Contraction Type Mappings on a 2-Metric Space

✍ Scribed by B. E. Rhoades

• Publisher: John Wiley and Sons
• Year: 1979
• Tongue: English
• Weight: 235 KB
• Volume: 91
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19790910112

✦ Synopsis

The concept of a 2-metric space hw been investigated by 5. G~ELER in a seriea of papers [6]-[S]. Other papers dealing with 2-metric spaces are [3]-[S], [lo], and [ 123.

In this note several fixed point theorems a m proved for contractive mappings in a 2-metric space. The contradive definitions used are extensions of thoae of [l] and [2] to 2-metric spaces. It hw been shown in [ll] that the contractive definitions of [l] and [2] are two of the most general for metric spaces, so the corresponding definitions for 2-metric spaces will be among the most general definitions possible. The result of [lo] is a special case of Theorem 5.

A 2-metric space is a space X in which, for each triple of points a, b, c, there exists a real-valued noa-negative function satisfying : (I a) for each pair of points a, b, a + b of X, there exists a point c € X such that @(a. 6 , a ) +O, ( 1 1)) @(a, 6, a ) =O when at least two of the points are equal,

(2)

and

(3)

A sequence (x,,} in X is called CAUCHY if lim e(x*, xm, a) =O for all a € X .

A sequence (x,,) in X is convergent and xEX is the limit of this sequence if lim e(x,,,2,a) =O for each aEX. A complete 2-metric space is one in which every CAUCHU sequence converges. Theorem 1. Let X be a c m p h e 2-metriC space, f : X +X 8atiSfying: there ex&k an h, O s h < 1 such that for each x, y , a € X , @(a*, b, c) =e(a, c, b) =e(b, c, a ) , @(a, b, c ) s e ( a , b, d)+e(a, d, c)+e(d, b, c ) .

(4) e ( W , f(y), a ) s h max {e@, Y , a), e(z, fb), a), e(y, f M a), e(z, f ( y ) , 4, e(y, f(4, a)> . Then f possesses a unique fixed pint z and lim f"(xo =z for each xo€,EX'. [2], one can show that, for integers n, m, n=-msO, Proof. Let z o € X and define (z*} by ~,,+~=f(z,,), n=O, 1, 2,. . . Then, a8 in (5) @(xm, Xm, a) mh"@(%, xk, a) Y