On Some Generalizations of Commuting Mappings

and Applied Analysis 3 that w fw gw, that is, w is a common fixed point for f, g . If z is any common fixed point for f, g (i.e., fz gz z , then, again by the uniqueness of POC, it must be z w. The following result is due to Ðorić et al. [21]. It shows that the results of Jungck and Rhoades are not generalizations of results obtained from Lemma 2.1. Proposition 2.3 (see [21]). Let a pair of mappings f, g have a unique POC. Then it is WC if and only if it is OWC. Proof. In this case, we have only to prove that OWC implies WC. Let w1 fx gx be the given POC, and let f, g be OWC. Let y ∈ C f, g , y / x. We have to prove that fgy gfy. Now w2 fy gy is a POC for the pair f, g . By the assumption, w2 w1, that is, fy gy fx gx. Since, by Lemma 2.2, w1 is a unique common fixed point of the pair f, g , it follows that w1 fw1 fgy and w1 gw1 gfy, hence fgy gfy. The pair f, g is WC. Proposition 2.4. Let d : X ×X → 0, ∞ be a mapping such that d x, y 0 if and only if x y. Let a pair of mappings f, g have a unique POC. If it is a pair of JH-operators, then it is WC. Proof. Let f, g be a pair of JH-operators. Then there is a point w fx gx in PC f, g such that d x,w ≤ δ PC f, g and d w,x ≤ δ PC f, g . Clearly PC f, g is a singleton. If not, then w1 fy gy is a POC for the pair f, g . By the assumption, w w1. As a result, we have δ PC f, g 0, which implies d x,w d w,x 0, that is, x fx gx. Consequently, we have gfx gx fx fgx and thus f, g is OWC. By Proposition 2.3, f, g is WC. It is worth mentioning that if iX is the identity mapping, then the pair f, iX is always WC, but it is a pair of JH-operators if and only if f has a fixed point. Proposition 2.5. Let d : X ×X → 0, ∞ be a mapping such that d x, y 0 if and only if x y. Suppose f, g is a pair of JH-operators satisfying d ( fx, fy ) ≤ adgx, gy bmaxdfx, gx, dfy, gy cmax { d ( gx, gy ) , d ( gx, fx ) , d ( gy, fy )} (2.1) for each x, y ∈ X, where a, b, c are real numbers such that 0 < a c < 1.Then f, g is WC. Proof. By hypothesis, there exists some x ∈ X such that w fx gx. It remains to show that f, g has a unique POC. Suppose there exists another pointw1 fy gy withw/ w1. Then, we have d w,w1 d ( fx, fy ) ≤ a c dfx, fy a c d w,w1 , (2.2) which is a contradiction since a c < 1. Thus f, g has a unique POC. By Proposition 2.4, f, g is WC. Proposition 2.6. Let d be symmetric on X. Let a pair of mappings f, g have a unique POC which belongs to F f . If it is a pair of occasionally weakly g-biased mappings, then it is WC. 4 Abstract and Applied Analysis Proof. Let f, g be a pair of occasionally weakly g-biased mappings. Then there exists some x ∈ X such that fx gx and d gfx, gx ≤ d fgx, fx . Since w fx gx belong to F f , then fw w, that is, fgx fx gx. Thus d gfx, gx ≤ d fgx, fx 0 and thus gfx gx fx fgx. Hence f, g is OWC. By Proposition 2.3, f, g is WC. Let φ : 0, ∞ → 0, ∞ be a nondecreasing function satisfying the condition φ t < t for each t > 0. Proposition 2.7. Let f, g be self-maps of symmetric space X and let the pair f, g be occasionally weakly g-biased. If for the control function φ, we have d ( fx, fy ) ≤ φmaxdgx, gy, dgx, fy, dgy, fx, dgy, fy, (2.3) for each x, y ∈ X, then f, g is WC. Proof. It remains to show that f, g has a unique POC which belongs to F f . Since f, g are occasionally weakly g-biased mappings, there exists some x ∈ X such that w fx gx and d gfx, gx ≤ d fgx, fx . If w1 fy gy and w/ w1, then d ( fx, fy ) ≤ φmaxdgx, gy, dgx, fy, dgy, fx, dgy, fy φ ( d ( fx, fy )) < d ( fx, fy ) , (2.4) which is a contradiction. Also if ffx / fx, we have d ( ffx, fx ) ≤ φmaxdgfx, gx, dgfx, fx, dgx, ffx, dgx, fx ≤ φmaxdfgx, fx, dfgx, fx, dgx, ffx, dgx, fx φ ( d ( ffx, fx )) < d ( ffx, fx ) , (2.5) which is a contradiction. By Proposition 2.6, f, g is WC. Remark 2.8. According to Propositions 2.4, 2.5, 2.6, and 2.7, it follows that results from [14]: (Theorems 2.8, 2.9, 2.10, 2.11, 2.12, 3.7, 3.9 and Corollary 3.8) are not generalizations (extensions) of some common fixed point theorems due to Bhatt et al. [9], Jungck and Rhoades [12, 13], and Imdad and Soliman [11]. Moreover, all mappings in these results are WC. Proposition 2.9. Let d be symmetric on X, and let a pair of mappings f, g have a unique CP , that is, C f, g is a singleton. If f, g is P-operator pair, then it is WC. Proof. According to (4), there is a point x ∈ X such that x ∈ C f, g {x} and d x, fx ≤ δ C f, g δ {x} 0. Hence, x fx gx is a unique POC of pair f, g and since gfx gx x fx fgx, f, g is OWC. By Proposition 2.3 it is WC. Remark 2.10. By Proposition 2.9, it follows that Theorem 2.1 from [19] is not a generalization result of [6], the main result of Jungck [1], and other results. Proposition 2.11. Let d be symmetric on X, and let a pair of mappings f, g have a unique POC. Then it is weakly g-biased if and only if it is occasionally weakly g-biased. Abstract and Applied Analysis 5and Applied Analysis 5 Proof. In this case, we have only to prove that (7) implies (6). Let w fx gx be the given POC. Let y ∈ C f, g , y / x. We have to prove that d gfy, gy ≤ d fgy, fy . Now w1 fy gy is a POC for the pair f, g . By the assumption, w w1, that is, fy gy fx gx. Further, we have gfy gfx and gy gx, which implies that d gfy, gy d gfx, gx ≤ d fgx, fx d fgy, fy , that is, the pair f, g satisfies (6). The following example shows that the assumption about the uniqueness of POC in Propositions 2.3, 2.4, 2.6, and 2.11 cannot be removed. Example 2.12. LetX 1, ∞ , d x, y |x−y|, fx 3x−2, gx x2 (see [21]). It is obvious that C f, g {1, 2}, the pair f, g is occasionally weakly g-biased, but it is not weakly g-biased. Also, f, g is occasionally weakly compatible, but it is not weakly compatible. However, the pair f, g has not the unique POC. Acknowledgments The authors are very grateful to the anonymous referees for their valuable comments and suggestions. The second author is thankful to the Ministry of Science and Technological Development of Serbia.