Some new generalizations of F − contraction type mappings that weaken certain conditions on Caputo fractional type di ﬀ erential equations

: In this paper, ﬁrstly, we introduce some new generalizations of F − contraction, F − Suzuki contraction, and F − expanding mappings. Secondly, we prove the existence and uniqueness of the ﬁxed points for these mappings. Finally, as an application of our main result, we investigate the existence of a unique solution of an integral boundary value problem for scalar nonlinear Caputo fractional di ﬀ erential equations with a fractional order (1,2).


Introduction
The Banach contraction principle generally known as Banach fixed-point theorem emerged in 1922 [1], and due to its coherence and effectiveness, it has turned out to be a very popular tool in several branches of mathematical analysis for solving the existence problems. Numerous researchers studied the Banach fixed point theorem in different directions and established the extensions, generalizations and the applications of their findings. Among them, Wardowski [2] provided very interesting extension of Banach's fixed point theorem. Definition 1.1. [2] Let a function F : (0, ∞) → R satisfy the following conditions: (F1) F is strictly increasing, i.e. for all α, β ∈ (0, ∞) such that α < β ⇒ F(α) < F(β). (F3) There exists k ∈ (0, 1) such that lim α→0 + α k F(α) = 0. The set of all functions F satisfying (F1) − (F3) will be denoted by F . Example 1.1. [2] Suppose that the functions F i : (0, ∞) → R, i = 1, 2, 3, 4 are defined by (1) F 1 : t → ln t.
Note that we have D(T u, T v) < D(u, v) for all u, v ∈ X with T u T v concluding that every F−contraction is a contractive mapping.
Wardowski [2] proved the existence of a unique fixed point in a complete metric space for every F−contraction mapping T . He also showed that F−contractions are the generalizations of Banach contractions. Theorem 1.1. [2] Let T be a self-mapping on a complete metric space (X, D). If T forms a F−contraction, then it possesses a unique fixed point u * . Moreover, for any u ∈ X the sequence {T n u} is convergent to u * .
After the development of F−contractions, several authors looked into the necessity of the conditions (F1) -(F3) and presented some weaken conditions by replacing or removing some of them.
We briefly present some existing cases. Secelean [3] suggested that the condition (F2) can be replaced by a simple one: (F2 ) inf F = −∞, or, also by (F2 ) there exists a sequence {α n } of positive real numbers such that lim n→∞ F(α n ) = −∞.
Secelean [4] also removed condition (F3) on the operator T by assuming some boundedness condition. Moreover he also"proved that (F3) can be dropped without any additional supposition on T . Piri and Kumam [5] replaced condition (F3) by the continuity of F. Vetro [6] replaced the constant τ with a function and generalized the F−contraction. Secelean and Wardowski [7] introduced ψF−contraction and weak ψF−contraction by weakening condition (F1) and introducing the class of increasing functions ψ. Further, Lukács and Kajántó [8] found some results in b−metric spaces of F−contraction by omitting condition (F2).
Alsulami [9] introduced generalized F−Suzuki contraction in b−metric spaces and established the existence of fixed points by using the conditions (F1) and (F2) only. Definition 1.3. Let (X, D) be a metric space. A mapping T : X → X is said to be a F-Suzuki contraction if there exist a real number τ > 0 such that for all u, v ∈ X, On the other hand, in 2017, Gornicki [10] introduced a new type of mappings called F-expanding mappings and proved some new fixed point results for this new kind of mapping, especially on a complete G−metric space. Definition 1.4. [10] Let (X, D) be a metric space. A mapping T : X → X is called F−expanding if there exist F ∈ F and τ > 0 such that for all u, v ∈ X Theorem 1.2. [10] Let (X, D) be a complete metric space and T : X → X be surjective and F−expanding. Then T has a unique"fixed point.
In 2001, James Merryfield [11] established a generalization of Banach contraction principal by following conjecture named as generalized Banach contraction conjecture. Theorem 1.3. [11] Let (X, D) be a complete metric space. Suppose T : X → X satisfies the following condition: There exists an integer p and a number k ∈ [0, 1) such that for all u, v ∈ X we have Then, T has exactly one fixed point.
