Fixed Point Results for F -Contractions in Cone Metric Spaces over Topological Modules and Applications to Integral Equations

: In this paper, the concept of F -contraction was generalized for cone metric spaces over topological left modules and some ﬁxed point results were obtained for self-mappings satisfying a contractive condition of this type. Some applications of the main result to the study of the existence and uniqueness of the solutions for certain types of integral equations were presented in the last part of the article, one of them being a fractional integral equation.


Introduction
The concept of cone metric space was introduced by Huang and Zhang [1] as a generalization of a metric space, proving that Banach's contraction theorem remains valid in this context. Afterwards, many authors have obtained fixed point results on cone metric spaces: Radenović and Rhoades [2], Rezapour and Hamlbarani [3], Kadelburg et al. [4], Du [5] and the references therein. Further, Liu and Xu [6] introduced the concept of cone metric space over a Banach algebra and proved some fixed point theorems for Lipschitz mappings. Later on, Xu and Radenović [7] extended the results of Liu and Xu [6], without the assumption of normality of the cone involved. Also, generalizations of Banach's theorem have been obtained in other directions. Wardowski [8] defined the class of F-contractions and proved a fixed point result as a extension of Banach contraction principle. After this, Wardowski and Van Dung [9] introduced the concept of F-weak contraction and obtained a new fixed point theorem. Cosentino and Vetro [10] obtained new fixed point theorems of Hardy-Rogers type for F-contractions in ordered metric spaces. Other results concerning F-contractions have been obtained by: Piri and Kumam [11], Minak et al. [12], Ahmad et al. [13], Kadelburg and Radenović [14], Dey et al. [15], Wardowski [16], Alfaqih et al. [17], Karapinar et al. [18] and the references therein.
Regarding the fractional differentiation and integration, many models have been proposed in the literature: the Riemann-Liouville fractional model [19], the Caputo model [20], the Atangana-Baleanu (or AB) fractional model [21,22], the generalised proportional fractional (or GPF) model [23], the Prabhakar fractional model [24,25], and others. Fernandez et al. [26] proposed a unified model of fractional calculus by using a general operator which includes many types of fractional operators. They consider some fractional differential equations and solve a general Cauchy problem in this new framework. Important results on nonlinear fractional differential equations were obtained by Agarwal et al. [27,28], Soradi-Zeid et al. [29], Almeida [30], Khan et al. [31], Keten et al. [32] and the references therein.
In this article, we generalize the notion of cone metric space for topological left module and we define the concept of F-contraction on this new space. Next, we obtain some fixed point results for self-mappings satisfying a contractive condition of this type. In the last part of the article, some applications of the main result to the study of the existence and Definition 5. Let (A, ⊕, , τ A , ) be a partially ordered topological ring and (E, +, ·, τ E ) a topological left A-module. A cone is a non-empty subset P of E satisfying the conditions: The cone P is solid if int(P) = ∅. We define the set P * = P \ {0 E }. Lemma 1. Let (A, ⊕, , τ A , ) be a partially ordered topological ring, with 1 A ∈ A + , (E, +, ·, τ E ) a topological left A-module and P a solid cone of E. Then, the following statements are valid: (i) P + P ⊆ P; (ii) the relation ≤ P over E, defined by x ≤ P y if and only if y − x ∈ P, is a partial order relation on E; (iii) if x ≤ P y and y P z, then x P z, for all x, y, z ∈ E; (iv) if x ≤ P y and u ≤ P v, then x + u ≤ P y + v, for all x, y, u, v ∈ E. Lemma 2. Let (A, ⊕, , τ A , ) be a partially ordered topological ring, (E, +, ·, τ E ) a topological left A-module and P a solid cone of E. Then, the following statements are valid: (iii) assuming that τ A is a Hausdorff topology, 0 A is an accumulation point of A + and a · int(P) ⊆ int(P) for all a ∈ A * + , if x ∈ P, x P ε + ε for all ε ∈ int(P), then x = 0 E ; (iv) if (x n ) n∈N , (y n ) n∈N ⊂ E, x, y ∈ E, x n n → x, y n n → y, and there is a number N ∈ N such that x n ≤ P y n for all n ≥ N, then x ≤ P y. Definition 6. Let (A, ⊕, , τ A , ) be a partially ordered topological ring, (E, +, ·, τ E ) a topological left A-module, P a solid cone of E and X a non-empty set. A cone metric on X is an application d : X × X → E which fulfills the conditions: (X, d) is called a cone metric space over the topological left A-module.
