Relation-Theoretic Nonlinear Almost Contractions with an Application to Boundary Value Problems

: This article investigates certain fixed-point results enjoying nonlinear almost contraction conditions in the setup of relational metric space. Some examples are constructed in order to indicate the profitability of our results. As a practical use of our findings, we demonstrate the existence of a unique solution to a specific first-order boundary value problem.


Introduction
The classical BCP and its applications are widely recognized.In recent years, this crucial result has been generalized by many researchers using different approaches (e.g., [1,2]).One of the natural generalizations of this result is almost contraction, which was introduced by Berinde [3].The almost contraction covers the usual (Banach) contraction, Kannan mapping [4], Chatterjea mapping [5], Zamfirescu contraction [6] and a certain class of quasicontractions [7].It is evident from this generalization that an almost contraction map does not necessarily possess a unique fixed point.Nonetheless, the convergence of the Picard iteration sequence can be used to calculate the fixed points of an almost contraction map.Alfuraidan et al. [8] presented a nonlinear formulation of almost contraction.For deeper investigation on almost contractions, we refer to [9][10][11][12][13].
In contrast, Alam and Imdad [14] presented an inevitable expansion of the BCP in a complete MS provisioned with an amorphous relation.In the past few years, multiple fixed-point results have been proven involving various contractivity conditions in relational MS, e.g., [15][16][17][18][19][20][21][22] and references therein.These outcomes comprised relation-preserving contractions that continue to be weaker than the ordinary contractions, which are indeed intended to verify the relation-preserving elements only.
The intent of this article is to investigate a fixed-point theorem employing nonlinear almost contraction in the setup of relational MS.The underlying relation in our results is amorphous (i.e., arbitrary), but the uniqueness theorem requires that the image of ambient space must be Λ s -directed.This indicates the worth of our main results ahead of the results of Berinde [3], Alam and Imdad [14], Algehyne et al. [21], Khan [22] and Alfuraidan et al. [8].We provide two illustrative examples that corroborate our results.In order to show the extent to the applicability of our results, we compute a unique solution of a first-order BVP.

Preliminaries
A relation Λ on a set V means any subset of V 2 .Assuming, V is a set, ζ is a metric on V, Λ is a relation on V and P : V → V is a function.

Definition 8 ([14]
). Λ is referred to as ζ-self-closed if every Λ-preserving convergent sequence in V permits a subsequence, every term of which remains Λ-comparative with the limit.
Following Bianchini and Grandolfi [25], we shall denote by Φ the family of the mono- Inspired by Berinde [3], Alfuraidan et al. [8] introduced the class of functions θ : In the following, we will denote this class by Θ.Using the symmetry of metric ζ, one can put forth the following assertion.Proposition 3. If φ ∈ Φ and θ ∈ Θ , then the contractivity conditions listed below are identical:

Main Results
Herein, we present the fixed-point results under a new contractivity condition depending on the auxiliary functions belonging to classes Φ and Θ in the setup of relational MS.
Theorem 1. Assume that (V, ζ) is an MS endowed with a relation Λ and P : V → V is a map.The following assumptions are also made: Then, P admits a fixed point.
Following assumption (b), the P-closedness of Λ and Proposition 2, we find Hence, {v ı } remains a Λ-preserving sequence. Denote Applying the condition (e) to (2) and utilizing (1), we find which, by simple induction and the incensing property of φ, becomes For every ı, ȷ ∈ N with ı < ȷ, using (3) and triangular inequality, we find This verifies that {v ı } is Cauchy.As {v ı } also remains an Λ-preserving sequence, . Now, we will conclude the proof by verifying that v * remains a fixed point of P. According to (d), first assume P is Λ-continuous.As {v ı } is a Λ-preserving sequence with Making use of the uniqueness of the limit, we find P (v * ) = v * .In the alternative, 4) and employing the items (i) and (iii) of Remark 2 and the definition of Θ, we find Hence, in each of these cases, v * serves as a fixed point of P.
Theorem 2. Assume that all premises of Theorem 1 are valid.Furthermore, if then P possesses a unique fixed point.
On utilizing ( 7), ( 9) and the triangular inequality, we conclude This concludes the proof.
Remark 3.Under trivial relational Λ = V 2 in Theorems 1 and 2, we obtain the nonlinear formulation of the result of Berinde [3], which runs as follows: then P possesses a unique fixed point.

Illustrative Examples
This section is devoted to furnishing some examples in support of Theorems 1 and 2.
Naturally, P is Λ-continuous and Λ is P-closed.Define the functions φ(t) = 2t/3 and θ(t) = 3t/4.Then, φ ∈ Φ and θ ∈ Θ.For any (v, w) ∈ Λ, we conclude i.e., P verifies premise (e) of Theorem 1.Therefore, all the hypotheses of Theorem 1 are satisfied.Similarly, we can verify all premises of Theorem 2; so P possesses a fixed point.Indeed, here, P admits a fixed point: (V, ζ) serves as a Λ-complete MS.Let P be considered as an identity map on V. Naturally, P is Λ-continuous and Λ is P-closed.

Definition 10 ([26]
).One says that υ ∈ C 1 [0, a] serves as a lower solution of (10) Definition 11 ([26]).One says that υ ∈ C 1 [0, a] serves as an upper solution of (10) In the following, we will prove a result which guarantees the existence of a unique solution to problem (10).
Proof.Express the problem (10) as Thus, ( 12) is identical to the integral equation where Ω(ℓ, τ) is a Green function defined by Consider the mapping P : V → V defined by Consequently, υ ∈ V continues to be a fixed point of P iff υ ∈ C 1 [0, a] becomes a solution of (13) and thereby (10).
On V, define a metric ζ and a relation Λ given as: and Now, we will approve each of the hypotheses of Theorem 1: be a lower solution of ( 10); then, one has Multiplying with e kℓ , one obtains According to ( 17) and ( 18 which yields (υ, P υ) ∈ Λ.

Conclusions
This manuscript comprised some fixed-point theorems under nonlinear almost contraction on an MS endowed with an amorphous relation.In the process, we also derived a nonlinear formulation of the Berinde fixed-point theorem [3].Still, by utilizing our results, we can obtain several existing fixed-point theorems, especially thanks to Alam and Imdad [14], Algehyne et al. [21], Khan [22] and Alfuraidan et al. [8].In future works, our results can be extended to nonlinear almost contractions by taking φ as a comparison function in the sense of Matkowski [27].This work concludes the feasible application of the results proven herewith to a BVP, provided a lower solution exists.In a similar manner, readers can find an analogous result in the existence of an upper solution.