Nonlinear Almost Relational Contractions via a Triplet of Test Functions and Applications to Second-Order Ordinary Differential Equations

Abstract

After the introduction of the relation-theoretic contraction principle, the branch of metric fixed-point theory has attracted much attention in this direction, and various fixed-point results have been proven in the framework of relational metric space via different approaches. The aim of this article is to establish some fixed-point outcomes in the framework of relational metric space verifying a generalized nonlinear contraction utilizing three test functions $\mathrm{\Phi }$, $\mathrm{\Psi }$ and $\mathrm{\Theta }$ satisfying the appropriate characteristics. The findings obtained herein expand, sharpen, improve, modify and unify a few well-known findings. To demonstrate the utility of our outcomes, several examples are furnished. We utilized our outcomes to investigate a unique solution of second-order ordinary differential equations prescribed with specific boundary conditions.

1. Introduction

Berinde [1] put out a creative expansion of the BCP in 2004 which is frequently described as “almost contraction”.

Definition 1

([1]). A map L on an MS $(U,\zeta )$ is referred to as almost contraction if ∃ $0<\kappa <1$ and $\mathsf{\ell }\in {\mathbb{R}}^{+}$ verifying

$$\zeta (L\mathsf{u},L\mathsf{w})\le \kappa ·\zeta (\mathsf{u},\mathsf{w})+\mathsf{\ell }·\zeta (\mathsf{u},L\mathsf{w}),\forall \mathsf{u},\mathsf{w}\in U.$$

The symmetric feature of $\zeta$ enables the above contraction condition to be the same as the following one:

$$\zeta (\mathrm{L}\mathsf{u},\mathrm{L}\mathsf{w})\le \kappa ·\zeta (\mathsf{u},\mathsf{w})+\mathsf{\ell }·\zeta (\mathsf{w},\mathrm{L}\mathsf{u}),\forall \mathsf{u},\mathsf{w}\in U.$$

Theorem 1

([1]). An almost contraction map on a CMS possesses a fixed point.

Though an almost contraction L is not necessarily continuous, it is, however, continuous on the fixed-point set of L (c.f. [2]). Including the ordinary contraction, a few existing generalized contractions are also covered by almost contraction. The concept of almost contractions has been developed by a number of researchers; for instance, see [3,4,5]. Khan [6], Khan et al. [7] and Algehyne and Khan [8] presented some fixed-point findings for almost contractions in the framework of relational MS.

The following class of almost contractions was proposed by Babu et al. [9] to investigate the uniqueness theorem:

Definition 2

([9]). A mapping L on an MS $(U,\zeta )$ is named strict almost contraction if ∃ $0<\kappa <1$ and $\mathsf{\ell }\in {\mathbb{R}}^{+}$ enjoying

$$\zeta (L\mathsf{u},L\mathsf{w})\le \kappa ·\zeta (\mathsf{u},\mathsf{w})+\mathsf{\ell }·min\{\zeta (\mathsf{u},L\mathsf{u}),\zeta (\mathsf{w},L\mathsf{w}),\zeta (\mathsf{u},L\mathsf{w}),\zeta (\mathsf{w},L\mathsf{u})\},\forall \mathsf{u},\mathsf{w}\in U.$$

Each strict almost contraction is almost contraction. But the converse is not valid, as shown by Example $2.6$ [9].

Theorem 2

([9]). Any strict almost contraction map on a CMS possesses a unique fixed point.

Alam and Imdad [10] proposed an exceptional and readily apparent variation of the BCP, while MS comprises a BR and the map keeps this BR. Afterward, Alam and Imdad [11] presented coincidence and common fixed-point outcomes in the framework of relational MS (see also [12]). Plenty of investigators have further developed and improved the relation-theoretic contraction principle [10] by implementing various contractivity requirements, (cf. [13,14,15,16,17,18,19,20]). These relational contractions are more comprehensive than Banach contractions since they are applied to elements that are linked via BR. These findings are all derived from their classical fixed-point counterparts under universal BR. These outcomes are deployed for recognizing certain kinds of BVPs.

The following contractivity criterion was introduced by Yan et al. [21] in 2012:

$$\mathrm{\Phi }(\zeta (\mathrm{L}\mathsf{u},\mathrm{L}\mathsf{w}))\le \mathrm{\Psi }(\zeta (\mathsf{u},\mathsf{w})).$$

(1)

It should be noted that both test functions meet the compatibility criteria: $\mathrm{\Psi }(\mathsf{s})<\mathrm{\Phi }(\mathsf{s})$, $\forall \mathsf{s}>0$. The fixed-point findings of Yan et al. [21] were improved by Alsulami et al. [22] and Su [23] by altering the characteristics of the test functions used in the contraction criterion (1). Subsequent to this, Sawangsup and Sintunavarat [24] demonstrated several outcomes in relational MS by employing a specific pair of test functions and using them to investigate a nonlinear matrix equation.

In this study, we study a generalized nonlinear contraction that encompasses the contraction conditions involved in the outcomes of Khan [6] and Sawangsup and Sintunavarat [24] and utilize and examine the same specific fixed-point outcomes in the context of relational metric space. In regard to our existence findings, the underlying BR must be locally L-transitive and L-closed. To validate the uniqueness outcome, however, an additional premise is required (i.e., the image of the map needs to be $\mathbf{S}$-directed). Our obtained findings expand, sharpen, improve, modify and unify a few well-known outcomes, especially those of Khan [6], Khan et al. [7], Alam et al. [12], Su [23], Sawangsup and Sintunavarat [24], Alfuraidan [25], Algehyne et al. [26] and similar authors. We give several exemplary instances to illustrate the key findings. We demonstrate our outcomes by resolving an observation about the existence and uniqueness of certain BVPs linked to a second-order ODE.

2. Preliminaries

As usual, $\mathbb{N}$ denotes the set of natural numbers, ${\mathbb{N}}_0$ denotes the set of whole numbers, $\mathbb{Q}$ denotes the set of rational numbers, ${\mathbb{Q}}^{+}$ denotes the set of nonnegative rational numbers, $\mathbb{R}$ denotes the set of real numbers, and ${\mathbb{R}}^{+}$ denotes the set of nonnegative real numbers. A BR $\mathbf{S}$ on a set U is defined to be a subset of ${\mathrm{U}}^2$. In subsequent definitions, U refers to the ambient set, $\zeta$ refers to the metric on U, $\mathbf{S}$ refers to the BR on U and $\mathrm{L}:\mathrm{U}\to \mathrm{U}$ refers to a map. We can say the following:

Definition 3

([10]). $\mathsf{u},\mathsf{w}\in U$ are $\mathbf{S}$-comparative, indicated as $[\mathsf{u},\mathsf{w}]\in \mathbf{S}$, if

$$(\mathsf{u},\mathsf{w})\in \mathbf{S}or(\mathsf{w},\mathsf{u})\in \mathbf{S}.$$

Definition 4

([27]). ${\mathbf{S}}^{-1}:=\{(\mathsf{u},\mathsf{w})\in U^2:(\mathsf{w},\mathsf{u})\in \mathbf{S}\}$ is the inverse of $\mathbf{S}$.

Definition 5

([27]). The BR ${\mathbf{S}}^s:=\mathbf{S}\cup {\mathbf{S}}^{-1}$ is the symmetric closure of $\mathbf{S}$.

Proposition 1

([10]). $(\mathsf{u},\mathsf{w})\in {\mathbf{S}}^s⟺[\mathsf{u},\mathsf{w}]\in \mathbf{S}.$

Proposition 2

([28]). If $\mathbf{S}$ is L-closed, then for every $\mathsf{n}\in {\mathbb{N}}_0$, $\mathbf{S}$ is $L^{\mathsf{n}}$-closed.

Definition 6

([27]). A BR on $M\subseteq U$ defined by

$${\mathbf{S}|}_M:=\mathbf{S}\cap M^2$$

is the restriction of $\mathbf{S}$ on M.