Definition 7. Let (A, ⊕, , τ A , ) be a partially ordered topological ring, (E, +, ·, τ E ) a topological left A-module, P a solid cone of E and (X, d) a cone metric space over the topological left A-module.
(1) A sequence (x n ) n∈N ⊂ X is called convergent to a point x ∈ X if it has the property: for every ε P 0 E there is a number n(ε) ∈ N such that for all n ≥ n(ε) we have d(x n , x) P ε; we also say that (x n ) n∈N ⊂ X converges to x ∈ X and we denote by x n d → x; (2) A sequence (x n ) n∈N ⊂ X is named a Cauchy sequence if it satisfies the condition: for every ε P 0 E there is a number n(ε) ∈ N such that for all m, n ≥ n(ε) we have d(x m , x n ) P ε; (3) The cone metric space (X, d) is called complete if: any Cauchy sequence of points in X is convergent in X.

Lemma 3.
Let (A, ⊕, , τ A , ) be a partially ordered Hausdorff topological ring, with 1 A ∈ A + , 0 A an accumulation point of A + , (E, +, ·, τ E ) a topological left A-module, P a solid cone of E, with a · int(P) ⊆ int(P) for all a ∈ A * + , and (X, d) a cone metric space over the topological left A-module. If the sequence (x n ) n∈N ⊂ X is convergent in X, then it has a unique limit. Definition 8. Let (E, τ E ) be a topological space. A subset S of E is called sequentially compact if any sequence in S has a convergent subsequence in S. Lemma 4. Let (A, ⊕, , τ A , ) be a partially ordered topological ring, with 1 A ∈ A + , (E, +, ·, τ E ) a Hausdorff topological left A-module, P a solid cone of E and (X, d) a cone metric space over the topological left A-module. If (x n ) n∈N ⊂ X is not a Cauchy sequence and there is a sequentially compact subset S x ⊆ P having the property {d(x m , x n ) | m, n ∈ N * , m < n} ⊆ S x , then the following statements are valid: (i) there exist ε 0 P 0 E and two subsequences (x m(k) ) k∈N , (x n(k) ) k∈N , where m, n : N → N are strictly increasing functions, with k < m(k) < n(k) for all k ∈ N, such that d(x m(k) , x n(k) ) P ε 0 , d(x m(k) , x n(k)−1 ) P ε 0 for all k ∈ N; (ii) moreover, if lim n→+∞ d(x n−1 , x n ) = 0 E , there exists a point x ∈ Fr(P) such that Proof. (i) Since (x n ) n∈N is not a Cauchy sequence, we deduce that there is ε 0 P 0 E and two subsequences (x m 1 (k) ) k∈N , (xñ 1 (k) ) k∈N , where m 1 ,ñ 1 : N → N are strictly increasing functions, with k < m 1 (k) <ñ 1 (k) for all k ∈ N, such that d(x m 1 (k) , xñ 1 (k) ) P ε 0 for all k ∈ N. For every k ∈ N we choose n 1 (k) to be the smallest integerñ 1 (k) which verifies m 1 (k) < n 1 (k) and d(x m 1 (k) , x n 1 (k) ) P ε 0 , thus d(x m 1 (k) , x n 1 (k)−1 ) P ε 0 . Hence, there exist ε 0 P 0 E and two subsequences (x m 1 (k) ) k∈N , (x n 1 (k) ) k∈N , where m 1 , n 1 : N → N are strictly increasing functions, with k < m 1 (k) < n 1 (k) for all k ∈ N, such that As S x is sequentially compact, any sequence in S x has a convergent subsequence in S x . According to the hypothesis we have {d(x m , x n ) | m, n ∈ N * , m < n} ⊆ S x . Therefore, for the Similarly, for the sequence (d( Applying the statement (1) for k := p(q(k)), k ∈ N, it follows that there exist ε 0 P 0 E and two subsequences ( (ii) As τ E is a Hausdorff topology, every convergent sequence in E has a unique limit in E.
