On the nonlinear matrix equation X p = A + ∑ i = 1 m M i ∗ ( B + X − 1 ) − 1 M i

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We establish several necessary and sufficient conditions for the existence and uniqueness of Hermitian positive definite (HPD) solutions to the general matrix equation X p = A + ∑ i = 1 m M i ∗ ( B + X − 1 ) − 1 M i , where p, m are positive integers, M i (i = 1, 2, …, m) are n × n nonsingular complex matrices, A and B are HPD matrices, and then give three algorithms to get the unique solution. Moreover, we give two numerical examples to illustrate the effectiveness of the theoretical results and the behavior of the considered methods.