Developments of some new results that weaken certain conditions of fractional type differential equations

We introduce double and triple F-expanding mappings. We prove related fixed point theorems. Based on our obtained results, we also prove the existence of a solution for fractional type differential equations by using a weaker condition than the sufficient small Lipschitz constant studied by Mehmood and Ahmad (AIMS Math. 5:385–398, 2019) and Hanadi et al. (Mathematics 8:1168, 2020). As applications, we ensure the existence of a unique solution of a boundary value problem for a second-order differential equation.


Introduction
In 2012, Wardowski [3] generalized the Banach contraction principle by introducing a new type of contractions, called F-contractions, and established a unique related fixed point theorem. This modification of BCP motivated many researchers to study further possibilities of its extensions . In 2017, Gornicki [26] presented some new fixed point results for F-expanding mappings. We modify this setting by introducing multiple F functions. The usage of multiple F functions permits to find solutions for an extensive range of integral equations. The nonlinear fractional differential equations have a valuable role in various fields of science, such as engineering, biology, fluid mechanics, physics, chemistry, bio-physics. For more details, see [21,22,[27][28][29][30][31][32][33][34][35][36][37]. After establishing the fixed point theorems for expanding type mappings, we provide some new sufficient conditions for the existence of solutions of an integral boundary value problem for a scalar nonlinear Caputo fractional differential equation with fractional order in (1,2). We also compare the obtained result with known ones in the literature. Furthermore, we use our obtained results to find a solution of an engineering problem, in which the transformed mathematical model of a problem representing activation of a spring affected by an external force is a boundary value problem for a second-order differential equation.
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Preliminaries
In this paper, N, N 0 , R, and R + denote the set of natural numbers, N ∪ {0}, real numbers, and positive real numbers, respectively. Throughout the paper, every set X taken into account is nonempty. Wardowski [3] defined the concept of F-contractions as follows.
Remark 2.1 From the conditions of F-contractions, it is easy to conclude that every Fcontraction mapping is necessarily continuous. Further, Wardowski [3] stated a modified version of the Banach contraction principle as follows.
Theorem 2.1 ([3]) Let (X, D) be a complete metric space and T : X → X be an Fcontraction. Then T has a unique fixed point, say x * ∈ X, and for every x ∈ X, the sequence {T n x} n∈N converges to x * .
For details on F-contraction mappings, see [8,9,38,39]. The concept of F-expanding mappings is given as follows.

Definition 2.2 ([26]
) Let (X, D) be a metric space. A mapping T : X → X is said to be F-expanding if there are F ∈ F and a real number τ > 0 such that, for all x, y ∈ X,

Main results
Firstly, we introduce two types of double F-expanding mappings that generalized Fexpanding mappings. Definition 3.1 Let (X, D) be a metric space. A mapping T : X → X is said to be double F-expanding of type I, if there exist a real number τ > 0 and F 1 , F 2 ∈ F such that, for all x, y ∈ X, we have where Ty)) .
Next, we introduce triple F-expanding mappings.

Definition 3.3
Let (X, D) be a metric space. A mapping T : X → X is said to be a triple F-expanding mapping, if there exist τ > 0 and F, F 1 , F 2 ∈ F such that, for all x, y ∈ X, we have D T 2 x, T 2 y > 0 and D(Tx, Ty) > 0 Example 3.1 Take F 1 (α) = ln α and F 2 (α) = ln kα, k > 0. Then F 1 and F 2 ∈ F . The double F-expanding mapping of type I will take the form Condition (F1) allows us to write ln min{kD T 2 x, T 2 y , D(Tx, Ty)) ≥ ln k α 2 D(x, y) α 1 +α 2 + τ .
By the assumption of the definition, we have α 1 + α 2 = 1, so Note that if we suppose (as a particular case) that, for all x, y ∈ X, D(Tx, Ty) < kD(T 2 x, T 2 y), then we have D(Tx, Ty) ≥ e τ D(x, y) with α 2 = 0. That is, T is an expanding mapping. Further, if for all x, y ∈ X, D(Tx, Ty) > kD(T 2 x, T 2 y), then we have D(T 2 x, T 2 y) ≥ e τ D(x, y). Hence, T is neither a contraction nor an expanding mapping.
With the usage of condition (F1), we can write min kD T 2 x, T 2 y , D(Tx, Ty) ≥ e τ kD(x, y) .
One may observe that, for all x, f (x) = y ∈ X, relations (6) and (7) produce the sequences in which the iterates of T may have several combinations of expansions and contractions.
Then, by the definition of a triple F-expanding mapping, we have for all x, y ∈ X Condition (F1) allows us to write Then either k 2 (D(T 2 x, T 2 y)) ≥ ke τ (D(x, y)) or k 1 (D(Tx, Ty)) ≥ ke τ (D(x, y)). We can define k 2 = 2α 2 , k 1 = 2α 1 , k 1 + k 2 = 2, where α 1 + α 2 = 1. So that we have either Both of the above inequalities can be written as That is a reversal of a mean Lipschitzian mapping, and so the fixed point of (3) will be the fixed point of T.
Theorem 3.1 Let (X, D) be a complete metric space. Suppose that a surjective continuous mapping T : X → X is a double F-expanding mapping of type I, and for all t 1 , t 2 ∈ R + , there are σ > 0 and τ > σ such that Then T has a unique fixed point in X, and for every x 0 ∈ X, the sequence {T m x 0 } ∞ m=1 converges in X.
Proof Consider a sequence {x 1 , x 2 , . . .} such that, for any x 0 ∈ X, we have x m+1 = Tx m = T m+1 x 0 for all m ∈ N 0 . If, for some m ∈ N, D(x m , Tx m ) = 0, T admits a fixed point. Let We will prove that lim m→∞ D(x m , Tx m ) = 0. For any m ∈ N, we can write Now, we will discuss the two possible cases (C) and (D): If (C) holds, then by the conditions of Definition 3.1, inequality (8) will take the form Relation (9) further yields Therefore, the possible existence of (C) implies the existence of (10). Similarly, if (D) holds, inequality (8) will take the form (11) can be written as If

