We also derive some new common fixed point theorems for ψ-graphic contractions as well as ordered contractions on preordered metric space.
But the following examples show that the notion of generalized convex contractions is a real generalization for the notions of convex contractions and ordered convex contractions which were provided, respectively, in [3] and [1].
Thus, T is not an ordered convex contraction of order 2 which has been used in Theorem 2.5 of [3], while by putting α ( x, y ) = 1 4 whenever x ≤ y and α ( x, y ) = 0 otherwise and a 1 = a 2 = b 1 = b 2 = 1 8, it is easy to check that the selfmap T is a generalized convex contraction of order 2. Recently, the notion of weakly Zamfirescu mappings was provided in [19] (see also Zamfirescu [20]).
Again, the following examples show that the notion of generalized convex contractions of order 2 is a real generalization for the notions of convex contractions of order 2 and ordered convex contractions of order 2, which were provided, respectively, in [1] and [3].
Since is an order contraction mapping, is order continuous.
From order contraction of it follows that is order continuous.
Since is an order contraction mapping, for a.e., we have (3.8).
If is an order contraction mapping, then there exists a unique random fixed point in.
Both (first-order) contraction mappings and the rth-order contraction mappings defined in Theorem 1.3 are special cases of the now-proven generalised Banach contraction conjecture (see Jachymski [8], Merryfield-Stein [9] and Arvanitakis [10]).
The main result of this subsection is as follows, which extends the conclusion of the Banach fixed point theorem (Theorem 1.2) to higher-order contraction mappings: (Higher-order contraction mapping theorem).
