Recently, Filipovi\'c et al. [M. Filipovi\'c, L. Paunovi\'c, S. Radenovi\'c, M. Rajovi\'c, Remarks on "Cone metric spaces and fixed point theorems of $T$-Kannan and $T$-Chatterjea contractive mappings", Math. Comput. Modelling. 54 (2011) 1467-1472] proved several fixed and periodic point theorems for solid cones on cone metric spaces. In this paper several fixed and periodic point theorems for $T$-contraction of two maps on cone metric spaces with solid cone are proved. The results of this paper extend and generalize well-known comparable results in the literature.