Generalized Fixed Point Results with Application to Nonlinear Fractional Differential Equations

: The main objective of this paper is to introduce the ( α , β ) -type ϑ -contraction, ( α , β ) -type rational ϑ -contraction, and cyclic ( α - ϑ ) contraction. Based on these deﬁnitions we prove ﬁxed point theorems in the complete metric spaces. These results extend and improve some known results in the literature. As an application of the proved ﬁxed point Theorems, we study the existence of solutions of an integral boundary value problem for scalar nonlinear Caputo fractional differential equations with a fractional order in ( 1, 2 ) .


Introduction
Fixed point theorems are useful tools in nonlinear analysis, the theory of differential equations, and many other related areas of mathematics. One of the most applicable method for various investigations is Banach's contraction principle [1]. Many researchers generalized and extended this theorem to different directions. For example, Boyd and Wong [2] elongated the main result of Banach and they replaced the constant in the contractive condition by an appropriate function. Recently, Samet et al. in [3] defined α-admissible and α-ψ-contractive type mappings and studied some of their properties in the framework of complete metric spaces. Later on, Salimi et al. in [4] introduced and investigated the twisted (α,β)-admissible mappings. Many extensions of the notion of α-ψ-contractive type mappings have been developed, see, for example, [5][6][7][8][9] and the references therein.
In all these investigations, the underlying space was complete metric space. There were some open problems for fixed point theorems in ordered metric spaces and cyclic representations of ϑ-contraction.
To solve the first problem, we define (α,β)-type ϑ-contraction with the help of control functions α and β. With this new notion, we not only generalize the main theorem of Wardowski [10] but also derive the results for ordered metric spaces by these control functions. We also introduce (α,β)-type rational ϑ-contraction which extend the notion of ϑ-contraction. Moreover, a cyclic (α-ϑ) contraction and cyclic ordered (α-ϑ) contraction are also introduced to solve the second problem.
To illustrate some of the applications of the fixed point theorems studied in this paper, we use the Caputo fractional differential equation. Note that nonlinear fractional differential equations play a very useful role in modeling in various fields of science, such as physics, engineering, bio-physics, fluid mechanics, chemistry, and biology [23,24]. In this paper, based on the proved fixed point theorems, we provide some new sufficient conditions for the existence of the solutions of an integral boundary value problem for a scalar nonlinear Caputo fractional differential equations with fractional order in (1,2). We also compare the obtained existence results with known ones in the literature.

Preliminaries
Let (Ω, ω) (Ω, for short) and C L (Ω) be the complete metric space Ω with a metric ω and the set of all non-empty closed subsets of Ω, respectively.
To be more precise and to be easier for readers to see the novelty of the results in this paper, we will initially give some that are known in the literature definitions.
We will give some examples of functions from the set ∆ which will be used later.

Fixed Point Results
We will introduce a new type of contraction mapping.
We will obtain some new fixed point results applying the introduced above types of mappings.
Then the mapping J has a fixed point in Ω, i.e., there exists a point l * ∈ Ω such that J (l * ) = l * .

The mapping J is continuous.
Then the mapping J has a fixed point in Ω.
In the case when the mapping J is not continuous we get the following result: Theorem 3. Let J : Ω → Ω be an (α,β)-type rational ϑ-contraction and the following condition be satisfied: Then the point l * from condition (c) is a fixed point of the mapping J .

The conditions (b) and (c) of Theorem 3 are fulfilled.
Then the point l * from condition (c) is a fixed point of the mapping J .
The fixed point result in Theorem 4 generalize the known in the literature result. Proof. The claim follows from the proof of Theorem 4 with α(l, κ)=β(l, κ) ≡ 1 for all l, κ ∈Ω.
Example 3. Consider the set Ω = l j : j ∈ N where the natural numbers: Let ω (l, κ) = |l − κ| for any l, κ ∈ Ω. Define the mapping J : Ω → Ω by, Let the functions α : According to Example 2 the function Θ ∈ ∆. Then the mapping J is (α,β)-type ϑ-contraction, with π = 12. or it is ϑ-contraction (see Remark 1). Consider the following three possible cases: and As ι > 1, so we get, From (17), we have, By (15) and (16), we have, and As ι > j > 1, we get: We know that, By (21), we have: By (19) and (20), we have: Thus all the hypotheses of Theorem 3 hold and therefore, the mapping J has a unique fixed point l 1 .
Then the mapping J has a fixed point in Ω, i.e., there exists a point l * ∈ Ω such that J (l * ) = l * .
Then the mapping J has a fixed point in Ω.
Now we prove a result for (α,β)-type rational ϑ-contraction when the mapping J is not continuous.
Then the point l * from condition (c) is a fixed point of the mapping J in Ω.
Similarly to case I, we get: ThusJ is (α, β)-type rational ϑ-contraction. Moreover all the assumptions of Theorem 6 are satisfied and 9 4 is a fixed point of J .

