Relational Generalized Nonlinear Contractions of Pant Type with an Application to Nonlinear Integral Equations

Abstract

The objective of this paper is to propose some fixed-point findings under a relational contraction of Pant type employing a pair of auxiliary functions and through a generalized class of transitive binary relations. Our outcomes extend, sharpen, modify and enrich many existing findings. To facilitate our research, we create a few instances that convey our findings. Through the use of our outcomes, we demonstrate the existence and uniqueness of solutions for a nonlinear integral equation.

1. Introduction

A substantial amount of nonlinear functional analysis depends on metric fixed-point theory, which offers significance due to its applications in numerous fields. For recent contributions on applications of metric fixed-point theory, readers are suggested to look at [1,2,3]. First introduced in 1922, metric fixed-point theory serves as an outcome of a review of the classical BCP. Fortunately, the BCP has contributed significantly to the creation of innovative methods for formulating the solutions of a wide range of equations, such as matrix equations, integral equations, and boundary value problems. The BCP has previously been expanded by a number of authors adopting suitable gauge functions to incorporate a wider class of contraction mappings. In this vein, Browder [4] proposed the conception of $\phi$-contractions, which was streamlined by Matkowski [5] and Boyd and Wong [6]. The class of $\phi$-contractions really utilizes an auxiliary function $\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$, which is employed as an alternative of the contraction constant. Employing two auxiliary functions, Dutta and Choudhury [7] proposed the conception of $(\varphi ,\psi )$-contractions. By improving the class of $(\varphi ,\psi )$-contractions, Alam et al. [8] explored a variation of the BCP.

Alam and Imdad [9] formulated the relational version of the BCP, which was refined by Alam et al. [10]. In actuality, relational contractions are indeed far more general than ordinary contractions since they are linked through a BR. A key characteristic of relational contractions is that only comparative elements should fulfill the contraction inequality instead of all elements. This fact demonstrates that the outcomes of relational contractions can be utilized to resolve many kinds of nonlinear matrix equations, boundary value problems, and nonlinear integral equations, while the outcomes of fixed points of abstract MS cannot be implemented. It turns out that numerous conclusions are drawn in the setting of relational MS employing the various types of existing contractions, such as: $F_{\Re }$-contractions [11], Boyd–Wong contractions [12,13], generalized nonlinear contractions [14], weak contractions [15], set-valued contractions [16,17,18], Suzuki-type implicit contractions [19], nonlinear almost contractions [20,21], Proinov-type contractions [22] and other similar types.

Pant [23] invented the subsequent non-unique fixed-point outcome.

Theorem 1.

Let $\mathcal{L}$ be a map from a complete MS $(\mathbf{S},\sigma )$ into itself such that ∃$0\le a<1$ satisfying

$$\begin{array}{c}\hfill \sigma (\mathcal{L}z,\mathcal{L}w)\le a·\sigma (z,w),\forall z,w\in \mathbf{S}with[z\ne \mathcal{L}(z)orw\ne \mathcal{L}(w\left)\right].\end{array}$$

Then, $\mathcal{L}$ admits a fixed point.

Theorem 1 was later enhanced for $\phi$-contractions by Pant [24]. Most recently, Alshaban et al. [25] revealed fixed-point achievements for almost nonlinear extended contractions over arbitrary BR, which were further developed by Filali et al. [26] and Filali and Khan [27] for extended contractions involving the locally $\mathcal{L}$-transitive BR.

The primary emphasis in this study is on the validity and uniqueness of fixed points for extended $(\varphi ,\psi )$-contractions associated with auxiliary functions in the setup of a relational MS. Our findings extend, modify, sharpen and enrich many well-known results, particularly those owing to Alam et al. [13], Sk et al. [14], Hossain et al. [15] and Pant [24]. We deliver a few exemplary instances to clarify the key findings. To assist with our insights, we address a finding dealing with the occurrence of a unique solution of a certain nonlinear FIE.

2. Preliminaries

Any subset of ${\mathbf{S}}^2$ is referred to as a BR on the set $\mathbf{S}$. Let $\mathbf{S}$ be a set, $\sigma$ a metric on $\mathbf{S}$, $\mathcal{L}$ a self-map on $\mathbf{S}$ a map, and $\zeta$ a BR on $\mathbf{S}$. We say the following.

