An Extension of Strict Almost Contractions Employing Control Function and Binary Relation with Applications to Boundary Value Problems

: This article comprises some ﬁxed point results for Boyd–Wong-type strict almost contractions using locally L -transitive binary relations. We provide several examples to illustrate our ﬁndings. On applying our results, we determine a unique solution of a special boundary value problem.


Introduction
One of the powerful and fundamental results of metric fixed point theory is the Banach contraction principle (abbreviation: BCP).Indeed, BCP guarantees a unique fixed point for a self-contraction on a complete metric space.This result also offers an iterative scheme to compute the unique fixed point.In the last century, this result has been extended by various researchers.In this direction, several authors enlarged the usual contraction to be a ψ-contraction by governing the contraction condition via suitable auxiliary function ψ : [0, ∞) → [0, ∞).By varying ψ suitably, various generalizations were obtained and this theme now has a considerable literature.A noted class of ψ-contraction is essentially due to Boyd and Wong [1] wherein the author improved the contraction condition by replacing Lipschitz constant c ∈ (0, 1) with a control function belonging to the following family: Theorem 1 ([1]).Assume that a self-map L on a complete metric space (Z, ρ) satisfies for some ψ ∈ Ω that ρ(Lv, Lz) ≤ ψ(ρ(v, z)), ∀ v, z ∈ Z.
Then L possesses a unique fixed point.
The above contractivity condition is called nonlinear contraction or ψ-contraction.Under the restriction ψ(s) = cs, 0 < c < 1, ψ-contraction reduces to usual contraction and Theorem 1 reduces to the BCP.
By symmetric property of ρ, the above condition is equivalent to the following one: Theorem 2 ([2]).An almost contraction self-map on a complete metric space owns a fixed point.
The notion of almost contraction has been developed by various researchers, e.g., see [4][5][6][7][8][9].Any almost contraction hasn't always a unique fixed point, but a sequence of Picard iterations remains convergent to a fixed point of the underlying mapping.To obtain a uniqueness theorem, Babu et al. [4] defined a slightly stronger class of almost contraction conditions.Definition 2 ([4]).A self-map L on a metric space (Z, ρ) is termed as strict almost contraction if Clearly, a strict almost contraction is an almost contraction.However, the converse is not generally true, see; Example 2.6 [4].

Theorem 3 ([4]
).A strict almost contraction on a complete metric space owns a unique fixed point.
A novel extension of BCP in relational metric space was investigated by Alam and Imdad [10].Since then, various fixed point theorems have been established employing different contractivity conditions in this context, e.g., [11][12][13][14][15][16][17][18][19].In such results, the contraction map is verified only for comparative pairs.Consequently, the relation-theoretic contractions remain weaker than usual contractions.The fixed point results obtained in the relationtheoretic setting are applicable into specific periodic BVPs (i.e., boundary value problems).
The aim of the present manuscript is to subsume two contractivity conditions, as mentioned earlier (i.e., ψ-contraction and strict almost contraction), and utilize this newly obtained contraction to establish relevant fixed-point theorems in a metric space with a locally L-transitive relation.We illustrate our results by adopting some examples.To validate our results, we adopt an application to a BVP, satisfying certain additional hypotheses.

Preliminaries
In the aftermath, the sets of natural, whole and real numbers will be denoted by N, N 0 and R, respectively.Recall that a subset of Z 2 is a binary relation (or, a relation) on the set Z.
Let us assume that Z is the given set, L : Z → Z is a mapping, S is a relation on Z and ρ remains a metric on Z. Definition 3 ([10]).The points v, z ∈ Z are called S-comparative if (v, z) ∈ S or (z, v) ∈ S. We denote such a pair by [v, z] ∈ S. Definition 4 ([20]).The relation S −1 := {(v, z) ∈ Z 2 : (z, v) ∈ S} is called inverse of S. Also, S s := S ∪ S −1 defines a symmetric relation on Z, often called symmetric closure of S.
Definition 5 ([21]).For a subset Q ⊆ Z, the set a relation on Q, is named as the restriction of S on Q. Definition 6 ([10]).S is referred to as L-closed if, for every pair v, z ∈ Z verifying (v, z) ∈ S, one has (Lv, Lz) ∈ S.
For each fixed v 0 ∈ P, the set O L (v 0 If we ignore the orbital properties in Definitions 9-11, we obtain the notions of 'S-complete metric space', 'S-continuous map' and 'ρ-self-closed relation', respectively (cf.[11]).

