New Fixed Point Theorem for generalized TF -contractive Mappings and its Application for Solving Some Polynomials

Let (X, d) be a complete metric space. In this paper, we study some new fixed point theorems for generalized TF -contractive mapping defined on complete metric spaces by using graph closed concept and we proved the existence and uniqueness of a fixed point. These conditions are analogous to Ćirić conditions. In this paper, we compare the two concepts of graph closed and sequentially convergent, and we show that the concept of sequentially convergent is a special case of the concept of graph closed. Also, we provide an counterexample for Dubey et. al. and provide an example in support of our main results. Finally, by using our main results, we present an application to solving some polynomials. Our results, extend several results on the topic in the corresponding literature.

In 2010, Moradi and Beiranvand [13] introduced a new class of contractive mappings and extend the Branciari's theorem as follows: 1), T, f : X −→ X be two mappings such that T is one-to-one and graph closed and f is a T F -contraction; that is: for all x, y ∈ X, where F : [0, +∞) −→ [0, +∞) is nondecreasing and continuous from the right with F −1 (0) = {0}; then f has a unique fixed point a ∈ X. Also for every x ∈ X, the sequence of iterates {T f n x} converges to T a.
In 2015, Mehmet Kir and Hukmi Kiziltunc [11], extended Kannan fixed point theorem by using T Fcontraction mappings. After that in 2017, Dubey et. al. [9], proved some fixed point theorems for T F type contractive conditions in the framework of complete metric spaces. Some of their results, as follows: Let (X, d) be a complete metric space and T, f : X −→ X be mappings such that T is continuous, one-to-one and subsequentially convergent. If a, b ∈ [0, 1) and x, y ∈ X where F : [0, +∞) −→ [0, +∞) is nondecreasing and continuous from the right with F −1 (0) = {0}. Then f has a unique fixed point a ∈ X. Also, if T is sequentially convergent then for every x 0 ∈ X the sequence of iterates {f n x 0 } converges to the fixed point. Theorem 1.5. [9] Let (X, d) be a complete metric space and T, f : X −→ X be mappings such that T is continuous, one-to-one and subsequentially convergent. If a, b, c ∈ [0, 1) and x, y ∈ X where F : [0, +∞) −→ [0, +∞) is nondecreasing and continuous from the right with F −1 (0) = {0}. Then f has a unique fixed point a ∈ X. Also, if T is sequentially convergent then for every x 0 ∈ X the sequence of iterates {f n x 0 } converges to the fixed point. Theorem 1.6. [9] Let (X, d) be a complete metric space and T, f : X −→ X be mappings such that T is continuous, one-to-one and subsequentially convergent. For all x, y ∈ X where a(x, y), b(x, y), c(x, y) ≥ 0 and sup[a(x, y) + 2b(x, y) + 2c(x, y)] ≤ λ < 1 and where F : [0, +∞) −→ [0, +∞) is nondecreasing and continuous from the right with F −1 (0) = {0}. Then f has a unique fixed point a ∈ X. Also, if T is sequentially convergent then for every x 0 ∈ X the sequence of iterates {f n x 0 } converges to the fixed point.
Remark 1.7. In the proof of above theorems, the boundedness of the sequence {T f n x 0 } is used by the authores, but not proved. Also the authores just considered a, b, c ∈ [0, 1) for Theorem 1.4 and Theorem 1.5.
In the following, we give counterexamples for these tow theorems. In the main resualts of this paper, we extend and correct the above theorems.
Example 1.8. Let X = {1, 2} endowed with the Euclidean metric and let f : It can be easily verified that, the condition of Theorems 1.4 and 1.5 are hold. But f has not fixed point.
In 2011, Samet and Vetro [19], generalized the Ćirić's fixed point theorem as follows: For all t ≥ 0, we define We assume that for every x, y ∈ X there exist non-negative numbers q(x, y), r(x, y), s(x, y) and t(x, y) such that sup holds for every x, y ∈ X. Then f admits a unique fixed point u ∈ X and for each x ∈ X, the sequence {f n x} converges to u.
In Section 3 we extend Ćirić, Branciari, Rhoades, Moradi and Beiranvand, Dubey et. al., Mehmet Kir and Hukmi Kiziltunc and Samet-Vetro's Theorems. In Section 4, as an application, we provide a solution to find an answer to some polynomials.

