f}^{2}{x}_{0}\right)\hfill \end{array}$

which is a contradiction. Thus, *fx*_{0} = *f*^{2}*x*_{0} and result follows from (2.1).

- (v)Suppose that
*fx*_{0}≠*f*^{2}*x*_{0}. Then,$\begin{array}{cc}\hfill d\left(f{x}_{0},{f}^{2}{x}_{0}\right)& <max\left\{d\left(T{x}_{0},Tf{x}_{0}\right)\right.,\left[d\left(f{x}_{0},T{x}_{0}\right)+d\left({f}^{2}{x}_{0},Tf{x}_{0}\right)\right]\u22152,\hfill \\ \left(\right)close="\}">\left[d\left({f}^{2}{x}_{0},T{x}_{0}\right)+d\left(f{x}_{0},Tf{x}_{0}\right)\right]\u22152\hfill \end{array}\le max\left\{d\left(f{x}_{0},{f}^{2}{x}_{0}\right),[d({f}^{2}{x}_{0},f{x}_{0})\right.+\left(\right)close="\}">d\left(f{x}_{0},{f}^{2}{x}_{0}\right)]\u22152\hfill =d(f{x}_{0},{f}^{2}{x}_{0})\hfill $

which is a contradiction. Hence, *fx*_{0} = *f* ^{2}*x*_{0} and so *fx*_{0} is the required common fixed point of *f* and *T*.

**Theorem 2.2**. *Let X be a metric space and f* : *X* → *X be a map. Suppose that T* : *X* → *C*(*X*) *is an f-Lipschitz map such that the pair* (*T, f*) *is weakly compatible (or R-weakly commuting) and cl*(*TX*) ⊂ *fX where fX is complete. If the pair* (*T, f*) *satisfies the property* (*E. A*), *then f and T have a common fixed point provided one of the conditions (i)-(v) in Theorem 2.1 holds*.

*Proof*. As the pair (

*T, f*) satisfies property (

*E. A*), there exists a sequence {

*x*

_{ n }} such that

*fx*

_{ n }→

*t*and

*t*∈ lim

*Tx*

_{ n }for some

*t*in

*X*. Since

*t*∈

*cl*(

*TX*) ⊂

*fX*so

*t*=

*fx*

_{0}for some

*x*

_{0}in

*X*. Further as

*T*is

*f*-Lipschitz, we obtain

Taking limit as *n* → ∞, we get lim *Tx*_{
n
} = *Tx*_{0} and hence *fx*_{0} ∈ *Tx*_{0}. The weak compatibility or R-weak commutativity of the pair (*T, f*) implies that (2.1) holds. The result now follows as in Theorem 1.2.

**Theorem 2.3**.

*Assume that X, f and T are as in Theorem 2.2 with the exception that T being f-Lipschitz, T satisfies the following inequality;*

*Then, conclusion of Theorem 2.2 holds*.

*Proof*. As the pair (

*T, f*) satisfies property (

*E. A*), there exists a sequence {

*x*

_{ n }} such that

*fx*

_{ n }→

*t*and

*t*∈ lim

*Tx*

_{ n }for some

*t*in

*X*. Since

*t*∈

*cl*(

*TX*) ⊂

*fX*so

*t*=

*fx*

_{0}for some

*x*

_{0}in

*X*. We claim that

*fx*

_{0}∈

*Tx*

_{0}. Assume that

*fx*

_{0}∉

*Tx*

_{0}, then we obtain

*n*→ ∞ yields,

As *fx*_{0} ∈ *A*, so *d*(*fx*_{0}, *Tx*_{0}) ≤ *H*(*A, Tx*_{0}) and hence *d*(*fx*_{0}, *Tx*_{0}) *< d*(*fx*_{0}, *Tx*_{0})*/* 2 which is a contradiction. Thus, *fx*_{0} ∈ *Tx*_{0}. The weak compatibility or *R*-weak commutativity of the pair (*T, f*) implies that (2.1) holds. The result now follows as in Theorem 1.2.

## 3. Applications

There are plenty of spaces which are not normable (see [[21], p. 113]). So it is natural to consider fixed point and approximation results in the context of a locally convex space. In this section, we show that the problem concerning the existence of common fixed points of *R*-subcommuting maps on sets not necessarily convex or compact in locally convex spaces has a solution.