Considering (i) we get d(x m(k) , x n(k) ) P ε 0 for all k ∈ N, hence ε 0 − d(x m(k) , x n(k) ) ∈ c(int(P)) for all k ∈ N. Since (d(x m(k) , x n(k) )) k∈N represents the sequence (d(x m 1 (p(q(k))) , x n 1 (p(q(k))) )) k∈N which is a subsequence of (d(x m 1 (p(k)) , x n 1 (p(k)) )) k∈N and taking into account that lim k→+∞ d(x m 1 (p(k)) , x n 1 (p(k)) ) = l 1 (according to the relation (2)), we deduce lim k→+∞ d(x m(k) , x n(k) ) = l 1 . We find lim is an open set, hence c(int(P)) is a closed set). Taking into account the properties of the cone metric d, the statement (i) and using Lemma 1 (iii) we obtain The properties of the cone metric d leads to x n(k) )) k∈N are subsequences of (d(x n−1 , x n )) n∈N and lim Passing to the limit for the sequences in relations (4), (5), we obtain Consequently, we proved that there exists a point x ∈ Fr(P) such that lim In the following, we define the operation * : (1) an increasing sequence if u n−1 ≤ P u n for all n ∈ N * ; (2) a decreasing sequence if u n−1 ≥ P u n for all n ∈ N * . Definition 10. Let (E, τ E ) be a topological space, S a subset of E and F : S → E a function.
(1) F is sequentially continuous at a point x ∈ S if: for every sequence (x n ) n∈N ⊂ S convergent to x, the sequence (F(x n )) n∈N ⊂ E is convergent to F(x); (2) F is named sequentially continuous on S if: it is sequentially continuous at every point x ∈ S. Definition 11. Let (A, ⊕, , τ A , ) be a partially ordered topological ring, (E, +, ·, τ E ) a topological left A-module and P a solid cone of E. We consider the set F of all functions F : P * → E which fulfill the conditions: is a decreasing sequence and for every ε P 0 E there is a number n(ε) ∈ N such that for all n ≥ n(ε) we have F(u n ) P −ε, then u n n → 0 E ; (3) if u, v ∈ P * , F(u) P F(v), then u ≤ P v; (4) F is sequentially continuous on P * . Definition 12. Let (A, ⊕, , τ A , ) be a partially ordered topological ring, (E, +, ·, τ E ) a topological left A-module, P a solid cone of E and (X, d) a cone metric space over the topological left A-module. An F-contraction corresponding to the function class F is a mapping T : X → X for which there exist τ P 0 E and a function F ∈ F such that τ + F(d(Tx, Ty)) ≤ P F(d(x, y)), for all x, y ∈ X, Tx = Ty.
Remark 1. The condition Tx = Ty from the property (6) implies x = y. Hence, d(x, y) ∈ P * whenever d(Tx, Ty) ∈ P * . Therefore, the function F ∈ F is defined for every x, y ∈ X satisfying the condition Tx = Ty. Theorem 1. Let (A, ⊕, , τ A , ) be a partially ordered Hausdorff topological ring, with 1 A ∈ A + , 0 A an accumulation point of A + , (E, +, ·, τ E ) a Hausdorff topological left A-module, P a solid cone of E, with a · int(P) ⊆ int(P) for all a ∈ A * + , 0 E / ∈ int(P), and (X, d) a cone metric space over the topological left A-module. We suppose that (X, d) is complete, T : X → X is an F-contraction corresponding to the function class F and for every x 0 ∈ X we consider the sequence (x n ) n∈N ⊂ X defined by x n = Tx n−1 for all n ∈ N * . If for any x 0 ∈ X for which (x n ) n∈N ⊂ X is not a Cauchy sequence, there is a sequentially compact subset S x 0 ⊆ P having the property {d(x m , x n ) | m, n ∈ N * , m < n} ⊆ S x 0 , then T has a unique fixed point x ∈ X and for every x 0 ∈ X the sequence (x n ) n∈N ⊂ X converges to x .
