Research article

Proinov-type relational contractions and applications to boundary value problems

  • Received: 14 January 2025 Revised: 07 May 2025 Accepted: 16 May 2025 Published: 11 June 2025
  • MSC : 34B15, 47H10, 54H25

  • This manuscript deals with proving certain metrical fixed-point findings for a class of generalized contractions involving a couple (ψ,φ) of test functions employing a local class of transitive relation. The outcomes investigated herein refine, modify, unify, and sharpen various existing outcomes. To attest the accuracy of our outcomes, we provide a few examples. By means of our outcomes, we impart an explanation of the reality of a solution to a boundary value problem.

    Citation: Abdul Wasey, Wan Ainun Mior Othman, Esmail Alshaban, Kok Bin Wong, Adel Alatawi. Proinov-type relational contractions and applications to boundary value problems[J]. AIMS Mathematics, 2025, 10(6): 13393-13408. doi: 10.3934/math.2025601

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  • This manuscript deals with proving certain metrical fixed-point findings for a class of generalized contractions involving a couple (ψ,φ) of test functions employing a local class of transitive relation. The outcomes investigated herein refine, modify, unify, and sharpen various existing outcomes. To attest the accuracy of our outcomes, we provide a few examples. By means of our outcomes, we impart an explanation of the reality of a solution to a boundary value problem.



    In the sequel, the following notations and acronyms will be utilized:

    N : the set of natural numbers

    N0:N{0}

    R : the set of real numbers

    R+:=(0,)

    R+0:=[0,)

    ● MS : metric space

    ● BVP: boundary value problem

    ● BCP : Banach contraction principle

    F(Q) : fixed-point set of function Q

    C(S) : the space of continuous functions on a set S

    C(S) : the space of continuously differentiable functions on a set S

    Fixed-point theory is one of the foremost prominent topics in nonlinear analysis, and subsequently, mathematics. Its findings can be extended to a wide range of integral equations, differential equations, and matrix equations to demonstrate the existence and uniqueness of various types of nonlinear problems. The classical BCP [1] is an extremely important conclusion of metric fixed point theory. The concept of BCP is crucial across many mathematical disciplines. It is successfully attempted to investigate solutions of Volterra as well as Fredholm integral equations, BVPs, nonlinear matrix equations, and nonlinear integro-differential equations, and to illustrate the convergence of algorithms in mathematical computing. Multiple variants of the BCP are accessible in the existing literature of metric fixed point theory; e.g., Boyd and Wong [2], Alber and Guerre-Delabriere [3], Ćirić [4], Kirk [5], Dutta and Choudhury [6], Jachymski [7] and related references. One of the noted classes of generalizations of BCP involves the contraction-inequality of the following form:

    ψ(σ(Qu,Qv))φ(σ(u,v)). (1.1)

    In recent years, various authors established fixed point theorems under contraction condition of the form (1.1) employing different perspectives, e.g., Amini-Harandi and Petruşel [8], Berzig [9], Proinov [10], Górnicki [11], Popescu [12], Olaru and Secelean [13], Roldán López de Hierro et al. [14], Găvruţa and Manolescu [15], and similar others. In follow-up evaluation, Ω refers to the class of the pair (ψ,φ) of the functions ψ,φ:R+R that verify:

    (a) ψ is monotonic increasing;

    (b) φ(t)<ψ(t), for every t>0;

    (c) Any one of the subsequent circumstances is valid:

    (c1) lim suptϵ+φ(t)<ψ(ϵ+), for every ϵ>0;

    or,

    (c2) lim suptϵφ(t)<lim inftϵ+ψ(t), for every ϵ>0;

    or,

    (c3) lim suptϵ+φ(t)<lim inftϵψ(t), for every ϵ>0.

    The above family of functions is considered by Górnicki [11] in order to obtain the analogues of some outcomes of Proinov [10] in the setup of preordered metric space.

    In 2015, Alam and Imdad [16] presented an inventive adaptation of the BCP to arbitrary binary relations. There are so many generalizations of this core results; however, we merely refer to the recent works due to [17], Alam and Imdad [18], Arif et al. [19], Algehyne et al. [20], and Alamer and Khan [21]. One of the most notable advantages of relational contractions is that no pair of elements is essential; the contraction inequality is enough to remain valid for comparable elements. Relational contractions remain somewhat weaker than their corresponding common contractions. Due to this, such outcomes are implemented to solve certain kinds of integral equations, BVPs and matrix equations, wherein the fixed point findings pertaining to ordinary MS are not laid down.

