On Fixed Point Theorem of C Class Functions-B Weak Cyclic Mappings

First; the class Φ is all non-decreasing mappings φ: [0,∞)→ [0,∞) characterized by φ(t) = 0 if and only if t = 0. Bilgili et al. (2014; Karapinar and Sadarangani, 2012; Du and Karapinar, 2013) discussed the concept of Φweakly cyclic contraction mappings and proved fixed point theorems for mappings on Banach spaces. While Harjani et al. (2013) considered cyclic weak Φcontraction on compact metric spaces, the considered mapping φ need not be continuous, another fixed point treating the concept is given in (Karapinar and Sadarangani, 2012). The fixed point theorem given in (Karapinar and Sadarangani, 2012) focused on a wider class of metric spaces. Jleli et al. (2014; Karapinar et al., 2012b) generalized the results to cyclic (φ, φ)-weak contractions in some other metric spaces. Many results have been proved in different situations and settings for the purpose of generalization of the Banach contraction principle for contraction mappings and for non-expansive mappings, (Hardy and Rogers, 1973; Gregus, 1980; Kaewcharoen and Kirk, 2006; Kannan, 1971; Kirk, 1965; Park, 1980; Rhoades, 1977; 2001; Sahar Mohamed Ali Abou Bakr, 2013; Wong, 1975; Rhoades, 2009; Ćirić, 2006). Recent results related to cyclic weak (φ-φ)contraction mappings appeared in (Sahar Mohamed Ali Abou Bakr, 2017) for mappings with weak cyclic representation in complete metric spaces and weakly complete normed spaces, the considered mapping φ need not be additive, the author gave some examples. On the other side, Morales and Rojas (2009) defined BZ type mappings or the B-Zamfirescu mappings, the mappings of any of the following types:


Introduction
First; the class Φ is all non-decreasing mappings ϕ: [0,∞)→ [0,∞) characterized by φ(t) = 0 if and only if t = 0. Bilgili et al. (2014;Karapinar and Sadarangani, 2012;Du and Karapinar, 2013) discussed the concept of Φweakly cyclic contraction mappings and proved fixed point theorems for mappings on Banach spaces. While Harjani et al. (2013) considered cyclic weak Φcontraction on compact metric spaces, the considered mapping φ need not be continuous, another fixed point treating the concept is given in (Karapinar and Sadarangani, 2012). The fixed point theorem given in (Karapinar and Sadarangani, 2012) focused on a wider class of metric spaces. Jleli et al. (2014;Karapinar et al., 2012b) generalized the results to cyclic (φ, ϕ)-weak contractions in some other metric spaces.
Recent results related to cyclic weak (φ-ϕ)contraction mappings appeared in (Sahar Mohamed Ali Abou Bakr, 2017) for mappings with weak cyclic representation in complete metric spaces and weakly complete normed spaces, the considered mapping φ need not be additive, the author gave some examples.
On the other side, Morales and Rojas (2009) defined BZ type mappings or the B-Zamfirescu mappings, the mappings of any of the following types: It is proved the existence of only one fixed point for such types of mapping for the continuous, one to one and sub-sequentially convergent mapping B.

Mathematical Preliminaries
First, in the sequel, (X, d) is the space X with a metric d and the class U is the class of all finite collections of nonempty closed subsets of The mappings B and S are self mappings on X. We have the following definitions.
Definition 1 (Bilgili and Karapinar, 2013) B is known as weak Φ-contraction if and only if there is a continuous function ϕ ∈ Φ such that: Definition 2 (Bilgili et al., 2014) The element Definition 3 (Jleli et al., 2014) B is U-cyclic Φ-weak contraction on X if and only if there are A∈U and a continuous function ϕ∈Φ satisfying the two conditions: for every x∈A k , y∈A k+1 , k = 1,2,..., j and A j+1 = A 1 .

Definition 4
B is U-cyclic φ-ϕ-weak contraction on X if and only if there are ∈ A U and φ, ϕ∈Φ, with φ continuous such that the following are true: ∀ x∈A k ; y∈A k+1 , k = 1,2,..., j and A j+1 = A 1 This paper generalizes the U-cyclic φ-ϕ-weak contraction types to new B-U-cyclic weak F-φ-ϕ contraction type, this new C-class of weak contraction mapping is defined step by step next.