Definition 7

([10]). $\mathbf{S}$ is L-closed if

$$(L\mathsf{u},L\mathsf{w})\in \mathbf{S},\forall \mathsf{u},\mathsf{w}\in U;(\mathsf{u},\mathsf{w})\in \mathbf{S}.$$

Proposition 3

([28]). $\mathbf{S}$ is $L^{\mathsf{n}}$-closed when $\mathbf{S}$ is L-closed.

Definition 8

([10]). A sequence $\{{\mathsf{u}}_{\mathsf{n}}\}\subset U$ is $\mathbf{S}$-preserving if $({\mathsf{u}}_{\mathsf{n}},{\mathsf{u}}_{\mathsf{n}+1})\in \mathbf{S}$ ∀ $\mathsf{n}\in {\mathbb{N}}_0$.

Definition 9

([11]). $(U,\zeta )$ is $\mathbf{S}$-complete MS if each $\mathbf{S}$-preserving Cauchy sequence remains convergent.

Definition 10

([11]). L is $\mathbf{S}$-continuous if for every $\mathsf{u}\in U$ and for every $\mathbf{S}$-preserving sequence $\{{\mathsf{u}}_{\mathsf{n}}\}\subset U$ along with ${\mathsf{u}}_{\mathsf{n}}\stackrel{\zeta }{⟶}\mathsf{u}$,

$$L({\mathsf{u}}_{\mathsf{n}})\stackrel{\zeta }{⟶}L(\mathsf{u}).$$

Definition 11

([10]). $\mathbf{S}$ is ζ-self-closed when the limit of an $\mathbf{S}$-preserving convergent sequence in $(U,\zeta )$ is $\mathbf{S}$-comparative with each term of a subsequence.

Definition 12

([29]). A subset $M\subseteq U$ is $\mathbf{S}$-directed if for each pair $\mathsf{u},\mathsf{w}\in M$ admits an element $\mathsf{v}\in U$ verifying $(\mathsf{u},\mathsf{v})\in \mathbf{S}$ and $(\mathsf{w},\mathsf{v})\in \mathbf{S}$.

Definition 13

([28]). $\mathit{S}$ is locally L-transitive if for any $\mathit{S}$-preserving sequence $\{{\mathsf{w}}_{\mathsf{n}}\}\subset L(U)$, ${\mathit{S}|}_M$ retains transitivity, where $M=\{{\mathsf{w}}_{\mathsf{n}}:\mathsf{n}\in \mathbb{N}\}$.

Definition 14

([30]). For $\mathsf{k}\in \mathbb{N}/\{1\}$, $\mathit{S}$ is $\mathsf{k}$-transitive if for all ${\mathsf{u}}_0,{\mathsf{u}}_1,\dots ,{\mathsf{u}}_{\mathsf{k}}\in U$,

$$({\mathsf{u}}_{i-1},{\mathsf{u}}_i)\in \mathit{S}\mathrm{for}\mathrm{each}i(1\le i\le \mathsf{k})\Rightarrow ({\mathsf{u}}_0,{\mathsf{u}}_{\mathsf{k}})\in \mathit{S}.$$

Thus far, the 2-transitive BR means the usual transitive BR.

Definition 15

([31]). $\mathit{S}$ is finitely transitive if we can determine $\mathsf{k}\in \mathbb{N}/\{1\}$ for which $\mathit{S}$ is $\mathsf{k}$-transitive.

Definition 16

([12]). $\mathit{S}$ is locally finitely L-transitive if for any $\mathit{S}$-preserving sequence $\{{\mathsf{w}}_{\mathsf{n}}\}\subset L(U)$, ${\mathit{S}|}_M$ remains finitely transitive, where $M=\{{\mathsf{w}}_{\mathsf{n}}:\mathsf{n}\in \mathbb{N}\}$.

Clearly, finitely transitive⟹ locally finitely L-transitive. Also, locally L-transitive ⟹ locally finitely L-transitive.

Definition 17

([32]). A sequence $\{{\mathsf{u}}_{\mathsf{n}}\}$ in an MS $(U,\zeta )$ is semi-Cauchy if

$$\underset{\mathsf{n}\to \infty }{lim}\zeta ({\mathsf{u}}_{\mathsf{n}},{\mathsf{u}}_{\mathsf{n}+1})=0.$$

Each Cauchy sequence is semi-Cauchy, but not conversely.

Lemma 1

([30]). Let $\{{\mathsf{u}}_{\mathsf{n}}\}$ be a non-Cauchy sequence in an MS $(U,\zeta )$. Then ∃${\epsilon }_0>0$ and subsequences $\{{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}}\}$ and $\{{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}}\}$ of $\{{\mathsf{u}}_{\mathsf{n}}\}$ along with the characteristics

(i)

$\mathsf{j}\le {\mathsf{m}}_{\mathsf{j}}<{\mathsf{n}}_{\mathsf{j}}forall\mathsf{j}\in \mathbb{N}$;

(ii)

$\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}})\ge {\epsilon }_{0}forall\mathsf{j}\in \mathbb{N}$;

(iii)

$\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\nu }_{\mathsf{j}}})<{\epsilon }_0$ for all ${\nu }_{\mathsf{j}}\in \{{\mathsf{m}}_{\mathsf{j}}+1,{\mathsf{m}}_{\mathsf{j}}+2,\dots ,{\mathsf{n}}_{\mathsf{j}}-2,{\mathsf{n}}_{\mathsf{j}}-1\}$.

Additionally, if $\{{\mathsf{u}}_{\mathsf{n}}\}$ is semi-Cauchy, then

$$\underset{\mathsf{j}\to \infty }{lim}\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}+\eta })={\epsilon }_0,\forall \eta \in {\mathbb{N}}_0.$$

Lemma 2

([31]). Let U be a set with a BR $\mathit{S}$. If $\{{\mathsf{u}}_{\mathsf{n}}\}\subset U$ is an $\mathit{S}$-preserving sequence and $\mathit{S}$ is a $\mathsf{k}$-transitive on $M=\{{\mathsf{u}}_{\mathsf{n}}:\mathsf{n}\in {\mathbb{N}}_0\}$, then

$$({\mathsf{u}}_{\mathsf{n}},{\mathsf{u}}_{\mathsf{n}+1+ϵ(\mathsf{k}-1)})\in \mathit{S},\forall \mathsf{n},ϵ\in {\mathbb{N}}_0.$$

We will use $\mathsf{Ϝ}$ to indicate the family of the pair $(\mathrm{\Phi },\mathrm{\Psi })$ of functions $\mathrm{\Phi },\mathrm{\Psi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ that possess the subsequent axioms:

(Ϝ₁)

$\mathrm{\Psi }(\mathsf{s})<\mathrm{\Phi }(\mathsf{s}),\forall \mathsf{s}>0$;

(Ϝ₂)

$\mathrm{\Phi }$ is an increasing, lower semicontinuous function and ${\mathrm{\Phi }}^{-1}(0)=0$;

(Ϝ₃)

$\mathrm{\Psi }$ is a right upper semicontinuous function and $\mathrm{\Psi }(0)=0$.

Turinici [32] (later Alfuraidan et al. [25]) proposed the following class of test functions to formulate a nonlinear framework of almost contraction.

$$\mathcal{G}=\{\mathrm{\Theta }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}:\underset{\mathsf{s}\to 0^{+}}{lim}\mathrm{\Theta }(\mathsf{s})=\mathrm{\Theta }(0)=0\}.$$

Proposition 4.

For $(\mathrm{\Phi },\mathrm{\Psi })\in \mathsf{Ϝ}$ and $\mathrm{\Theta }\in \mathcal{G}$, (A) and (B) are equivalent:

(A)

$\mathrm{\Phi }(\zeta (L\mathsf{u},L\mathsf{w}))\le \mathrm{\Psi }(\zeta (\mathsf{u},\mathsf{w}))+min\{\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{u}))\},$

$\forall \mathsf{u},\mathsf{w}\in Uwith(\mathsf{u},\mathsf{w})\in \mathit{S}.$

(B)

$\mathrm{\Phi }(\zeta (L\mathsf{u},L\mathsf{w}))\le \mathrm{\Psi }(\zeta (\mathsf{u},\mathsf{w}))+min\{\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{u}))\},$

$\forall \mathsf{u},\mathsf{w}\in Uwith[\mathsf{u},\mathsf{w}]\in \mathit{S}.$

Proof. 