    This paper is an attempt to address outcomes on fixed points of a contraction map of the form (1.1) under the family Ω involving a locally Q-transitive relation. A few examples are furnished for attesting to the credibility of our findings. We entail the availability of a unique solution of a periodic BVP solution by utilizing our findings.

    This section covers some essential concepts and preliminary findings for the discussions to follow. In upcoming descriptions, let U be a nonempty set, σ a metric on U, Q a function on U, and S a binary relation on U, (i.e., SU2). We say that

    Definition 2.1. [16] The points u and v of U remain S-comparative if (u,v)S or (v,u)S. We write it as [u,v]S.

    Definition 2.2. [22] S1={(u,v)U2:(v,u)S} is the inverse of S.

    Definition 2.3. [22] Ss:=SS1 is the symmetric closure of S.

    Proposition 2.1. [16] (u,v)Ss[u,v]S.

    Definition 2.4. [16] A sequence {un}U is S-preserving if

    (un,un+1)S,nN0.

    Definition 2.5. [16] S is Q-closed if

    (u,v)S(Qu,Qv)S.

    Proposition 2.2. [17] If S is Q-closed, then Ss is also Q-closed.

    Proposition 2.3. [18] If S is Q-closed, then for each nN0, S is also Qn-closed.

    Definition 2.6. [17] (U,σ) is S-complete if each S-preserving Cauchy sequence in U is convergent.

    Definition 2.7. [17] Q is S-continuous at a point uU if for any S-preserving sequence {un} verifying unσu, we have Q(un)σQ(u). Moreover, Q is S-continuous if it is S-continuous at each point of U.

    Definition 2.8. [16] S is σ-self-closed if each S-preserving sequence {un} verifying unσu contains a subsequence {unk}with[unk,u]S,kN0.

    Definition 2.9. [23] Given a pair u,vU, a path of length l in S from u to v is a finite ordered set {w0,w1,w2,...,wl}U that verifies:

    (ⅰ) w0=uandwl=v;

    (ⅱ) (wi,wi+1)S, for each i(0il1).

    Definition 2.10. [17] A subset W of U is S-connected if any two elements of W enjoy a path.

    Definition 2.11. [18] S is Q-transitive if for all u,v,wU,

    (Qu,Qv),(Qv,Qw)S(Qu,Qw)S.

    Definition 2.12. [18] S is locally transitive if for every S-preserving sequence {un}U (with range W={un:nN}), the restriction S|W is transitive.

    Definition 2.13. [18] S is locally Q-transitive if for every S-preserving sequence {un}Q(U) (with range W={un:nN}), the restriction S|W is transitive.

    The following outcomes achieves the variation of 'locally Q-transitivity' over other concepts of 'transitivity':

    Proposition 2.4. [18] We have

    (ⅰ) S is Q-transitive S|Q(U) is transitive;

    (ⅱ) S is locally Q-transitive S|Q(U) is locally transitive;

    (ⅲ) S is transitive S is Q-transitive S is locally Q-transitive;

    (ⅳ) S is transitive S is locally transitive S is locally Q-transitive.

    Definition 2.14. [24] A sequence {un}P is semi-Cauchy if

    limnσ(un,un+1)=0,nN.

    Obviously, each Cauchy sequence is semi-Cauchy, but the converse is not true, as demonstrated by the following example.

    Example 2.1. Consider P=R with metric σ(u,v)=|uv|, for all u,vP. Then, the sequence {un}P defined by un=nk=11k is semi-Cauchy but not Cauchy.

    Lemma 2.1. [12] Let {un} be a semi-Cauchy sequence in a MS (U,σ). If {un} is not Cauchy, then we can determine a couple of subsequences {unk} and {umk} of {un} and a positive real number ϵ with

    limkσ(unk+1,umk+1)=limkσ(unk,umk)=ϵ+. (2.1)

    For a relation S and a function Q on a set U, the subsequent annotations will be utilized:

    SQ:={(u,v)S:Q(u)Q(v)},

    U(Q,S):={uU:(u,Qu)S}.

    Obviously, (u,v)SQQ(u)Q(v)uv.

