In this manuscript, a class of self-mappings on cone Banach spaces which have at least one fixed point is considered. More precisely, for a closed and convex subset C of a cone Banach space with the norm ∥ x ∥ P = d ( x , 0 ) , if there exist a , b , s and T : C → C satisfies the conditions 0 ≤ s + | a | − 2 b < 2 ( a + b ) and 4 a d ( T x , T y ) + b ( d ( x , T x ) + d ( y , T y ) ) ≤ s d ( x , y ) for all x , y ∈ C , then T has at least one Fixed point.