@ Appl. Gen. Topol. 20, no. 1 (2019), 81-95doi:10.4995/agt.2019.9949 c© AGT, UPV, 2019 Fixed point results concerning α-F-contraction mappings in metric spaces Lakshmi Kanta Dey a, Poom Kumam b and Tanusri Senapati c a Department of Mathematics, National Institute of Technology Durgapur, India (lakshmikdey@yahoo.co.in) b Department of Mathematics, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand. (poom.kum@kmutt.ac.th) c Department of Mathematics, Indian Institute of Technology Guwahati, Assam, India (senapati.tanusri@gmail.com) Communicated by I. Altun Abstract In this paper, we introduce the notions of generalized α-F-contraction and modified generalized α-F-contraction. Then, we present sufficient conditions for existence and uniqueness of fixed points for the above kind of contractions. Necessarily, our results generalize and unify sev- eral results of the existing literature. Some examples are presented to substantiate the usability of our obtained results. 2010 MSC: 47H10; 54H25. Keywords: metric space; fixed point; generalized α-F-contraction; modified generalized α-F-contraction. 1. Introduction and Preliminaries Throughout this paper we denote by R+, R, N and N0, the set of positive real numbers, set of real numbers, set of natural numbers and set of nonnegative in- tegers respectively. It is widely known that the Banach contraction principle [1] is the first metric fixed point theorem and one of the most powerful and versa- tile results in the field of functional analysis. Due to its significance and several applications, over the years, it has been generalized in different directions by Received 12 April 2018 – Accepted 11 November 2018 http://dx.doi.org/10.4995/agt.2019.9949 L. K. Dey, P. Kumam, T. Senapati several mathematicians (for example, see ([2, 3, 4, 5, 7, 10, 17, 18, 15, 16, 19]) and references therein). Before stating our main results, at first we recollect some useful definitions and results in the comparable literature which will be needed throughout the study. So, we start by presenting the concept of α-admissible mappings and triangular α-admissible mappings as follows: Definition 1.1 ([14]). A mapping g : X → X is said to be an α-admissible mapping if there exists a function α : X × X → R+ such that for all x, y ∈ X α(x, y) ≥ 1 ⇒ α(gx, gy) ≥ 1. Definition 1.2 ([11]). A mapping g : X → X is said to be a triangular α-admissible mapping if there exists a function α : X × X → R+ such that (1) for all x, y ∈ X, α(x, y) ≥ 1 ⇒ α(gx, gy) ≥ 1, (2) for all x, y, z ∈ X, α(x, y) ≥ 1, α(y, z) ≥ 1 ⇒ α(x, z) ≥ 1. Note 1.3. [11] Let g be a triangular α-admissible mapping. If (xn) is any sequence defined by xn+1 = gxn and α(xn, xn+1) ≥ 1, then for all n, m ∈ N, we get α(xn, xm) ≥ 1. In 2012, Wardowski [19] introduced the concept of F-contractions which plays a crucial part in the recent trend of research in fixed point theory. After that, Wardowski and Dung [20] and Dung and Hang [6] extended the con- cept of F-contractions to F-weak contractions and generalized F-contractions respectively. By mixing up the concept