The implication (B)⇒(A) is trivial. Conversely, suppose that (A) holds. Assume that $\mathsf{u},\mathsf{w}\in U$ with $[\mathsf{u},\mathsf{w}]\in \mathbf{S}$. Then, in case $(\mathsf{u},\mathsf{w})\in \mathbf{S}$, (A) yields (B). Otherwise, we conclude $(\mathsf{w},\mathsf{u})\in \mathbf{S}$. In this case, by symmetry of metric $\zeta$ and (A), we find

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(\zeta (\mathrm{L}\mathsf{u},\mathrm{L}\mathsf{w}))& =& \mathrm{\Phi }(\zeta (\mathrm{L}\mathsf{w},\mathrm{L}\mathsf{u}))\hfill \\ & \le & \mathrm{\Psi }(\zeta (\mathsf{w},\mathsf{u}))+min\{\mathrm{\Theta }(\zeta (\mathsf{w},\mathrm{L}\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{u},\mathrm{L}\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{w},\mathrm{L}\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{u},\mathrm{L}\mathsf{w}))\}\hfill \\ & =& \mathrm{\Psi }(\zeta (\mathsf{u},\mathsf{w}))+min\{\mathrm{\Theta }(\zeta (\mathsf{u},\mathrm{L}\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{w},\mathrm{L}\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{u},\mathrm{L}\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{w},\mathrm{L}\mathsf{u}))\}.\hfill \end{array}$$

It follows that (A)⇒(B). □

3. Main Results

We reveal the following findings on fixed points for relational almost nonlinear contraction.

Theorem 3.

Let $(U,\zeta )$ be an MS along with a BR $\mathit{S}$, and $L:U\to U$ be a map. Also,

(i)

$(U,\zeta )$ is $\mathit{S}$-complete;

(ii)

$\exists {\mathsf{u}}_0\in U$ with $({\mathsf{u}}_0,L{\mathsf{u}}_0)\in \mathit{S}$;

(iii)

$\mathit{S}$ is locally finitely L-transitive and L-closed;

(iv)

L is $\mathit{S}$-continuous, or $\mathit{S}$ is ζ-self-closed;

(v)

∃$(\mathrm{\Phi },\mathrm{\Psi })\in \mathsf{Ϝ}$ and $\mathrm{\Theta }\in \mathcal{G}$ that enjoys

$$\begin{array}{c}\hfill \mathrm{\Phi }(\zeta (L\mathsf{u},L\mathsf{w})\le \mathrm{\Psi }(\zeta (\mathsf{u},\mathsf{w}))+min\{\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{u}))\},\\ \hfill \forall (\mathsf{u},\mathsf{w})\in \mathit{S}.\end{array}$$

Then, L has at least one fixed point.

Proof. 

The proof will be accomplished in five steps.

Step 1. 

Starting with ${\mathsf{u}}_0\in \mathrm{U}$, define the following sequence $\{{\mathsf{u}}_{\mathsf{n}}\}\subset \mathrm{U}$:

$${\mathsf{u}}_{\mathsf{n}}:=\mathrm{L}({\mathsf{u}}_{\mathsf{n}-1})={\mathrm{L}}^{\mathsf{n}}({\mathsf{u}}_0),\forall \mathsf{n}\in \mathbb{N}.$$

(2)

From $(ii)$, L-closedness of $\mathbf{S}$ and Proposition 2, we attain

$$({\mathrm{L}}^{\mathsf{n}}{\mathsf{u}}_0,{\mathrm{L}}^{\mathsf{n}+1}{\mathsf{u}}_0)\in \mathbf{S},$$

which, according to (2), reduces to

$$({\mathsf{u}}_{\mathsf{n}},{\mathsf{u}}_{\mathsf{n}+1})\in \mathbf{S},\forall \mathsf{n}\in {\mathbb{N}}_0.$$

(3)

Therefore, $\{{\mathsf{u}}_{\mathsf{n}}\}$ is $\mathbf{S}$-preserving.

Step 2. 

Denote ${\zeta }_{\mathsf{n}}:=\zeta ({\mathsf{u}}_{\mathsf{n}},{\mathsf{u}}_{\mathsf{n}+1})$, $\mathsf{n}\in {\mathbb{N}}_0$. If there is ${\mathsf{n}}_0\in {\mathbb{N}}_0$ verifying ${\zeta }_{{\mathsf{n}}_0}=\zeta ({\mathsf{u}}_{{\mathsf{n}}_0},{\mathsf{u}}_{{\mathsf{n}}_0+1})=0$, then from (2), we conclude $\mathrm{L}({\mathsf{u}}_{{\mathsf{n}}_0})={\mathsf{u}}_{{\mathsf{n}}_0}$. Hence, ${\mathsf{u}}_{{\mathsf{n}}_0}$ is a fixed point of L and so the task is complete. In the case of ${\zeta }_{\mathsf{n}}>0,\forall \mathsf{n}\in {\mathbb{N}}_0$, we will then continue with Step 3.

Step 3. 

We will exhibit that $\{{\mathsf{u}}_{\mathsf{n}}\}$ is semi-Cauchy, i.e., $\underset{\mathsf{n}\to \infty }{lim}\zeta ({\mathsf{u}}_{\mathsf{n}},{\mathsf{u}}_{\mathsf{n}+1})=0$. Using $(v)$, (2) and (3), we find

$$\begin{array}{ccc}\hfill \mathrm{\Phi }({\zeta }_{\mathsf{n}})& =& \mathrm{\Phi }(\zeta ({\mathsf{u}}_{\mathsf{n}},{\mathsf{u}}_{\mathsf{n}+1}))=\mathrm{\Phi }(\zeta (\mathrm{L}{\mathsf{u}}_{\mathsf{n}-1},\mathrm{L}{\mathsf{u}}_{\mathsf{n}}))\hfill \\ & \le & \mathrm{\Psi }(\zeta ({\mathsf{u}}_{\mathsf{n}-1},{\mathsf{u}}_{\mathsf{n}}))+min\{\mathrm{\Theta }(\zeta ({\mathsf{u}}_{\mathsf{n}-1},{\mathsf{u}}_{\mathsf{n}})),\mathrm{\Theta }(\zeta ({\mathsf{u}}_{\mathsf{n}},{\mathsf{u}}_{\mathsf{n}+1})),\mathrm{\Theta }(\zeta ({\mathsf{u}}_{\mathsf{n}-1},{\mathsf{u}}_{\mathsf{n}+1})),\mathrm{\Theta }(0)\},\hfill \end{array}$$

which, employing the characteristic of $\mathrm{\Theta }$, reduces to

$$\begin{array}{c}\hfill \mathrm{\Phi }({\zeta }_{\mathsf{n}})\le \mathrm{\Psi }({\zeta }_{\mathsf{n}-1}),\forall \mathsf{n}\in {\mathbb{N}}_0.\end{array}$$

(4)

From the property (${\mathsf{Ϝ}}_1$), we conclude

$$\mathrm{\Phi }({\zeta }_{\mathsf{n}})\le \mathrm{\Psi }({\zeta }_{\mathsf{n}-1})<\mathrm{\Phi }({\zeta }_{\mathsf{n}-1}).$$

Since $\mathrm{\Phi }$ is monotone-increasing, we conclude

$${\zeta }_{\mathsf{n}}<{\zeta }_{\mathsf{n}-1},\forall \mathsf{n}\in \mathbb{N}.$$

This demonstrates that the real sequence $\{{\zeta }_{\mathsf{n}}\}$ remains decreasing, which is already bounded below by 0. Consequently, ∃ $\tau \ge 0$ verifying ${\zeta }_{\mathsf{n}}\stackrel{\mathbb{R}}{⟶}{\tau }^{+}$ as $\mathsf{n}\to \infty$.

