D. R. Smart in his book entitled “Fixed Point Theorems” raised some problems to describing the fixed point properties of non-expansive mapping and shrinking mapping. In this paper, we have introduced and solved these fixed point problems connecting to non-expansive mapping and shrinking mapping. Here we have also introduced and solved two other fixed point problems of non-expansive mapping where the mapping is defined on the Hilbert space.

fixed point, contraction mapping, non-expansive mapping, shrinking mapping.