Following the Wardowski's idea along with the conjecture presented by James Merryfield [11], in this paper, we introduce some new generalizations of F−contraction, F-Suzuki contraction and F−expanding mappings and prove the existence of their unique fixed points. Moreover, as an application of our main result, we investigate the existence of unique solution of the nonlinear Caputo fractional differential equations.
Proinov established the following fixed point theorem for a self-mapping T on a complete metric space (X, D). Theorem 1.4. [12] Let (X, D) be a metric space and T : X → X be a mapping such that where, the functions F 1 , F : (0, ∞) → R satisfy the following conditions. i) F 1 is nondecreasing; ii) F(t) < F 1 (t) for t > 0; iii) lim sup t→ε+ F(t) < F 1 (ε+) for any ε > 0. Then T has a unique fixed point u * ∈ X and the iterative sequence {T n u} converges to u * for every u ∈ X.
Proinov also obtained the following improvement of Wardowski's fixed point theorem. Theorem 1.5. Let (X, D) be a metric space and T : X → X be a mapping such that where, τ > 0 and the function F : (0, ∞) → R is nondecreasing. Then T has a unique fixed point u * ∈ X and the sequence {T n u} converges to u * for every u ∈ X.
In this paper, we establish a fixed point theorem using a certain condition that generalizes the main contractive-type conditions used by Wardowski and Proinov.

Main results
We start this section by introducing some new types of generalized F−contraction and generalized F−Suzuki contraction mappings. Let F : (0, ∞) → R satisfy the following conditions: Let us denote by F the set of all functions F satisfying the conditions (F * 1 ), (F * 2 ), (F * 3 ) and denote by T the set of all functions F satisfying the conditions (F * 1 ), (F * 2 ), (F * 3 ).
Note that F is strictly increasing with an upper bound F(t) = 1. Therefore, for some t > 0, 0 < F(t) < t, and 0 < α < β, we have F(α) < F(β) ≤ F(α) + F(t) and F ∈ F. Further, we can easily check that the conditions (F * 1 ), (F * 2 ) and (F * 3 ) are also satisfied, hence F ∈ T. Definition 2.1. Let (X, D) be a metric space. A mapping T : X → X is said to be a generalized F-contraction if there exist τ > 0 and an integer p > 1 such that for all u, v ∈ X, where"F ∈ F. Definition 2.2. Let (X, D) be a metric space. A mapping T : X → X is said to be a generalized F-Suzuki contraction if there exist τ > 0 and an integer p > 1 such that for all u, v ∈ X, with u v where, F ∈ T. Theorem 2.1. Let (X, D) be a complete metric space and T : X → X be continuous generalized Fcontraction mapping with p = 2. Then T has a unique fixed point u * ∈ X and for every u 0 ∈ X the sequence {T n u 0 } ∞ n=1 converges to the fixed point. Proof. Let u 0 ∈ X be an arbitrary point and define a sequence {u n } ⊆ X by u n+1 = T u n = T n+1 u 0 , for all n ∈ N. Now, we will prove that lim n→∞ D(u n , T u n ) = 0. If u n 0 = T u n 0 for some n 0 ∈ N, then D(u n , T u n ) = D(u n+1 , T u n+1 ) = · · · = 0, for all n ≥ n 0 and so D(u n , T u n ) converges to 0, as n → ∞. Assume that u n u n+1 , for all n ∈ N. By the contraction assumption on T , there exits τ > 0 such that τ + F(min{D(T 2 u n−1 , T 2 u n ), D(T u n−1 , T u n )}) ≤ F(D(u n−1 , u n )), or F(min{D(T 2 u n−1 , T 2 u n ), D(T u n−1 , T u n )}) ≤ F(D(u n−1 , u n )) − τ. (2.2) If D(u n−1 , u n ) = min{D(T u n−1 , T u n ), D(u n−1 , u n )}, the inequality (2.2) will take the form as follows which also can be written as If D(T u n−1 , T u n ) = min{D(T u n−1 , T u n ), D(u n−1 , u n )}. Then, by condition (F * 1 ), we have F(D(u n−1 , u n )) ≤ F(D(T u n−1 , T u n )) + F(τ).
