Investigation of positive definite solution of nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i$$

In this paper, we consider the nonlinear matrix equation $$X^{p}=Q +\sum \nolimits _{i=1}^m A_i^*X^{\delta }A_i,$$ where $$A_i (i=1,2,\ldots ,m)$$ are $$n\times n$$ nonsingular complex matrices, Q is a $$n\times n$$ Hermitian positive definite (HPD) matrix, $$p\ge 1, m\ge 1$$ are positive integers, and $$\delta \in (0,1)$$ . We discuss the solution of this equation via properties of Thompson metric and two fixed point theorems in ordered Banach spaces and estimate the bounds of the HPD solution. Furthermore, perturbation analysis is investigated.