of α-admissible mappings with F-contractions [19] and F-weak contractions [20], Gopal et al. [8] introduced the concept of α-type F-contractions and α-type F-weak contractions as follows: Definition 1.4 ([8]). Let (X, d) be a metric space and g : X → X be a mapping. Suppose α : X × X → {−∞} ∪ (0, ∞) be a function. The function g is said to be an α-type F-contraction if there exists τ > 0 such that for all x, y ∈ X, d(gx, gy) > 0 ⇒ τ + α(x, y)F(d(gx, gy)) ≤ F(d(x, y)). Definition 1.5 ([8]). Let (X, d) be a metric space and g : X → X be a self- mapping. Let α : X × X → {−∞} ∪ (0, ∞) be a function. The function g is said to be an α-type F-weak contraction if there exists τ > 0 such that for all x, y ∈ X, d(gx, gy) > 0 implies that τ +α(x, y)F(d(gx, gy)) ≤ F ( max { d(x, y), d(x, gx), d(y, gy), d(x,gy)+d(y,gx) 2 }) . In the above definitions, the function F belongs to the family F of mappings from (0, ∞) → R satisfying the following conditions: (F1) F is a strictly increasing function, i.e., for all x, y ∈ R+ with x < y, F(x) < F(y); (F2) For each sequence (αn) of positive numbers, lim n→∞ αn = 0 ⇐⇒ lim n→∞ F(αn) = −∞; c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 82 Fixed point theorems for α-F -contraction mappings (F3) There exists a k ∈ (0, 1) such that lim α→0+ αkF(α) = 0. In this sequel, the authors of [8] established some fixed point results and finally they presented an application to nonlinear fractional differential equations. Subsequently, Piri and Kumam [13] established some new fixed point results by taking a weaker family of functions as well as by weakening the contraction condition given by: Definition 1.6 ([13]). Let (X, d) be a metric space and let g : X → X be a mapping. The function g is said to be a modified generalized F-contraction of type (A) if there exists τ > 0 such that for all x, y ∈ X, d(gx, gy) > 0 ⇒ τ + F(d(gx, gy)) ≤ F(Ng(x, y)), where, Ng(x, y) = max { d(x, y), d(x, gy) + d(y, gx) 2 , d(g2x, x) + d(g2x, gy) 2 , d(g2x, gx), d(g2x, y), d(gx, y) + d(y, gy), d(g2x, gy) + d(x, gx) } and F satisfies the following conditions: (1) F is strictly increasing, (2) F is continuous. In a similar fashion, they also defined modified generalized F-contraction of type (B) by considering different class of functions satisfying the above con- tractive condition along with the following properties: (1) F is strictly increasing; (2) There exists a k ∈ (0, 1) such that lim α→0+ αkF(α) = 0. Using the notions of modified generalized F-contraction of type (A) and type (B), the authors presented some new fixed point results which generalized and extended several related results discussed in Wardowski [19], Piri and Kumam [12], Dung and Hang [6] and Wardowski and Dung [20]. For the sake of completeness of our paper, we need to recall the definition of α-complete metric spaces and α-continuous mappings. Definition 1.7 ([9]). Let (X, d) be a metric space and α : X × X → [0, ∞) be a function. The metric space (X, d) is said to be an α-complete metric space if and only if every Cauchy sequence with α(xn, xn+1) ≥ 1, for all n ∈ N0, converges in X. Definition 1.8 ([9]). Let (X, d) be a metric space. Let g be a self-map defined on X and α : X × X → [0, ∞) be a function. Then g is said to be an α- continuous mapping if for every x ∈ X and sequence (xn) ∈ X with (xn) converging to x, α(xn, xn+1) ≥ 1, for all n ∈ N0 ⇒ gxn → gx. Here, we provide an example of an α-continuous mapping which is not con- tinuous. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 83 L. K. Dey, P. Kumam, T. Senapati Example 1.9. Let X = [0, ∞) and d(x, y) = |x − y|, for all x, y ∈ X. We define α : X × X → [0, ∞) by α(x, y) = { 1, for all x, y ∈ [0, 1]; 1 2 , otherwise and the mapping g : X → X by gx =    x 2 , for all x ∈ [0, 1]; 2x, 1 < x ≤ 3; x2, otherwise Clearly, g is not continuous as x = 1 and x = 3 are points of discontinuity but g is an α-continuous map. Remark 1.10. Every complete metric space is α-complete and every continuous map is α-continuous but in both the cases, the converse does not hold in general. In this article, by F, we denote the following family of functions given by F = {F/F : (0, ∞) → R} satisfying the following conditions: (F ′) F is a strictly increasing function, i.e., for all x, y ∈ R+ with x < y, F(x) < F(y); (F ′′) There exists a k ∈ (0, 1) such that lim α→0+ αkF(α) = 0. The aim of this article is to present some new fixed point results in α- complete metric spaces and show that our obtained results generalize several existing results in the literature. For this, we introduce the concept of gener- alized α-type F-contractions and modified generalized α-type F-contractions. For simplicity, we call these contractions as generalized α-F-contractions and modified generalized α-F-contractions respectively. Finally, we construct some non-trivial examples to validate the potential of our results. 2. Main Results We begin with this section by presenting the new concept of generalized α-F-contractions and modified generalized α-F-contractions respectively. Definition 2.1. Let (X, d) be a metric space and g : X → X be a mapping. Let α : X × X → [0, ∞) be a function and F ∈ F. The function g is said to be a generalized α-F-contraction mapping if there exists τ > 0 such that for all x, y ∈ X, d(gx, gy) > 0 ⇒ τ + α(x, y)F(d(gx, gy)) ≤ F(Mg(x, y)) where, Mg(x, y) = max { d(x, y), d(x, gx), d(y, gy), d(x, gy) + d(y, gx) 2 , d(g2x, x) + d(g2x, gy) 2 , d(g2x, gx), d(g2x, y), d(g2x, gy) } . c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 84 Fixed point theorems for α-F -contraction mappings Definition 2.2. Let (X, d) be a metric space and g : X → X be a mapping. Let α : X × X → [0, ∞) be a function and F ∈ F. The function g is said to be a modified generalized α-F-contraction mapping if there exists τ > 0 such that for all x, y ∈ X, d(gx, gy) > 0 ⇒ τ + α(x, y)F(d(gx, gy)) ≤ F(Ng(x, y)) where, Ng(x, y) = max { d(x, y), d(x, gy) + d(y, gx) 2 , d(g2x, x) + d(g2x, gy) 2 , d(g2x, gx), d(g2x, y), d(gx, y) + d(y, gy), d(g2x, gy) + d(x, gx) } . Remark 2.3. Every modified generalized F-contraction (respectively, gener- alized F-contraction) is a modified