Definition 12 ([12]
). S is referred to as locally L-transitive if, for each S-preserving sequence {v n } ⊂ L(Z) possessing the range E = {v n : n ∈ N}, S| E remains transitive.
Making use of the symmetric property of metric ρ, we get the following: Proposition 2. For a given control function ψ ∈ Ω and K ≥ 0, the following conditions are equivalent:

Main Results
We are going to prove the results about existence and uniqueness of fixed points for relational strict almost ψ-contractions.Theorem 4. Suppose that (Z, ρ) is metric space endued with a relation S and L : Z → Z is a map.Moreover, Then, L possesses a fixed point.
Proof.Given v 0 ∈ Z (by (iii)).Construct a sequence {v n } ⊂ Z, as follows: By assumption (iii), L-closedness of S and Proposition 1, we get which, due to availability of (1), reduces to Hence, {v n } is a S-preserving sequence.
Let us denote ) = 0 for some n 0 ∈ N 0 , then in lieu of (1), one has L(v n 0 ) = v n 0 .Thus, v n 0 is a fixed point of L; hence, we have completed the solution.
In case ρ n > 0, ∀ n ∈ N 0 , employing assumption (v), ( 1) and ( 2), we get Employing the property of ψ in (3), we have This embraces that {ρ n } is a monotonically decreasing sequence of positive reals.Further, {ρ n } remains bounded below by '0'.Consequently, ∃ δ ≥ 0 such that We assert δ = 0. Assuming, to contrary, that δ > 0. Invoking to limit superior in (3), employing (4) and the property of Ω, one finds This contradiction implies that δ = 0. Thus, we have Now, we assert that {v n } is Cauchy.Assuming, to contrary, that {v n } is not Cauchy.Then, by Lemma 1, ∃ subsequences {v n k } and {v l k } of {v n } and ∃ 0 > 0, satisfying As {v n } is S-preserving (due to (2)) and {v n } ⊂ L(Z) (due to (1)), using locally L-transitivity of S, we find (v l k , v n k ) ∈ S. Therefore, by using the contractivity condition (v), we obtain Letting the upper limit in (6) and making use of Lemma (1) and definition of Ω, one finds which gives rise to a contradiction.Thus, {v n } remains Cauchy.Since{v n } is also L-orbital and S-preserving, therefore, by (O, S)-completeness of Z, ∃ v ∈ Z verifying v n ρ −→ v. Finally, we conclude the proof using the assumption (iv).Suppose that the mapping If ρ(v n k 0 , v) = 0 for some k 0 ∈ N, then we find ρ(Lv n k 0 , L v) = 0 so that ρ(v n k 0 +1 , L v) = 0; hence, (7) holds for such k 0 ∈ N. In either case, we have ρ 7) holds for all k ∈ N. On letting the limit of ( 7) and employing v n k ρ −→ v, we get v n k +1 ρ −→ L( v).By uniqueness property of limit, we find L( v) = v, so that v remains a fixed point of L.
Theorem 5. Along with the hypotheses of Theorem 4, if L(Z) is S-directed, then L possesses a unique fixed point.
As v, z ∈ L(Z), by our hypothesis, ∃ ω ∈ Z, satisfying Denote n := ρ( v, L n ω).Using ( 8) and ( 9) and assumption (v), one obtains using the definition of Ω, (10) reduces to n < n−1 .Hence, in both cases, we have Using the arguments similar to Theorem 4, the above inequality gives rise to Similarly, one can find lim By using ( 11) and ( 12) and the triangular inequality, one has Therefore, L possesses a unique fixed point.

