IJPAM: Volume 109, No. 3 (2016)

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

Dipankar Das$^1$, Nilakshi Goswami$^2$
$^{1,2}$Department of Mathematics
Gauhati University
Guwahati, 14, Assam, INDIA

Abstract. Let $X$ and $Y$ be Banach spaces and $P$ and $Q$ be two subsets of $X$ and $Y$ respectively. Let $T_1:X\otimes_\gamma Y \to X$ and $T_2:X\otimes_\gamma Y \to Y$ be two mappings and $S$ be a self mapping on $P\otimes Q$. Using $T_1$ and $T_2$ we define a self mapping $T$ on $X\otimes_\gamma Y$. Different conditions under which $T+TS+S$ has a fixed point in $P\otimes Q$ are established here. Analogous results are also established taking the pair $(T_1,T_2)$ as $(k,k^/)$ contraction mappings. Again considering $X\otimes_\gamma Y$ as a reflexive Banach space. We derive the conditions for $\dfrac{1}{m}(T+ST+S),\;m>2,\;m\in N,$ for having a fixed point in $P\otimes Q$. Some iteration schemes converging to a fixed point of $T+ST+S$ in $P\otimes Q$ are also presented here.

Received: May 5, 2016

Revised: August 27, 2016

Published: October 1, 2016

AMS Subject Classification: 46B28, 47A80, 47H10

Key Words and Phrases: projective tensor norm, reflexive tensor product, demiclosed mapping
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DOI: 10.12732/ijpam.v109i3.13 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 109
Issue: 3
Pages: 651 - 663


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