Application of Fixed Point Results on Rational F -Contraction Mappings to Solve Boundary Value Problems

: The notion of rational F -contractions using α -admissibility of type- S in b -metric-like spaces is introduced and the new ﬁxed and periodic point theorems are proved for such mappings. Numerical examples are illustrated to check the efﬁciency and applicability of our fresh ﬁndings. It is also observed that some of the works reported in the literature are the particular cases of the present study.


Introduction
The notion of F-contraction mapping was introduced by Wardowski [1] in fixed point theory and proved the related results. These results are the generalization of Banach contraction mapping principle as well as various fixed point theorems appearing in the literature, for instance [2]. On the other hand, Alghamdi et al. [3] found existence and uniqueness of fixed points for the mappings in b-metric-like and partially ordered b-metric-like spaces.
The notion of α-admissible maps was introduced that provided a beautiful class of mapping by Samet et al. [4] to observe the existence as well as uniqueness of fixed point. Using the same concept or slight modifications, a lot of work has been done in that direction. Sintunavarat [5] introduced the concept of α-admissible type-S in partial b-metric space and derived based fixed point results.
In the present paper, we introduce different types of rational F-contraction with α-admissibility type-S and examine the existence and uniqueness of fixed points in b-metric-like spaces.
Throughout this paper, R, R + and N are denoted as real numbers, nonnegative real numbers and positive integers, respectively.

Prerequisites
Definition 1 ([1]). Let F be the family of all functions F : (0, ∞) → R such that (F 1 ) F is strictly increasing, i.e., for all u, v ∈ R + such that u < v, F(u) < F(v); Then, (U, d) is said to be a b-metric space. k ≥ 1 is the coefficient of (U, d).
Definition 5 ([7]). Let U be a nonempty set, a mapping σ : Then (U, σ) is said to be a metric-like space. Definition 6 ( [3]). Let U be a nonempty set and a real number k ≥ 1 be given. A function σ b : U × U → R + such that the following assertions hold ∀ u, v, w ∈ U : Then, (U, σ b ) is said to be a b-metric-like space.
Ref. [8] recommended that the converses of the below facts need not be held.

•
Let U be a nonempty set and σ b is b-metric-like on U such that the pair (U, σ b ) be a b-metric-like space.

•
In a b-metric-like space (U, σ b ), if u, v ∈ U and σ b (u, v) = 0, then u = v, and σ b (u, u) may be positive for u ∈ U. • It can be easily observed that every b-metric and partial b-metric spaces are b-metric-like spaces with the same k.
Every b-metric-like σ b on U generates a topology τ σ b whose base is the family of all open σ b -balls {B σ b (u, δ) : u ∈ U, δ > 0}, where {B σ b (u, δ) = {v ∈ U : |σ b (u, v) − σ b (u, u)| < δ}, ∀ u ∈ U and δ > 0. Definition 7 ([3]). Let (U, σ b ) be a b-metric-like space with coefficient k, let {u n } be any sequence in U and u ∈ U. Then, (i) {u n } is called convergent to u w.r.t. τ σ b , if lim n→∞ σ b (u n , u) = σ b (u, u), It can be noted that the limit of a sequence may not be unique in b-metric-like spaces.
Let us discuss the notion of b-convergence, b-Cauchy sequence, b-continuity and b-completeness in b-metric-like spaces.
Each b-convergent sequence is b-Cauchy with a unique limit in b-metric-like spaces. The following lemma is necessary to prove main results. Lemma 1 ([9]). Let (U, σ b ) be a b-metric-like space with coefficient s ≥ 1 and let {u n } and {v n } be b-convergent to points u, v ∈ U, respectively. Then,

Remark 1 ([10]
). Let (U; σ b ) be a b-metric-like space and let T : U → U be a continuous mapping. Then,
The idea of α-admissibility was studied by [4] for the first time. After that, Ref. [5] extended this concept as α-admissibility type-S in the light of metric spaces and b-metric spaces, respectively. Definition 10 ( [4,5]). For a nonempty set U, let α : U × U → [0, ∞) and f : U → U are mappings. Then, ∀u, v ∈ U, (i) we say that the mapping T is α-admissible mapping if and is denoted by the symbol P (U, α).

Definition 11
(i) T is said to be a weak α-admissible mapping if and is denoted by W P (U, α) (ii) T is said to be a weak α-admissible mapping of type S if α(u, Tu) ≥ s ⇒ α(Tu, TTu) ≥ s, and is denoted by W P s (U, α), where s ≥ 1.
Ref. [5] presented some examples to show that the class of [] α-admissible mappings and the class of α-admissible mappings of type S are independent; that is, P (X, α) = P s (U, α).