Preliminaries
In this paper (X, d) denotes a complete metric space.
Definition 2.2. [12] Let (X, d) be a metric space. A mapping T : X −→ X is said to be sequentially convergent if we have, for every sequence {y n } , if {T y n } is convergent then {y n } also is convergent. T is said to be subsequentially convergent if we have, for every sequence {y n } , if {T y n } is convergent then {y n } has a convergent subsequence.
Let Ψ denotes the class of all nondecreasing and continuous maps F : if there exists F ∈ Ψ and one-to-one and graph closed mapping T : X −→ X such that for all x, y ∈ X and some α ∈ For the main results of this paper, we prove the following usful lemma.
Lemma 2.4. Let (X, d) be a complete metric space and let T : X −→ X be a mapping such that T is continuous and subsequentially convergent. Then T is a graph closed map.
Proof. Suppose that {x n } is a sequence such that lim n→∞ T x n = a. Since T is subsequentially convergent, then there exists a subsequence {x n(k) } such that lim k→∞ x n(k) = b. Since T is continuous and lim n→∞ T x n = a, then we conclude that T b = a. This completes the proof.
Remark 2.5. In 2012 Aydi et. al. [3] proved that the main results of some papers; that consider the sequentially convergent; are particular results of previous existing theorems in the literature. We can not conclude that, every graph closed map is subsequentially convergent. For example, suppose that X = R endowed with the Euclidean metric and T : X −→ X defined by, T x := sin x. Obviousley, T is continuous and graph closed, but T is not subsequentially convergent. Because the sequence {sin(2nπ)} is convergent, but the sequence {2nπ} has not any convergent subsequence. In this paper we consider the graph closed mappins for the main results.

Main Results
The following theorem is the main result of this paper.
Theorem 3.1. Let (X, d) be a complete metric space and let f : X −→ X be a mapping such that, for all x, y ∈ X and some α ∈ [0, 1) (i.e., generalized T F -contractive) where F ∈ Ψ and T : X −→ X is a one-to-one and graph closed map. Then f has a unique fixed point b ∈ X and for every x ∈ X the sequence of iterates {T f n x} converges to T b. Also if T is sequentially convergent then for every x ∈ X the sequence of iterates {f n x} converges to b (the fixed point of f ).
Let x ∈ X. Define x n = T f n x.
Therefore {x n } is Cauchy.
Step 4. f has a fixed point. proof. Since (X, d) is complete and {x n } is Cauchy, there exists a ∈ X such that Since T is graph closed, there exists b ∈ X such that T b = a. Now we show that b is a fixed point of f . From F ∈ Ψ we conclude that where Hence Hence, from (14) and this shows that Obviousely, if T is sequentially convergent then for every x ∈ X the sequence of iterates {f n x} converges to b and this completes the proof. The following example shows that (3) is indeed a proper extension of (1).
Example 3.3. Let X = [1, +∞) endowed with the Euclidean metric. We consider a mapping S : X −→ X defined by Sx = 4 √ x. Obviously S has a unique fixed point b = 16. If (1) holds, then for every x, y ∈ X such that x = y, we must have |Sx − Sy| < m(x, y).
But by taking x = 1 and y = 4 we have, |Sx − Sy| = m(x, y) = 4. Therefore we can not use the Rhoades theorem (Theorem 1.2). Now we define T : X −→ X by T x = ln(e.x). Obviously T is one-to-one and graph closed. By taking F (t) = t, all conditions of Theorem 3.1 are hold and therefore S has a unique fixed point. Since (2) holds and F is nondecreasing, ≤ q(x, y)F (N (x, y)) + r(x, y)F (N (x, y)) + s(x, y)F (N (x, y)) ≤ λF (N (x, y)).
holds for every x, y ∈ X. So by letting T x = x, all conditions in Theorem 3.1 are hold. Hence by using Theorem 3.1 f has a unique fixed point. Therefore Theorem 3.1 is a generalization of Theorem 1.9.
In the following, we extend the Theorem 1.6.
Corollary 3.5. Let (X, d) be a complete metric space and let f : X −→ X be a mapping such that, where a(x, y), b(x, y), c(x, y) ≥ 0 for all x, y ∈ X and sup x,y∈X [a(x, y) + 2b(x, y) + 2c(x, y)] ≤ λ < 1 for some λ ∈ [0, 1), and where F ∈ Ψ and T : X −→ X is a one-to-one and graph closed map. Then f has a unique fixed point b ∈ X and for every x ∈ X the sequence of iterates {T f n x} converges to T b. Also if T is sequentially convergent then for every x ∈ X the sequence of iterates {f n x} converges to b (the fixed point of f ).
Proof. One can esealy shows that a(x, y)[F (d(T x, T y))] + b(x, y)[F (d(T x, T f y)) + F (d(T y, T f xy))] +c(x, y)[F (d(T x, T f x)) + F (d(T x, T f y))] ≤ λF (N (x, y)), for all x, y ∈ X. Now by using Theorem 3.1, the result is obtained.

Application to solving polynomials
As an application of the main theorem of this paper we conclude the existence of solution of some polynomials.
has a unique solution on [ n √ c, +∞).
Proof. Let 0 < ε < b n √ c be arbitrary. Put α = c + ε. It is enough to show that the problem (14) has a unique solution on [ n √ α, +∞). There exists β > 0 such that ln(α − c) + β ≥ α. Hence f is generalized T F -contractive. So f has a unique fixed point z on [α, +∞) and the sequence of iterates {T f n (c + 1)} converges to T z and therefore, the sequence of iterates {f n (c + 1)} converges to z. Therefore the equation x = b n √ x + c has a unique solution on [α, +∞). Also there exists a unique y > 0 such that y n = z. Obviously y ∈ [ n √ α, +∞). Hence from z = b n √ z + c we have y n = by + c and this completes the proof.