Abstract:
Banach contraction principle [2] can be seen as the pivot of the theory of fixed
point and its applications. The theory of fixed point plays an important role
in nonlinear functional analysis and it is very useful for showing the existence
and uniqueness theorems for nonlinear differential and integral equations. The
importance of the Banach contraction principle cannot be over emphasized in
the study of fixed point theory and its applications. The Banach contraction
principle have been extended and generalized by researchers in this area by
considering classes of nonlinear mappings and spaces which are more general
than the class of a contraction mappings and metric spaces (see [1,7,10,14–16,
19, 22] and the references therein). For example, Geraghty [11] introduced a
generalized contraction mapping called Geraghty-contraction and established
the fixed point theorem for this class of contraction mappings in the frame
work of metric spaces. We recall that for a metric space (X, d), a mapping
T : X → X is said to be an α-contraction if there exists α ∈ [0, 1) such that
(1.1) d(T x, T y) ≤ αd(x, y), ∀ x, y ∈ X.
