Common Fixed Points of Generalized Cyclic C Class -- Weak Nonexpansive Mappings

This paper shows that if S and T are two joint generalized cyclic F-Ψ-ɸ-Λ weak nonexpansive type mappings, then they have only one common fixed point. In particular, every generalized cyclic C class Ψ-ɸ-Λ weak nonexpansive mapping has a unique fixed point. Hence it extends the results of the attached references of this paper.


Introduction and Preliminaries
Since 1922 till now many generalizations of Banach contraction principle (Banach, 1922) have been achieved.
For cyclic - mappings, we refer to the references below.
In particulr; Sahar Mohamed Ali Abou Bakr (2013) proved the existence of only one fixed point for both {a, b, c}-ntype and {a, b, c}-ctype types of mappings defined on closed convex weakly Cauchy subset C of a normed space X.

Definition 1
Let C be a subset of a normed space X and T be a mapping from C into C satisfying: Then: (1) T is said to be {a, b, c}-ntype mapping, if 0 < a < 1, 0 < b, 0  c < 1 = 2 and a + b + c = 1 (2) T is said to be {a, b, c}-ctype mapping, if 0 c < 1/2 and a + b + c < 1 Sahar Mohamed Ali Abou Bakr and Ansari ( (1) U is a T-cyclic representation of X with respect to S: That is; T(S(A1))  A2, T(S((A2)))  A3,...,T(S(Aj-1))  Aj and T(S((Aj)))  A1 (2) The following contractivity condition is satisfied: In this study; we define the real valued function S,(abc): X  X  R + as follows:

Definition 4
Let S: XX fulfill the condition:
S abc S abc , , S abc x y ad x y x y X     that is if b=c=0, then we have the usual contraction or nonexpansive mapping acoording to the value of a, a<1 or not. One can see some related fixed point theorems proved in the attached references below.
In the light of the particular cases; F(u, v) = u-v and = Id; the identity mapping, we noticed the following:

■■ (1) The class of all (A,B) generalized cyclic F--ϕ  weak non-expansive is wider than the class of all (A, B) generalized cyclic F--ϕ- weak contraction. (2) The class of all (A, B) generalized cyclic F--ϕ-
weak nonexpansive is wider than the class of all (A, ( In this study, the real valued function S,T,(abc): X  XR + is defined as: where, S, T: XX are two self mappings and a, b, c are three real numbers.
We introduced the following fascinating definition for joint-cyclic mapping:

Definition 7
Let (X, d) be a metric space with AB , S, T: XX be two self mappings and a, b, c  [0, 1] be three real numbers satisfying: (1) The cyclic condition: S(A) B and T(B) A (2) The contractivity condition: Then S and T are said to be joint (A, B) generalized cyclic: (1) ϕ- weak contraction types iff a + c + b < 1 (2) ϕ- weak nonexpansive types iff a + c + b = 1

Definition 8
Let (X, d) be a metric space with X A B  , S, T: XX be two self mappings and a, b, c[0, 1], b  0 be three real numbers satisfying: (1) The cyclic condition: S(A) B and T(B)  A (2) The contractivity condition: where,  and ϕ are non-decreasing functions , ϕ: and (0) = 0, ϕ(0) = 0,  is continuous and ϕ is lower semi-continuous.

Definition 9
Let (X, d) be a metric space with X A B  , S,T: XX be two self mappings and a, b, c[0,1] b  0 be three real numbers satisfying: (1) The cyclic condition: S(A) B and T(B)  A (2) The following contractivity condition:

Remarks
( This paper shows that if S and T are two joint generalized cyclic F--ϕ- weak nonexpansive types mappings, then they have only one common fixed point.
In particular, every cyclic C class generalized -ϕ- weak nonexpansive mapping has a unique fixed point.
The existing functions F,  and ϕ give extensions of many results of the references attached in this study.

Main Results
We have: First, suppose n is an odd natural number. Then: n n n n n n n n S T abc S T abc Since  is non-decreasing, we see that: , , n n n n n n S T abc Continuing gives: (2.4)

■■
Second; by a similar method when n is an even natural number, we obtain the same conclusion as inequalities (2.4). Hence the sequences {d(vn+1, vn)}nN and {S,T,(abc)(vn+1, vn)}nN are monotonic non-increasing and bounded below by 0, thus their limit exist, each equals its infimum.
On the other side; they have the same infimum because of the inequalities (2.4), therefore if their infimum is r, then: Notice that the converse is also true, if x0 = y0 = S(x0) = T(x0) = S(y0) = T(y0), then S,T,(abc)(x0, y0) = 0 is clear. On the other side, this showed that x0 = y0A∩B. If there exists another point vA∩B such that with v  y0, then we get: Hence; the following is a contradiction:

Proof
Let v be the unique common fixed point of S and T, in addition suppose that limn vn = u with v  u. Then there is n N such that:

Conclusion
This paper shows that if S and T are two joint generalized cyclic F--ϕ- weak nonexpansive type mappings, then they have only one common fixed point.
In particular, every generalized cyclic C class -ϕ- weak nonexpansive mapping has a unique fixed point.
Hence continuing restrictions of F,  and ϕ to be taken special cases gives extensions of many fixed point in the filed of fixed point theory. In particular, it extends the results of attached references in this study.