Results
In this section, we investigate some fixed point results for rational F-contractions mapping with α-admissibility type-S and for the classes of W P s (U, α) and P s (U, α) : In addition, for each elements u and v in a b-metric-like space (U, where T is a self-mapping on U, we write ∆(u, v) instead of ∆ s (u, v) when s = 1, i.e., , Definition 12. Let (U, σ b ) be a b-metric-like space with coefficient s ≥ 1, let α : U × U → [0, ∞) be given mappings. Then, T : U → U is called rational F-contraction if the following condition holds: We denote by Θ s (U, α, F) the collection of all rational F-contractions on a b-metric-like space (U, σ b ) with coefficient s ≥ 1.
and T : U → U be given mappings. Suppose that the following conditions hold: Then, Fix(T) = ∅.
Define the sequence {u n } by u n+1 = Tu n . If there exists n 0 ∈ N, such that u n 0 = u n 0 +1 , then u n 0 ∈ Fix(T) and hence the proof is completed. Thus, we assume that u n = u n+1 , for all n ∈ N.
It follows that Hence, we have 1 2s σ b (u n , Tu n ) < σ b (u n , Tu n ), ∀ n ∈ N.
Now, we need to prove that lim It follows from T ∈ W S s (U, α) and α(u 0 , Tu 0 ) ≥ s that By induction, we obtain As we have it follows from T ∈ Θ s (U, α, F) that the inequalities (6) and (11) imply that for all n ∈ N. Note that, for each n ∈ N, we have If ∆ s (u n , Tu n ) = σ b (Tu n , T 2 u n ) for some n ∈ N, then inequality (12) implies that From (12), we have for all n ∈ N. Therefore, the above inequality becomes which is equivalent to Iteratively, we find that Using the method of contradiction, let us prove that {u n } is a b-Cauchy sequence in U. Assume that there exists 0 > 0 and sequences and q(k) is the smallest number such that (18) holds: By (σ b3 ), (18) and (19), we get Owing to (17), there exists N 1 ∈ N such that which together with (20) shows hence From (17), (18) and (21), we get Using the triangular inequality, we deduce that Passing to the limit k → +∞ in (25), by (16) yields i.e., σ b (Tu p(k) , Tu q(k) ) > 0. Using the transitivity property type S of α, we get By (6), (21), (23), and (25), we obtain for k > max{N 1 , N 2 }. Passing to the k → +∞ in (28) and using (27), we obtain which contradicts τ > 0. Therefore, {u n } is a b-Cauchy sequence in U. Now, (U, σ b ) is a b-completeness b-metric-like space; there exists u * ∈ U such that By b-continuity of T, we get lim n→∞ σ b (Tu n , Tu * ) = 0.
From the triangle inequality, we have Passing to the limit as n → ∞ in the above inequality, we obtain Then, Tu * = u * . This shows that Fix(T) = ∅.
Considering different cases of condition (6) in Theorem 1, we have the following contraction results.
In addition, we can see that T is b-continuous and there is u 0 = 1 such that Hence, all the conditions of Theorem 1 are fulfilled and Fix(T) = ∅. This example is verified that 0 ∈ Fix(T). Now, we need to prove that T ∈ Θ s (U, α, F).
In addition, we can see that T is b-continuous and there is u 0 = 1 such that All the requirements of Theorem 1 are satisfied. Hence, it can be concluded that Fix(T) = ∅. In this example, it shows that that 0 ∈ Fix(T).
In the following theorem, we derive fixed point results by replacing assumption (S 4 ) of Theorem 1 by α s -regularity of U.
and T : U → U be a rational F contraction mapping with α-admissibility type-S. Again, assume the following conditions: for all n ∈ N and u n → u ∈ U as n → ∞, then α(u n , u) ≥ s, for all n ∈ N.
Proof. Following the proof in Theorem 1, we obtain that {u n } is a b-Cauchy sequence in the b-complete b-metric-like space (U, σ b ). By b-completeness of U, there exists u ∈ U such that that is, u n → u as n → ∞. By α s -regularity of U, we have where ∆ s (u n , u) = max Taking the limit supremum as n → ∞ in (48) and using Lemma (1), we get which is a contradiction since τ > 0, which is possible only if F(σ b (u, Tu)) = 0. It follows that σ b (u, Tu) = 0, equivalently, u = Tu and thus Fix(T) = ∅. This completes the proof.
Next, we use Remark 2 to establish the following results for the class S s (U, α).

Periodic Point Results
Now, we discuss periodic point theorems for self-mappings on a b-metric-like space for which the following definition is required. ([14]). A mapping T : U → U is said to have the property-(P) if Fix(T n )= Fix(T), for every n ∈ N.

Definition 13
and T : U → U be given mappings. Suppose that the following conditions hold: there exists u 0 ∈ U such that α(u 0 , Tu 0 ) ≥ s; (S 3 ) α has a transitive property type S, (S 4 ) T is b-continuous; (S 5 ) If z ∈ Fix(T n ) and z / ∈ Fix(T), then α(T n−1 z, T n z) ≥ s.

Application to First-Order Periodic Boundary Value Problem
Consider the first-order periodic boundary value problem where T > 0 and T : I × R → R is a continuous function. We prove an existence theorem for the solution of (54) as an application of Theorem 1. Consider the space Obviously, (F , σ b , 2) is a b-complete b-metric like space. Then, (F , σ b , 2) is a b-complete b-metric like space. This problem (54) is equivalent to the integral equation where G(t, s) is the Green function given by Define the mapping T : C(I, R) → C(I, R) by Note that, if u * ∈ C 1 (I, R) is a fixed point of T, then u * ∈ C 1 (I, R) is a solution of (54). Next, we give the following notions which are required to complete this section.
Then, the existence of a lower solution for (54) implies the existence of a unique solution of (54). Then, u * ∈ C(I, R) is a solution of the integral Equation (56).

Remark 3.
Similarly, we can get the upper solution of (54) if we prove the upper condition in place of a lower condition.

Conclusions
The notion of rational F-contractions using α-admissibility of type-S is considered in b-metric-like spaces and the new fixed point and periodic point results are studied for such mappings. Some new theorems have been established on existence of solutions for rational F-contractions mapping with α-admissibility type-S, for the classes W P s (X, α) and P s (X, α). Numerical examples are illustrated in order to check the effectiveness and applicability of results. Furthermore, as an application to our results, the solution of first-order periodic boundary value problem is discussed.
Author Contributions: All the authors have contributed equally to this paper. All the authors have read and approved the final manuscript.
Funding: Not applicable.