generalized α-F-contraction (respectively, generalized α-F-contraction). The reverse implications do not hold. We illustrate this by presenting an example. Example 2.4. Let X = {0, 1, 2, 3, 4} and we define the distance function d as follows d(x, y) =    0, iff x = y; 5 2 , (x, y) ∈ {(0, 3), (3, 0)}; 3 2 , otherwise. Also, we define a mapping g : X → X by g(0) = g(3) = 1; g(1) = g(4) = 3; g(2) = 0. Therefore, we get d(gx, gy) > 0 ⇐⇒ [x ∈ {0, 3}∧y ∈ {1, 4}; x ∈ {0, 3}∧y = 2; x ∈ {1, 4}∧y = 2]. Now, we are interested to find Ng(x, y). For this purpose, we consider the following cases: Case-I. Let x ∈ {0, 3} and y ∈ {1, 4}. Then for any (x, y) ∈ {(0, 1), (0, 4), (3, 1), (3, 4)}, we get d(gx, gy) = d(1, 3) = 3 2 . Let (x, y) = (0, 1). Then, we have Ng(0, 1) = max{d(0, 1), d(0, g1) + d(1, g0) 2 , d(g20, 0) + d(g20, g1) 2 , d(g20, g0), d(g20, 1), d(g0, 1) + d(1, g1), d(g20, g1) + d(0, g0)} = max { 3 2 , 5 4 } = 3 2 . c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 85 L. K. Dey, P. Kumam, T. Senapati For (x, y) = (0, 4), we get Ng(0, 4) = max { d(0, 4), d(0, g4) + d(4, g0) 2 , d(g20, 0) + d(g20, g4) 2 , d(g20, g0), d(g20, 4), d(g0, 4) + d(4, g4), d(g20, g4) + d(0, g0) } = max { 3 2 , 5 4 , 2 } = 2. For (x, y) = (3, 1), we obtain Ng(3, 1) = max { d(3, 1), d(3, g1) + d(1, g3) 2 , d(g23, 3) + d(g23, g1) 2 , d(g23, g3), d(g23, 1), d(g3, 1) + d(1, g1), d(g23, g1) + d(3, g3) } = max { 3 2 , 0 } = 3 2 and for (x, y) = (3, 4), also have Ng(3, 4) = max { d(3, 4), d(3, g4) + d(4, g3) 2 , d(g23, 3) + d(g23, g4) 2 , d(g23, g3), d(g23, 4), d(g3, 4) + d(4, g4), d(g23, g4) + d(3, g3) } = max { 3 2 , 3 4 , 0, 3 } = 3. Case-II. Let x ∈ {0, 3} and y = 2. Then for (x, y) ∈ {(0, 2), (3, 2)}, d(gx, gy) = d(1, 0) = 3 2 . Then, for (x, y) = (0, 2), we have Ng(0, 2) = max { d(0, 2), d(0, g2) + d(2, g0) 2 , d(g20, 0) + d(g20, g2) 2 , d(g20, g0), d(g20, 2), d(g0, 2) + d(2, g2), d(g20, g2) + d(0, g0) } = max { 3 2 , 3 4 , 5 2 , 3, 4 } = 4. For (x, y) = (3, 2), we get Ng(3, 2) = max { d(3, 2), d(3, g2) + d(2, g3) 2 , d(g23, 3) + d(g23, g2) 2 , d(g23, g3), d(g23, 2), d(g3, 2) + d(2, g2), d(g23, g2) + d(3, g3) } = max { 3 2 , 3 4 , 2, 3, 4 } = 4. Case-III. Let x ∈ {1, 4} and y = 2 . Then (x, y) ∈ {(1, 2), (4, 2)}, d(gx, gy) = d(3, 0) = 5 2 . c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 86 Fixed point theorems for α-F -contraction mappings Let (x, y) = (1, 2). Then Ng(1, 2) = max { d(1, 2), d(1, g2) + d(2, g1) 2 , d(g21, 1) + d(g21, g2) 2 , d(g21, g1), d(g21, 2), d(g1, 2) + d(2, g2), d(g21, g2) + d(1, g1) } = max { 3 2 , 3 4 , 3 } = 3. For (x, y) = (4, 2), we have Ng(4, 2) = max { d(4, 2), d(4, g2) + d(2, g4) 2 , d(g24, 4) + d(g24, g2) 2 , d(g24, g4), d(g24, 2), d(g4, 2) + d(2, g2), d(g24, g2) + d(4, g4) } = max { 3 2 , 3 } = 3. From the above cases, we observe that whenever (x, y) ∈ {(0, 1), (3, 1)}, d(gx, gy) = Ng(x, y). Since F is increasing, we can’t find any τ > 0 such that τ + F(d(gx, gy)) ≤ F(Ng(x, y)). This shows that g is not a modified generalized F-contraction. Hence, g can not be an F-contraction, F-weak contraction and generalized F-contraction. Let