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Let C be a closed convex weakly Cauchy subset of a normed space X. Then we define a new {a,b,c} type nonexpansive and {a,b,c} type contraction mapping T from C into C. These types of mappings will be denoted respectively by {a,b,c}-ntype and {a,b,c}-ctype. We proved the following: 1. If T is {a,b,c}-ntype mapping, then {\|T(x)-x\|:xC}=0, accordingly T has a unique fixed point. Moreover, any sequence {x_n}_nN in C with _n→ ∞\|T(x_n)-x_n\|=0 has a subsequence strongly convergent to the unique fixed point of T. 2. If T is {a,b,c}-ctype mapping, then T has a unique fixed point. Moreover, for any xC the sequence of iterates {T^n(x)}_nN has subsequence strongly convergent to the unique fixed point of T. This paper extends and generalizes some of the results given in [2,4,7] and [13]. , number=3, journal=Analysis in Theory and Applications, year=2013, month=Jul., pages=280–288