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Stability of iterative procedures for multivalued maps in metric spaces

Singh S., Bhatnagar C., Mishra S.

Demonstratio Mathematica

2005

Vol. 38, nr 4

905-916

Let (X,d) be a metric space and T a self-map of X. Let Xn+i = f(T,Xn) denote some iterative procedure. Let {xn} be convergent to a fixed point u of T and {yn} be an arbitrary sequence in X. Set En=d[yn+i, f(T, yn)], n = 0,1,2,..., then the iterative procedure f(T,Xn) is T-stable provided that limEn = 0 implies that limnyn = u. This definition has been extended by Singh and Chadha [34] to discuss the problem of stability for multivalued operators on metric spaces. The purpose of this paper is to present a fixed point theorem for generalized multivalued contractions on a setting more general than metric spaces. The same is utilized to discuss the problem of stability of iterative procedures in multivalued analysis. Some special cases due to Stefan Czerwik and others are discussed as special cases.