Definition 1

([9]). z, w $\in \mathbf{S}$ are ζ-comparative if $(z,w)\in \zeta$ or $(w,z)\in \zeta$. We denote this by $[z,w]\in \zeta$.

Definition 2

([28]). The inverse BR of ζ is ${\zeta }^{-1}:=\{(z,w)\in {\mathbf{S}}^2:(w,z)\in \zeta \}$. Also, the symmetric closure of ζ is ${\zeta }^s:=\zeta \cup {\zeta }^{-1}$.

Remark 1

([9]). $(z,w)\in {\zeta }^s⟺[z,w]\in \zeta .$

Definition 3

([9]). ζ is an $\mathcal{L}$-closed BR when

$$(z,w)\in \zeta \Rightarrow (\mathcal{L}z,\mathcal{L}w)\in \zeta .$$

Proposition 1

([12]). If ζ remains $\mathcal{L}$-closed, then ζ is ${\mathcal{L}}^n$-closed for every $n\in {\mathbb{N}}_0$.

Definition 4

([9]). A sequence $\left\{z_n\right\}\subset \mathbf{S}$ with $(z_n,z_{n+1})\in \zeta$, for every $n\in \mathbb{N}$, is ζ-preserving.

Definition 5

([29]). If $\mathbf{M}\subseteq \mathbf{S}$, then the BR

$${\zeta |}_{\mathbf{M}}:=\zeta \cap {\mathbf{M}}^2,$$

(on $\mathbf{M}$), is a restriction of ζ in $\mathbf{M}$.

Definition 6

([12]). ζ is locally $\mathcal{L}$-transitive if for each ζ-preserving sequence $\left\{w_n\right\}\subset \mathcal{L}\left(\mathbf{S}\right)$, ${\zeta |}_{\mathcal{Z}}$ remains transitive, whereas $\mathcal{Z}=\{w_n:n\in \mathbb{N}\}$.

Definition 7

([30]). Given $\nu \in \mathbb{N}\backslash \left\{1\right\}$, ζ is ν-transitive if for any $z_0,z_1,\dots ,z_{\nu }\in \mathbf{S}$,

$$(z_{i-1},z_i)\in \zeta \mathrm{for}\mathrm{each}i(1\le i\le \nu )\Rightarrow (z_0,z_{\nu })\in \zeta .$$

Thus, by 2-transitive BR, we mean the usual transitive BR.

Definition 8

([31]). ζ is finitely transitive if we can find $\nu \in \mathbb{N}\backslash \left\{1\right\}$ for which ζ is ν-transitive.

Definition 9

([13]). ζ is locally finitely $\mathcal{L}$-transitive if for each ζ-preserving sequence $\left\{z_n\right\}\subset \mathcal{L}\left(\mathbf{S}\right)$ with range $\mathcal{Z}=\{z_n:n\in \mathbb{N}\}$, ${\zeta |}_{\mathcal{Z}}$ remains finitely transitive.

Clearly, finite transitivity⟹ locally finite $\mathcal{L}$-transitivity. Also, local $\mathcal{L}$-transitivity⟹ locally finite $\mathcal{L}$-transitivity.

Definition 10

([9]). ζ is σ-self-closed if every ζ-preserving convergent sequence in $\mathbf{S}$ has a subsequence containing ζ-comparative terms with a convergence limit.

Definition 11

([32]). The MS $(\mathbf{S},\sigma )$ is ζ-complete if every ζ-preserving Cauchy sequence in $\mathbf{S}$ is convergent.

Definition 12

([32]). The map $\mathcal{L}$ is ζ-continuous if for all $z\in \mathbf{S}$ and for all ζ-preserving sequences $\left\{z_n\right\}\subset \mathbf{S}$ with $z_n\stackrel{\sigma }{⟶}z$,

$$\mathcal{L}\left(z_n\right)\stackrel{\sigma }{⟶}\mathcal{L}\left(z\right).$$

Definition 13

([33]). A subset $\mathbf{M}\subseteq \mathbf{S}$ is ζ-directed if for every pair $z,w\in \mathbf{M}$, ∃$v\in \mathbf{S}$ with $(z,v)\in \zeta$ and $(w,v)\in \zeta$.