Proceeding with the limit inferior in (4) and by lower semicontinuity of $\mathrm{\Phi }$ and right upper semicontinuity of $\mathrm{\Psi }$, we find

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(\tau )& \le & \underset{\mathsf{n}\to \infty }{lim\; inf}\mathrm{\Phi }({\zeta }_{\mathsf{n}})\hfill \\ & \le & \underset{\mathsf{n}\to \infty }{lim\; inf}\mathrm{\Psi }({\zeta }_{\mathsf{n}})\hfill \\ & \le & \underset{\mathsf{n}\to \infty }{lim\; sup}\mathrm{\Psi }({\zeta }_{\mathsf{n}})\hfill \\ & \le & \mathrm{\Psi }(\tau ),\hfill \end{array}$$

which, in view of (${\mathsf{Ϝ}}_1$), yields $\tau =0$. Hence, we conclude

$$\begin{array}{c}\hfill \underset{\mathsf{n}\to \infty }{lim}{\zeta }_{\mathsf{n}}=0.\end{array}$$

(5)

Step 4. 

We will exhibit that $\{{\mathsf{u}}_{\mathsf{n}}\}$ is Cauchy. Let, by contrast, $\{{\mathsf{u}}_{\mathsf{n}}\}$ be not Cauchy. Utilizing Lemma 1, ∃ ${\epsilon }_0>0$ and subsequences $\{{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}}\}$ and $\{{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}}\}$ of $\{{\mathsf{u}}_{\mathsf{n}}\}$ that verify

$$\mathsf{j}\le {\mathsf{m}}_{\mathsf{j}}<{\mathsf{n}}_{\mathsf{j}},\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}})\ge {\epsilon }_0>\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\nu }_{\mathsf{j}}}),\forall \mathsf{j}\in \mathbb{N},{\nu }_{\mathsf{j}}\in \{{\mathsf{m}}_{\mathsf{j}}+1,{\mathsf{m}}_{\mathsf{j}}+2,\dots ,{\mathsf{n}}_{\mathsf{j}}-2,{\mathsf{n}}_{\mathsf{j}}-1\}.$$

By (5) and Lemma 1, we attain

$$\begin{array}{c}\hfill \underset{\mathsf{j}\to \infty }{lim}\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}+\eta })={\epsilon }_0,\mathrm{for}\mathrm{all}\eta \in {\mathbb{N}}_0.\end{array}$$

(6)

By (2), we arrive at $\mathrm{M}:=\{{\mathsf{u}}_{\mathsf{n}}:\mathsf{n}\in {\mathbb{N}}_0\}\subset \mathrm{L}(\mathrm{U})$. By locally finite L-transitivity of $\mathbf{S}$, ∃ $\mathsf{k}\in \mathbb{N}/\{1\}$, for which ${\mathbf{S}|}_{\mathrm{M}}$ retains $\mathsf{k}$-transitivity.

As ${\mathsf{m}}_{\mathsf{j}}<{\mathsf{n}}_{\mathsf{j}}$ and $\mathsf{k}-1>0$, by a division algorithm, we find

$$\begin{array}{cc}& {\mathsf{n}}_{\mathsf{j}}-{\mathsf{m}}_{\mathsf{j}}=(\mathsf{k}-1)({\mathsf{a}}_{\mathsf{j}}-1)+(\mathsf{k}-{\mathsf{b}}_{\mathsf{j}})\\ & {\mathsf{a}}_{\mathsf{j}}-1\ge 0,0\le \mathsf{k}-{\mathsf{b}}_{\mathsf{j}}<\mathsf{k}-1\\ ⟺& \left\{\begin{array}{c}{\mathsf{n}}_{\mathsf{j}}+{\mathsf{b}}_{\mathsf{j}}={\mathsf{m}}_{\mathsf{j}}+1+(\mathsf{k}-1){\mathsf{a}}_{\mathsf{j}}\hfill \\ {\mathsf{a}}_{\mathsf{j}}\ge 1,1<{\mathsf{b}}_{\mathsf{j}}\le \mathsf{k}.\hfill \end{array}\right.\hfill \end{array}$$

As ${\mathsf{b}}_{\mathsf{j}}\in (1,\mathsf{k}]$, the subsequences $\{{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}}\}$ and $\{{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}}\}$ of $\{{\mathsf{u}}_{\mathsf{k}}\}$ (satisfying (6)) may be determined in such a way in which ${\mathsf{b}}_{\mathsf{j}}=\mathsf{b}$ retains a constant. Thus, we conclude

$$\begin{array}{c}\hfill {\mathsf{m}}_{\mathsf{j}}^{\prime }={\mathsf{n}}_{\mathsf{j}}+\mathsf{b}={\mathsf{m}}_{\mathsf{j}}+1+(\mathsf{k}-1){\mathsf{a}}_{\mathsf{j}}.\end{array}$$

(7)

Utilizing (6) and (7), we conclude

$$\begin{array}{c}\hfill \underset{\mathsf{j}\to \infty }{lim}\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})=\underset{\mathsf{j}\to \infty }{lim}\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}+\mathsf{b}})={\epsilon }_0.\end{array}$$

(8)

From triangle inequality, we conclude

$$\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}+1},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1})\le \zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}+1},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}})+\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})+\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1})$$

and

$$\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})\le \zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}+1})+\zeta ({\mathsf{u}}_{{\zeta }_{\mathsf{j}+1}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1})+\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }}).$$

Hence, we find

$$\begin{array}{c}\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})-\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}+1})-\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})\le \zeta ({\mathsf{u}}_{{\zeta }_{\mathsf{j}+1}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1})\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \le \zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}+1},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}})+\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})+\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1}).\end{array}$$

Proceeding with the limit as $\mathsf{j}\to \infty$ and employing (6) and (8), the foregoing inequality implies that

$$\begin{array}{c}\hfill \underset{\mathsf{j}\to \infty }{lim}\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}+1},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1})={\epsilon }_0.\end{array}$$

(9)

Utilizing (7) and Lemma 1, we attain $({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})\in \mathbf{S}.$

This denotes ${\lambda }_{\mathsf{j}}:=\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})$. Applying $(v)$, we find

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}+1},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1}))& =& \mathrm{\Phi }(\zeta (\mathrm{L}{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},\mathrm{L}{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }}))\le \mathrm{\Psi }(\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }}))\hfill \\ & & +min\{\mathrm{\Theta }(\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},\mathrm{L}{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}})),\mathrm{\Theta }(\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }},\mathrm{L}{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})),\mathrm{\Theta }(\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},\mathrm{L}{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})),\mathrm{\Theta }(\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }},\mathrm{L}{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}}))\}\hfill \end{array}$$

so that

$$\mathrm{\Phi }(\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}+1},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1}))\le \mathrm{\Psi }({\lambda }_{\mathsf{j}})+min\{\mathrm{\Theta }({\zeta }_{{\mathsf{m}}_{\mathsf{j}}}),\mathrm{\Theta }({\zeta }_{{\mathsf{m}}_{\mathsf{j}}^{\prime }}),\mathrm{\Theta }(\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }+1})),\mathrm{\Theta }(\zeta ({\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}^{\prime }},{\mathsf{u}}_{{\mathsf{m}}_{\mathsf{j}}+1}))\}.$$

(10)

Using (5) and a characteristic of $\mathrm{\Theta }$, we attain

$${\displaystyle \underset{\mathsf{j}\to \infty }{lim}\mathrm{\Theta }({\zeta }_{{\mathsf{m}}_{\mathsf{j}}})=\underset{\mathsf{j}\to \infty }{lim}\mathrm{\Theta }({\zeta }_{{\mathsf{m}}_{\mathsf{j}}^{\prime }})=\underset{\mathsf{s}\to 0^{+}}{lim}\mathrm{\Theta }(\mathsf{s})=0.}$$

(11)

Moving to the inferior limit in (10) and utilizing (8), (9), (11), lower semicontinuity of $\mathrm{\Phi }$ and right upper semicontinuity of $\mathrm{\Psi }$, we conclude

$$\mathrm{\Phi }({\epsilon }_0)\le \mathrm{\Psi }({\epsilon }_0).$$

This, along with (${\mathsf{Ϝ}}_1$), gives ${\epsilon }_0=0$, a contradiction. Consequently, $\{{\mathsf{u}}_{\mathsf{n}}\}$ is Cauchy. As $\{{\mathsf{u}}_{\mathsf{n}}\}$ is also $\mathbf{S}$-preserving and U is $\mathbf{S}$-complete, ∃$\overline{\mathsf{u}}\in \mathrm{U}$ verifying ${\mathsf{u}}_{\mathsf{n}}\stackrel{\zeta }{⟶}\overline{\mathsf{u}}$.

