 Research
 Open Access
 Published:
Common fixed points and best approximations in locally convex spaces
Fixed Point Theory and Applications volume 2011, Article number: 99 (2011)
Abstract
We extend the main results of Aamri and El Moutawakil and Pant to the weakly compatible or Rweakly commuting pair (T, f) of maps, where T is multivalued. As applications, common fixed point theorems are obtained for new class of maps called Rsubcommuting 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.
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 Rweakly 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 singlevalued and multivalued maps. Among others Kaneko [3] extended wellknown result of Nadler [4] to multivalued fcontraction 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 fcontraction 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, Rweakly commutative and compatible maps (see [1, 2, 6–11]). The following more general common fixed point theorem for 1subcommutative maps was proved in [12].
Theorem 1.2. Let M be a nonempty τbounded, τsequentially complete and qstarshaped subset of a Hausdorff locally convex space (E, τ). Let T and I be selfmaps of M. Suppose that T is Inonexpansive, I(M) = M, Iq = q, I is nonexpansive and affine. If T and I are 1subcommutative maps, then T and I have a common fixed point provided one of the following conditions holds:

(i)
M is τsequentially 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 Rweakly 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 Rweakly 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, Rsubcommuting 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 wellknown results on fixed points and common fixed points of best approximation for Rsubcommutative maps.
Throughout this article, 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
Although 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
and let P_{ M }(u) = {y ∈ M : p(y  u) = d_{ p }(u, M), for all p ∈ P} be the set of best Mapproximations to u ∈ X. For any mapping f : M → X, we define (cf. [6])
Let 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
The map T is called nonexpansive if for each x, y ∈ M and p ∈ P,
Let f : M → M be a singlevalued map. Then, T : M → K(M) is called an fcontraction if there exists k_{ p }, 0 < k_{ p } < 1 such that for each x, y ∈ M and for each p ∈ P, we have
If we have the Lipschitz constant k_{ p } = 1 for all p ∈ P, then T is called an fnonexpansive 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 Rweakly 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 singlevalued, the definitions of compatibility and Rweak 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 qstarshaped 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 Rsubcommuting maps and proved that this class of maps contains properly the class of commuting maps.
We extend this notion to the pair (T, f) of maps when T is not necessarily singlevalued. Suppose q ∈ F(I), M is qstarshaped with T(M) ⊂ M and f(M) ⊂ M. Then, f and T are Rsubcommutative 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 Rsubcommutativity (which in turn implies Rweak 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 Rsubcommuting but not commuting.
The mapping 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 fcontraction such that the pair (T, f) is weakly compatible (or Rweakly 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 righthand 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 righthand 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 righthand 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 righthand 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 fcontractiveness of T implies that
Hence, 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
for any 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}.
Thus, by the weak compatibility of f and T,
If the pair (T, f) is Rweakly commuting, then
implies that (2.1) holds.

(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}.
That is, 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).
The conditions (iii) and (iv) imply (ii) (see [2] for details).