Proof. Let us suppose that T has two distinct fixed points, so there exist x 1 , x 2 ∈ X such that Tx 1 = x 1 = x 2 = Tx 2 . Because T : X → X is an F-contraction corresponding to the function class F we deduce that there exist τ P 0 E and a function F ∈ F such that the property (6) is verified. Applying this property we get τ + F(d(Tx 1 , Tx 2 )) ≤ P F(d(x 1 , x 2 )), thus τ + F(d(x 1 , x 2 )) ≤ P F(d(x 1 , x 2 ))), so τ ≤ P 0 E , hence τ ∈ (−P). It follows that τ ∈ int(P) ∩ (−P). On the other hand, int(P) ∩ (−P) = ∅ (according to Lemma 2 (i)), consequently τ ∈ ∅, which is in contradiction with τ P 0 E . Therefore, T has at most one fixed point.
We choose x 0 ∈ X be an arbitrary element and let us define the sequence (x n ) n∈N ⊂ X by the recurrence relation x n = Tx n−1 for all n ∈ N * . To prove the existence of a fixed point of the operator T we distinguish the following cases: Considering that T has at most one fixed point, we get x := x N−1 ∈ X is the unique fixed point of T. Moreover, x n = x N−1 for all n ≥ N, which means that the sequence (x n ) n∈N ⊂ X converges to x .
On the other hand, F ∈ F , hence from Definition 11 (1) we obtain for every ε P 0 E there is a number n(ε, d(x 0 , x 1 ), τ) ∈ N such that for all n ≥ n(ε, d(x 0 , x 1 ), τ) we have F(d(x 0 , x 1 )) − n * τ P −ε. Consequently, from the previous affirmation and using the property (7), via Lemma 1 (iii) we find: for every ε P 0 E there is a number n(ε, d(x 0 , x 1 ), τ) ∈ N such that for all n ≥ n(ε, d(x 0 , Let us consider n ∈ N * be an arbitrary number. Using the property (6) we get τ + F(d(Tx n−1 , Tx n )) ≤ P F(d(x n−1 , x n )), so F(d(x n , x n+1 )) ≤ P F(d(x n−1 , x n )) − τ. Since τ P 0 E it follows that F(d(x n , x n+1 )) P F(d(x n−1 , x n )). As F ∈ F , from Definition 11 (3) we obtain d(x n , x n+1 ) ≤ P d(x n−1 , x n ), thus (d(x n , x n+1 )) n∈N is a decreasing sequence. Considering Definition 11 (2) and the property (8), we deduce d(x n , x n+1 ) n → 0 E , so d(x n−1 , x n ) n → 0 E . In the following, we prove that (x n ) n∈N ⊂ X is a Cauchy sequence. Let us suppose that (x n ) n∈N ⊂ X is not a Cauchy sequence. Since for x 0 ∈ X for which (x n ) n∈N ⊂ X is not a Cauchy sequence, there is a sequentially compact subset S x 0 ⊆ P having the property {d(x m , x n ) | m, n ∈ N * , m < n} ⊆ S x 0 and considering d(x n−1 , x n ) n → 0 E , using Lemma 4 (i), (ii) we deduce there exist ε 0 P 0 E , two subsequences (x m(k) ) k∈N , (x n(k) ) k∈N , where m, n : N → N are strictly increasing functions, with k < m(k) < n(k) for all k ∈ N, and a point x ∈ Fr(P) such that As ε 0 ∈ int(P), x ∈ Fr(P) and according to Lemma 2 (ii), int(P) ∩ Fr(P) = ∅, we get ε 0 − x = 0 E . Considering relation (9), it follows that there is a number K ∈ N such that d(x m(k) , x n(k) ) = 0 E for all k ≥ K, hence x m(k) = x n(k) for all k ≥ K. Using the property (6) we find τ + F(d(Tx m(k)−1 , Tx n(k)−1 )) ≤ P F(d(x m(k)−1 , x n(k)−1 )) for all k ≥ K, hence On the other hand, as F ∈ F , from Definition 11 (4) we have F is sequentially continuous on P * , hence F is sequentially continuous at ε 0 − x ∈ P * . Considering the property (9) we obtain Passing to the limit in inequality (10) and considering the relation (11), we deduce τ + F(ε 0 − x) ≤ P F(ε 0 − x), thus τ ≤ P 0 E , so τ ∈ (−P). It follows that τ ∈ int(P) ∩ (−P). On the other hand, int(P) ∩ (−P) = ∅ (according to Lemma 2 (i)), therefore τ ∈ ∅ which is in contradiction with τ P 0 E . Consequently, (x n ) n∈N ⊂ X is a Cauchy sequence. Since (X, d) is a complete cone metric space, we deduce that there exists an element x ∈ X such that the sequence (x n ) n∈N ⊂ X converges to x ∈ X.