Step 5. 

We will exhibit that $\overline{\mathsf{u}}$ retains a fixed point of L with the help of hypothesis $(iv)$. Let L be $\mathbf{S}$-continuous. Then, ${\mathsf{u}}_{\mathsf{n}+1}=\mathrm{L}({\mathsf{u}}_{\mathsf{n}})\stackrel{\zeta }{⟶}\mathrm{L}(\overline{\mathsf{u}})$, thereby yielding $\mathrm{L}(\overline{\mathsf{u}})=\overline{\mathsf{u}}$.

When $\mathbf{S}$ is $\zeta$-self-closed, we may deduct a subsequence $\{{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}}\}$ of $\{{\mathsf{u}}_{\mathsf{n}}\}$ which enjoys the property $[{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}},\overline{\mathsf{u}}]\in \mathbf{S},\forall \mathsf{j}\in \mathbb{N}.$ From $(v)$, Proposition 4, $[{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}},\overline{\mathsf{u}}]\in \mathbf{S}$ and a characteristic of $\mathrm{\Theta }$, we find

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(\zeta ({\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}+1},\mathrm{L}\overline{\mathsf{u}}))& =& \zeta (\mathrm{L}{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}},\mathrm{L}\overline{\mathsf{u}})\hfill \\ & \le & \mathrm{\Psi }(\zeta ({\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}},\overline{\mathsf{u}}))+min\{\mathrm{\Theta }(\zeta ({\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}},{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}+1})),\mathrm{\Theta }(0),\mathrm{\Theta }(\zeta ({\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}},\overline{\mathsf{u}})),\mathrm{\Theta }(\zeta (\overline{\mathsf{u}},{\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}+1}))\}\hfill \\ & =& \mathrm{\Psi }(\zeta ({\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}},\overline{\mathsf{u}}))\hfill \end{array}$$

so that

$$\mathrm{\Phi }(\zeta ({\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}+1},\mathrm{L}\overline{\mathsf{u}}))\le \mathrm{\Psi }(\zeta ({\mathsf{u}}_{{\mathsf{n}}_{\mathsf{j}}},\overline{\mathsf{u}})).$$

(12)

Moving to the inferior limit in (12) and employing lower semicontinuity of $\mathrm{\Phi }$ and right upper semicontinuity of $\mathrm{\Psi }$, we find

$$\mathrm{\Phi }(\zeta (\overline{\mathsf{u}},\mathrm{L}\overline{\mathsf{u}}))\le \underset{k\to \infty }{lim\; inf}\mathrm{\Phi }(\zeta ({\mathsf{u}}_{m_k+1},\mathrm{L}\overline{\mathsf{u}}))\le \underset{k\to \infty }{lim\; sup}\mathrm{\Psi }(\zeta ({\mathsf{u}}_{m_k},\overline{\mathsf{u}}))\le \mathrm{\Psi }(0)$$

so that

$$\mathrm{\Phi }(\zeta (\overline{\mathsf{u}},\mathrm{L}\overline{\mathsf{u}}))=\mathrm{\Psi }(0)$$

which, due to the property (${\mathsf{Ϝ}}_3$), yields

$$\mathrm{\Phi }(\zeta (\overline{\mathsf{u}},\mathrm{L}\overline{\mathsf{u}}))=0.$$

Employing the property (${\mathsf{Ϝ}}_2$), the last equation reduces to $\zeta (\overline{\mathsf{u}},\mathrm{L}\overline{\mathsf{u}})=0$ so that $\mathrm{L}(\overline{\mathsf{u}})=\overline{\mathsf{u}}$. Thus, in all, $\overline{\mathsf{u}}$ retains a fixed point as desired. □

Theorem 4.

In contrast to the assumptions of Theorem 3, if $L(U)$ is ${\mathit{S}}^s$-directed, then L possesses a unique fixed point.

Proof. 

By Theorem 3, L admits at least one fixed point. If $\overline{\mathsf{u}}$ and $\overline{\mathsf{w}}$ are two fixed points of L, then

$$\mathrm{L}(\overline{\mathsf{u}})=\overline{\mathsf{u}}\mathrm{and}\mathrm{L}(\overline{\mathsf{w}})=\overline{\mathsf{w}}.$$

(13)

As $\overline{\mathsf{u}},\overline{\mathsf{w}}\in \mathrm{L}(\mathrm{U})$, $\mathrm{L}(\mathrm{U})$ being ${\mathbf{S}}^s$-directed ensures the existence of $\mathsf{v}\in \mathrm{U}$ verifying

$$[\overline{\mathsf{u}},\mathsf{v}]\in \mathbf{S}\mathrm{and}[\overline{\mathsf{w}},\mathsf{v}]\in \mathbf{S}.$$

(14)

This denotes ${\omega }_{\mathsf{n}}:=\zeta (\overline{\mathsf{u}},{\mathrm{L}}^{\mathsf{n}}\mathsf{v})$. By (13), (14) and $(v)$, we conclude

$$\begin{array}{ccc}\hfill \mathrm{\Phi }({\omega }_{\mathsf{n}})=\mathrm{\Phi }(\zeta (\overline{\mathsf{u}},{\mathrm{L}}^{\mathsf{n}}\mathsf{v}))& =& \mathrm{\Phi }(\zeta (\mathrm{L}\overline{\mathsf{u}},\mathrm{L}({\mathrm{L}}^{\mathsf{n}-1}\mathsf{v})))\le \mathrm{\Psi }(\zeta (\overline{\mathsf{u}},{\mathrm{L}}^{\mathsf{n}-1}\mathsf{v}))\hfill \\ & & +min\{\mathrm{\Theta }(0),\mathrm{\Theta }(\zeta ({\mathrm{L}}^{\mathsf{n}-1}\mathsf{v},{\mathrm{L}}^{\mathsf{n}}\mathsf{v})),\mathrm{\Theta }(\zeta (\overline{\mathsf{u}},{\mathrm{L}}^{\mathsf{n}}\mathsf{v})),\mathrm{\Theta }(\zeta ({\mathrm{L}}^{\mathsf{n}-1}\mathsf{v},\overline{\mathsf{u}}))\}\hfill \\ & =& \mathrm{\Psi }({\omega }_{\mathsf{n}-1})\hfill \end{array}$$

i.e.,

$$\mathrm{\Phi }({\omega }_{\mathsf{n}})\le \mathrm{\Psi }({\omega }_{\mathsf{n}-1}).$$

(15)

From the property (${\mathsf{Ϝ}}_1$), we find

$$\mathrm{\Phi }({\omega }_{\mathsf{n}})\le \mathrm{\Psi }({\omega }_{\mathsf{n}-1})<\mathrm{\Phi }({\omega }_{\mathsf{n}-1}).$$

Since $\mathrm{\Phi }$ is monotone-increasing, we conclude

$${\omega }_{\mathsf{n}}<{\omega }_{\mathsf{n}-1},\forall \mathsf{n}\in \mathbb{N}.$$

Thus, real sequence $\{{\omega }_{\mathsf{n}}\}$ is decreasing, which is already bounded below by 0. Consequently, ∃$\vartheta \ge 0$ verifying ${\omega }_{\mathsf{n}}\stackrel{\mathbb{R}}{⟶}{\vartheta }^{+}$ as $\mathsf{n}\to \infty$.