(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 \n \n \n =\n d\n \n (\n \n \n f\n \n \n x\n \n \n 0\n \n \n ,\n \n \n f\n \n \n 2\n \n \n \n \n x\n \n \n 0\n \n \n \n )\n \n \n
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 fLipschitz map such that the pair (T, f) is weakly compatible (or Rweakly 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 fLipschitz, we obtain
Taking limit as n → ∞, we get lim Tx_{ n } = Tx_{0} and hence fx_{0} ∈ Tx_{0}. The weak compatibility or Rweak 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 fLipschitz, 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
Letting 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 Rweak 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 Rsubcommuting 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 fnonexpansive map and the pair (T, f) is Rsubcommutative. 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
Obviously each T_{ n } is fcontraction map. Note that
which implies that (T_{ n }, f) is Rweakly 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 singlevalued 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 Rsubcommutative 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 fnonexpansive 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 fnonexpansive on D ⋃ {u} and f is nonexpansive on P_{ M }(u) ⋃ {u}. If f and T are Rsubcommutative 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 fnonexpansiveness of T we get
As 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 fnonexpansive map T where the pair (T, f) is assumed to be Rsubcommutative. 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]).
References
Jungck G: Compatible mappings and common fixed points. Int J Math Math Sci 1986, 9: 771–779. 10.1155/S0161171286000935
Pant RP: Common fixed points of noncommuting mappings. J Math Anal Appl 1994, 188: 436–440. 10.1006/jmaa.1994.1437
Kaneko H: Singlevalued and multivalued fcontractions. Boll Un Mat Ital 1985, 6: 29–33.
Nadler SB Jr: Multivalued contraction mappings. Pacific J Math 1969, 30: 475–488.
Latif A, Tweddle I: On multivalued f nonexpansive maps. Demonstratio Mathematica 1999, XXXII: 565–574.
AlThagafi MA: Common fixed points and best approximation. J Approx Theory 1996, 85: 318–323. 10.1006/jath.1996.0045
Aamri M, El Moutawakil D: Some new common fixed point theorems under strict contractive conditions. J Math Anal Appl 2002, 270: 181–188. 10.1016/S0022247X(02)000598
Baskaran E, Subrahmanyam LV: Common fixed points in closed balls. Atti Sem Mat Fis Univ Modena 1988, 36: 1–5.
Beg I, Azam A: Fixed points of asymptotically regular multivalued mappings. J Austral Math Soc (Ser A) 1991, 47: 1–13.
Kaneko H: A common fixed point of weakly commuting multivalued mappings. Math Japon 1988,33(5):741–744.
Sessa S, Rhoades BE, Khan MS: On common fixed points of compatible mappings in metric and Banach spaces. Int J Math Math Sci 1988,11(2):375–395. 10.1155/S0161171288000444
Hussain N, Khan AR: Common fixed point results in best approximation theory. Appl Math Lett 2003,16(4):575–580. 10.1016/S08939659(03)000399
Davies RO: Another version of a common fixed point theorem. Publ Math Debrecen 1991, 38: 237–243.
Jungck G: On a fixed point theorem of Fisher and Sessa. Int J Math Math Sci 1990, 13: 497–500. 10.1155/S0161171290000710
Dotson WJ: Fixed point theorems for nonexpansive mappings on starshaped subsets of Banach spaces. J London Math Soc 1972, 4: 408–410. 10.1112/jlms/s24.3.408
Jungck G, Sessa S: Fixed point theorems in best approximation theory. Math Japon 1995,42(2):249–252.
Lami Dozo E: Multivalued nonexpansive mappings and Opial's condition. Proc Amer Math Soc 1973, 38: 286–292.
Latif A, Tweddle I: Some results on coincidence points. Bull Austral Math Soc 1999, 59: 111–117. 10.1017/S0004972700032652
Ko HM, Tsai YH: Fixed point theorems for point to set mappings in locally convex spaces and a characterization of complete metric spaces. Bull Academia Sinica 1979, 7: 461–470.
Shahzad N: Noncommuting maps and best approximations. Rad Mat 2001, 10: 77–83.
Fabian M, Habala P, Hajek P, Santalucia VM, Pelant J, Zizler V: Functional Analysis and Infinitedimensional Geometry. Springer, New York; 2001.
Singh KL, Chen Y: Fixed points for nonexpansive multivalued mapping in locally convex spaces. Math Japon 1991,36(3):423–425.
Kim IS: Fixed point theorems of quasicompact multivalued mappings in general topological vector spaces. Bull Korean Math Soc 2001,38(1):113–120.
Khan AR, Bano A, Latif A, Hussain N: Coincidence point results in locally convex spaces. Int J Pure Appl Math 2002,3(4):413–423.
Khan AR, Hussain N: Random coincidence point theorem in Frechet spaces with applications. Stoch Anal Appl 2004, 22: 155–167.
Khan AR, Hussain N, Khan LA: A note on Kakutani type fixed point theorems. Int J Math Math Sci 2000,24(4):231–235. 10.1155/S0161171200004191
Sahab SA, Khan MS, Sessa S: A result in best approximation theory. J Approx Theory 1988, 55: 349–351. 10.1016/00219045(88)901013
Sahney BN, Singh KL, Whitfield JHM: Best approximation in locally convex spaces. J Approx Theory 1983, 38: 182–187. 10.1016/00219045(83)901259
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
AlMezel, S.A. Common fixed points and best approximations in locally convex spaces. Fixed Point Theory Appl 2011, 99 (2011). https://doi.org/10.1186/16871812201199
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/16871812201199
Keywords
 best approximations
 common fixed points
 locally convex spaces
 Rsubcommuting maps
 Rweakly commuting.