Further, we show that x is a fixed point of T. For this, we consider the set U = {n ∈ N | x n = Tx } and we distinguish the following subcases: II.1: U is an infinite set. We can choose a subsequence (x m(k) ) k∈N of (x n ) n∈N , where m : N → N is a strictly increasing function, with k < m(k) for all k ∈ N, such that (x m(k) ) k∈N converges to Tx . However, we show that the sequence (x n ) n∈N converges to x . Considering Lemma 3 we obtain that the sequence (x n ) n∈N has a unique limit, hence Tx = x , so x is a fixed point of T.
II.2: H is a finite set. We find that there exists a number N ∈ N such that x n = Tx for all n ≥ N. Let us consider n ≥ N be an arbitrary element. Using the property (6) we obtain τ + F(d(Tx n−1 , Tx )) ≤ P F(d(x n−1 , x )), thus F(d(x n , Tx )) ≤ P F(d(x n−1 , x )) − τ. Since τ P 0 E it follows that F(d(x n , Tx )) P F(d(x n−1 , x )). As F ∈ F , from Definition 11 (3) we obtain d(x n , Tx ) ≤ P d(x n−1 , x ) for all n ≥ N.
Further, the properties of the cone metric d lead to Taking into account the relations (12), (13) we get We choose ε ∈ int(P) be an arbitrary element. Since (x n ) n∈N ⊂ X converges to x ∈ X, we deduce that there is a number n(ε) ∈ N such that for all n ≥ n(ε) we have d(x n , x ) P ε. It follows that d(x n , x ) + d(x n−1 , x ) P ε + ε for all n ≥ n(ε) + 1.
Let us consider n ≥ max{n(ε) + 1, N} to be a natural number. Using the relations (14), (15) and taking into account Lemma 1 (iii) we obtain d(x , Tx ) P ε + ε. Therefore, d(x , Tx ) ∈ P and d(x , Tx ) P ε + ε for all ε ∈ int(P). Applying Lemma 2 (iii) it follows that d(x , Tx ) = 0 E , thus x = Tx , hence x is a fixed point of T.
Consequently, in both subcases II.1 and II.2 we showed that x ∈ X is a fixed point of T. Considering that T has at most one fixed point, we obtain that x ∈ X is the unique fixed point of T. Also, we proved that (x n ) n∈N converges to x . (A, ⊕, , τ A , ) be a partially ordered Hausdorff topological ring, with 1 A ∈ A + , 0 A an accumulation point of A + , (E, +, ·, τ E ) a Hausdorff topological left A-module, P a solid cone of E, with a · int(P) ⊆ int(P) for all a ∈ A * + , 0 E / ∈ int(P), and (X, d) a cone metric space over the topological left A-module. We suppose that (X, d) is complete, T : X → X is an F-contraction corresponding to the function class F and for every x 0 ∈ X we consider the sequence (x n ) n∈N ⊂ X defined by x n = Tx n−1 for all n ∈ N * . If there is a sequentially compact subset S ⊆ P having the property {d(Tx, Ty) | x, y ∈ X} ⊆ S, then T has a unique fixed point x ∈ X and for every x 0 ∈ X the sequence (x n ) n∈N ⊂ X converges to x .

