J. Fixed Point Theory Appl. 17 (2007) 1–32 DOI 10.1007/s11784-016-0291-2 © Springer International Publishing 2016 Journal of Fixed Point Theory and Applications Positive definite solution of a nonlinear matrix equation Snehasish Bose, Sk Monowar Hossein and Kallol Paul Abstract. Using fixed point theory, we present a sufficient condition for the existence of a positive definite solution of the nonlinear matrix equa- tion X = Q ± ∑ m i=1 A i * F (X)A i , where Q is a positive definite matrix, A i ’s are arbitrary n × n matrices and F is a monotone map from the set of positive definite matrices to itself. We show that the presented con- dition is weaker than that presented by Ran and Reurings [Proc. Amer. Math. Soc. 132 (2004), 1435–1443]. In order to do so, we establish some fixed point theorems for mappings satisfying (ψ,φ)-weak contractivity conditions in partially ordered G-metric spaces, which generalize some existing results related to (ψ,φ)-weak contractions in partially ordered metric spaces as well as in G-metric spaces for a given function f . We conclude, by presenting an example, that our fixed point theorem cannot be obtained from any existing fixed point theorem using the process of Jleli and Samet [Fixed Point Theory Appl. 2012 (2012), Article ID 210]. Mathematics Subject Classification. 15A24, 47H09, 47H10. Keywords. Matrix equation, fixed point, weak contraction, lower semi- continuous function, partially ordered set, G-metric space. 1. Introduction Consider the nonlinear matrix equation X = Q ± m ∑ i=1 A i * F (X)A i , (1.1) where Q is a positive definite matrix, A i ’s are arbitrary n×n matrices and F is a continuous monotone map from the set of positive definite matrices to itself. This kind of matrix equation closely resembles the discrete algebraic Riccati equation and therefore the existence of positive definite solutions of (1.1) arises in a number of applications such as in system theory, control theory, dynamic programming, stochastic filtering, ladder networks, statistics and in many other diverse fields; see [7, 24] and references therein. Thus, it has