then condition (A) allows us to write
Combining inequalities (12) and (13), The above inequality can be written as Therefore, the existence of (D) implies the existence of inequality (14).
Both cases (C) and (D) yield inequalities (10) and (13) that can be written in the combined form as follows: where ς m is either 0 or 1.
The above inequality can also be written as Repeating this process, we have Next, we will show that T is bijective. So, which implies that Both of the above inequalities are contradictions, and so we have D(x, y) = 0 if and only if x = y. Therefore, T is bijective.
Consider a mapping such that T = T = I, where I is the identity mapping.
Now, equation (16) implies that Therefore, from (21) we get Now, we will prove that the sequence {u m } ∞ m=1 is a Cauchy sequence. On the contrary, suppose that there exist ε > 0 and sequences {g(m)} ∞ n=1 and {h(m)} ∞ m=1 of natural numbers such that We further suppose that h(m) is greater than g(m) by l(m). Now, we can write That is, The above inequality along with (19) yields Further, from (17) Next, we claim that On the contrary, suppose that there exists r ≥ N such that It follows from (18), (20), and (22) that ε ≤ D(u g(r) , u h(r) ) ≤ 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) ) That is a contradiction. Next, we suppose that, for some Therefore, relation (21) with the assumption of the theorem gives Now, we will deal with two possible cases of (23): Both of the above inequalities will take the form So that we have the following contradictions: F 2 (ε) ≥ F 2 (ε) + τ or F 1 (ε) ≥ F 1 (ε) + τ . Therefore, {u m } ∞ n=1 is a Cauchy sequence. The completeness of (X, D) proves that {u m } ∞ n=1 converges to some point u * in X. Now, the continuity of implies that Therefore, has a fixed point u * in X and u * = u * so that u * = Tu * . Now, for the uniqueness, let us suppose that T has more than one fixed point. That is, there exist two distinct or Both relations (24) and (25) are the contradictions, and so we have a unique fixed point.

Theorem 3.2
Let (X, D) be a complete metric space. Suppose that a surjective continuous mapping T : X → X is a double F-expanding mapping of type II, and for all t, t 1 , t 2 ∈ R + , there is σ > 0 such that σ < τ and Then T has a unique fixed point in X, and for every x 0 ∈ X, the sequence {T m x 0 } ∞ m=1 converges to a definite number.
Proof If, for all x, y ∈ X, relation (R2) holds, then the analysis of the previous theorem yields The above inequality can be written as follows: Repeating this process, we have Now, the analysis similar to the previous theorem yields If, for all x, y ∈ X, relation (R1) holds, then we can write The above inequality will take the form That can be written as Repeating this process, we have Since σ < τ , we have Therefore, relations (26) and (27) imply that or, Now, using the analysis of the previous theorem, relations (R1) and (R2) yield that As τ > 0, relations (28) and (29) are contradictions. Now, we consider relation (30) Using condition (B'), we have It is a contradiction, as τ > σ . Therefore, has a fixed point u * in X and u * = u * . So that u * = Tu * . Now, for uniqueness, let us suppose that T has more than one fixed point. That is, there exist two distinct u, v ∈ X such that Tu = u = v = Tv. Therefore, D(u, v) = D(Tu, Tv) = D(T 2 u, T 2 v) > 0, and the assumption of the theorem leads to the following four possibilities: All the four relations (32)- (35) are contradictions, and so we have a unique fixed point. We will prove that lim m→∞ D(x m , Tx m ) = 0. For any m ∈ N, we have If F (D(x m-1 , x m )) ≥ F 1 (D(x m-1 , x m )), inequality (36) can be written as follows: If F(D(x m-1 , x m )) < F 1 (D(x m-1 , x m )), condition (A) allows us to write Then inequality (36) can be written as Combining inequalities (37) and (38), we have where ς i is either 0 or -1. Next, we will consider the following two possible cases: or If (40) is true, inequality (39) will take the form If (41) is true, we have F 2 (D(Tx m-1 , Tx m )) < F 1 (D(x m-1 , x m )). So that relation (41) can be written as Moreover, condition (41) will change the above inequality in the following form: Combining (42) and (43), we have That is equivalent to Repeating this process, we have Now, the analysis similar to the previous theorem yields that As τ > σ , we have Therefore, relation (44) implies that Using the analysis of the previous theorem, we will have That yields either F 2 (ε) ≥ F(ε) + τ or F 1 (ε) ≥ F(ε) + τ . Now, with the usage of condition (B), we can write F 2 (ε) ≥ F(ε) + τ or F 2 (ε) > F(ε) + σ , which implies, F 2 (ε) > F 2 (ε). It is a contradiction. Similarly, F 1 (ε) ≥ F(ε) + τ . Also, it yields a contradiction. Thus, {u m } ∞ m=1 is a Cauchy sequence. The completeness of (X, D) proves that {u m } ∞ m=1 converges to some point u * in X. Now, the continuity of implies Therefore, has a fixed point u * in X, that is, u * = u * . So that u * = Tu * . Now, for uniqueness, let us suppose that T has more than one fixed point. That is, there exist two distinct or Since both relations (45) and (46) are contradictions, the mapping T has a unique fixed point.

