In this article, we introduce a new type of non-expansive mapping, namely weakly K-nonexpansive mapping, which is weaker than non-expansiveness and stronger than quasi-nonexpansiveness and prove some weak and strong convergence results, using weakly K-nonexpansive mappings. Also we define weakly (α;K)-nonexpansive mapping and using it prove one stability result for JF-iteration process. Some prominant examples are presented which illustrats the facts. A numerical example is presented to compare the convergence behavior of some known iterative algorithms, for weakly K-nonexpansive mappings. Moreover we show by an example that the class of α-nonexpansive mappings due to Aoyama and Kohsaka and the class of generalized α-nonexpansive mappings due to Pant and Shukla are independent. Finally, our fixed point theorem is applied to obtain solution of a nonlinear fractional differential equation.