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Results appearing in this paper can be used to establish the solvability of nonlinear discrete time systems subject to generalized nonlinear boundary conditions. Two separate sets of results are established, each of which can be used to establish existence of solutions to problems for which previous related work proves inconclusive. The first set of results imposes conditions on the sizes of nonlinearities, while the second framework requires geometric properties of the nonlinearity in the dynamics. [ABSTRACT FROM AUTHOR]

In the renormalization analysis of critical phenomena in quasi-periodic systems, a fundamental role is often played by fixed points of functional recurrences of the formwhere theare affine contractions and eachis eitheror. We develop a general theory of these fixed points by regarding them as fixed points of ‘composition sum operators’, and apply this theory to test for fixed points in classes of complex analytic functions with various key types of singularities. Finally we demonstrate the construction of the full space of fixed points of one important class, arising from the much studied operatorMdefined byThe construction reveals previously unknown solutions. [ABSTRACT FROM PUBLISHER]