us consider F(x) = ln x for all x ∈ (0, ∞). Clearly, F ∈ F. Now, we define a function α : X × X → [0, ∞) by α(x, y) = { 1 2 , (x, y) ∈ {(0, 1), (3, 1)}; 1 otherwise. Then, we can find τ > 0 such that τ + α(x, y)F(d(gx, gy)) ≤ F(Ng(x, y)), whenever d(gx, gy) > 0. In particular, when α(x, y) = 1 2 one can choose τ ∈ (0, 1 5 ). Therefore g is a modified generalized α-F-contraction. In the following, we present an example to show that the class of modified generalized α-F-contraction mappings is larger than that of generalized α-F- contraction mappings. Example 2.5. Let X = {−1, 0, 1} and g be a self-mapping on X defined by g(−1) = g(0) = 0, g(1) = −1. We define a distance function d on X by d(x, y) =    0, x = y; 1 2 , (x, y) ∈ {(1, −1), (−1, 1)}; 1 otherwise. So, (X, d) is a complete metric space. Now d(gx, gy) > 0 for (x, y) = (0, 1) and (x, y) = (−1, 1). Therefore we consider the following two cases. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 87 L. K. Dey, P. Kumam, T. Senapati Case-I. Let (x, y) = (0, 1). Then, d(g0, g1) = d(0, −1) = 1 Mg(0, 1) = max { d(0, 1), d(0, g0), d(1, g1), d(0, g1) + d(1, g0) 2 , d(g20, 0) + d(g20, g1) 2 , d(g20, g0), d(g20, 1), d(g20, g1) } = max { 1, 0, 1 2 } = 1. and Ng(0, 1) = max { d(0, 1), d(0, g1) + d(1, g0) 2 , d(g20, 0) + d(g20, g1) 2 , d(g20, g0), d(g20, 1), d(g0, 1) + d(1, g1), d(g20, g1) + d(0, g0) } = max { 1, 0, 1 2 , 3 2 } = 3 2 . Case-II. Let (x, y) = (−1, 1). Then d(g(−1), g1) = d(0, −1) = 1 and Mg(−1, 1) = max {d(−1, 1), d(−1, g(−1)), d(1, g1), d(−1, g1) + d(1, g(−1)) 2 , d(g2(−1), −1) + d(g2(−1), g1) 2 , d(g2(−1), g(−1)), d(g2(−1), 1), d(g2(−1), g1)} = max { 1, 1 2 , 0 } = 1. Ng(−1, 1) = max { d(−1, 1), d(−1, g1) + d(1, g(−1)) 2 , d(g2(−1), −1) + d(g2(−1), g1) 2 , d(g2(−1), g(−1)), d(g2(−1), 1), d(g(−1), 1) + d(1, g1), d(g2(−1), g1) + d(−1, g(−1)) } = max { 1, 0, 1 2 , 2, 3 2 } = 2. If we choose F(x) = ln(x) for all x ∈ (0, ∞) and α(x, y) ≥ 0, then g can not be a generalized α-F-contraction, since τ + α(0, 1)F(d(g0, g1)) ≤ F(Mg(0, 1)) ⇒ τ + α(0, 1) ln(1) ≤ ln(1) ⇒ τ ≤ 0. If we choose Ng(0, 1) instead of Mg(0, 1), one can check that g is a modified generalized F-contraction and hence modified generalized α-F-contraction. In a similar fashion, for case-II, it can be shown that g is a modified gener- alized α-F-contraction but not generalized α-F-contraction. Now, we are in a position to state our main results. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 88 Fixed point theorems for α-F -contraction mappings Theorem 2.6. Let (X, d) be an α-complete metric space and g : X → X be a modified generalized α-F-contraction where F ∈ F. Assume that the following conditions hold: (1) g is α-admissible, α-continuous mapping; (2) there exists x0 ∈ X such that α(x0, gx0) ≥ 1. Then g has a fixed point. Proof. By the hypothesis, there exists a point x0 ∈ X such that α(x0, gx0) ≥ 1. Now we define a sequence (xn) by xn+1 = gxn, for all n ∈ N0. If for some n ∈ N, xn = gxn, then xn is a fixed point of g and the proof is complete. So we assume that there exists no such integer n for which xn = gxn. Now α(x0, gx0) ≥ 1 ⇒ α(x0, x1) ≥ 1. Since g is an α-admissible mapping, for all n ∈ N0, we get α(xn, xn+1) ≥ 1. As d(gxn−1, gxn) > 0 and g is a modified generalized α-F-contraction, for some τ > 0, we have F(d(xn, xn+1)) = F(d(gxn−1, gxn)) ≤ τ + α(xn−1, xn)F(d(gxn−1, gxn)) ≤ F(Ng(xn−1, xn)).