Some Common Fixed Point Results for Modified Subcompatible Maps and Related Invariant Approximation Results

and Applied Analysis 3 Example 13. LetX = Rwith the usual norm andM = [0,∞). Define S, T : M → M by S (x) = {{ {{ { x 2 , 0 ≤ x < 1


Introduction and Preliminaries
From the last five decades, fixed point theorems have been used in many instances in invariant approximation theory.The idea of applying fixed point theorems to approximation theory was initiated by Meinardus [1] where he employs a fixed point theorem of Schauder to establish the existence of an invariant approximation.Later on, Brosowski [2] used fixed point theory to establish some interesting results on invariant approximation in the setting of normed spaces and generalized Meinardus's results.Singh [3], Habiniak [4], Sahab et al. [5], and Jungck and Sessa [6] proved some similar results in the best approximation theory.Further, Al-Thagafi [7] extended these works and proved some invariant approximation results for commuting self-maps.Al-Thagafi results have been further extended by Hussain and Jungck [8], Shahzad [9][10][11][12][13][14] and O'Regan and Shahzad [15] to various class of noncommuting self-maps, in particular to R-subweakly commuting and R-subcommuting self-maps.Recently, Akbar and Khan [16] extended the work of [7][8][9][10][11][12][13][14][15] to more general noncommuting class, namely, the class of subcompatible self-maps.
In this paper, we improve the class of subcompatible selfmaps used by Akbar and Khan [16] by introducing a new class of noncommuting self-maps called modified subcompatible self-maps which contain commuting, R-subcommuting, Rsubweakly, commuting, and subcompatible maps as a proper subclass.For this new class, we establish some common fixed point results for some families of self-maps and obtain several invariant approximation results as applications.The proved results improve and extend the corresponding results of [3][4][5][6][7][8][10][11][12][13][14][15].
Before going to the main work, we need some preliminaries which are as follows.
Definition 2. Let  be a subset of a metric space (, ) and ,  be self-maps of .A point  ∈  is a coincidence point (common fixed point) of  and  if  =  ( =  = ).
For a useful discussion on these classes, that is, the class of commuting, R-weakly commuting, compatible, and weakly compatible maps, see also [20].
Definition 3. Let  be a linear space and let  be a subset of .The set  is said to be star-shaped if there exists at least one point  ∈  such that the line segment [, ] joining  to  is contained in  for all  ∈ ; that is,  + (1 − ) ∈  for all  ∈ , where 0 ≤  ≤ 1.

Definition 4.
Let  be a linear space and let  be a subset of .A self-map  :  →  is said to be (i) affine [21] if  is convex and (ii) q-affine [21] if  is -star-shaped and Here we observe that if  is -affine then  = .
Remark 5. Every affine map  is -affine if  =  but its converse need not be true even if  = , as shown by the following examples.

Common Fixed Point for Modified Subcompatible Self-Maps
First we introduce the notion of modified subcompatible maps.
The following general common fixed point result is a consequence of Theorem 5.1 of Jachymski [22], which will be needed in the sequel.
Theorem 17.Let  and  be self-maps of a complete metric space (, ) and either  or  is continuous.Suppose {  } ∈N∪{0} is a sequence of self-maps of  satisfying the following.
(3) For each  ∈ N and, for any ,  ∈       0  −        ≤ max  (, ) , where then all the   ( ∈ N ∪ {0}),  and  have a common fixed point provided one of the following conditions hold.
(a)  is sequentially compact and   is continuous for each  ∈ N ∪ {0}.
Proof.For each  ∈ N ∪ {0}, define    :  →  by for all  ∈  and a fixed sequence of real numbers   (0 <   < 1) converging to 1.Then,    is a self-map of  for each  ∈ N ∪ {0} and for each  ≥ 1.
Also, using ( 18) and ( 20 where which is a contradiction.Therefore,  =  0  and, hence, In Theorems 18 and 19, if we take   =  for each  ∈ N ∪ {0}, we obtain the following corollary which generalizes Theorems 2.2 and 2.3 of Hussain and Jungck [8], respectively.Corollary 20.Let  be a nonempty q-star-shaped subset of a normed space , and let  and  be continuous and affine self-maps of .Let  be a self-map of  satisfying the following. (1) () ⊆ () ∩ ().
(3) For all ,  ∈ ,      −      ≤ max  (, ) , where Then S, T, and  have a common fixed point provided one of the following conditions hold.
(a)  is sequentially compact and  is continuous.
As we have done in Theorem

Examples
Now, we present some examples which demonstrate the validity of the proved results.
Definition 10.Let  be a Banach space.A map  :  ⊆  →  is said to be demiclosed at 0 whenever {  } is a sequence in  such that   converges weakly to  ∈  and   converges strongly to 0 ∈ ; then 0 = .