The above inequality together with inequality (2.1) yields that which is equivalent to, Repeating this process we get, (2.6) Therefore, the results of (2.6) and condition (F * 2 ) implies, Next, we claim that {u n } is a Cauchy sequence. Arguing by contradiction, we assume that there exist > 0 and sequence {n(k)} ∞ k=1 and {m(k)} ∞ k=1 of natural numbers such that It follows from (2.8) and the above inequality that On the other hand, from (2.8) there exits N ∈ N, such that Next, we claim that Arguing by contradiction, there exists l ≥ N such that It follows from (2.9), (2.11) and (2.13) that which is a contradiction. Therefore, it follows from (2.12) and the assumptions of"the theorem that (2.14) From condition (F * 3 ), (2.10) and (2.14), we get which shows that {u n } is a Cauchy sequence. From the completeness of"(X, D), {u n } converges to some point u * ∈ X.
Finally, the continuity of T yields that For uniqueness, we assume that u is another fixed point such that T u * = u * u = T u . Then we have which is a contradiction. Therefore, T has a unique fixed point in X. Theorem 2.2. Let (X, D) be a complete metric space and T : X → X be a generalized F-Suzuki contraction mapping with p = 2. Then T has a unique fixed point in X and for every u 0 ∈ X the sequence {T n u 0 } ∞ n=1 converges to the fixed point. Proof. Let u 0 ∈ X be an arbitrary point and define a sequence {u n } ⊆ X by u n+1 = T u n = T n+1 u 0 , for all n ∈ N. Now, we will prove that lim n→∞ D(u n , T u n ) = 0. If u n 0 = T u n 0 for some n 0 ∈ N, then D(u n , T u n ) = D(u n+1 , T u n+1 ) = · · · = 0, for all n ≥ n 0 and so D(u n , T u n ) converges to 0, as n → ∞. Assume that u n u n+1 , for all n ∈ N. Since, for all n ∈ N , so, from the contraction assumption on T , there exits τ > 0 such that As in the proof of Theorem 2.1, the above inequality gives that Moreover, analysis similar to that in the proof of Theorem 2.1 shows the sequence {u n } ∞ n=1 is a Cauchy sequence. Since, (X, D) is complete, then the sequence {u n } ∞ n=1 converges to some point u * ∈ X, that is, Now, we will claim that Suppose, on the contrary, that there exists a m ∈ N satisfying the following three inequalities, Now, (2.17) along with triangular inequality gives Also, (2.20) along with (2.18) gives Similarly, (2.18) yields From (2.19) and (2.22), we have Now, by the contraction assumption on T , there exists τ > 0 such that which yields, From condition (F * 1 ), we have which is a contradiction. Hence, (2.16) holds true. Again, from (2.16), we have Using ( Hence, u * is a fixed point of T . For uniqueness, let us suppose that T has another fixed point u , such that T u * = u * u = T u .
, from the contraction assumption on T , there exists τ > 0 such that, which is a contradiction. Therefore, T has a unique fixed point.
In the next definition, we introduce some new notions of generalized F−expanding mappings. Definition 2.3. Let (X, D) be a metric space. A mapping T : X → X is said to be a generalized F−expanding mapping of type (A), if there exists τ > 0 such that for all u, v ∈ X, where F ∈ F. Definition 2.4. Let (X, D) be a metric space. A mapping T : X → X is said to be a generalized F−expanding mapping of type (B), if there exists τ ≥ 0 such that for all u, v ∈ X, where F ∈ F. Theorem 2.3. Let (X, D) be a complete metric space and let T : X → X be continuous surjective and generalized F−expanding of type (A). Then T has a unique fixed point in X and for every u 0 ∈ X the sequence {T n u 0 } ∞ n=1 converges to the fixed point. Proof. Firstly, we will show that T is bijective, which only needs to show that If D(u, v) > 0, from the assumption on T , there exits τ > 0 such that The second above inequality together with condition (F * 1 ) implies, which yields that 0 ≥ D(u, v), a contradiction. So, we have T u T v or D(T u, T v) > 0, hence T is injective and then bijective. Consider a mapping S such that T S = S T = I u , where I u is identity mapping on X.
