A coupled common fixed point theorem for a family of mappings

Abstract. In this paper we introduce the concept of coincidentally commuting pair in the context of coupled fixed point problems. It is established that an arbitrary family of mappings has a coupled common fixed point with two other functions under certain contractive inequality condition where two specific members of the family are assumed to be coincidentally commuting with these two functions respectively. The main result has certain corollaries. An example shows that the main theorem properly contains one of its corollaries.

An important category in fixed point theory is the common fixed point problems.An early result was established by Jungck under commuting conditions [17].The concept of commuting has been generalized in various directions and in several ways over the years.One such notion is "coincidentally commuting", also known as "weak compatibility" which was introduced in [18].Several works have been done on fixed points of "coincidentally commuting" mappings as, for instances, in [19][20][21].
Coupled common fixed point and coincidence point problems were first addressed by Lakshmikantham and Ciric [9] in which the authors extended the work of Bhaskar and Lakshmikantham [1].Following this result other coupled coincidence point results appeared in [3] and [5].
c Vilnius University, 2013 In this paper we define the concept of "coincidentally commuting" for two mappings F : X × X → X and g : X → X and establish some coupled common fixed point results.Our main result is for family of mappings which is not necessarily countable.It has several corollaries and an illustrative example.This example shows that the corollaries are effectively included in the main theorem.We have also shown with an example that the concept of "coincidentally commuting" is strictly weaker than the prior concept of "commuting" given in [9].
Definition 3. (See [9].)An element (x, y) ∈ X × X, is called a coupled common fixed point of the mappings F : X × X → X and g : X → X if F (x, y) = gx = x and F (y, x) = gy = y.Definition 4. (See [9].)Let X be a non-empty set and F : X × X → X and g : X → X.We say F and g are commutative if gF (x, y) = F (gx, gy) for all x, y ∈ X. Definition 5. Let X be a non-empty set and F : X × X → X and g : X → X. F and g are said to be coincidentally commuting if they commute at their coupled coincidence points; that is, if gx = F (x, y) and gy = F (y, x) for some (x, y) ∈ X × X, then gF (x, y) = F (gx, gy) and gF (y, x) = F (gy, gx).
Example 1.Let X = [0, ∞).Let F : X × X → X and g : X → X be defined respectively as follows: Here, the functions g and F commute at their only coupled coincidence point (0, 0).Therefore, the pair of functions (g, F ) is coincidentally commuting.But the pair of functions (g, F ) is not commuting.
In view of the above example we have the following observation.
Remark 1.Every commuting pair is a coincidentally commuting pair but its converse is not true.

Main result
Theorem 1.Let (X, d) be a complete metric space.Let {F α : X × X → X: α ∈ Λ} be a family of mappings and h, g : X → X be two self mappings such that: (A) h(X) and g(X) are closed subsets of X, and (B) there exist α 0 , β 0 ∈ Λ such that: Then, there exists a unique (x, y) ∈ X × X such that x = hx = gx = F α (x, y) and y = hy = gy = F α (y, x) for all α ∈ Λ, that is, h, g and {F α : α ∈ Λ} have a unique coupled common fixed point in X.Moreover, any coupled common fixed point of h, g, F α0 and F β0 is a coupled common fixed point of h, g and {F α : α ∈ Λ}.
From ( 13) and ( 14), we have It follows from ( 12) and ( 15) that Now, we show that both {p n } and {q n } are Cauchy sequences.For each m n, we have Therefore, Since δ < 1, we have It follows that {p n } and {q n } are Cauchy sequences in X.From the completeness of X, there exist x, y ∈ X such that lim n→∞ p n = x and lim n→∞ q n = y.
Therefore, from ( 8), ( 9) and ( 16), we have and Since F α0 (X × X) ⊆ g(X), F β0 (X × X) ⊆ h(X) and h(X), g(X) are closed subsets of X, from ( 17) and ( 18), it is clear that x, y ∈ h(X) ∩ g(X).Then there exist r, s ∈ X such that hr = x and hs = y and there exist w, z ∈ X such that gw = x and gz = y.
From the condition (iii), we have ad(hr, gx 2n+1 ) + bd(hs, gy 2n+1 ) Taking n → ∞ in the above inequality, using ( 17) and ( 18), we have Therefore, we have Similarly, we can prove Again, from the condition (iii), we have Taking n → ∞ in the above inequality, using ( 17) and ( 18), we have Therefore, we have Similarly, we can prove y = gz = F β0 (z, w).
From ( 29) and (30), we have Therefore, (x, y) is a coupled common fixed point of h, g, F α0 and F β0 .By what we have already proved, (x, y) is the unique coupled common fixed point of h, g and {F α : α ∈ Λ}.
Note.The contractive inequality (iii) of Theorem 1 is an analogue of Condition B which was introduced by Babu, Sandhya and Kameswari [22] in 2008.
Since every commuting pair of functions is a coincidentally commuting pair, we have the following corollary.
Then, there exists a unique (x, y) ∈ X × X such that x = hx = gx = F α (x, y) and y = hy = gy = F α (y, x) for all α ∈ Λ, that is, h, g and {F α : α ∈ Λ} have a unique coupled common fixed point in X.Moreover, any coupled common fixed point of h, g, F α0 and F β0 is a coupled common fixed point of h, g and {F α : α ∈ Λ}.Corollary 2. Let (X, d) be a complete metric space.Let {F α : X × X → X: α ∈ Λ} be a family of mappings and h, g : X → X be two self mappings such that: (A) h(X) and g(X) are closed subsets of X, and (B) there exist α 0 , β 0 ∈ Λ such that: (ii) the pairs (h, F α0 ) and (g, F β0 ) are coincidentally commuting, ad(hx, gu) + bd(hy, gv) for all α ∈ Λ and for all x, y, u, v ∈ X, where a and b are non-negative real numbers with a + b < 1.
Then, there exists a unique (x, y) ∈ X × X such that x = hx = gx = F α (x, y) and y = hy = gy = F α (y, x) for all α ∈ Λ, that is, h, g and {F α : α ∈ Λ} have a unique coupled common fixed point in X.Moreover, any coupled common fixed point of h, g, F α0 and F β0 is a coupled common fixed point of h, g and {F α : α ∈ Λ}.
Then, there exists a unique (x, y) ∈ X × X such that x = hx = gx = F α (x, y) and y = hy = gy = F α (y, x) for all α ∈ Λ, that is, h, g and {F α : α ∈ Λ} have a unique coupled common fixed point in X.Moreover, any coupled common fixed point of h, g, F α0 and F β0 is a coupled common fixed point of h, g and {F α : α ∈ Λ}.
Then there exists a unique (x, y) ∈ X × X such that x = hx = gx = F (x, y) = G(x, y) and y = hy = gy = F (y, x) = G(y, x); that is, h, g, F and G have a unique coupled common fixed point in X.
Corollary 5. Let (X, d) be a complete metric space.Let F : X × X → X and G : X × X → X be two mappings.Suppose there exist non-negative real numbers a, b and L with a + b < 1 such that Remark 2. Corollary 5 is an extension of Theorem 2.1 [10] to a pair of maps, and Corollary 6 is just Theorem 2.1 and 2.2 [1] in metric space setting.Remark 3. In Example 2, F α ⊆ g(X) for only α = 1 but the pair (h, F 1 ) is not commuting so that this example is not applicable to Corollary 1. Hence Theorem 1 properly contains Corollary 1.