Equivalent Contractive Conditions in Metric Spaces ∗

In this note, we introduce some contractive conditions which complete the corresponding contractive conditions appeared in references [1]-[11]. By these conditions, we prove that twenty-two contractive conditions are equivalent. Then we use these conditions to get some common fixed point theorems for noncompatible maps and some weakly compatible mappings. There are a number of papers dealing with fixed point, common fixed points for compatible maps or noncompatible maps. About these results, one can refer to [1][11]. J. Jachymski divided these results into two categories. The first category assumes that the maps employed satisfy some (ε − δ)-type conditions introduced by Meir and Keeler. The second category assumes that the maps satisfy some inequalities involving a contractive gauge function. Such a class of functions started from [2]. But many of these conditions are equivalent. Some equivalent conditions have been discussed in [3] and [4]. In this note, we introduce a limit type contractive condition (C1) and several other conditions (C2)-(C9). By these conditions, we prove that twenty-two contractive conditions are equivalent, this completes the work in [3] and [4]. These conditions were frequently used to prove existence theorems in common fixed point for compatible maps. However, the study of common fixed points of noncompatible mappings is also very interesting. In this note, by using these conditions and property (E.A) in [1], we prove some new common fixed point theorems for noncompatible maps and some weakly compatible mappings. Let (X,d) be a metric space, A,B, S and T be selfmaps of a setX. For any x, y ∈ X, we define M(x, y) = max{d(Sx, Ty), d(Sx,Ax), d(Ty,By), [d(Sx,By) + d(Ty,Ax)]/2}. We list the following contractive conditions: (C1) limn→∞M(xn, yn) = L > 0 implies limn→∞d(Axn, Byn) < L, for any sequences {xn}, {yn} ⊂ X. (C2) There exists a function δ : (0,∞)→ (0,∞) such that for any ε > 0, limt→εδ(t) > ε, and for any x, y ∈ X, ε ≤M(x, y) < δ(ε) implies d(Ax,By) < ε. ∗Mathematics Subject Classifications: 47H10 †Department of Mathematical, Xuzhou Normal University, Xuzhou 221116, P. R. China ‡Department of Mathematics, Gyeongsang National University, Chinju 660-701, South Korea §Department of Mathematics, Gyeongsang National University, Chinju 660-701, South Korea