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This paper studies the distributed fusion estimation problem for a class of time-varying systems with bounded nonlinearities, where nonlinear terms are considered under two different cases. By using matrix analysis approach, each local estimator with time-varying gain is designed by minimizing an upper bound of the square error of the estimator. Based on the obtained local estimators, the distributed fusion estimators are designed by establishing a class of convex optimization problems. Notice that the designed local estimators and fusion estimators are stable, and the established convex optimization problems can be easily solved by standard software packages. Localization of a mobile robot is given to demonstrate the effectiveness of the proposed methods.