B S A A B S A A B S A A and B S A A
Then the self mapping S on X is B-cyclic weak φ-ϕ contraction mapping on X if and only if there are φ, ϕ∈Φ with φ continuous such that the following are true: • A is a TS-cyclic representation of X

Remark
The weak type contraction mapping defined in (6) generalizes the definition of cyclic weak ϕ-contraction of Erdal Karapinar, Kishin Sadarangani, cyclic weak (φ-ϕ) contraction of Sahar Mohamed Ali Abou Bakr and TB contraction mappings of Jose R. Morales, Edixon Rojas (Karapinar and Sadarangani, 2012;Sahar Mohamed Ali Abou Bakr, 2017;Morales and Rojas, 2009) respectively, because these are a particular cases corresponding to taking B and φ identities.
Finally, we have the following: Definition 8 (Ansari, 2014;Ansari et al., 2016) The real valued mapping F: it is continuous and satisfying the axioms: is a C-class function, in particular we have the following: In addition to the following: Now; the generalized C-class of TS cyclic weak (φ, ϕ)-contraction mappings are defined as: continuous and F∈C satisfying: for every x∈A k , y∈A k+1 ,k = 1, 2,...,j and A j+1 = A 1

Remark
The contraction type mapping defined in definition (9) is a generalization of the contraction type defined in definition (6), because it is a particular case when taking

Main Results
The results of this work are depending on Propositions (1) and (2) below.

Proposition 1
Let S be U-B-cyclic F-φ-ϕ-weak contraction on X. Then: Proof Choose x 0 ∈ X and focus on the iterated sequence:  If there is a natural number n 0 such that S(x n+1 ) = S(x n )∀n ≥ n 0 , then S(x n ) is fixed of S. This insures that such infimum is zero. Suppose that B(S(x n+1 )) ≠ B(S(x n )) for all n = 0, 1, 2,.... Then, the contraction condition yields: Since φ is non-decreasing function, we see that: This proves that the sequence {d(B(x n ), B(x n+1 ))} n∈N is a non-decreasing, hence the limit: The limit of the inequalities (3.1) as n→∞ gives: Letting n →∞ in the last inequalities, we get the following contradiction:

Proposition 2
Let S be U-B-cyclic F-φ-ϕ-weak contraction mapping on X. Then the iterated sequence {B(x n ) = B(S n (x 0 ))} n∈N is Cauchy.

Proof
We determine for a given ϵ>0 a natural number n 0 ∈N satisfying; if m, n>n 0 with x n ∈A i and x m ∈A i+1 for some i∈{1, 2,... j} (n-m ≡ 1(j)) then d(B(x n ),B(x m ))<ϵ gives a contradiction.
Finally; we have:

Theorem 1
If (X, d) is complete, B is one to one continuous self sequentially convergent mapping on X and S is U-Bcyclic F-φ-ϕ-weak contraction mapping on X. Then S owns fixed points. Moreover; we have the following: , because each A i , i = 1, 2,..., j contains in finitely many members of the infinite sequence {B(S n (x 0 ))} n∈N , hence y is a limit point for Ai for each i = 1, 2,..., j, giving that Ai is closed for each i = 1, 2,..., j shows that y∈A i for each i = 1, 2,...,j (as closed set contains all its limit points), thus: Using (3.6) and (3.7) proves that B(z) = y. We will see that such a z is fixed of S. In fact; there is i∈{1, 2,..., j} such that z∈A i , for the number i +1 we get n i ∈N such that:   Using the continuity of φ and taking the limit of the inequalities (3.9) as i→∞ prove the following: Using (3.11) in (3.8) after taking the limit as i→∞ gives: Hence d(B(S(z)), B(z)) = 0, therefore, B(S(z)) = B(z), since B is one to one, we get S(z) = z and hence z is fixed of S.
To show that two consecutive sets of { } 1 j i i A = cannot contain two different fixed points, by contrary assume that w and z are two different fixed of S, S(w) = w and S(z) = z those are lying in two consecutive sets, we have the following:  (d(B(w),B(z))) = 0, or ϕ(d(B(w),B(z))) = 0, that is; d(B(w),B(z)) = 0, consequently B(w) = B(z), since B is one to one, w = z. This completes the proof. We also have:

Theorem 2
Let (X, ∥.∥) be weakly complete normed space, C be a

Proof
Using Proposition (2) the sequence of iterates {B(S n (x 0 ))} n∈N is Cauchy, using the weak completeness assumption of X there exists x∈X such that: Since C is closed convex subset of X, the sequence {B(S n (x 0 ))} n∈N converges strongly to x and x∈C. For the other parts of the proof use Theorem (1).

Conclusion
This paper suggests new C-class of TS cyclic weak (φ, ϕ)-contraction mappings and proved the existence of unique fixed point for such types of mappings.