Comments on some Fixed Point Theorems

The purpose of this paper is to show that a number of fixed point theorems are special cases of two theorems of Jungck and the author [6].

then there is a unique point w ∈ X such that f w = Sw = w and a unique point z ∈ X such that gz = Tz = z Moreover, z = w, so that there is a unique common fixed point of f , g, S, and T.
Then T, f , and g have a coincidence point in M. Also, if the pairs (g, T) and (g, f ) are weakly compatible, then T, f and g have a unique common fixed point in X.
Two maps are weakly compatible if they commute at every coincidence point.Therefore weakly commuting implies owc.The conclusion of Theorem 2 now follows from Theorem 1.
Theorem 3. Let A, B, S, and T be selfmaps of a complete metric space (X, d) satisfying for all x, y ∈ X.Then we have the following: which is a special case of (1), since ψ ∈ Ψ.
Using (i), if (B, T) is weakly compatible, then it is owc.If z is a common fixed point of A and S, then z = Az = Sz, and (A, S) is owc.The conclusion now follows from Theorem 1. Case (ii) is handled in a similar manner.
Theorems 2.5 and 2.6 of [1] are also special cases of Theorem 1.
The result of [2] was generalized by [13], and was further generalized as Theorem 2.1 of [14].The following is Theorem 2.1 of [14].
Theorem 4. Let f and g be self-maps of a complete metric space X satisfying f (X) ⊂ g(X) and the inequality which is a special case of (1).
The condition ( f , g) is compatible implies that ( f , g) are owc.The result now follows from Theorem 1.
The following is Theorem 2.1 of [5] Theorem 5. Let A, B, S, and T be continuous self mappings of a metric space (X, d).Suppose that ' for all x, y ∈ X.Then there is a unique point z ∈ X such that Az = Bz = Sx = Tz = z.
Two maps are said to be faintly compatible if and only if the pair is conditionally compatible and commutes on a nonempty subset of coincidence points.Thus (A, S) and (B, T) are owc and the conclusion follows from Theorem 1.
Note that the condition of continuity of A, B, S, and T is not needed.
The following is Theorem 2.3 of [1].which is a special case of (1).
Since (A, S) and (B, T) are weakly compatible, they are owc, and the conclusion follows from Theorem 1.
Theorems 2.4, 2.5, and 2.6 of [1] are proved in the same way.
The following is Theorem 3.1 of [3] Theorem 7. Let X be a set with a symmetric d.Let f , g and h be three self mappings of (X, d) and ψ ∈ Ψ satisfying for all x, y ∈ X, and the pair ( f , h) or (g, h) is owc.Then f , g and h have a unique common fixed point.
which is a special case of (1).
Suppose that ( f , h) is owc.Then there exists a point z ∈ X such that f z = hz and f hz = h f z.From (7), and f z = gz.
Now substituting x = f z, y = gz into (7) one obtains and ( f , g) are owc.The conclusion now follows from Theorem 1.
Using the same argument, the assumption f z = gz implies that ( f , h) are owc, and the conclusion again follows from Theorem 1.
The following is Theorem 1 of [9] Theorem 8. Let f , g be selfmaps of a complete metric space with f continuous.Suppose that f and g commute and g(X) ⊂ f (X).Suppose that g satisfies d(gx, gy) ≤a 1 d(gx, f x) + a 2 d(gy, f y) + a 3 d(gx, f y) where a i ≥ 0, ∑ 5 i=1 a i < 1.Then f and g have a unique common fixed point in X.
Since f and g commute, they are owc.Inequality (9)  The following is Theorem 1 of [11].Theorem 9. Let (X, d) be a compact metric space, A, T : X −→ X such that Note that condition (ii) is not needed.
The following is Theorem 1 of [12] Theorem 10.Let S, I and T, J be two pairs of weakly commuting mappings of a complete metric space (X, d) into itself satisfying the inequality [d(Sx, Ty)] 3 ≤ αd(Ix, Jy)d(Ix, Sx)d(Jy, Ty), (10) for all x, y ∈ X, where α ∈ (0, 1).If the range of I contains the range of T and the range of J contains the image of S, and if one of S, T, I and J is continuous, then S, T, I and J have a unique common fixed point z.Further, z is the unique common fixed point of S and I and I and of T and J.
From (10), which is a special case of (1).
Since (S, I) and (T, J) are weakly commuting, they are owc, and the conclusion follows from Theorem 1.
The following is Theorem 1 of [15] Theorem 12. Let P, Q, T be selfmaps of a complete metric space X such that PT = TP, QT = TQ, P(X) ∪ Q(X) ⊆ T(X).If T is continuous and there exists an h ∈ (0, 1) such that d(Px, Qy) ≤ h max{d(Tx, Ty), d(Px, Tx), d(Qy, Ty), for all x, y ∈ X, then P, Q, T have a unique common fixed point.
The following is Theorem 2.1 of [17] Theorem 13.Let X be a nonempty set with symmetric d and f , g, f and r be self-maps on (X, d) satisfying any two of the following three inequalities, where φ ∈ ψ: , for all x, y ∈ X.If any of the pairs ( f , r), (g, r) or (h, r) is owc, then f , g, h and r have a unique common fixed point.
Condition (13) implies that Therefore Theorem 13 is a special case of Theorem 7.
The following is Theorem 2.1 of [7].
Theorem 14.Let f and g be self-maps of a G-metric space (X, G) satisfying f (X) ⊆ g(X), where α ∈ [0, 1/2), and one of f or g is continuous.
Then f and g have a unique common fixed point in X, provided f and g are compatible maps.