which together with (2.25) implies that Using the inverse mapping S , the above inequality takes the form  From condition (F * 1 ) we have, Combining (2.29) and (2.30), together with the assumption on T , we have which shows that S is the generalized F−contraction defined in Theorem 2.1. From the conclusion of Theorem 2.1, S has a unique fixed point, so does T . Theorem 2.4. Let (X, D) be a complete metric space and let T : X → X be continuous surjective and generalized F−expanding of type (B). Then T has a unique fixed point in X and for every u 0 ∈ X the sequence {T n u 0 } ∞ n=1 converges to the fixed point. Proof. Firstly, we will show that T is bijective, which only needs to show that If D(u, v) > 0, from the assumption on T , there exits τ ≥ 0 such that The second inequality together with condition (F * 1 ) implies, which yields that 0 ≥ D(u, v), a contradiction. So, we have T u T v or D(T u, T v) > 0, hence T is injective and then bijective. Consider a mapping S such that T S = S T = I u , where I u is identity mapping on X. Since T is bijective, we have Combining (2.32) and (2.33), together with the assumption on T , we have which is equivalently stated as there exists τ > 0 such that which shows that"S is the generalized F−contraction defined in Theorem 2.1. From the conclusion of Theorem 2.1, S has a unique fixed point, so does T . Theorem 2.5. Let (X, D) be a complete metric space. Suppose a continuous mapping T : X → X satisfy where, non-decreasing functions F, F 1 ∈ F and for all t, t 1 ∈ R + , there exist υ > 0, τ > 2υ, such that Then T has a unique fixed point in X and for every u 0 ∈ X, the sequence {T m u 0 } +∞ m=1 converges to the fixed point. Proof. As, F, F 1 : (0, ∞) → R + are non-decreasing functions, so that we can write If inequality (2.38) holds true, the inequality (2.37) will take the form If inequality (2.39) is true, we have F 1 (D(T u m−1 , T u m )) < F 1 (D(u m−1 , u m )). From condition (A), we have (2.41) Using inequality (2.41) in (2.37), we can write Moreover, from (2.39), we have The above inequality can be written as Repeating this process, we have So that Since, τ > 2υ and So that we can write That is, Further, from (2.45) there exists N ∈ N such that for all m ≥ N, Next we claim that for all m ≥ N, ≤ D(u g(r) , u g(r)+1 ) + D(u g(r)+1 , u h(r) ) ≤ D(u g(r) , u g(r)+1 ) + D(u g(r)+1 , u h(r)+1 ) + D(u h(r)+1 , u h(r) ) = D(u g(r) , T u g(r) ) + D(u g(r)+1 , u h(r)+1 ) + D(u h(r) , T u h(r) ) Which is a contradiction. Therefore, (2.49) together with the assumption of the theorem gives τ + min{F 1 (D(T 2 u g(m) , T 2 u h(m) )), F 1 (D(T u g(m) , T u h(m) ))} (2.51) ≤ F(D(u g(m) , u h(m) ).
That yields a contradiction as τ > 2υ. The completeness of (X, D) proves that {u m } +∞ m=1 converges to some point u * in X. Now, the continuity of T implies Therefore, T has a unique fixed point u * .
Here is an example to show the validity of Theorem 2.1. Example 2.2. [20] Let B be closed unit ball in l 1 space of all absolutely summable sequence u = (u 1 , u 2 , · · · ) with a metric inherited from the standard norm u = It is easy to observe that, for all w 1 , w 2 ∈ [−1, 1], we have Further, let us define a surjective mapping T : B → B by T u = T (u 1 , u 2 , · · · ) = (h(u 2 ), 2 3 u 3 , u 4 , u 5 , · · · ).