• Moving to the inferior limit in (15) and employing lower semicontinuity of $\mathrm{\Phi }$ and right upper semicontinuity of $\mathrm{\Psi }$, we attain

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(\vartheta )& \le & \underset{\mathsf{n}\to \infty }{lim\; inf}\mathrm{\Phi }({\omega }_{\mathsf{n}})\hfill \\ & \le & \underset{\mathsf{n}\to \infty }{lim\; inf}\mathrm{\Psi }({\omega }_{\mathsf{n}})\hfill \\ & \le & \underset{\mathsf{n}\to \infty }{lim\; sup}\mathrm{\Psi }({\omega }_{\mathsf{n}})\hfill \\ & \le & \mathrm{\Psi }(\vartheta ),\hfill \end{array}$$

which, by utilizing the property axiom (${\mathsf{Ϝ}}_1$), indicates that $\vartheta =0$. Hence, we arrive at

$${\displaystyle \underset{\mathsf{n}\to \infty }{lim}{\omega }_{\mathsf{n}}=\underset{\mathsf{n}\to \infty }{lim}\zeta (\overline{\mathsf{u}},{\mathrm{L}}^{\mathsf{n}}\mathsf{v})=0.}$$

(16)

Similarly, we find

$${\displaystyle \underset{\mathsf{n}\to \infty }{lim}\zeta (\overline{\mathsf{w}},{\mathrm{L}}^{\mathsf{n}}\mathsf{v})=0.}$$

(17)

Employing (16), (17) and triangle inequality, we conclude

$$\zeta (\overline{\mathsf{u}},\overline{\mathsf{w}})=\zeta (\overline{\mathsf{u}},{\mathrm{L}}^{\mathsf{n}}\mathsf{v})+\zeta ({\mathrm{L}}^{\mathsf{n}}\mathsf{v},\overline{\mathsf{w}})\to 0,\mathrm{as}\mathsf{n}\to \infty .$$

Thus, $\overline{\mathsf{u}}=\overline{\mathsf{w}}$; i.e., L enjoys a unique fixed point. □

Remark 1.

In what follows, we list several existing results that are deduced from our outcomes as consequences.

• Setting $\mathrm{\Phi }(\mathsf{s})=\mathsf{s}$ and $\mathrm{\Theta }(\mathsf{s})=\mathsf{\ell }·\mathsf{s}$ (where $\mathsf{\ell }\in {\mathbb{R}}^{+}$), we deduce the corresponding results of Khan et al. [7].
• For $\mathrm{\Phi }(\mathsf{s})=\mathsf{s}$ and $\mathrm{\Psi }(\mathsf{s})=\kappa ·\mathsf{s}$ (where $0<\kappa <1$), our outcomes reduce to the main results of Alfuraidan [25].
• If $\mathbf{S}$ is a partially ordered BR and $\mathrm{\Theta }(\mathsf{s})=0$ in Theorem 3, then we get the main result of Su [23].
• If we set $\mathrm{\Theta }(\mathsf{s})=0$ in Theorem 3, then we get the main result of Algehyne et al. [26].
• Setting $\mathrm{\Phi }(\mathsf{s})=\mathsf{s}$ and $\mathrm{\Theta }(\mathsf{s})=0$, we obtain the corresponding outcomes of Alam et al. [12].

4. Illustrative Examples

We deliver the subsequent instances to illuminate our findings.

Example 1.

Take $U=\mathbb{N}\cup [0,1]$ with the following metric ζ:

$$\zeta (\mathsf{u},\mathsf{w})=\left\{\begin{array}{cc}0,\hfill & if\mathsf{u}=\mathsf{w};\hfill \\ \mathsf{u}+\mathsf{w},\hfill & if\mathsf{u}\ne \mathsf{w}and(\mathsf{u},\mathsf{w})\notin {[0,1]}^2;\hfill \\ |\mathsf{u}-\mathsf{w}|,\hfill & if\mathsf{u}\ne \mathsf{w}and(\mathsf{u},\mathsf{w})\in {[0,1]}^2.\hfill \end{array}\right.$$

Define a BR $\mathbf{S}$ on U by

$$\mathbf{S}=\{(\mathsf{u},\mathsf{w})\in U^2:\mathsf{u}>\mathsf{w}\}.$$

Clearly, $\mathbf{S}$ is a ζ-self-closed BR and $(U,\zeta )$ is an $\mathbf{S}$-complete MS.

Define a map $L:U\to U$ by

$$L(\mathsf{u})=\left\{\begin{array}{cc}\mathsf{u}/2,\hfill & \mathrm{if}\mathsf{u}\in [0,1),\hfill \\ 2/3,\hfill & \mathrm{if}\mathsf{u}\in \mathbb{N}.\hfill \end{array}\right.$$

Here, $\mathbf{S}$, being strictly ordered, is transitive and hence it remains locally finitely L-transitive. Clearly, $\mathbf{S}$ is also an L-closed BR . Moreover, ${\mathsf{u}}_0=1\in U$ (and hence $L({\mathsf{u}}_0)=2/3$) verifies $({\mathsf{u}}_0,L{\mathsf{u}}_0)\in \mathbf{S}$.

Define the pair $(\mathrm{\Phi },\mathrm{\Psi })\in \mathsf{Ϝ}$ of test functions by

$$\mathrm{\Phi }(\mathsf{s})=\left\{\begin{array}{cc}log(5\mathsf{s}+1),\hfill & \mathrm{if}\mathsf{s}\in [0,1]\hfill \\ \hfill & \\ log4\mathsf{s},\hfill & \mathrm{if}\mathsf{s}>1\hfill \end{array}\right.$$

and

$$\mathrm{\Psi }(\mathsf{s})=\left\{\begin{array}{cc}log(3\mathsf{s}+1),\hfill & \mathrm{if}\mathsf{s}\in [0,1]\hfill \\ \hfill & \\ log3\mathsf{s},\hfill & \mathrm{if}\mathsf{s}>1.\hfill \end{array}\right.$$

Also, $\mathrm{\Theta }\in \mathcal{G}$ is taken arbitrarily.

Let $\mathsf{u},\mathsf{w}\in U$ with $(\mathsf{u},\mathsf{w})\in \mathbf{S}$; then, $\mathsf{u}>\mathsf{w}$. Now, the following two cases arise:

Case 1: 

When $\mathsf{u}\in [0,1]$, we conclude

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(\zeta (L\mathsf{u},L\mathsf{w}))& =& log(5\zeta (L\mathsf{u},L\mathsf{w})+1)=log\left({\displaystyle \frac{5}{2}}(\mathsf{u}-\mathsf{w})+1\right)\hfill \\ & <& log(3(\mathsf{u}-\mathsf{w})+1)\hfill \\ & <& \mathrm{\Psi }(\zeta (\mathsf{u},\mathsf{w}))+min\{\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{u}))\}.\hfill \end{array}$$

Case 2: 

If $\mathsf{u}\in \mathbb{N}\setminus \{1\}$, then for $\mathsf{w}\in [0,1)$, we have

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(\zeta (L\mathsf{u},L\mathsf{w}))& =& \mathrm{\Phi }(|2/3-\mathsf{w}/2|)=log(5|2/3-\mathsf{w}/2|+1)\hfill \\ & <& log{\displaystyle \frac{13}{3}}\hfill \\ & <& log(3(\mathsf{u}+\mathsf{w}))\hfill \\ & \le & \mathrm{\Psi }(\zeta (\mathsf{u},\mathsf{w}))+min\{\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{u}))\}.\hfill \end{array}$$

Otherwise, if $\mathsf{w}\in \mathbb{N}$, then we have

$$\begin{array}{ccc}\hfill \mathrm{\Phi }(\zeta (L\mathsf{u},L\mathsf{w}))& =& \mathrm{\Phi }(0)\hfill \\ & \le & \mathrm{\Psi }(\zeta (\mathsf{u},\mathsf{w}))+min\{\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{u},L\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{w},L\mathsf{u}))\}.\hfill \end{array}$$

Thereby, the contraction condition $(v)$ of Theorem 3 is fulfilled. Finally, we can easily show that $L(U)$ is ${\mathbf{S}}^s$-directed. Thus, all the assumptions in Theorems 3 and 4 hold. Consequently, L possesses a unique fixed point ($\overline{\mathsf{u}}=0$).