(2.1) Now, by simple computations, we have Ng(xn−1, xn) = max { d(xn−1, xn), d(xn−1, gxn) + d(xn, gxn−1) 2 , d(g2xn−1, xn−1) + d(g 2xn−1, gxn) 2 , d(g2xn−1, gxn−1), d(g 2xn−1, xn), d(gxn−1, xn) + d(xn, gxn), d(g2xn−1, gxn) + d(xn−1, gxn−1) } = max { d(xn−1, xn), d(xn, xn+1), d(xn−1, xn+1) 2 } . If max{d(xn−1, xn), d(xn, xn+1)} = d(xn, xn+1), then (2.1) shows that τ + α(xn−1, xn)F(d(xn, xn+1)) ≤ F(d(xn, xn+1)) which is impossible. We must have max{d(xn−1, xn), d(xn, xn+1)} = d(xn−1, xn). Therefore, (2.1) implies that F(d(xn, xn+1)) ≤ α(xn−1, xn)F(d(xn, xn+1)) ≤ F(d(xn−1, xn)) − τ(2.2) ⇒ F(d(xn, xn+1)) < F(d(xn−1, xn)) as τ > 0 ⇒ d(xn, xn+1) < d(xn−1, xn). This shows that (xn) is a decreasing sequence of nonnegative real numbers. We claim that lim n→∞ d(xn+1, xn) = 0. If possible, let lim n→∞ d(xn+1, xn) = δ for some c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 89 L. K. Dey, P. Kumam, T. Senapati δ > 0. Therefore, for every n ∈ N, we have d(xn, xn+1) ≥ δ. By (F ′) and (2.3), we have F(δ) ≤ F(d(xn, xn+1)) ≤ α(xn+1, xn)F(d(xn, xn+1)) < F(d(xn−1, xn)) − τ < F(d(xn−2, xn−1)) − 2τ ... < F(d(x0, x1)) − nτ.(2.3) As lim n→∞ (F(d(x0, x1)) − nτ) = −∞, so we can find some m ∈ N such that F(d(x0, x1))−nτ < F(δ) for all n > m, which contradicts the above equation. Therefore, we must have lim n→∞ d(xn, xn+1) = 0. Next, we claim that (xn) is a Cauchy sequence. By (F ′′), there exists k ∈ (0, 1) such that (2.4) lim n→∞ (αkn)F(αn) = 0, where lim n→∞ αn = lim n→∞ d(xn, xn+1) = 0. Again, from (2.3) and (2.4), we can obtain lim n→∞ (αkn)(F(αn) − F(α0)) ≤ lim n→∞ − (αkn)nτ ≤ 0 ⇒ lim n→∞ {nαkn} = 0 as τ > 0.(2.5) So, we can find some n0 ∈ N such that n(αn) k ≤ 1, for all n ≥ n0 ⇒ αn ≤ 1 n 1 k , for all n ≥ n0.(2.6) In view of (2.6), for all m > n > n0, we have d(xn, xm) ≤ d(xn, xn+1) + d(xn+1, xn+2) + . . . . + d(xm−1, xm) < Σ∞j=1αj ≤ Σ ∞ j=1 1 j 1 k . As 1 k > 1, the above series is convergent. This implies that lim n,m→∞ d(xn, xm) = 0, i.e., (xn) is a Cauchy sequence. Since, (X, d) is an α-complete metric space and (xn) is a Cauchy sequence with α(xn, xn+1) ≥ 1 for all n ∈ N, we can find some x ∈ X such that xn → x whenever n → ∞. Now, we claim that x is a fixed point of g. Since xn → x as n → ∞ and α(xn, xn+1) ≥ 1, for all n ∈ N0, the α-continuity property of g implies that gxn → gx as n → ∞. Finally, we have xn+1 = gxn ⇒ lim n→∞ xn+1 = lim n→∞ gxn ⇒ x = gx. Hence x is a fixed point of g. � c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 90 Fixed point theorems for α-F -contraction mappings Notice that the condition of α-continuity of g in Theorem 2.6 can actually be replaced by another weaker condition. In the sequel, we present the following result. Theorem 2.7. Let (X, d) be an α-complete metric space and let g : X → X be a