Then for i ≥ 2, we have For each u = (u 1 , u 2 , · · · ), v = (v 1 , v 2 , · · · ) ∈ B, we have and for i ≥ 2, Then, we have for all u = (u 1 , which implies that, Therefore, F(α) = ln α represents generalized F-contraction mapping, hence Theorem 2.1 guarantees the existence of a unique fixed point of T .
Note that, F(α) = ln α does not contract, whenever max{ T u − T v , Therefore, T does not represent F-contraction mapping defined in [2]. Hence Theorem 1.1 does not guarantee the existence of a fixed point. Similarly, for F (α) = ln α + α, we can write so, This can be written as, Therefore, for inequality (2.52) shows that T is a generalized F-contraction mapping.

Applications to Caputo fractional differential equations
As an application of our work, we will study the existence of solutions to Caputo fractional differential equations of the fractional order in (1,2) and the integral boundary condition. The main condition in the problems studied in [21,22] is associated with sufficient small Lipschitz constant. We will use a less restrictive condition than the Lipschitz condition by applying our obtained fixed point theorems.
The proof of Lemma 3.1 is based on the presentation of the solution given in [23].
Next, we will define a mild solution of (3.1) and (3.2).
Moreover, one can easily observe that the use of multiple functions in the generalized F-contraction also allows us to define a function u ∈ C([ϑ 1 , ϑ 2 ] × R, R) in Theorem 3.1 with a weaker condition , where m, r ∈ (0, ∞). Example 3.1. Consider the nonlinear Caputo fractional differential equation where,  (3.15) and (3.16) are also studied in [22] (see Example 5 therein) and [24] (see Example 3.3 therein). Based on the obtained fixed points theorems we used the weaker conditions for the right hand side part of the equation and found the existence of fixed point for K > 0 and Λ > 1. Remark 3.3. Wardowski obtained some fixed point theorems (see; Theorem 1.1) assuming that T satisfies the following contractive-type condition τ + F(D(T x, T y)) ≤ F(D(x, y)), (3.17) where, F : (0, ∞) → R is nondecreasing. Whereas, the condition that we used in Theorem 2.1 is of the following form τ + F min (D(T x, T y)) , D(T 2 x, T 2 y) ≤ F (D(x, y)) . (3.18) One can easily observe that relation (3.18) represents a generalization of (3.17). Moreover, the following Proinov's condition represents a generalization of Wardowski's contraction condition.

Conclusions
Although, Proinov [12] claimed that being a special case of Skof's result [19], the F− contraction type mappings and their generalizations do not add a valuable work in the literature anymore, we found some new generalizations that extend Wardowski [2], Skof [19] as well as Proinov's idea [12] of F−contraction type mappings. Moreover, with the use of multiple functions and the idea of generalized Banach contraction principal [11], we applied less restrictive conditions on Caputo fractional differential equations than the sufficient small Lipschitz constant studied by Mehmood [22] and Hanadi [24]. The new generalizations of F-contraction, F-expanding type mappings and the corresponding results will break open new grounds for the research workers as they will be able to find the existence of solution to an extensive range of differential equations (see [25][26][27][28][29][30][31]) with some weaker conditions.

Questions
In this research, the new generalizations of F-contraction mapping, F-Suzuki contraction mapping, F-expanding mapping and the corresponding results will provide a new direction of metric fixed point theory for the research workers. They may try to find the existence of fixed point for the further extensions of certain generalized mappings. 1) One may find the above results with p > 2, for the generalizations of F-contraction, F-Suzuki contraction and F-expanding mappings. 2) One may work on the idea of introducing new generalizations of F-contraction, F-Suzuki contraction and F-expanding mappings.
3) There may exist the possibility of finding fixed points for these generalized mappings in other generalized metric spaces.