Example 2.

Take $U={\mathbb{R}}^{+}$ with Euclidean metric ζ. Construct a BR $\mathbf{S}$ on U by $\mathbf{S}:=\le$. Consider the map $L:U\to U$ defined by

$$L(\mathsf{u})=\left\{\begin{array}{c}{\mathsf{u}}^2,\mathrm{if}0\le \mathsf{u}<1/2\hfill \\ 0,\mathrm{if}1/2\le \mathsf{u}\le 1.\hfill \end{array}\right.$$

Clearly, $\mathbf{S}$ is locally finitely L-transitive and an L-closed BR on U. Moreover, the MS $(U,\zeta )$ is $\mathbf{S}$-complete. L however is not $\mathbf{S}$-continuous, yet $\mathbf{S}$ is ζ-self-closed.Define the auxiliary functions

$$\mathrm{\Phi }(\mathsf{s})=\mathsf{s},\mathrm{\Psi }(\mathsf{s})=\mathsf{s}/2\mathrm{and}\mathrm{\Theta }(\mathsf{s})=2\mathsf{s}.$$

Then $(\mathrm{\Phi },\mathrm{\Psi })\in \mathsf{Ϝ}$ and $\mathrm{\Theta }\in \mathcal{G}$. The contraction inequality $(v)$ of Theorem 3 is also met. Similarly, the leftover presumptions of Theorems 3 and 4 are met. It follows that L possesses a unique fixed point ($\overline{\mathsf{u}}=0$).

Example 3.

Take $U={\mathbb{R}}^{+}$ with Euclidean metric ζ. Construct a BR $\mathbf{S}$ on U by $\mathbf{S}:={\mathbb{Q}}^{+}\times {\mathbb{R}}^{+}$. Consider $L:U\to U$ as an identity map. Then, the BR $\mathbf{S}$ is locally finitely L-transitive and L-closed. Also, the MS $(U,\zeta )$ is $\mathbf{S}$-complete and the map L is $\mathbf{S}$-continuous.

Fix $\alpha \ge 1$ and $\beta \in (0,\alpha )$. Define the pair $(\mathrm{\Phi },\mathrm{\Psi })\in \mathsf{Ϝ}$ of auxiliary functions by

$$\mathrm{\Phi }(\mathsf{s})=\left\{\begin{array}{cc}\mathsf{s},\hfill & if\mathsf{s}\in [0,1)\hfill \\ \alpha {\mathsf{s}}^2,\hfill & if\mathsf{s}\ge 1\hfill \end{array}\right.$$

and

$$\mathrm{\Psi }(\mathsf{s})=\left\{\begin{array}{cc}{\mathsf{s}}^2,\hfill & if\mathsf{s}\in [0,1)\hfill \\ \beta \mathsf{s},\hfill & if\mathsf{s}\ge 1.\hfill \end{array}\right.$$

Also, let $\mathrm{\Theta }\in \mathcal{G}$ be arbitrary. Then, inequality $(v)$ of Theorem 3 is also met. Similarly, the leftover presumptions of Theorem 3 are also met.

Herein, $L(U)$ is not $\mathbf{S}$-directed; consequently, Theorem 4 is not applicable for this example. Each point of the domain serves as a fixed point of L.

5. Applications to ODE

Consider the second-order ODE of the form

$$\left\{\begin{array}{c}{\mathsf{u}}^{″}+\mathrm{H}(\vartheta ,\mathsf{u})=0,\vartheta \in [0,1],\\ \mathsf{u}(0)=\mathsf{u}(1)=0.\end{array}\right.$$

(18)

Theorem 5.

Along with the BVP (18), if $H:[0,1]\times \mathbb{R}\to {\mathbb{R}}^{+}$ is continuous and monotonic increasing in second variable and $\exists 0\le \hslash \le 8$ enjoying

$$H(\vartheta ,b)-H(\vartheta ,a)\le \hslash \sqrt{log\left[{(b-a)}^2+1\right]},\forall a,b\in \mathbb{R};a\le b,$$

(19)

then the BVP (18) admits a unique nonnegative solution.

Proof. 

Note that $\mathsf{u}\in {\mathcal{C}}^2[0,1]$ serves as a solution of (18) iff $\mathsf{u}\in \mathcal{C}[0,1]$ solves the equation

$$\mathsf{u}(\vartheta )={\int }_0^1\mathbf{G}(\vartheta ,\tau )H\left(\tau ,\mathsf{u}(\tau )\right)d\tau ,\mathrm{for} \mathrm{each}\vartheta \in [0,1],$$

where in $\mathbf{G}(\vartheta ,\tau )$ is a Green function defined by

$$\mathbf{G}(\vartheta ,\tau )=\left\{\begin{array}{cc}\vartheta (1-\tau ),& 0\le \vartheta \le \tau \le 1\\ \tau (1-\vartheta ),& 0\le \tau \le \vartheta \le 1.\end{array}\right.$$

Consider the cone

$$\mathrm{U}=\left\{\mathsf{u}\in \mathcal{C}[0,1]:\mathsf{u}(\vartheta )\ge 0,\mathrm{for} \mathrm{each}\vartheta \in [0,1]\right\}.$$

On U, consider the following metric $\zeta$:

$$\zeta (\mathsf{u},\mathsf{w})=sup\left\{|\mathsf{u}(\vartheta )-\mathsf{w}(\vartheta )|:\vartheta \in [0,1]\right\}.$$

On U, define the following BR:

$$\mathbf{S}=\left\{(\mathsf{u},\mathsf{w})\in {\mathrm{U}}^2:\mathsf{u}(\vartheta )\le \mathsf{w}(\vartheta ),\mathrm{for} \mathrm{each}\vartheta \in [0,1]\right\}.$$

Let $\mathrm{L}:\mathrm{U}\to \mathrm{U}$ be a map defined by

$$(\mathrm{L}\mathsf{u})(\vartheta )={\int }_0^1\mathbf{G}(\vartheta ,\tau )H(\tau ,\mathsf{u}(\tau ))d\tau ,\forall \mathsf{u}\in \mathrm{U}.$$

We will confirm all the conditions of Theorems 3 and 4.

(i)

Obviously, the MS $(\mathrm{U},\zeta )$ is $\mathbf{S}$-complete.

(ii)

As H and $\mathbf{G}$ are both nonnegative functions, zero operator $0\in \mathrm{U}$ verifies for all $\vartheta \in [0,1]$ that

$$0(\vartheta )=0\le {\int }_0^1\mathbf{G}(\vartheta ,\tau )\mathrm{H}(\tau ,0)d\tau =(\mathrm{L}0)(\vartheta )$$

so that

$$(0,\mathrm{L}0)\in \mathbf{S}.$$

(iii)

$\mathbf{S}$, being a partially ordered BR, is locally finitely L-transitive. Let $\mathsf{u},\mathsf{w}\in \mathrm{U}$ such that $(\mathsf{u},\mathsf{w})\in \mathbf{S}$. Then, for every $\vartheta \in [0,1]$, we have $\mathsf{u}(\vartheta )\le \mathsf{w}(\vartheta )$. Employing the increasing property of H for the second variable, for every $\vartheta \in [0,1]$, we attain

$$(\mathrm{L}\mathsf{u})(\vartheta )={\int }_0^1\mathbf{G}(\vartheta ,\tau )\mathrm{H}(\tau ,\mathsf{u}(\tau ))d\tau \le {\int }_0^1\mathbf{G}(\vartheta ,\tau )\mathrm{H}(\tau ,\mathsf{w}(\tau ))d\tau =(\mathrm{L}\mathsf{w})(\vartheta )$$

thereby implying $(\mathrm{L}\mathsf{u},\mathrm{L}\mathsf{w})\in \mathbf{S}$. Thus, $\mathbf{S}$ is L-closed.