Applications to Caputo fractional differential equations
As applications of our work, we will study the existence of solutions of Caputo fractional differential equations of the fractional order in (1,2) with an integral boundary condition. The main condition in the problems studied in [1] (see Theorem 3.2 therein) and [2] (see Theorem 12 therein) is associated with sufficient small Lipschitz constant. We will use a less restrictive condition than the Lipschitz condition by applying our obtained fixed point theorem. For 1 < r < 2 and a Caputo fractional derivative C α D r t x(t) = 1 1-r x (s) ds, consider a nonlinear Caputo fractional differential equation with an integral boundary condition where, for some λ ∈ (α, β), x(λ) ∈ R and α, β are given real numbers such that 0 ≤ α < β.
In order to assure the existence of solution of nonlinear Caputo fractional differential equation (47), we consider the following fractional differential equation: Kilbas [40] proved that the following function represents the solution of boundary value problem (48) and (49) for g ∈ , based on the following presentation of the solution: Next, we will define a mild solution of (47) and (48).

Definition 4.1
The function x ∈ is a mild solution of boundary value problem (47) and (48) if it satisfies For any function u ∈ , we define a surjective mapping ϒ : → by for t ∈ [α, β]. Now, we establish the existence result as follows.
Proof Note that any fixed point of the mapping ϒ is a mild solution of boundary value problem (47) and (48). Now, let x, y ∈ be such that W(x, y) > 0. By condition (i), we obtain Combining relations (57) and (59), we have Therefore, χ : → is a triple F-expanding mapping, and the operator χ has a fixed point in . That is, there exists a function x * ∈ C([α, β], R) such that x * = χ(x * ). This implies that (x * ) = x * . The function x * is a mild solution of boundary value problem (47) and (48).
Example 4.1 Consider the nonlinear Caputo fractional differential equation with the integral boundary conditions Here, r = 1.75. Also, Note that > 1. Therefore, Theorem 4.1 guarantees the existence of a solution of boundary value problem (60) and (61).
Remark 4.1 Note that boundary value problem (60) and (61) is also studied in [1] (see Example 5 therein) and [2] (see Example 3.3 therein). Based on the obtained fixed point theorems, we used the weaker conditions for the right-hand side part of the equation and found the existence of a fixed point for K > 1 and > 1.

An application to integral equations
As another application of our work, we will consider an engineering problem in which the transformed mathematical model of a problem representing activation of spring affected = ke rτ +τ v τ 2rτ e τ rrτe τ r + 1 = ke 2τ r e τ v τ 2rτrτ e -τ r -1 + e -τ r .
That is, Likewise, for w ∈ X, we can find that g(w)) τ ≥ ke τ w τ .
That is, Thus, We deduce Define F 1 (x) = ln x k , F 2 (x) = ln x k 2 , and F(x) = ln x. So that inequalities (65) and (66) take the form Therefore, the fixed point of the above triple F-contraction is the solution of problem (62).

Conclusion
In this article, we generalized F-expanding mappings by using multiple F functions and certain conditions on the mapping. These conditions allow us to deal with the class of mappings whose iterates expand in general, but some of their iterates may contract as well. Moreover, with the usage of multiple F functions, we presented an idea that allows to use weaker conditions for several fractional type differential equations. The new generalizations of F-expanding mappings and the corresponding results will break open new grounds for the researchers working in the field as they will be able to find the existence of solutions of an extensive range of differential equations.