B. E. Rhoades
For a discussion of the basic properties of G-metric spaces the reader may consult [10].In [18] it was shown that, if one defines for any x, y in G, then then d G (x, y) becomes a metric.
Using ( 16) one obtains which is a special case of (1).Since f and g are compatible, they are owc.The conclusion follows from Theorem 1.
Theorems 3.5 and 4.3 of [7] are also special cases of Theorem 1.
The following is Theorem 3.1 of [16].
Theorem 15.Let A, B, S and T be selfmappings from a complete metric space (X, d) into itself satisfying the following conditions A(X) ⊆ T(X) and B(X) ⊆ S(X) Further, if the sequence Ax 0 , Bx 1 , Ax 2 , Bx 3 , . . ., Ax 2n , Bx 2n+1 , . . .converges to z ∈ X, then A, B, S and T have a unique common fixed point z in X Theorem 15 will use Theorem 4 of [6].
The following is Theorem 4 of [6].
Theorem 16.Let X be a set, r a symmetric on X.Let f , g, S, T be selfmaps of X satisfying f (x) ⊆ T(X), g(X) ⊆ S(X), and r( f x, f y) ≤ g(r(Sx, Ty), r( f x, Sx), r(gy, Ty), r( f x, Ty), r(gy, Sx)) for all x, y ∈ X, where g ∈ G.If { f , S} and {g, T} are owc, then f , g, S, T have a unique common fixed point.
The functions defined by inequality (18) satisfy conditions (g 1 ) − (g 3 ).Compatible mappings of type E are owc.Therefore the result follows from Theorem 16.The following is Theorem 2.1 of [8] Theorem 17.Let (X, d) be a complete Takahashi metric space with convex structure W which is continuous in the third variable.Let K be a nonempty closed subset of X.Let δK be the boundary of K with δK = φ.Let the mappings A, B, S, T : K −→ K Assume that A, B, S, T satisfy the following conditions: Ax)d(Ty,By) d(Sx,Ty)+d(Sx,By)+d(Ty,Ax) , d(Sx, Ty) , if d(Sx, Ty) + d(Sx, By) + d(Ty, Ax) = 0 0, if d(Sx, Ty) + d(Sx, By) + d(Ty, Ax) = 0 (3) and the pair (B, T) is weakly compatible, and if z is a common fixed point of A and S then z is a common fixed point of A, B, S and T and it is unique.(ii)If B(X) ⊆ S(X) and the pair (A, S) is weakly compatible, and if z is a common fixed point of B and T then z is a common fixed point of A, B, S, and T and it is unique.Since d(Sx, Ax) ≤ d(Sx, Ty) + d(Ty, AX), (3) implies d(Ax, By) ≤ max{d(Ty, By), d(Sx, Ty)},
(i) d(Tx, Ty) < 1 2 [d(Ax, TAx) + d(Ay, TAy], for all x, y ∈ X with Tx = Ty.(ii) d(Ax, Ay) ≤ d(x, y).(iii) TAx = ATx.Then A and T have a unique common fixed point.By (iii), A and T commute.Then A(TA) = (AT)A and A and AT commute Also T(TA) = T(AT) = (TA)T, and T and TA commute.Therefore (A, T), (A, TA), and (T, AT) are owc.Inequality (1) implies that d(Tx, Ty) < max{d(Ax, TAx), d(Ay, TAy)}.Thus T, A and TA have a unique common fixed point by Theorem 1.