(iv)

Let $\{{\mathsf{u}}_{\mathsf{n}}\}\subset \mathrm{U}$ be an $\mathbf{S}$-preserving sequence converging to $\tilde{\mathsf{u}}\in \mathrm{U}$. Thus, for every $\vartheta \in [0,1]$, we conclude ${\mathsf{u}}_{\mathsf{n}}(\vartheta )\le \tilde{\mathsf{u}}(\vartheta ),\forall \mathsf{n}\in \mathbb{N}$. Hence, $({\mathsf{u}}_{\mathsf{n}},\tilde{\mathsf{u}})\in \mathbf{S},\forall \mathsf{n}\in \mathbb{N}$ so that $\mathbf{S}$ is $\zeta$-self-closed.

(v)

Let $\mathsf{u},\mathsf{w}\in \mathrm{U}$ such that $(\mathsf{u},\mathsf{w})\in \mathbf{S}$. Thus, for every $\vartheta \in [0,1]$, we conclude $\mathsf{u}(\vartheta )\le \mathsf{w}(\vartheta )$. Employing (19), we attain

$$\begin{array}{cc}\hfill \zeta (\mathrm{L}\mathsf{w},\mathrm{L}\mathsf{u})& =\underset{\vartheta \in [0,1]}{sup}\left|(\mathrm{L}\mathsf{w})(\vartheta )-(\mathrm{L}\mathsf{u})(\vartheta )\right|\hfill \\ \hfill & =\underset{\vartheta \in [0,1]}{sup}\left((\mathrm{L}\mathsf{w})(\vartheta )-(\mathrm{L}\mathsf{u})(\vartheta )\right)\hfill \\ \hfill & =\underset{\vartheta \in [0,1]}{sup}{\int }_0^1\mathbf{G}(\vartheta ,\tau )(\mathrm{H}(\tau ,\mathsf{w}(\tau ))-\mathrm{H}(\tau ,\mathsf{u}(\tau )))d\tau \hfill \\ \hfill & \le \underset{\vartheta \in [0,1]}{sup}{\int }_0^1\mathbf{G}(\vartheta ,\tau )\hslash \sqrt{log\left[{(\mathsf{w}-\mathsf{u})}^2+1\right]}\hfill \\ \hfill & \le \underset{\vartheta \in [0,1]}{sup}{\int }_0^1\mathbf{G}(\vartheta ,\tau )\hslash \sqrt{\left[log\left|\right|\mathsf{w}-{\mathsf{u}\left|\right|}^2+1\right]}d\tau \hfill \\ \hfill & =\hslash \sqrt{log\left[\left|\right|\mathsf{w}-{\mathsf{u}\left|\right|}^2+1\right]}\underset{\vartheta \in [0,1]}{sup}{\int }_0^1\mathbf{G}(\vartheta ,\tau )d\tau .\hfill \end{array}$$

(20)

It can be easily proven that

$${\int }_0^1\mathbf{G}(\vartheta ,\tau )d\tau ={\displaystyle \frac{-{\vartheta }^2}{2}}+{\displaystyle \frac{\vartheta }{2}}$$

so that

$$\underset{\vartheta \in [0,1]}{sup}{\int }_0^1\mathbf{G}(\vartheta ,\tau )d\tau ={\displaystyle \frac{1}{8}}.$$

Utilizing the above, (20) becomes

$$\begin{array}{cc}\hfill \zeta (\mathrm{L}\mathsf{w},\mathrm{L}\mathsf{u})& \le {\displaystyle \frac{\hslash }{8}}\sqrt{log\left[\left|\right|\mathsf{w}-{\mathsf{u}\left|\right|}^2+1\right]}\hfill \\ \hfill & \le \sqrt{log\left[\left|\right|\mathsf{w}-{\mathsf{u}\left|\right|}^2+1\right]}(\mathrm{by} \mathrm{the} \mathrm{hypothesis}0\le \hslash \le 8)\hfill \\ \hfill & =\sqrt{log\left[\zeta {(\mathsf{u},\mathsf{w})}^2+1\right]}\hfill \end{array}$$

so that

$$\zeta {(\mathrm{L}\mathsf{w},\mathrm{L}\mathsf{u})}^2\le log\left[\zeta {(\mathsf{u},\mathsf{w})}^2+1\right].$$

This defines $\mathrm{\Phi }(\mathsf{s})={\mathsf{s}}^2$ and $\mathrm{\Psi }(\mathsf{s})=log({\mathsf{s}}^2+1)$. Thus, we attain $(\mathrm{\Phi },\mathrm{\Psi })\in \mathsf{Ϝ}$. Also, let $\mathrm{\Theta }\in \mathcal{G}$ be arbitrary. Then, the foregoing inequality reduces to

$$\mathrm{\Phi }(\zeta (\mathrm{L}\mathsf{u},\mathrm{L}\mathsf{w})\le \mathrm{\Psi }(\zeta (\mathsf{u},\mathsf{w}))+min\{\mathrm{\Theta }(\zeta (\mathsf{u},\mathrm{L}\mathsf{u})),\mathrm{\Theta }(\zeta (\mathsf{w},\mathrm{L}\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{u},\mathrm{L}\mathsf{w})),\mathrm{\Theta }(\zeta (\mathsf{w},\mathrm{L}\mathsf{u}))\}.$$

Therefore, all the hypotheses of Theorem 3 are established. Consequently, L possesses a fixed point.

Take arbitrary $\mathsf{u},\mathsf{w}\in \mathrm{U}$ so that $\mathrm{L}(\mathsf{u}),\mathrm{L}(\mathsf{w})\in \mathrm{L}(\mathrm{U})$. This denotes $z:=max\{\mathrm{L}\mathsf{u},\mathrm{L}\mathsf{w}\}$. Thus, we conclude $(\mathrm{L}\mathsf{u},z)\in \mathbf{S}$ and $(L\mathsf{w},z)\in \mathbf{S}$ so that $\mathrm{L}(\mathrm{U})$ is ${\mathbf{S}}^s$-directed. Thus, by Theorem 4, L enjoys a unique fixed point, say $\overline{\mathsf{u}}$. Due to $\overline{\mathsf{u}}\in \mathrm{U}$, we can conclude that $\overline{\mathsf{u}}$ is to be a unique (nonnegative) solution of (18). □

6. Conclusions

Using a triplet of test functions, we addressed specific findings in an MS by carrying out a locally finitely $\mathsf{Ϝ}$-transitive BR for relational almost contraction. We also included an application to a second-order BVP to reinforce the value of the theoretical framework and the depth of our findings. The research findings included an optimum contraction requirement that only applies to comparative element pairs, not all elements. This exhibits the merits of our research over a few established findings from an inventory of the recent literature.

We came up with three distinct examples to illustrate our findings. Examples 1 and 2 demonstrated Theorem 4, which in turn validates two different proposals (either L remains $\mathbf{S}$-continuous, or $\mathbf{S}$ serves as $\zeta$-self-closed). Example 3, on the other hand, merely meets the premise of the existence finding (i.e., Theorem 3) in regard to failing to demonstrate uniqueness.

Recognizing the importance of the relation-theoretic fixed-point approach, we take into account the following possible lines of investigation for future studies:

• Enhancing the features of test functions;
• Expanding our findings to a pair of self-maps by demonstrating the common fixed-point theorems;
• Strengthening our findings in the setting of fuzzy MS along the lines of [33,34];
• Adapting our insights to integral equations, nonlinear matrix equations and first-order periodic BVPs.