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Let X be a Banach algebra and D be a nonempty subset of X. Let (T 1, T 2) be a pair of self mappings on D satisfying some specific conditions. Here we discuss different situations for existence of solution of the operator equation u = T 1 uT 2 u in D. Similar results are established in case of reflexive Banach algebra X with the subset D. Again, considering a bounded, open and convex subset B in a uniformly convex Banach algebra X with three self mappings T 1 ,T 2 ,T 3 on B, we derive the conditions for existence of solution of the operator equation u = T 1 uT 2 u + T 3 u in B. Application of some of these results to the tensor product is also shown here with some examples.
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