Definition 14

([34]). A sequence $\left\{z_n\right\}$ in an MS $(\mathbf{S},\sigma )$ is semi-Cauchy if

$$\underset{n\to \infty }{lim}\sigma (z_n,z_{n+1})=0.$$

Each Cauchy sequence remains semi-Cauchy.

Lemma 1

([30]). Let $\left\{z_n\right\}$ be a non-Cauchy sequence in an MS $(\mathbf{S},\sigma )$. Then ∃${ϵ}_0>0$ and subsequences $\left\{z_{n_{\mathsf{ȷ}}}\right\}$ and $\left\{z_{m_{\mathsf{ȷ}}}\right\}$ of $\left\{z_n\right\}$ with

(i)

$\mathsf{ȷ}\le m_{\mathsf{ȷ}}<n_{\mathsf{ȷ}},\forall \mathsf{ȷ}\in \mathbb{N}$;

(ii)

$\sigma (z_{m_{\mathsf{ȷ}}},z_{n_{\mathsf{ȷ}}})\ge {ϵ}_0,\forall \mathsf{ȷ}\in \mathbb{N}$;

(iii)

$\sigma (z_{m_{\mathsf{ȷ}}},z_{{\nu }_{\mathsf{ȷ}}})<{ϵ}_0$, $\forall {\nu }_{\mathsf{ȷ}}\in \{m_{\mathsf{ȷ}}+1,m_{\mathsf{ȷ}}+2,\dots ,n_{\mathsf{ȷ}}-2,n_{\mathsf{ȷ}}-1\}$.

• Moreover, if $\underset{n\to \infty }{lim}\sigma (z_n,z_{n+1})=0$, then

  $$\underset{\mathsf{ȷ}\to \infty }{lim}\sigma (z_{m_{\mathsf{ȷ}}},z_{n_{\mathsf{ȷ}}+\gamma })={ϵ}_0,\forall \gamma \in {\mathbb{N}}_0.$$

Lemma 2

([31]). Let $\mathbf{S}$ be a set composed with a BR ζ. Suppose that $\left\{z_n\right\}\subset \mathbf{S}$ is a ζ-preserving sequence and ζ is an ν-transitive on $\mathcal{Z}=\{z_n:n\in {\mathbb{N}}_0\}$; then

$$(z_n,z_{n+1+\lambda (\nu -1)})\in \zeta ,\forall n,\lambda \in {\mathbb{N}}_0.$$

In what follows, $\Phi$ denotes the collection of functions $\varphi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying

• ${\Phi }_1:\varphi$ remains right continuous;
• ${\Phi }_2:\varphi$ remains increasing.
• $\Psi$ denotes the collection of functions $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying
• ${\Psi }_1:\psi \left(\mathrm{s}\right)>0,\forall \mathrm{s}>0$;
• ${\Psi }_2:\underset{\mathrm{s}\to \mathrm{r}}{lim\; inf}\psi \left(\mathrm{s}\right)>0,\forall \mathrm{r}>0$.

Proposition 2

([8]). If $\varphi \in \Phi$ and $\psi \in \Psi$ verify

$$\varphi \left(t\right)\le \varphi \left(s\right)-\psi \left(s\right),\forall t\in {\mathbb{R}}^{+},s>0,$$

then $t<s.$

Proposition 3.

Given $\varphi \in \Phi$ and $\psi \in \Psi$, (A) and (B) are equivalent:

(A)

$\varphi \left(\sigma \right(\mathcal{L}z,\mathcal{L}w\left)\right)\le \varphi \left(\sigma \right(z,w\left)\right)-\psi \left(\sigma \right(z,w\left)\right),\forall (z,w)\in \zeta with[z\ne \mathcal{L}(z)$ or $w\ne \mathcal{L}\left(w\right)].$

(B)

$\varphi \left(\sigma \right(\mathcal{L}z,\mathcal{L}w\left)\right)\le \varphi \left(\sigma \right(z,w\left)\right)-\psi \left(\sigma \right(z,w\left)\right),\forall [z,w]\in \zeta with[z\ne \mathcal{L}(z)$ or $w\ne \mathcal{L}\left(w\right)].$

Proof. 