    Now, we shall prove the fixed point theorem via a locally Q-transitive relation employing the pair of functions belonging to the family Ω.

    Theorem 3.1. Assuming that (U,σ) is a MS endowed with a relation S and Q a function on U. Also,

    (ⅰ) U(Q,S),

    (ⅱ) S is Q-closed and locally Q-transitive,

    (ⅲ) (U,σ) is S-complete,

    (ⅳ) Q is S-continuous or S is σ-self-closed,

    (ⅴ) (ψ,φ)Ω with

    (u,v)SQψ(σ(Qu,Qv))φ(σ(u,v)),

    Then, U has a fixed point.

    Proof. The proof distinguishes into several steps:

    Step 1. By assumption (ⅰ), u0U(Q,S). Define

    un:=Qn(u0),nN0. (3.1)

    so that

    un+1=Q(un),nN0.

    Thus, {un} is the Picard sequence based at the initial point u0.

    Step 2. We assert that {un} is an S-preserving sequence. As (u0,Qu0)S, using Q-closedness of S and Proposition 2.3, we get

    (Qnu0,Qn+1u0)S,

    so that

    (un,un+1)S,nN0. (3.2)

    Therefore, the sequence {un} is S-preserving.

    Step 3. We consider the case: σ(un0,un0+1)=0 for some n0N0. Then, employing (3.1), we get un0=un0+1=Q(un0)=0. This yields that un0F(Q), and hence, our task is accomplished. In either case (when σ(un,un+1)>0, for all nN0), we'll continue the succeeding steps.

    Step 4. We prove that {un} is semi-Cauchy, i.e.,

    limnσ(un,un+1)=0. (3.3)

    Denote σn:=σ(un,un+1)>0. Clearly, (un,un+1)SQ. Employing assumption (v), we attain, for all nN0 that

    ψ(σn+1)φ(σn),

    which by using axiom (b) of Ω reduces to

    ψ(σn+1)φ(σn)<ψ(σn). (3.4)

    It follows from (3.4) and axiom (a) that σn+1<σn for each nN0. Thus, the sequence {σn}R+ is monotonically decreasing which is also bounded below. Consequently, δ0 with σnRδ.

    Let δ>0. Next, we use property (c) of Ω. If (ψ,φ) satisfies axiom (c1), then, employing limit superior in (3.4), we attain

    ψ(δ+)=limnψ(σn+1)lim supnφ(σn)lim suptδ+ψ(t),

    which contradicts axiom (c1). Thus, we conclude that limnσn=0.

    Second, assume that the pair (ψ,φ) satisfies axiom (c2). Letting the limit inferior in (3.4), we attain

    lim inftδ+ψ(t)lim infnψ(σn+1)lim supnφ(σn)lim suptδφ(t),

    which contradicts axiom (c2). Hence, we conclude that limnσn=0.

    Finally, assume that (ψ,φ) satisfy axiom (c3). Letting the limit inferior as n in (3.4), we attain

    lim inftδψ(t)lim infnψ(σn+1)lim supnφ(σn)lim suptδ+φ(t),

    which contradicts axiom (c3). Hence, we conclude that limnσn=0. Thus, in each of the cases, (3.3) is verified.

    Step 5. We assert that {un} is Cauchy. If {un} is not Cauchy, then using Lemma 2.1, we can determine two subsequences {unk} and {umk} of {un} and ϵ>0 for which (2.1) holds. In view of (3.1), we have {un}Q(U). Using locally Q-transitivity of S, we get (unk,umk)S. From (2.1), we find σ(unk+1,umk+1)>ϵ for all kN; consequently, we have (unk,umk)SQ. Applying condition (v) for these points, we get

    ψ(σ(unk+1,umk+1))φ(σ(unk,umk)),kN. (3.5)

    Using axiom (b), we obtain

    ψ(σ(unk+1,umk+1))φ(σ(unk,umk))<ψ(σ(unk,umk)),

    which, using monotonicity of ψ, gives rise

    σ(unk+1,umk+1)<σ(unk,umk).

    Now, we shall employ property (c) of Ω. First, assume that the pair (ψ,φ) satisfies axiom (c1). Employing the limit superior in (3.5), we obtain

    ψ(ϵ+)=limkψ(σ(unk+1,umk+1))lim supkφ(σ(unk,umk))lim suptϵ+φ(t),

    which contradicts to axiom (c1).

