Existence of common fixed point and best proximity point for generalized nonexpansive type maps in convex metric space

Here, we extend the notion of (E.A.) property in a convex metric space defined by Kumar and Rathee (Fixed Point Theory Appl 1–14, 2014) by introducing a new class of self-maps which satisfies the common property (E.A.) in the context of convex metric space and ensure the existence of common fixed point for this newly introduced class of self-maps. Also, we guarantee the existence of common best proximity points for this class of maps satisfying generalized non-expansive type condition. We furnish an example in support of the proved results.

A normed linear space X and each of its convex subset are simple examples of convex metric spaces with W given by W (x, y, ) = x + (1 − )y for all x, y ∈ X and 0 ≤ ≤ 1 . Also, Property (I) is always satisfied in a normed linear space. There are many convex metric spaces which are not normed linear space, for details (see Guay et al. 1982;Takahashi 1970).
Definition 5 Let (X, d) be a convex metric space and M be a subset of X. A mapping I : M → X is said to be (1) affine (Al-Thagafi and Shahzad 2006;Huang and Li 1996), if M is convex and I(W (x, y, )) = W (Ix, Iy, ) for all x, y ∈ M and ∈ [0, 1].
Definition 6 Let (X, d) be a metric space, M a nonempty subset of X and let I and T be self-maps of M. A point x ∈ M is a coincidence point (common fixed point) of I and T if Ix = Tx(Ix = Tx = x). The pair {I, T } is called compatible (Jungck 1986) if lim n→∞ d(ITx m , TIx m ) = 0, whenever {x n } is a sequence in X such that lim n→∞ Ix n = lim n→∞ Tx n = t ∈ X.
For more details about these classes, one can refer to (see Agarwal et al. 2014). In 1998, Pant (1998) defined the concept of reciprocal continuity as follows.
Definition 7 (Pant 1998) Let (X, d) be a metric space and I, T : X → X. Then the pair (I, T) is said to be reciprocally continuous if whenever {x n } is a sequence in X such that lim n→∞ Ix n = lim n→∞ Tx n = t ∈ X.
It is easy to see that if I and T are continuous, then the pair (I, T) is reciprocally continuous but the converse is not true in general (see Imdad et al. 2011, Example 2.3). Moreover, in the setting of common fixed point theorems for compatible pairs of self-mappings satisfying some contractive conditions, continuity of one of the mappings implies their reciprocal continuity.
Definition 8 (Bouhadjera and Godet-Thobie 2009) Let I and T be two self-maps of a metric space (X, d). Then the pair (I, T) is said to be subcompatible if there exists a sequence {x n } such that Obviously, compatible maps which satisfy (E.A.) property are subcompatible but the converse statement does not hold in general (see Rouzkard et al. 2012, Example 2.5) Definition 9 (Kumar and Rathee 2014) Let M be a q-starshaped subset of a convex metric space (X, d) and let I, T : M → M with q ∈ F (I). The pair (I, T) is said to satisfy (E.A.) property with respect to q if there exists a sequence {x n } in M such that for all where T x = W (Tx, q, ).
Obviously, if the pair (I, T) satisfy (E.A.) property with respect to q, then I and T satisfy (E.A.) property but converse assertion is not necessarily true (see Kumar and Rathee 2014, Example 12).

Main results
We start to this section with following definition.
Definition 10 Let M be a q-starshaped subset of a convex metric space (X, d) and let A, B, S and T : M → M. Two pairs (A, S) and (B, T) are said to satisfy common property (E.A.) with respect to q if there exist two sequences {x n } and {y n } in M such that for all where S x = W (Sx, q, ) and T y = W (Ty, q, ) Remark 11 In Definition 10, if A = B and S = T, then Definition 9 can be obtained as a particular case of Definition 10. Therefore the common property (E.A.) defined here extends the notion of (E.A.) property in convex metric space defined by Kumar and Rathee (2014).
The following Lemma is particular case of the Theorem 4.1 of Chauhan and Pant (2014).