The conditions (b) and (c) of Theorem 6 are fulfilled.
Then the point l * from condition (c) is a fixed point of the mapping J .
The above inequality is a contradiction because π > 0. Hence, l * is unique. Now we define cyclic (α-ϑ) contraction and derive some results from our main theorems.
Then the mapping J has a fixed point in S 1 ∩ S 2 .
Then the mapping J has a fixed point in S 1 ∩ S 2 .

Corollary 6.
Let the function ϑ ∈ ∆, the sets S 1 , S 2 ∈ C L (Ω), and J : S 1 ∪ S 2 → S 1 ∪ S 2 with J S 1 ⊆ S 2 and J S 2 ⊆ S 1 is continuous and the inequality: holds for all l ∈ S 1 and κ ∈ S 2 . Then the mapping J has a fixed point in S 1 ∩ S 2 .
Now we define cyclic ordered (α-ϑ) contraction and derive some results from our main theorems.

Applications to Caputo Fractional Differential Equations
Recently, many researchers have studied the existence of solutions of varies types of fractional differential equations. In this paper we will emphasize our study of Caputo fractional differential equations of the fractional order in (1, 2) and the integral boundary condition. Note that similar problems are studied in [25][26][27] but the main condition is connected with enough small Lipschitz constant of the right hand side part of the equation. Based on the obtained fixed points theorems we can use weaker conditions for the right hand side part of the equation (see Example 5).
We will apply some of the proved above Theorems to investigate the existence of the solutions of the nonlinear Caputo fractional differential equation: with the integral boundary condition: t a (t − s) 1−q x (s)ds represents the Caputo fractional derivative, and a, b : 0 ≤ a < b are given real numbers. Let For any x, y ∈ Ω we define ω(x, y) = x − y [a,b] .
Consider the linear fractional differential equation: with the integral boundary condition (27) where g ∈ Ω.
Lemma 1. Let g ∈ Ω. Then the boundary value problem (28), (27) has a solution: The proof of Lemma 1 is based on the presentation of the solution given in [28].
Based on the presentation (29) we will define a mild solution of (26) and (27).

Definition 8.
The function x ∈ Ω is a mild solution of the boundary value problem (26) and (27) if it satisfies: For any function u ∈ Ω, we define the mapping J : Ω → Ω by: Now, we establish the existence result as follows.
Then the boundary value problem (26), (27) has a mild solution.
According to Theorem 3 the operator J has a fixed point in Ω, i.e., there exists a function x * ∈ C([a, b], R) such that x * = J (x * ). This function x * is a mild solution of the boundary value problem for (26) and (27).

Remark 2.
Note that the condition (i) of Theorem 12 for the function f (t, x) is less restrictive than the Lipschitz condition used in many existence results (see, for example [25]). Now we will provide an example to demonstrate the existence result.
Remark 3. Note that the boundary value problem (33) and (34) is studied in [25], but the absolute value is missing under the square root. Also, the function f (t, x) is assumed as Lipschitz, but it is not (see Figure 1 for the particular value t = 2.2 ∈ (2, 3)). At the same time the function f (t, x) satisfies the condition 1 with k = 0.25 (see Figure 2 for the particular value t = 2.2 ∈ (2, 3)), and by one of the fixed point theorems proved in this paper the existence of the solution follows.

Discussion
In fixed point theory, the contractive inequality and underlying space play a significant role. A pioneer result in this theory is a Banach contraction principle that consists of compete metric space (Ω, ω) as underlying space and the following contractive inequality: ω(J (l), J (κ)) ≤ πω(l, κ) in which J is a self mapping and π ∈ [0, 1). Over the years, many mathematicians have generalized and extended above contractive inequality in different ways.
As it is pointed out in [10] the introduced mapping and inequality (36) are a generalization of Banach contraction (35) with ϑ(l) = ln(l), for l > 0.
As a partial case of some of our results, we obtained known results in the literature. For example, if α(l, κ) = β(l, κ) = 1 in Theorem 2 then we obtain Theorem 1 ([10]) by which one can derive the result of [1].

Conclusions
In the present paper, we introduced two new types of contractions: (α,β)-type ϑ-contraction and (α,β)-type rational ϑ-contraction. Based on their applications we proved new fixed points theorems. These results generalized some known ones from fixed point theory. To support our results, we provided two non trivial examples. The obtained results are noteworthy contributions to the current results of literature in the theory of fixed points. In this field, one can establish (α,β)-type ϑ-contraction and (α,β)-type rational ϑ-contraction for the multivalued mappings in the perspective of complete metric spaces and generalized metric spaces. To illustrate the application of the new fixed point theorems, we considered an integral boundary value problem for a Caputo fractional scalar equation of order from the interval (1,2) and proved the existence of the solution.