    Second, assume that ψ and φ satisfy axiom (c2). Employing the limit inferior in (3.5), we get

    lim inftϵψ(t)=lim infkψ(σ(unk+1,umk+1))lim supkφ(σ(unk,umk))lim suptϵ+φ(t),

    which contradicts to axiom (c2).

    Finally, assume that ψ and φ satisfy axiom (c3). Employing the limit inferior as k in (3.5), we attain

    lim inftϵψ(t)=lim infkψ(σ(unk+1,umk+1))lim supkφ(σ(unk,umk))lim suptϵ+φ(t),

    which contradicts to axiom (c3). Therefore, in each of the cases, {un} is Cauchy, which is also S-preserving. Employing S-completeness of (U,σ), ¯uU with unσ¯u.

    Step 6. We verify that ¯u is a fixed point of U employing the hypothesis (ⅳ). Assume that Q is S-continuous. As {un} is S-preserving with unσ¯u, using S-continuity of Q, we obtain un+1=Q(un)σQ(¯u). Therefore, we conclude Q(¯u)=¯u, i.e., ¯u is a fixed point of U. Alternatively, in case S to be σ-self-closed, there is a subsequence {unk} of {un} with [unk,¯u]SQ, for all kN. Now two cases arise:

    Case (ⅰ): If for infinitely many values of k, [unk,¯u]SQ, then we have σ(Qunk+1,Q¯u)=0 yielding thereby

    σ(¯u,Q¯u)σ(¯u,unk+1)+σ(unk+1,Q¯u)=σ(¯u,unk+1)0ask,

    so that U(¯u)=¯u and hence the proof is completed.

    Case (ⅱ): Assume that [unk,¯u]SQ for infinitely many values of k. In view of the symmetric property of metric σ, the contraction condition (ⅴ) will be satisfied for all [u,v]SQ. Thus, we obtain

    ψ(unk+1,Q¯u)=ψ(Qunk,Q¯u)φ(unk,¯u)<ψ(unk,¯u),

    so that

    ψ(σ(unk+1,Q¯u))<ψ(σ(unk,¯u)).

    Using monotonicity of ψ above equality give rise to

    σ(unk+1,Q¯u)<σ(unk,¯u)0ask,

    so that Q(¯u)=¯u, and hence, ¯u is a fixed point of Q.

    Theorem 3.2. In alliance with the predictions of Theorem 3.1, if

    (u) Q(U) is Ss-connected,

    then, Q owns a unique fixed point.

    Proof. Due to Theorem 3.1, F(Q). Choose u,vF(Q), then for each nN0, we arrive at

    Qn(u)=uandQn(v)=v.

    Clearly u,vQ(U). By Ss-connectedness of Q(U), we determine a path w0,w1,w2,...,wl between u and v; so,

    w0=u,wl=vand[wi,wi+1]S,i=0,1,...,l1.

    As S is Q-closed, we have

    [Qnwi,Qnwi+1]S,nN0andi=0,1,...,l1.

    Denote

    δni:=σ(Qnwi,Qnwi+1)nN0andi=0,1,...,l1.

    We show that

    limnδin=0. (3.6)

    For every fixed i, consider the two possible cases:

    Case (ⅰ): Assume that

    δin0=σ(Qn0wi,Qn0wi+1)=0,forsomen0N0,

    thereby implying Qn0(wi)=Qn0(wi+1). By (3.1), we attain Un0+1(wi)=Qn0+1(wi+1); so, δin0+1=0. Using induction, we get δin=0nn0 so that limnδin=0.

    Case (ⅱ): If for every nN0, δin>0, then we have (Qnwi,Qnwi+1)SsQ. From (ⅴ), we attain

    ψ(δin+1)=ψ(σ(Qn+1wi,Qn+1wi+1))=ψ(σ(Q(Qnwi),Q(Qnwi+1)))φ(σ(Qnwi,Qnwi+1))=φ(δin),

    so that

    ψ(δin+1)φ(δin). (3.7)

    Using axiom (b) of Ω, (3.7) reduces to

    ψ(δin+1)φ(δin)<ψ(δin),nN0,

    which, in view of axiom (a), reduces to δin+1<δin for all nN0. Hence, proceeding with the proof of Theorem 3.1, we can determine δi0 satisfying δinRδi.