Lemma 12 Let A, B, S and T be self-maps of a metric space (X, d). If the pairs (A, S) and
(B, T) are subcompatible, reciprocally continuous and satisfy for some ∈ (0, 1) and all x, y ∈ X. Then S and T have a unique common fixed point in X. Now, we start with the following theorem. (7) T n (y) = W (Ty, q, n ) and S n (x) = W (Sx, q, n ) for t ∈ M, where Since n ∈ (0, 1), by using Eqs. (7) and (8)       (12) (13) d(T n By m , BT n y m ) ≤ n d(TBy m , BTy m ). Taken into account Eqs. (11) and (14), it follows that (T n , B) and (S n , A) are subcompatible for each n ∈ N. Since A, B, S and T are continuous for each n ∈ N, the pair (S n , A) and (T n , B) are reciprocally continuous.
By using equation (6) and Property (I), we get that for each x, y ∈ M and n ∈ (0, 1). By Lemma 12, for each n ∈ N, there exists x n ∈ M such that Now by taking the compactness of M, we know continuous image of compact set is compact so T(M) and S(M) are compact and every compact set is sequentially compact. Therefore there exist subsequences {Tx m } of {Tx n } and {Sx m } of {Sx n } such that lim m→∞ Tx m = z and lim m→∞ Sx m = y. Now, we have to prove that y = z.
On the contrary suppose that y � = z, then we have This implies that the sequence {x m } converges to two points which is contradiction. Hence y = z. Since x m → z as m → ∞ and the mappings A, B, S and T are continuous, it follows So, z is common fixed point of A, B, S and T.

Corollary 14 Let M be a nonempty q-starshaped subset of a convex metric space (X, d) with Property (I) and let A, B, S and T be continuous self-maps on M such that the pair (A, S) and (B, T) satisfying common property (E.A.) w.r.t. q. Assume that A and B are q-affine, M is compact. If A, B, S and T are compatible and satisfy the inequality
≤ n max d(Ax, By), d(Ax, S n x), d(By, T n y), d(Ax, T n y), d(By, S n x) Ax n = S n x n = Bx n = T n x n = x n .

Corollary 15 Let M be a nonempty q-starshaped subset of a convex metric space (X, d) with Property (I) and let A, B, S and T be continuous self-maps on M such that the pair (A, S) and (B, T) satisfying common property (E.A.) w.r.t. q. Assume that A and B are q-affine, M is compact. If A, B, S and T are R-subweakly commuting and satisfy the inequality
Now we present an example in support of our theorem.
Example 16 Let X = R endowed with usual metric and let M = −1, 2 3 . Define A, B, S and T : M → M by: We have to check the following: The pair (A, S) and (B, T) satisfying common property (E.A.) w.r.t. q = 1 3 . (iii) A, B, S and T are compatible.
Thus, A W x, 1 3 , = W Ax, 1 3 , for all x ∈ M and hence A is q-affine with q = 1 3 . Now we shall prove that B is q-affine with q = 1 3 .
Subcase (iv): if x ∈ 1 3 , 2 3 and y ∈ −1, 1 3 , then So, we have d(Sx, Ty) ≤ d(Ax, By). Thus, for each x, y ∈ M, the mappings A, B, S and T satisfying the inequality (6). Also M is compact and A, B, S and T are continuous. Thus we conclude that A, B, S and T satisfying all the conditions of Theorem 13 and consequently  (2001)).

Application to invariant approximation
For a nonempty subset M of a metric space (X, d) and p ∈ X, an element y ∈ M is called a best approximation to p if d(p, y) = dist(p, M), where dist(p, M) = inf {d(p, z) : z ∈ M} . The set of all best approximations to p is denoted by P M (p).
As an application of Theorem 13, we present an invariant approximation theorems.  Proof Let x ∈ P M (p). Then for all ∈ (0, 1), we have Therefore W (x, p, ) / ∈ M for any ∈ (0, 1) and hence x ∈ δM ∩ M. Thus, as S(δM ∩ M) ⊆ M and T (δM ∩ M) ⊆ M, we have Tx ∈ M and Sx ∈ M. Also, since Ax ∈ P M (p) and p ∈ F (S) ∩ F (T ) ∩ F (A) ∩ F (B), by using Eq. (18), we get and Thus, Tx ∈ P M (p) and Sx ∈ P M (p). So A, B, S and T are self-maps on P M (p). In view of Theorem 13, we can say that A, B, S and T have a common fixed point in P M (p). Proof Let x ∈ D. Then by following the steps as we have done in Theorem 18, we get that Tx ∈ P M (p) and Sx ∈ P M (p). Since the maps A and B are nonexpansive and p ∈ F (S) ∩ F (T ) ∩ F (A) ∩ F (B), by using Eq. 19, we have and That imply ATx and BTx ∈ P M (p) and hence Tx ∈ C A,B M (p). Similarly we can show that Sx ∈ C A,B M (p). Thus we can say A, B, S and T are self-maps on D and so Theorem 13 guarantees the existence of z ∈ P M (p) such that z is a common fixed point of A, B, S and T.

