Theorem 3.2 Let ( X, d ) be a metric space and T be a generalized convex contraction of order 2 on X with the based mapping α.
In this section, we introduce the concepts of ((alpha, beta ))-generalized convex contraction and ((alpha, beta))-generalized convex contraction of order 2 and prove the approximate fixed point theorems for such mappings.
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]).
Thus, T is not a convex contraction of order 2, 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 see that T is a generalized convex contraction of order 2. Example 3.4 Let X = { 1, 3, 5 }, ≤ = { ( 1, 1 ), ( 3, 3 ), ( 5, 5 ), ( 1, 3 ) }, d ( x, y ) = | x − y | and T be a selfmap on X defined by T 1 = 3, T 3 = 1 and T 5 = 5.
Also, we say that the selfmap T on X is a generalized convex contraction of order 2 whenever there exist a mapping α : X × X → [ 0, ∞ ) and a 1, a 2, b 1, b 2 ∈ [ 0, 1 ) with a 1 + a 2 + b 1 + b 2 < 1 such that α ( x, y ) d ( T 2 x, T 2 y ) ≤ a 1 d ( x, T x ) + a 2 d ( T x, T 2 x ) + b 1 d ( y, T y ) + b 2 d ( T y, T 2 y ). for all x, y ∈ X. Other useful references: [12 15].
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].
Recently, Miandaragh et al. [3, 4] introduced two more general concepts of convex contractions, which are called generalized convex contractions and generalized convex contractions of order 2 and they also discussed some approximate fixed point results for such mappings.
The purpose of this paper is to formulate the concepts of generalized convex contractions and generalized convex contractions of order 2 in general terms and prove the existence results of approximate fixed points for these mappings on a complete metric space by using the idea of cyclic ((alpha, beta))-admissible mappings due to Alizadeh et al. [5].
(G_{i}(0,0,t,mu)=O(mu^{n+2})), (G_{j}(0,t,mu)=O(mu^{n+2})); (G_{i} eta,w,t,mu)) is a contraction operator with contraction coefficient of order (O(mu)) for η and w of order (O(mu)); (G_{j} zeta,t,mu)) is a contraction operator with contraction coefficient of order (O(mu)) for ζ of order (O(mu)).
(G_{i} eta,w,t,mu)) is a contraction operator with contraction coefficient of order (O(mu)) for η and w of order (O(mu)); (G_{j} zeta,t,mu)) is a contraction operator with contraction coefficient of order (O(mu)) for ζ of order (O(mu)). .
end{aligned} (3.15) The operator (q(lambda,zeta,t,mu)) is a contraction with a contraction coefficient of order (O(mu)) for λ and ζ of order (O(mu)), and (q 0,0,t,mu)=0); (Q lambda,eta,zeta,t,mu)) is for contraction operators with contraction coefficient of order (O(mu)) for λ, η and ζ of order (O(mu ^{2})), and (Q 0,0,0,t,mu)=O(mu^{n+2})).