    In view of property (c), let us assume that (ψ,φ) satisfies axiom (c1). Employing limit superior in (3.7), we obtain

    ψ(δi+)=limnψ(δin+1)lim supnφ(δin)lim supδinδi+ψ(δin),

    which is a contradiction to axiom (c1). Hence, we conclude that limnδin=0.

    Second, assume that the pair (ψ,φ) satisfies axiom (c2). Letting the limit inferior in (3.7), we obtain

    lim inftδi+ψ(t)lim infnψ(δin+1)lim supnφ(δin)lim suptδiφ(t),

    which is a contradiction to axiom (c2). Hence, we conclude that limnδin=0.

    Finally, assume that (ψ,φ) satisfies axiom (c3). Letting the limit inferior in (3.7), we attain

    lim inftδiψ(t)lim infnψ(δin+1)lim supnφ(δin)lim suptδi+φ(t),

    which is a contradiction to axiom (c3). Hence, we conclude that limnδin=0.

    Hence, (3.6) is proved. Using the triangle inequality, we find

    σ(u,v)=σ(Qnw0,Qnwk)δ0n+δ1n++δk1n0asn;

    so, u=v. Thus, Q admits a unique fixed point.

    In the following, we provide two instances to substantiate the relevance of Theorems 3.1 and 3.2.

    Example 4.1. Let U=[2,4] be a MS with usual metric σ. On U, consider the relation S={(2,2),(2,3),(3,2),(3,3),(0,4)}. Then, (U,σ) is a complete MS. Define a function Q on U

    Q(u)={2if2u3,3if3u4.

    Thus, S is Q-closed. Assuming that {un}U is S-preserving sequence and unσu. Consequently, we conclude (un,un+1)S, for every nN. Note that (un,un+1){(2,4)}, implying thereby (un,un+1){(2,2),(2,3),(3,2),(3,3)},nN; so, {un}{2,3}. Closedness of {2,3} yields that [un,u]S. Hence, S is σ-self-closed. Define the functions ψ,φ:R+R by

    ψ(t)=t2andφ(t)=t2(t2+1).

    Then, (ψ,φ)Ω and the contraction-inequality (ⅴ) of Theorem 3.1 is verified for (ψ,φ). Moreover, the remaining hypotheses of Theorems 3.1 and 3.2 are also verified. This concludes that Q owns a unique fixed point (namely: ˉu=2).

    Example 4.2. Take U=R+ with Euclidean metric σ. Construct a relation S on U by

    S:={(u,v)U2:u2+2u=v2+2v}.

    Clearly, the MS (U,σ) forms an S-complete. Define a function Q on U by

    Q(u)=ln(u2+2u+1),uU.

    Then, S is a locally finitely Q-transitive and Q-closed relation, while Q is S-continuous. Also, U(Q,S) as (0,Q0)S.

    Take (u,v)S. Then, we have

    Q(u)=ln(u2+2u+1)=ln(v2+2v+1)=Q(v)

    yielding thereby

    (Qu)2+2Qu=(Qv)2+2Qv.

    This implies that (Qu,Qv)S, and hence, S is Q-closed. Define the pair (ψ,φ)Ω such that

    ψ(t)={ln(t+1),ift1,3t4,ift>1, (4.1)

    and φ(t)=2t/3. Then, for all (u,v)S, we can easily verify the following condition:

    ψ(σ(Qu,Qv))φ(σ(u,v)).

    Thus far, the requirements of Theorems 3.1 and 3.2 are all fulfilled. Thus, Q owns a unique fixed point (namely: ˉu=0).

    Consider the following first-order periodic BVP:

    {ω(ξ)=(ξ,ω(ξ)),for eachξ[0,L],ω(0)=ω(L), (5.1)

    where :[0,L]×RR is a continuous function.

    Definition 5.1. [25] ω_C(1)[0,L] is named as a lower solution of (5.1) if

    {ω_(ξ)(ξ,ω_(ξ)),for eachξ[0,L],ω_(0)ω_(L).

    Definition 5.2. [25] ¯ωC(1)[0,L] is named as an upper solution of (5.1) if

    {¯ω(ξ)(ξ,¯ω(ξ)),for eachξ[0,L],¯ω(0)¯ω(L).

    We now present the outcome, insuring a solution to Problem (5.1).

