Some Fixed and Periodic Points in Abstract Metric Spaces

and Applied Analysis 3 Let X, d be an abstract metric space. The following properties are often used, particularly in the case when the underlying cone is nonnormal. The only assumption is that the cone K is solid. For details about these properties see, for example, 11 . p1 If a ha where a ∈ K and h ∈ 0, 1 , then a θ. p2 If θ u c for each c, θ c, then u θ. p3 If u v and v w, then u w. p4 If c ∈ int K, θ xn, and xn → θ, then there exists k ∈ N such that, for all n > k we have xn c. Note that, in general, the converse is not true. Indeed, in Example 1.1, xn θ, but xn c for n sufficiently large. In generalizing some theorems of Huang-Zhang 7 , Abbas and Rhoades proved the following result in abstract metric spaces over normal cones. Theorem 1.4 see 9 . Let X, d be a complete cone metric space over a normal cone. Suppose that f, g : X → X are two self-maps satisfying d ( fx, gy ) αdx, y βdx, fx dy, gy γdx, gy dy, fx, 1.2 for all x, y ∈ X, where α, β, γ ≥ 0, and α 2β 2γ < 1. Then f and g have a unique common fixed point in X. Moreover, any fixed point of f is a fixed point of g and conversely. Sing et al. extended this result of Abbas-Rhoades to four maps. They proved the following theorem. Theorem 1.5 see 27 . Let X, d be a complete cone metric space over a normal cone. Suppose that the mappings f , g, S, and T are four selfmaps on X such that fX ⊂ TX and gX ⊂ SX and satisfying d ( fx, gy ) αdSx, Ty βdSx, fx dTy, gy γdSx, gy dTy, fx, 1.3 for all x, y ∈ X, where α, β, γ ≥ 0 and α 2β 2γ < 1. Suppose that the pairs {f, S} and {g, T} are weakly compatible. Then f , g, S, and T have a unique common fixed point. In 1977, Rhoades proved the following interesting result. Theorem 1.6 see 28 . Let X, d be a complete metric space. Let f : X → X, and suppose that there exist decreasing functions αi : 0, ∞ → 0, 1 , i 1, . . . , 5, such that ∑5 i 1 αi t < 1 for each t ∈ 0, ∞ and satisfying d ( fx, fy ) ≤ α1 ( d ( x, y )) d ( x, y ) α2 ( d ( x, y )) d ( x, fx ) α3 ( d ( x, y )) d ( y, fy ) α4 ( d ( x, y )) d ( fy, x ) α5 ( d ( x, y )) d ( fx, y ) , 1.4 for all x, y ∈ X, x / y. Then f has a unique fixed point z and for each x0 ∈ X the sequence {fx0} converges to z. We generalize in this paper Theorems 1.4 and 1.5 by removing normality condition in their formulations. An example will show that these generalizations are proper. Further, 4 Abstract and Applied Analysis some results of Abbas and Rhoades about periodic points of selfmaps from 29 are extended to abstract metric spaces. Theorem 1.6 is also presented in this new setting, with a slightly shorter proof. Note that it was shown in 15, 30, 31 that some of the fixed point results in abstract metric spaces can be directly reduced to the respective metric results. However, the results of the present paper do not fall into this category, since some of them are new even in the context of metric spaces. 2. Fixed Point Theorems In this section we will prove Theorems 1.4 and 1.5 by omitting the assumption of normality in the results. We use only the definition of convergence in terms of the relation “ ”. The only assumption is that K is a solid cone, so we use neither continuity of the vector metric d, nor Sandwich Theorem. We begin with the following. Theorem 2.1. Let X, d be an abstract metric space over a solid coneK. Suppose that f , g, S, and T are four self-maps on X such that fX ⊂ TX and gX ⊂ SX and suppose that at least one of these four subsets of X is complete. Let d ( fx, gy ) αdSx, Ty βdSx, fx dTy, gy γdSx, gy dTy, fx, 2.1 for all x, y ∈ X, where α, β, γ ≥ 0 and α 2β 2γ < 1. Then the pairs f, S and g, T have a unique common point of coincidence. If, moreover, pairs f, S and g, T are weakly compatible, then f , g, S, and T have a unique common fixed point. For definitions of terms like “point of coincidence” and “weakly compatible pair” see, for example, 11 . Remark 2.2. In the papers 9 and 27 , the coneK is supposed to be normal and solid. In that case the proof is essentially the same as in the setting of usual metric spaces. We now give the proof of Theorem 2.1. Proof. Suppose x0 ∈ X is an arbitrary point, and define the sequence {yn} by y2n fx2n Tx2n 1, y2n 1 gx2n 1 Sx2n 2, n 0, 1, 2, . . .. Now, as in 27 , by 2.1 , we have d ( y2n, y2n 1 ) d ( fx2n, gx2n 1 ) αd Sx2n, Tx2n 1 β [ d ( Sx2n, fx2n ) d ( Tx2n 1, gx2n 1 )] γ [ d ( Sx2n, gx2n 1 ) d ( Tx2n 1, fx2n )] α β γdy2n−1, y2n ) ( β γ ) d ( y2n, y2n 1 ) , 2.2 which implies that d y2n, y2n 1 δd y2n−1, y2n , where δ α β γ / 1 − β γ < 1. Similarly it can be shown that d ( y2n 1, y2n 2 ) δdy2n, y2n 1 ) . 2.3 Abstract and Applied Analysis 5and Applied Analysis 5 Therefore, for all n, d ( yn, yn 1 ) δdyn−1, yn ) · · · δndy0, y1 ) . 2.4