Abstract
In this paper, we present common fixed point theorems for commuting operators which generalize Darbo's and Sadovski's fixed point theorems. As example and application, we study the existence of common solutions of equations in Banach spaces using the measure of noncompactness. MSC: 47H10; 47H09
Highlights
In, Darbo [ ] proved the fixed point property for α-set contraction (i.e., α(S(A)) ≤ kα(A) with k ∈ ], [) on a closed, bounded and convex subset of Banach spaces
It should be noted that any α-set contraction is a condensing function, but the converse is not true
In, we have proved in [ ] the existence of a common fixed point for commuting mappings satisfying α S(A) ≤ k sup α Ti(A), ( )
Summary
In , Darbo [ ] proved the fixed point property for α-set contraction (i.e., α(S(A)) ≤ kα(A) with k ∈ ] , [) on a closed, bounded and convex subset of Banach spaces. Let be a convex closed bounded subset of X, I be a set of index, and {Ti}i∈I and S be two continuous functions from into such that:. Let be a convex, closed and bounded subset of X, I be a given set of index, and {Ti}i∈I , S be continuous functions from into such that:. The nonempty set F = {x ∈ : T x = x} is convex, closed and bounded subset of , for T being continuous and affine. Can be used to show that any affine operator T that commutes with S and satisfies Darbo’s condition [ ] has a common fixed point with S. Let X be a Banach space, be a nonempty convex, closed and bounded subset of X, T and S be continuous functions from into such that:.
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