Int. Journal of Math. Analysis, Vol. 3, 2009, no. 8, 385 - 392
New Contractive Conditions and Fixed Point S. S. Chauhan Applied Science Dept., Punjab College of Engg.and Technology, Lalru Mandi Distt. Mohali, Punjab, India Mailing address: # 59,Badal Colony, Zirakpur, Distt. Mohali, Punjab, India
[email protected] Abstract Let X be a metric space and if one of A(X),B(X),C(X),D(X),R(X),S(X),T(X) and U(X) is complete subspace of X, then a coincidence theorem and a fixed point theorem for four pairs of weakly compatible mappings under a new contractive condition without using continuity have been proved. Also an example is illustrated in the support of this theorem to further strengthen the concept. Consequently some corresponding results related to fixed point have been improved, especially the result of Jha et al. [8], Chug and Kumar [10] and Popa [17]. Mathematical Subject Classification: 47H10, 54H25 Keywords: Fixed point, metric space, weak compatible mappings
1. Introduction The study of common fixed point of mappings satisfying contractive type conditions have been very active field of research during last two decades. Many researchers like Jungck [4], Pathak [5], Jachymski [6,7], Pant [11,12,13], and Pant et al. [14,15] are playing constructive role to enrich this field. Jungck [2] proved a common fixed point theorem for commuting mappings which generalizes the Banach’s fixed point theorem, according to which “Let (X,d) be a complete metric space. If T satisfies d(Tx,Ty) ≤ kd(x,y) for each x,y ∈ X where 0 ≤ k 0 such that ε ≤ m(x,y) < ε + δ implies d(Ax,By) ≤ ε (1.2) or a φ-contractive condition of the form d(Ax,By)≤φ(m(x,y)) (1.3) where φ: R+ → R+ is a gauge contractive function such that φ(t) < t for each t > 0. Various conditions on φ in addition to (1.3) have been used by different authors for the existence of a fixed point such as follow (A) φ(t) is non decreasing and lim φn(t) = 0 for each t >0 ([3],[7]), n →∞
(B) φ is non decreasing and continuous from right [16] (C) φ is upper semi continuous ( [7] ,[9], 12],[13]) and (D) φ(t) is non decreasing and t/ ( t-f(t)) is non increasing ( [1]). It is now known that (see [7], [14]) if any of the conditions (A), (B), (C) or (D) is assumed on φ then a φ contractive condition (1.3) implies an analogous (ε, δ) – contractive condition (1.2) and both the contractive condition hold simultaneously. Similarly, a Meir-Keeler type (ε, δ) - contractive condition does not ensure the existence of a fixed point. The following example illustrate that an (ε, δ) –contractive condition of type (1.2) neither ensures the existence of a fixed point nor implies an analogous φ- contractive condition (1.3). Example1 ([14]): Let X = [0,2] and d be the Euclidean metric on X. Define f: X → X by f(x) = (1+x)/2 if x < 1; f(x) = 0 if x ≥ 1.Then it satisfies the contractive condition ε ≤ max {d(x,y),d(x,fx),d(y,fy),[d(x,fy)+d(fx,y)] / 2} < ε + δ implies d(fx,fy) < ε ,with δ(ε ) =1 for ε ≥ 1 and δ(ε ) = 1- ε for ε < 1 but f does not have a fixed point. Also f does not satisfy the contractive condition d(fx,fy) ≤ φ(max {d(x,y),d(x,fx),d(y,fy),[d(x,fy)+d(fx,y)] / 2}) ,since φ(t) is not defined at t=1 . Hence the two type contractive conditions (1.2) and (1.3) are independent of each other. Thus to ensure the existence of common fixed point under the condition (1.2), the following conditions on the function δ have been introduced and used by various authors (E) δ is non decreasing ([11], [12]) (F) δ is lower semi continuous ([3], [4]) Jha et al.[8] assumed contractive condition (1.2) together with the following condition of the form d(Ax,By) < max {k1[d(Sx,Ty)+d(Ax,Sx)+d(By,Ty),k2[d(Sx,By)+d(Ax,Ty)] / 2}
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for 0 ≤ k1 0 there exist δ > 0 such that for all x,y in X ε ≤ M (x, y) < ε + δ implies d(Ax,By)≤ε (2.1) and d(Ax ,By) < M(x ,y) whenever M (x,y) > 0 (2.2) where M(x ,y) = max {d(Sx ,Ty),d(Ax ,Sx),d(By,Ty),[d(Sx,By)+d(Ax,Ty)] / 2} Then for each x0 in X the sequence yn in X defined by the rule y2n = Ax2n = Tx2n+1 ; y2n+1 = Bx2n+1 = Sx2n+2 is a Cauchy sequence. Lemma [8]: Let (A, S) and (B, T) be compatible pairs of self mappings of a complete metric space (X, d) such that (a) AX ⊂ TX, BX ⊂ SX (b) given ε > 0 there exist δ > 0 such that for all x, y in X ε ≤ M (x, y) < ε + δ implies d(Ax ,By) ≤ ε , (c)d(Ax, By) < max {k1[d(Sx,Ty)+d(Ax,Sx)+d(By,Ty)], k2[d(Sx,By)+d(Ay,Tx)] /2}, 0 ≤ k1 0 there exist δ > 0 such that for all x ,y in X ε ≤ M(x ,y) < ε + δ implies d(Ax ,By) ≤ ε , (iii)d(Ax,By)