    Theorem 5.1. Along with the problem (5.1), if λ,α>0 with

    α(2λ(eλL1)L(eλL+1))12,

    such that for l,mRwithlm,

    0(ξ,l)+λl[(ξ,m)+λm]αln[(lm)2+1]. (5.2)

    If (5.1) admits a lower solution, then it possesses a unique solution.

    Proof. Rewrite Problem (5.1) as

    {ω(ξ)+λω(ξ)=(ξ,ω(ξ))+λω(ξ),forξ[0,L],ω(0)=ω(L), (5.3)

    Equation (5.3) is equivalent to the integral equation

    ω(ξ)=L0ϝ(ξ,τ)[(τ,ω(τ))+λω(τ)]dτ, (5.4)

    where the Green function is

    ϝ(ξ,τ)={eλ(L+τξ)eλL1,0τ<ξL;eλ(τξ)eλL1,0ξ<τL.

    Denote U:=C[0,L]. Define a function Q:UU by

    (Qω(ξ)=L0ϝ(ξ,τ)[(τ,ω(τ))+λω(τ)]dτ, (5.5)

    Thus, θU is a fixed point of Q if and only if, θC1[0,L] forms a solution of (5.4), and hence, of (5.1). On U, endow a relation

    S={(ω,ν)U×U:ω(ξ)ν(ξ),ξ[0,L]}; (5.6)

    and a metric

    σ(ω,ν)=supξ[0,L]|(ω(ξ)ν(ξ)|,ω,νU. (5.7)

    Now, we check all the presumptions of Theorem 3.2.

    (ⅰ) Assuming that ω_(ξ) is a lower solution for (5.1). We conclude

    ω_(ξ)+λω_(ξ)(ξ,ω_(ξ))+λω_(ξ),forξ[0,L].

    Taking the product with eλξ, we attain

    (ω_(ξ)eλξ)[(ξ,ω_(ξ))+λω_(ξ)]eλξ,forξ[0,L],

    or

    ω_(ξ)eλξω_(0)+ξ0[(τ,ω_(τ))+λω_(τ)]eλτdτ,forξ[0,L].

    As ω_(0)ω_(L), the last inequality gives us

    ω_(0)eλξω_(L)eλLω_(0)+L0[(τ,ω_(τ))+λω_(τ)]eλτdτ,

    so that

    ω_(0)L0eλτeλL1[(τ,ω_(τ))+λω_(τ)]dτ,

    which, using (5.6), gives rise

    ω_(ξ)eλξξ0eλ(L+τ)eλL1[(τ,ω_(τ))+λω_(τ)]dτ+LξeλτeλL1[(τ,ω_(τ))+λω_(τ)]dτ,

    and consequently,

    ω_(ξ)ξ0eλ(L+τξ)eλL1dτ+ξ0eλ(τξ)eλL1[(τ,ω_(τ))+λω_(τ)]dτ=L0ϝ(ξ,τ)[(τ,ω_(τ))+λω_(τ)]dτ=(Qω_)(ξ),forξ[0,L].

    (ⅱ) Take (ω,ν)S. Then, for each τ[0,L], we have ω(τ)ν(τ). Consequently, using (5.2), we obtain

    (τ,ω(τ)+λω(τ)(τ,ν(τ))+λν(τ),

    which yields that

    (Qω(ξ)=L0ϝ(ξ,τ)[(τ,ω(τ))+λω(τ)]dτL0ϝ(ξ,τ)[(τ,ω(τ))+λω(τ)]dτ=(Qν(ξ).

    It follows that (Qω,Qν)S so that S is Q-closed. Also, S being transitive is locally Q-transitive.

    (ⅲ) The MS (U,σ) being complete is S-complete.

    (ⅳ) Let {ωn}U be S-preserving sequence converging to ωU. Hence, for every ξ[0,L],{ωn(ξ)} is an increasing sequence in R converging to ω(ξ), and so, nN and τ[0,L], we conclude ωn(ξ)ω(ξ). Again, due to (5.6), it follows that (ωn,ω)S,nN. Thus, S is σ-self-closed.