The assessment (B)⇒(A) is readily apparent. Contrariwise, we proceed to say that (A) is accurate. Assume that $[z,w]\in \zeta$. Subsequently, in scenario $(z,w)\in \zeta$, (A) implies (B). Otherwise, we attain $(w,z)\in \zeta$. In this scenario, by symmetry of metric $\sigma$ and (A), we arrive at

$$\begin{array}{ccc}\hfill \varphi \left(\sigma \right(\mathcal{L}z,\mathcal{L}w\left)\right)& =& \varphi \left(\sigma \right(\mathcal{L}w,\mathcal{L}z\left)\right)\hfill \\ & \le & \varphi \left(\sigma \right(w,z\left)\right)-\psi \left(\sigma \right(w,z\left)\right)\hfill \\ & =& \varphi \left(\sigma \right(z,w\left)\right)-\psi \left(\sigma \right(z,w\left)\right).\hfill \end{array}$$

It follows that (A)⇒(B). □

3. Main Results

We bring forth the following findings on fixed points of relational expanded $(\varphi ,\psi )$–contractions.

Theorem 2.

Let $(\mathbf{S},\sigma )$ be an MS equipped with a BR ζ and $\mathcal{L}:\mathbf{S}\to \mathbf{S}$ a map. Also,

(a) 

$(\mathbf{S},\sigma )$ remains ζ-complete;

(b) 

∃$z_0\in \mathbf{S}$ with $(z_0,\mathcal{L}{z}_0)\in \zeta$;

(c) 

ζ remains locally finitely $\mathcal{L}$-transitive and $\mathcal{L}$-closed;

(d) 

$\mathbf{S}$ remains ζ-continuous, or ζ remains σ-self-closed;

(e) 

∃$\varphi \in \Phi$ and $\psi \in \Psi$ with

$$\begin{array}{c}\hfill \varphi \left(\sigma \right(\mathcal{L}z,\mathcal{L}w\left)\right)\le \varphi \left(\sigma \right(z,w\left)\right)-\psi \left(\sigma \right(z,w\left)\right),\forall (z,w)\in \zeta with[z\ne \mathcal{L}(z)orw\ne \mathcal{L}(w\left)\right].\end{array}$$

Then, $\mathcal{L}$ possesses a fixed point.

Proof. 

The proof will be dealt with in six steps.

• Step–I. Define sequence $\left\{z_{\mathrm{n}}\right\}\subset \mathbf{S}$ of a Picard iteration initiating with $z_0\in \mathbf{S}$; i.e.,

  $$z_{\mathrm{n}}:={\mathcal{L}}^{\mathrm{n}}\left(z_0\right)=\mathcal{L}\left(z_{\mathrm{n}-1}\right),\forall \mathrm{n}\in \mathbb{N}.$$

  (1)
• Step–II. We show that $\left\{z_{\mathrm{n}}\right\}$ remains $\zeta$-preserving. Making use of $\left(b\right)$, the $\mathcal{L}$-closedness of $\zeta$ and Proposition 1, we arrive at

  $$({\mathcal{L}}^{\mathrm{n}}{z}_0,{\mathcal{L}}^{\mathrm{n}+1}{z}_0)\in \zeta ,$$

  which owing to (1) becomes

  $$(z_{\mathrm{n}},z_{\mathrm{n}+1})\in \zeta ,\forall \mathrm{n}\in {\mathbb{N}}_0.$$

  (2)
• Step–III. Denote ${\sigma }_{\mathrm{n}}:=\sigma (z_{\mathrm{n}},z_{\mathrm{n}+1})$. If ∃${\mathrm{n}}_0\in {\mathbb{N}}_0$ for which ${\sigma }_{{\mathrm{n}}_0}=0$, then from (1), we find $z_{{\mathrm{n}}_0}=z_{{\mathrm{n}}_0+1}=\mathcal{L}\left(z_{{\mathrm{n}}_0}\right)$; thereby $z_{n_0}\in Fix\left(\mathcal{L}\right)$ and so we are done. Unless we establish that ${\sigma }_{\mathrm{n}}>0$, ∀$\mathrm{n}\in {\mathbb{N}}_0$, we move on to Step–IV.
• Step–IV. We emphasize that $\left\{z_{\mathrm{n}}\right\}$ is a semi-Cauchy sequence; i.e., $\underset{\mathrm{n}\to \infty }{lim}\sigma (z_{\mathrm{n}},z_{\mathrm{n}+1})=0$. Making use of $\left(e\right)$, (1) and (2), we obtain