Best proximity point
First we discuss the concept of best proximity. Let T : A → B be a map where A and B are two nonempty subsets of a metric space (X, d) and let A and B are disjoint subsets of a metric space then the equation Tx = x might have no solution. Therefore in case of nonself-maps we are not sure about the existence of fixed point. In such a case we try to minimize the distance d(x, Tx) and a point x for which d(x, Tx) is minimum is called a best proximity point. In the recent years there have been many interesting best proximity point theorems are proved, for example, see De la Sen et al. (2013), Eldred and Veeramani (2006), Prolla (1983), Reich (1978), and Sankar Raj (2011), Sehgal and Singh (1988). In the present section we prove a new best proximity theorem for four maps but before this we recall some definitions which are required in the sequel.
Definition 20 Let (X, d) be a convex metric space and A, B be two nonempty subsets Definition 21 Let A and B be two nonempty subsets of convex metric space (X, d).
Let A be a p-starshaped set and B be a q starshaped set. Let f, g, S, and T be four nonself-maps from A to B. Two pairs (f, S) and (g, T) are said to satisfy common property (E.A.) with respect to q if there exists two sequences {x n } and {y n } in A such that for all ∈ [0, 1] where S x = W (Sx, q, ) and T y = W (Ty, q, ).

Definition 22
Let (X, d) be a convex metric space and A and B be two nonempty subsets of X such that B is q-starshaped set. A pair (f, S) of two nonself-maps from A to B is said to be proximally commuting if for some ∈ [0, 1] whenever d(x, W (Su, q, )) = d(y, fu) = d(A, B) =⇒ W (Sy, q, ) = fx.
If A and B are two nonempty subsets of a metric space (X, d), we define the following two sets.
Definition 23 (Sankar Raj, preprint) If A 0 � = φ, then the pair (A, B) is said to have P-property if and only if for any x 1 , x 2 ∈ A 0 and y 1 , y 2 ∈ B 0 Now we presents a best proximity point theorem:

Theorem 24 Let (A, B) be a pair of nonempty, closed subsets of a convex metric space (X, d). Suppose that A is p-starshaped and B is q-stasrshaped set with Property (I). Also suppose that A 0 is closed. Let f, g, S and, T be continuous nonself maps from A to B satisfying the conditions:
(i) Two pairs (f, S) and (g, T) satisfying common property (E.A.) w.r.t q and proximally commuting; pair (A, B)  Proof For each n ∈ N, we define sequences T n : A → B and S n : A → B by T n y = W (Ty, q, n ) and S n x = W (Sx, q, n ) for all x, y ∈ A and n is a sequence in (0, 1) such that n → 1 Consider By using Property (I) for the set B =⇒ d(S n x, T n y) ≤ n max{d(fx, gy), dist(fx, S n x), dist(gy, T n y), 1 2 [dist(fx, T n y) + dist(gy, S n x)]} Now T (A) ⊆ f (A) we can prove that T n (A) ⊆ f (A). For this purpose, consider T is pq-affine and A is p-starshaped set Similarly it can be proved that S n (A) ⊆ f (A). Now T n (A) ⊆ f (A) so for fixed x 0 ∈ A, there exists an element x 1 ∈ A such that T n x 0 = fx 1 similarly a point x 2 ∈ A can be chosen such that S n x 1 = gx 2 , continuing in process, we can obtain a sequence {x 2n } ∈ A such that d(S n x, T n y) = d(W (Sx, q, n ), W (Ty, q, n )). d(W (Sx, q, n ), W (Ty, q, n )) ≤ n d(Sx, Ty) ≤ n max{d(fx, gy), dist(fx, [Sx, q] ≤ n max{d(fx, gy), dist(fx, S n x), dist(gy, T n y), 1 2 [dist(fx, T n y) + dist(gy, S n x)]} y ∈ T n (A) y = T n x for some x ∈ A y = W (Tx, q, n ).
=⇒ d(u m , u n ) → 0 when m → ∞ this implies {u n } is a Cauchy sequence. Since {u n } ⊂ A 0 and A 0 is closed subset of the complete metric space (X, d), we can find u ∈ A 0 such that lim n→∞ u n = u.
Since f (A 0 ) ⊆ B 0 , there exists x ∈ A 0 such that As (f , S n ) and (g, T n ) proximally commuting, so Taking limit n → ∞ in Eqs. (29) and (31) we have Since f (A 0 ) ⊆ B 0 , there exists z ∈ A 0 such that Because the pair (A, B) has P-property so d(x, z) = d(S n u, T n x) This implies that (1 − n )d(x, z) ≤ 0. So, x = z and hence Suppose that y is another best proximity point of the mappings f, g, S and T such that Using Eqn. (29) and P-property for the pair (A, B), we get that x = y.

Conclusion
In this note, we defined the common property (E.A.) in the context of convex metric space that means here we assign the algebraic structure to the common property (E.A.) that is already exists in metric space. Due to this, we have been able to obtained a set of common fixed point theorems in which to ensure the existence of common fixed points the condition of range of one set is contained in other is not required. Thus, this newly introduced concept plays a great role in solving many kinds of physical sciences problems which can be recast in terms of common fixed point problems.