SOME FIXED POINT THEOREMS ON THE SUM AND PRODUCT OF OPERATORS IN TENSOR PRODUCT SPACES

Abstract

Let X and Y be Banach spaces and P and Q be two subsets of X and Y respectively. Let T1 : X ⊗ Y → X and T2 : X ⊗ Y → Y be two mappings and S be a self mapping on P⊗Q. Using T1 and T2 we define a self mapping T on X⊗Y. Different conditions under which T +TS+S has a fixed point in P ⊗Q are established here. Analogous results are also established taking the pair (T1,T2) as (k,k / ) contraction mappings. Again considering X ⊗ Y as a reflexive Banach space. We derive the conditions for 1 m (T + ST + S), m > 2, m ∈ N, for having a fixed point in P ⊗ Q. Some iteration schemes converging to a fixed point of T + ST + S in P ⊗ Q are also presented here.