Abstract. An analogue of Banach's fixed point theorem in partially ordered sets is proved in this paper, and several applications to linear and nonlinear matrix equations are discussed.

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In this paper a definition is given for an unbounded Toeplitz-like operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.2010 Mathematics Subject Classification. Primary 47B35, 47A53; Secondary 47A68.