**Remark 3.1**. *Theorem 2.1 (i) holds in the setup of a Hausdorff complete locally convex space X (the same proof holds with the exception that we take T* : *X* → *K*(*X*) *and apply Theorem 1* [22]*instead of Nadler's fixed point theorem to obtain a fixed point of the multivalued contraction J)*.

**Theorem 3.1**.

*Let M be a weakly compact subset of a Hausdorff complete locally convex space X which is starshaped with respect to q*∈

*M. Let f*:

*M*→

*M be an affine weakly continuous map with f*(

*M*) =

*M, f*(

*q*) =

*q, T*:

*M*→

*K*(

*M*)

*be an f-nonexpansive map and the pair*(

*T, f*)

*is R-subcommutative. Suppose the following conditions hold:*

- (a)
*fx*≠*f*^{2}*x implies λfx*+ (1 -*λ*)*q*∉*Tx, λ*≥ 1 (*cf*. [23]), - (b)
*either f*-*T is demiclosed at*0*or X is an Opial's space*.

*Then, f and T have a common fixed point*.

*Proof*. For each real number

*h*

_{ n }with 0

*< h*

_{ n }

*<*1 and

*h*

_{ n }→ 1 as

*n*→

*∞*, we define

*T*

_{ n }is

*f*-contraction map. Note that

*T*

_{ n }

*, f*) is

*R*-weakly commutative pair for each

*n*. Next, we show that if

*fx*≠

*f*

^{2}

*x*, then

*fx*∉

*T*

_{ n }

*x*for all

*n*≥ 1. Suppose that

*fx*∈

*T*

_{ n }

*x*=

*h*

_{ n }

*Tx*+ (1 -

*h*

_{ n })

*q*. Then,

*fx*=

*h*

_{ n }

*u*+ (1 -

*h*

_{ n })

*q*for some

*u*∈

*Tx*which implies that (

*h*

_{ n })

^{-1}[

*fx*- (1 -

*h*

_{ n })

*q*] ∈

*Tx*and this contradicts hypothesis (a). By Remark 3.1 each pair (

*T*

_{ n }

*, f*) has a common fixed point. That is, there is

*x*

_{ n }∈

*M*such that

The set *M* is weakly compact, we can find a subsequence still denoted by {*x*_{
n
}} such that *x*_{
n
} converges weakly to *x*_{0} ∈ *M*. Since *f* is weakly continuous so *fx*_{
n
} converges weakly to *fx*_{0}. Since *X* is Hausdorff so *x*_{0} = *fx*_{0}. As *fx*_{
n
} ∈ *T*_{
n
}*x*_{
n
} = *h*_{
n
}*Tx*_{
n
} + (1 - *h*_{
n
})*q* so there is some *u*_{
n
} ∈ *Tx*_{
n
} such that *fx*_{
n
} = *h*_{
n
}*u*_{
n
} + (1 - *h*_{
n
})*q* which implies that *fx*_{
n
} - *u*_{
n
} = ((1 - *h*_{
n
})/*h*_{
n
})(*q* - *fx*_{
n
}) converges to 0 as *n* → *∞*. Hence, by the demiclosedness of *f* - *T* at 0, we get that 0 ∈ (*f* - *T*)*x*_{0}. Thus, *x*_{0} = *fx*_{0} ∈ *Tx*_{0} as required.

In case *X* is an Opial's space, Lemma 2.5 [24] or Lemma 3.2 [25] implies that *f* - *T* is demiclosed at 0. The result now follows from the above argument.

If *T* : *M* → *M* is single-valued in Theorems 3.1, we get the following analogue of Theorem 6 [16] for a pair of maps which are not necessarily commutative in the set up of Hausdorff locally convex spaces.

**Theorem 3.2**. *Let M be a weakly compact subset of a Hausdorff complete locally convex space X which is starshaped with respect to q* ∈ *M. Suppose f and T are R-subcommutative selfmaps of M. Assume that f is continuous in the weak topology on M, f is affine, f*(*M*) = *M, f*(*q*) = *q, T is f-nonexpansive map and fx* ≠ *f*^{2}*x implies λfx* + (1 *- λ*)*q* ≠ *Tx for ×* ∈ *M and λ ≥* 1. *Then, there exists a* ∈ *M such that a* = *fa* = *Ta provided that either (i) f - T is demiclosed at* 0, *or (ii) × satisfies Opial's condition*.