modified generalized α-F-contraction, where F ∈ F. Assume that the following conditions hold: (1) g is α-admissible; (2) there exists x0 ∈ X such that α(x0, gx0) ≥ 1; (3) if (xn) is a sequence in X with α(xn, xn+1) ≥ 1, for all n ∈ N0 and xn → x as n → ∞, we have α(xn, x) ≥ 1, for all n ∈ N0. Then g has a fixed point. Proof. Following the proof of Theorem 2.6, we know that (xn) is a Cauchy sequence with α(xn, xn+1) ≥ 1, for all n ∈ N0 and it converges to some point x ∈ (X, d). By the hypothesis (3), we have α(xn, x) ≥ 1, for all n ∈ N0. We claim that x is a fixed point of g. On the contrary, suppose that gx 6= x ⇒ d(x, gx) > 0. We can find a number n ∈ N such that d(xm, gx) > 0, for all m ≥ n ⇒ d(gxm−1, gx) > 0. So by the condition of the theorem and by the property of F , we can find some τ > 0 such that τ + α(xm−1, x)F(d(gxm−1, gx)) ≤ F(Ng(xm−1, x)) ⇒ F(d(gxm−1, gx)) < F(Ng(xm−1, x)), [as α(xm−1, x) ≥ 1; τ > 0] ⇒ d(gxm−1, gx) < Ng(xm−1, x) ⇒ lim m→∞ d(xm, gx) < lim m→∞ Ng(xm−1, x).(2.7) Now, we compute Ng(xm−1, x) = max { d(xm−1, x), d(xm−1, gx) + d(x, gxm−1) 2 , d(g2xm−1, x) + d(g 2x, gx) 2 , d(g2xm−1, gxm−1), d(g 2xm−1, gx), d(g2xm−1, gx) + d(xm−1, gxm−1), d(gxm−1, x) + d(x, gx) } . Using this in the above inequality, we get lim m→∞ d(xm, gx) < max{d(x, x), d(x, gx)} which leads to a contradiction. Hence, our assumption was wrong. We must have d(x, gx) = 0, i.e., x is a fixed point of g. � In the following theorem, we present a fixed point result for a modified generalized α-F-contraction where the function F satisfies only (F ′) property. Theorem 2.8. Let (X, d) be an α-complete metric space and let g : X → X be a modified generalized α-F-contraction where F is strictly increasing function on (0, ∞). Assume that the following conditions hold: c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 91 L. K. Dey, P. Kumam, T. Senapati (1) g is triangular α-admissible; (2) there exists x0 ∈ X such that α(x0, gx0) ≥ 1; (3) if (xn) is a sequence in X with α(xn, xn+1) ≥ 1, for all n ∈ N0 and xn → x as n → ∞, we have α(xn, x) ≥ 1, for all n ∈ N0. Then g has a fixed point. Proof. Following the proof of Theorem 2.6, we have lim n→∞ d(xn, xn+1) = 0. Now, we prove that (xn) is a Cauchy sequence. If possible, suppose by contradiction that (xn) is not a Cauchy sequence. Then for some ǫ > 0, we can find sequences p(n) and q(n) of natural numbers such that p(n) > q(n) > n, d(xp(n), xq(n)) ≥ ǫ and d(xp(n)−1, xq(n)) < ǫ,(2.8) for all n ∈ N. Therefore, we have ǫ ≤ d(xp(n), xq(n)) ≤ d(xp(n), xp(n)−1) + d(xp(n)−1, xq(n)) < d(xp(n), xp(n)−1) + ǫ which implies that lim n→∞ d(xp(n), xq(n)) = ǫ.(2.9) Again, from (2.8), we can find n0 ∈ N such that d(xp(n), gxp(n)) < ǫ 4 and d(xq(n), gxq(n)) < ǫ 4 , for all n ≥ n0 ∈ N.(2.10) Now, we claim that d(gxp(n), gxq(n)) > 0. Indeed, if no, then there exists m ≥ n0 such that d(gxp(m), gxq(m)) = d(xp(m)+1, xq(m)+1) = 0. From (2.10), it follows that ǫ ≤ d(xp(m), xq(m)) ≤ d(xp(m), xp(m)+1) + d(xp(m)+1, xq(m)) ≤ d(xp(m), xp(m)+1) + d(xp(m)+1, xq(m)+1) + d(xq(m)+1, xq(m)) = d(xp(m), gxp(m)) + d(xp(m)+1, xq(m)+1) + d(xq(m), gxq(m)) ≤ ǫ 4 + 0 + ǫ 4 = ǫ 2 which is a contradiction. Therefore, we get d(gxp(m), gxq(m)) > 0, for all m ∈ N. From (2.9), we get lim m→∞ d(gxp(m), gxq(m)) = lim m→∞ d(xp(m)+1, xq(m)+1) = ǫ. