Best Proximity Pair Theorems for Multifunctions with Open Fibres
✍ Scribed by S. Sadiq Basha; P. Veeramani
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 145 KB
- Volume
- 103
- Category
- Article
- ISSN
- 0021-9045
No coin nor oath required. For personal study only.
✦ Synopsis
Let A and B be non-empty subsets of a normed linear space, and f : A Ä B be a single valued function. A solution to the functional equation fx=x, (x # A) will be an element x o in A such that fx o =x o (i.e., such that d( fx, x)=0). In the case of non-existence of a solution to the equation fx=x, it is natural to explore the existence of an optimal approximate solution that will fulfill the requirement to some extent. In other words, an element x o in A should be found in such a way that d
Thus, the crux of finding an optimal approximate solution to the aforesaid equation fx=x boils down to ascertaining a solution to the optimization problem Min[d(x, fx): x # A]. But, d(x, fx) d(A, B) for all x # A. So, in the case of seeking an optimal approximate solution to the aforesaid equation fx=x, it should be contemplated to find an element x o in A such that d(x o , fx o )=d(A, B). Indeed, given a multifunction T: A Ä 2 B with open fibres, best proximity pair theorems, furnishing the sufficient conditions for the existence of an element x o # A such that d(x o , Tx o )=d(A, B), are proved in this paper.