Consequences
Outlined, by making use of our findings, we shall obtain few familiar fixed-point theorems from the literature.In particular, for the universal relation, S = Z 2 , Theorem 5 reduces to the following corollary: then L owns a unique fixed point.
For K = 0, Theorem 4 reduces to an enhanced variant of the fixed-point theorem of Alam and Imdad [12], given below.Corollary 2. Suppose that (Z, ρ) is metric space endued with a relation S and L : Z → Z is a map.Moreover, Then, L possesses a fixed point.

Examples
Intending to illustrate Theorems 4 and 5, we undertake some examples.
s+1 and choose K ≥ 0 arbitrarily.Then, for all (v, z) ∈ S, we have Thus, the map L satisfies the condition (v) of Theorem 4. Similarly, rest assumptions of Theorems 4 and 5 can be verified.In turn, L owns a unique fixed point, namely, v = 0.
Example 2. Consider Z = [0, 1) along with the metric ρ(v, z) = |v − z|.Let L : Z → Z be a map defined by Take a relation S := {(v, z) ∈ Z 2 : v ≥ z}.Clearly, (Z, ρ) is (O, S)-complete.Also, S is locally L-transitive and L-closed binary relation on Z. Here, L is not (O, S)-continuous.However, S is (O, ρ)-self-closed.Also, L satisfies the contractivity condition (v) for the auxiliary function ψ(s) = 2s/3 and for the constant K = 1.Similarly, rest assumptions of Theorems 4 and 5 can be verified.In turn, L owns a unique fixed point, namely, v = 0.
Proof.Clearly, ( 13) can be written as which remains equivalent to the Fredholm integral equation: Define the two relations on Z as follows: and In lieu of one of the hypotheses, let w ∈ C [0, l] be a lower solution of (13).Now, we shall show that (w, Lw) ∈ S. One has By multiplying to both of the sides with e r , we find Employing ( 18) and ( 19), we find so that (w, Lw) ∈ S. Similarly, if w ∈ C [0, l] is an upper solution of ( 13), then we can prove that (w, Lw) ∈ S.
Next, we shall verify that S is L-closed.Choose w, v ∈ Z such that (w, v) ∈ S. Making use of ( 14), we find By ( 16) and (20) and N( , ξ) > 0, ∀ , ξ ∈ [0, l], we obtain which in view of (6) yields that (Lw, Lv) ∈ S; hence, the conclusion is immediate.Similarly, we can verify that S is also L-closed.
Endow the following metric ρ on Z: Clearly, the metric space (Z, ρ) is (O, S)-complete as well as (O, S)-complete.To verify the contraction condition, take w, v ∈ Z such that (w, v) ∈ S. Making use of ( 14), ( 16) and ( 21 where K ≥ 0 is arbitrary.A similar contraction condition can be verified analogously for the relation S. Let {w n } ⊂ Z be an L-orbital S-preserving sequence converging to w ∈ Z.Then, we have w n ( ) ≤ w( ), ∀ n ∈ N and ∀ ∈ [0, l].By (6), we have (w n , w) ∈ S, ∀ n ∈ N. In turn, S is (O, ρ)-self-closed.Similarly, we can verify that S is also (O, ρ)-self-closed.
Thus conditions (i)-(v) of Theorem 4 are verified for the relational metric spaces (Z, ρ, S) and (Z, ρ, S).Consequently, L has a fixed point.
Take arbitrary w, v ∈ Z so that L(w), L(v) ∈ L(Z).Set u := max{Lw, Lv}, thereby implying (Lw, u) ∈ S and (Lv, u) ∈ S.This shows that the set L(Z) is S-directed.Similarly, L(Z) is also S-directed.Thus, using Theorem 5, L ows a unique fixed point, which in turns remains the unique solution of (13).

Funding:
The work received no external funding.Data Availability Statement: Not applicable.