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In this paper we point out the following two points. Firstly, the result of Madjid Eshaghi Gordji et.al [ Madjid Eshaghi Gordji, Maryam Ramezani1, Yeol Je Cho and Saeideh Pirbavafa, A generalization of Geraghty's theorem in partially ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory and Applications 2012, 2012:74] can be easily obtained from the result of [ Amini-Harandi. A, Emami, H: A fixed point theorem for contraction type maps in partially ordered differential equations. Nonlinear Anal. 72, 2238-2242 (2010)] , so that the longer proof of the result is not necessary. Secondly, the existence of solution for a initial-value problem does not needed the application of the result of Madjid Eshaghi Gordji et.al, we can also apply the result of AminiHarandi. A et.al to get the the existence of solution for this initial-value problem and to get more generalized results. Keywords-Contraction mapping principle; Partially ordered metric spaces; Fixed point; The initial-value problem;Application.

Review for Fixed Point Theory and Applications 2012, 2012:74 (By Madjid Eshaghi Gordji et.al )”