Corollary 1. Let
Proof. Let x 0 ∈ X for which (x n ) n∈N ⊂ X is not a Cauchy sequence, be an arbitrary element. Because for every m, n ∈ N * , m < n, we have x m = Tx m−1 , x n = Tx n−1 , it follows that {d(x m , x n ) | m, n ∈ N * , m < n} ⊆ {d(Tx, Ty) | x, y ∈ X}. According to the hypothesis, there is a sequentially compact subset S ⊆ P having the property {d(Tx, Ty) | x, y ∈ X} ⊆ S, hence {d(x m , x n ) | m, n ∈ N * , m < n} ⊆ S. Therefore, for any x 0 ∈ X for which (x n ) n∈N ⊂ X is not a Cauchy sequence, there is a sequentially compact subset S x 0 := S ⊆ P having the property {d(x m , x n ) | m, n ∈ N * , m < n} ⊆ S x 0 . Consequently, the hypotheses of Theorem 1 are fulfilled, hence T has a unique fixed point x ∈ X and for every x 0 ∈ X the sequence (x n ) n∈N ⊂ X converges to x .
In the following, we determine some conditions for the existence and uniqueness of a solution of the following integral equation for all t, s ∈ [a, b], x, y ∈ C([a, b], R n ), i = 1, n, then the integral Equation (16) has a unique solution in C([a, b], R n ).
Proof. Let (R, +, ·, τ R , ≤) be the partially ordered Hausdorff topological ring and (R, +, ·, τ R ) the Hausdorff topological left R-module, where τ R is the Euclidean topology. We consider the solid cone P = R + , with int(P) = R * + . We define the set C( where (|x i (t)|e −τ(t−a) ). We remark that is a complete cone metric space over the topological left R-module. Next, we consider the function which belongs to the class F .
A function x ∈ C([a, b], R n ) is a solution of the integral Equation (16) if and only if it is a fixed point of the operator T.
The definition of the operator T leads to Considering the hypothesis we find τd(x, y) + 1 τd(x, y) + 1 ds τd(x, y) + 1 .
It follows that, for every x, y ∈ C([a, b], R n ), T(x) = T(y), we have .
Considering the relation (19), we find Consequently, the hypotheses of Corollary 1 are satisfied, hence the operator T has a unique fixed point x ∈ C([a, b], R n ). Further, for every x 0 ∈ C([a, b], R n ) the sequence (x n ) n∈N ⊂ C([a, b], R n ), defined by x n = Tx n−1 for all n ∈ N * , converges to x . It follows that the integral Equation (16) has a unique solution in C([a, b], R n ).
In the sequel, we will apply Corollary 1 to a fractional integral equation. Following the study of Fernandez et al. [26] we consider [a, b] a real interval, α ≥ 0, β ≥ 0, R > 0 satisfying R > (b − a) β and A a real analytic function on the interval (−R, R), defined by the locally uniformly convergent power series where a k (α, β), k ∈ N, are real coefficients. Next, we determine some conditions for the existence and uniqueness of a solution of the following fractional integral equation defined by Fernandez et al. [26], the above fractional integral equation can be rewritten as for all t, s ∈ [a, b], s ≤ t, x, y ∈ C([a, b], R), then the fractional integral Equation (23)  (|x(t) − y(t)|e −τ(t−a) ). (27) We remark that (C([a, b], R), ρ) is a complete cone metric space over the topological left R-module. Moreover, the function belongs to the class F .
A function x ∈ C([a, b], R) is a solution of the integral Equation (23) if and only if it is a fixed point of the operator T.

Conclusions
In this paper, we define the notion of F-contraction for cone metric spaces over topological left modules and we establish some fixed point results for self-mappings satisfying a contractive condition of this type. Applications of the main result to the study of the existence and uniqueness of the solutions for certain types of integral equations were presented in the last part of the article, one of them being an integral equation of fractional type.