    (ⅴ) Take (ω,ν)SQ. Then, for each τ[0,L], we attain ω(τ)ν(τ). Consequently, using (5.2), we obtain

    σ(ω,Qν)=supξ[0,L]|(Qω(ξ)(Qν(ξ)|=supξ[0,L]((Qν(ξ)(Qω(ξ))=supξ[0,L]L0ϝ(ξ,τ)[(τ,ν(τ))+λν(τ)(τ,ω(τ))λω(τ)]dτsupξ[0,L]L0ϝ(ξ,τ)αln[(ω(τ)ν(τ))2+1]dτ.

    Employing Cauchy-Schwarz inequality, we attain

    Loϝ(ξ,τ)αln[(ω(τ)ν(τ))2+1]dτ(L0ϝ(ξ,τ)2dτ)12(L0α2ln[(ω(τ)ν(τ))2+1])12.

    The first integral reduces to

    L0ϝ(ξ,τ)2dτ=ξ0ϝ(ξ,τ)2dτ+Lξϝ(ξ,τ)2dτ=ξ0e2λ(L+τξ)(eλL1)2dτ+Lξe2λ(τξ)(eλL1)2dτ=12λ(eλL1)2e2λL1=eλL+12λ(eλL1).

    The second integral can be estimated as

    L0α2ln[(ω(τ)ν(τ))2]α2ln[||ωv||2+1]L=α2ln[σ(ω,ν)2+1]L.

    Taking into account, we conclude

    σ(Qω,Qν)supξ[0,L](eλL+12λ(eλL1))12(α2ln[σ(ω,ν)2+1]L)12=(eλL+12λ(eλL1))12αL(ln[σ(ω,ν)2+1])12,

    and from the last inequality, we obtain

    σ(Qω,Qν2(eλL+12λ(eλL1))α2Lln[σ(ω,ν)2+1],

    or equivalently,

    2λ(eλL1)σ(Qω,Qν2(eλL+1)α2Lln[σ(ω,ν)2+1].

    Using the hypothesis:

    L(2λ(eλL1)QeλL+1)12,

    the last inequality reduces to

    2λ(eλL1)σ(Qω,Qν22λ(eλL1)ln[σ(ω,ν)2+1],

    and hence,

    σ(Qω,Qν2ln[σ(ω,ν)2+1].

    Put ψ(ξ)=ξ2 and φ(ξ)=ln(ξ2+1). Then, we have (ψ,φ)Ω. Thus, (5.7) reduces to

    ψ(d(Qω,Qν)φ(σ(ω,ν)),(ω,ν)SQ.

    Let ω,νU be arbitrary. Then, one has ϑ:=max{Qω,Qν}U. As (Qω,ϑ)S and (Qν,ϑ)S, {Qω,ϑ,Qν} is a path in Ss between Q(ω) and Q(ν). Thus, Q(U) is Ss-connected, and so by Theorem 3.2, Q owns a unique fixed point, which forms the unique solution of Problem (5.1).

    Intending to illustrate Theorem 5.1, we consider the following numerical example.

    Example 5.1. Let (ξ,ω(ξ))=cosξ for 0ξπ; then is a continuous function. Note that ω_=0 is a lower solution for ω(ξ)=cosξ. Therefore, Theorem 5.1 can be applied for the given problem, and hence, ω(ξ)=sinξ forms the unique solution.

    We investigated metrical fixed-point findings for a relational contraction map under generalized contraction via a pair of test functions, which, under the preordered (reflexive and transitive) relation, deduce the corresponding outcomes of Górnicki [11]. To demonstrate our outcomes, we furnished a few examples. From an application point of view, we discussed an existence and uniqueness theorem for certain BVP under the availability of a lower solution. Analogously, we can also study the existence and uniqueness of the BVPs whenever an upper solution exists. As a future plane, we can improve our outcomes to a couple of self-maps by establishing coincidence and common fixed point theorems.

    Abdul Wasey: Methodology, original draft writing, conceptualization, editing, funding; Wan Ainun Mior Othman: Review, supervision, funding; Esmail Alshaban: Funding, reviewing, and editing; Kok Bin Wong: Review, supervision, funding; Adel Alatawi: Funding, reviewing and editing. All authors have read and approved the final version of the manuscript for publication.

    The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

    We would like to express our sincere gratitude to the four referees for their thoughtful and constructive critiques. Their insightful comments, detailed observations, and valuable suggestions played a significant role in helping us refine our arguments and ultimately improve the overall quality of our work.

    The authors declare no conflict of interest.



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