If *f* is the identity on *M*, then Theorem 3.2 (i) gives the conclusion of Theorem 2 of Dotson [15] for Hausdorff locally convex spaces. A result similar to Theorem 3.2 (ii) for closed balls of reflexive Banach spaces appeared in [8].

Finally, we consider an application of Theorem 3.2 to best approximation theory; our result sets an analogue of Theorem 3.2 [6] for the maps which are not necessarily commuting in the setup of locally convex spaces and extends the corresponding results of Shahzad [20] to locally convex spaces.

**Theorem 3.3**. *Let T and f be selfmaps of a Hausdorff complete locally convex space X and M* ⊂ *X such that T*(∂*M*) ⊂ *M, where* ∂*M is the boundary of M in X. Let u* ∈ *F*(*T*) ⋂ *F*(*f*),$D={D}_{M}^{f}\left(u\right)$ be nonempty weakly compact and starshaped with respect to q ∈ *F*(*f*), *f is affine and weakly continuous, f*(*D*) = *D, and fx* ≠ *f*^{2}*x implies λfx* + (1 *- λ*)*q* ≠ *Tx for ×* ∈ *D and λ ≥* 1. *Suppose that T is f-nonexpansive on D* ⋃ {*u*} *and f is nonexpansive on P*_{
M
}(*u*) ⋃ {*u*}. *If f and T are R-subcommutative on D, then T, f have a common fixed point in P*_{
M
}(*u*) *under each one of the conditions (i)-(ii) of Theorem 3.2*.

*Proof*. Let

*y*∈

*D*. Then,

*fy*∈

*D*because

*f*(

*D*) =

*D*and hence

*f*(

*y*) ∈

*PM*(

*u*). By the definition of

*D, y*∈ ∂

*M*and since

*T*(∂

*M*) ⊂

*M*, it follows that

*Ty*∈

*M*. By

*f*-nonexpansiveness of

*T*we get

*fu*=

*u*and

*fy*∈

*P*

_{ M }(

*u*) so for each

*p*∈

*P, p*(

*Ty - u*) ≤

*p*(

*fy - u*) =

*d*

_{ p }(

*u, M*) and hence

*Ty*∈

*P*

_{ M }(

*u*). Further as

*f*is nonexpansive on

*P*

_{ M }(

*u*) ⋃ {

*u*}, so for every

*p*∈

*P*, we obtain

Thus, *fTy* ∈ *P*_{
M
}(*u*) and hence $Ty\in {C}_{M}^{f}\left(u\right).$ Consequently, *Ty* ∈ *D* and so *T, f* : *D* → *D* satisfy the hypotheses of Theorem 3.2. Thus, there exists *a* ∈ *P*_{
M
}(*u*) such that *a* = *fa* = *Ta*.

**Remark 3.2**. (i)

*Theorem 3.2 extends Theorem 1.2 to multivalued f-nonexpansive map T where the pair*(

*T, f*)

*is assumed to be R-subcommutative. Here we have also relaxed the nonexpansiveness of the map f*.

- (ii)
*Theorem 3.3 extends Theorem 3.3*[12],*which is itself a generalization of several approximation results*. - (iii)
*If f*(*P*_{ M }(*u*)) ⊆*P*_{ M }(*u*),*then*$PM\left(u\right){C}_{M}^{f}\left(u\right)$*and so*${D}_{M}^{f}\left(u\right)={P}_{M}\left(u\right)$ (*cf*. [1]).*Thus, Theorem 3.3 holds for D*=*P*_{ M }(*u*).*Hence, Theorem 3.1*[12],*Theorem 7*[16],*Theorem 2.6*[26],*Theorem 3*[27],*Corollaries 3.1, 3.3, 3.4, 3.6 (i), 3.7 and 3.8 of*[28]*and many other results are special cases of Theorem 3.3 (see also Remarks 3.2*[12]).

## Declarations

## Authors’ Affiliations

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