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 92 Fixed point theorems for α-F -contraction mappings Since g is a modified generalized α-F-contraction, we can find τ > 0 such that τ + α(xp(n), xq(n))F(d(gxp(n), gxq(n))) ≤F(Ng(xp(n), xq(n))), for all n ≥ n0 ⇒ α(xp(n), xq(n))F(d(gxp(n), gxq(n))) ≤F(Ng(xp(n), xq(n))) − τ. Since, α(xp(n), xq(n)) ≥ 1, for all n ∈ N0; τ > 0 and F is strictly increasing, we have F(d(gxp(n), gxq(n))) < F(Ng(xp(n), xq(n))) ⇒ d(gxp(n), gxq(n)) < Ng(xp(n), xq(n)), ∀n ∈ N ⇒ lim n→∞ d(gxp(n), gxq(n)) < lim n→∞ Ng(xp(n), xq(n)).(2.11) Now, we observe that lim n→∞ Ng(xp(n), xq(n)) = max{ lim n→∞ {d(xp(n), xq(n)), d(xp(n), xq(n)+1) + d(xq(n), xp(n)+1) 2 , d(xp(n)+2, xp(n)) + d(xp(n)+2, xq(n)+1) 2 , d(xp(n)+2, xp(n)+1), d(xp(n)+2, xq(n)), d(xp(n)+2, xq(n)+1) + d(xp(n), xp(n)+1), d(xp(n)+1, xq(n)) + d(xq(n), xq(n)+1)}}. Using the triangle inequality and by some simple computations, one can easily check that lim n→∞ Ng(xp(n), xq(n)) = ǫ. Using this in (2.11), we have ǫ = lim n→∞ d(gxp(n), gxq(n)) < ǫ which implies that our assumption was wrong. So (xn) must be a Cauchy sequence with the property α(xn, xn+1) ≥ 1, hence it converges to some point x̃ in X as (X, d) is an α-complete metric space. Next, we show that x̃ is a fixed point of g. By the hypothesis of the theorem, we have α(xn, x̃) ≥ 1. Again, by the property of F , we obtain F(d(xn, gx̃)) ≤ τ + α(xn−1, x)F(d(gxn−1, gx̃)) ≤ F(Ng(xn−1, x̃)) ⇒ d(xn, gx̃) ≤ Ng(xn−1, gx̃) ⇒ lim n→∞ d(xn, gx̃) ≤ lim n→∞ Ng(xn−1, x̃) ⇒ d(x̃, gx̃)) = 0. This shows that x̃ is a fixed point of g. � Now, we present an additional condition to ensure the uniqueness of fixed point. Theorem 2.9. Let g be a modified generalized α-F-contraction. If g has two fixed points x, y ∈ X with α(x, y) ≥ 1, then we must have x = y. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 93 L. K. Dey, P. Kumam, T. Senapati Proof. Given x, y ∈ Fix(g) with x 6= y ⇒ gx 6= gy ⇒ d(gx, gy) > 0. For any n ∈ N, we have gnx = x and gny = y. As g is an α-F-contraction with d(gx, gy) > 0, there exists some τ > 0 such that τ + α(x, y)F(d(gx, gy)) ≤ F(Ng(x, y)) ⇒ τ + α(x, y)F(d(gx, gy)) < F(d(x, y)) ⇒ F(d(gx, gy)) < F(d(x, y)), [as α(x, y) ≥ 1; τ > 0] ⇒ F(d(x, y)) < F(d(x, y)). This contradiction shows that x = y. � Remark 2.10. Notice that the above theorems establish the existence and then uniqueness of fixed point of the function g without assuming the continuity property of F as well as the continuity property of g. Remark 2.11. Our results generalize several fixed point results in the existing literature. For instance, taking α(x, y) = 1, we can obtain the main results of Piri and Kumam [13] and Dung and Hang [6] as a corollary of our main results. Most importantly, our results are the generalized versions of the fixed point results given by Gopal et al. [8]. 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