Abstract
Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach’s spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.
Highlights
Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces (X, d), which are Banach spaces if X is a vector space on a certain field F and the metric is homogeneous and translation-invariant, are discussed through the paper
It is proven in this paper that Banach contraction principle [1,2,3,4] guarantees the existence of unique fixed points under contractive conditions fulfilled by some relevant strips of composite operators within in the whole composite sequence of operators
We can consider a sequence of projection operators {PMk } with PMk : X → Mk such that Pk = PMk so that X = Im Pk ⊕ Ker Pk and z = Pkx ∈ Im Pk is in Mk for any x ∈ X and z = x − z = (I − Pk)x ∈ Ker Pk for k ∈ N0 = N ∪ {0}
Summary
Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces (X, d), which are Banach spaces if X is a vector space on a certain field F (usually R or C) and the metric is homogeneous and translation-invariant, are discussed through the paper. (ii) Assume that the self-mappings on X in the subsequence {Tk}k≥n0 are contractive for some n0 ∈ N0, that the sequence of operators {Tk} converges to T : X → X, and that the projection operator P : X → M ⊂ X is constant and bounded (i.e., if it is not orthogonal, i.e., it is oblique, its norm exceeds one and it is finite) Property (i) holds.
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