  $$\varphi \left(\sigma (z_{\mathrm{n}},z_{\mathrm{n}+1})\right)=\varphi \left(\sigma (\mathcal{L}{z}_{\mathrm{n}-1},\mathcal{L}{z}_{\mathrm{n}})\right)\le \varphi \left(\sigma (z_{\mathrm{n}-1},z_{\mathrm{n}})\right)-\psi \left(\sigma (z_{\mathrm{n}-1},z_{\mathrm{n}})\right)$$

  (3)

  so that

  $$\varphi \left({\sigma }_{\mathrm{n}}\right)\le \varphi \left({\sigma }_{\mathrm{n}-1}\right),\forall \mathrm{n}\in \mathbb{N}.$$

  Through axiom ${\Phi }_2$, we obtain ${\sigma }_{\mathrm{n}}<{\sigma }_{\mathrm{n}-1}$, $\forall \mathrm{n}\in \mathbb{N}$. Therefore, $\left\{{\sigma }_{\mathrm{n}}\right\}\subset {\mathbb{R}}^{+}\backslash \left\{0\right\}$ remains a decreasing sequence that is also bounded below. So ∃$\overline{\sigma }\ge 0$ verifying

  $$\begin{array}{c}\hfill \underset{\mathrm{n}\to \infty }{lim}{\sigma }_{\mathrm{n}}=\overline{\sigma }.\end{array}$$

  (4)

Let $\overline{\sigma }>0$. Taking the upper limit in (3), we arrive at

$$\begin{array}{ccc}\hfill \underset{\mathrm{n}\to \infty }{lim\; sup}\varphi \left({\sigma }_{\mathrm{n}}\right)& \le & \underset{\mathrm{n}\to \infty }{lim\; sup}\varphi \left({\sigma }_{\mathrm{n}-1}\right)+\underset{\mathrm{n}\to \infty }{lim\; sup}[-\psi \left({\sigma }_{\mathrm{n}-1}\right)]\hfill \\ & \le & \underset{\mathrm{n}\to \infty }{lim\; sup}\varphi \left({\sigma }_{\mathrm{n}-1}\right)-\underset{\mathrm{n}\to \infty }{lim\; inf}\psi \left({\sigma }_{\mathrm{n}-1}\right).\hfill \end{array}$$

From ${\Phi }_1$, we arrive at

$$\begin{array}{c}\hfill \varphi \left(\overline{\sigma }\right)\le \varphi \left(\overline{\sigma }\right)-\underset{\mathrm{n}\to \infty }{lim\; inf}\psi \left({\sigma }_{\mathrm{n}-1}\right)\end{array}$$

so that

$$\begin{array}{c}\hfill \underset{\mathrm{s}\to \mathrm{r}>0}{lim\; inf}\psi \left(\mathrm{s}\right)=\underset{\mathrm{n}\to \infty }{lim\; inf}\psi \left({\sigma }_{\mathrm{n}-1}\right)\le 0\end{array}$$

which contradicts ${\Psi }_2$. It follows that

$$\underset{\mathrm{n}\to \infty }{lim}{\sigma }_{\mathrm{n}}=\underset{\mathrm{n}\to \infty }{lim}\sigma (z_{\mathrm{n}},z_{\mathrm{n}+1})=0.$$

(5)

• Step–V. We emphasize that $\left\{z_{\mathrm{n}}\right\}$ is Cauchy. If $\left\{z_{\mathrm{n}}\right\}$ is not Cauchy, then by Lemma 1, ∃${ϵ}_0>0$ and subsequences $\left\{z_{{\mathrm{n}}_{\mathsf{ȷ}}}\right\}$ and $\left\{z_{{\mathrm{m}}_{\mathsf{ȷ}}}\right\}$ of $\left\{z_{\mathrm{n}}\right\}$ that verify

  $$\mathsf{ȷ}\le {\mathrm{m}}_{\mathsf{ȷ}}<N_{\mathsf{ȷ}},\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{n}}_{\mathsf{ȷ}}})\ge {ϵ}_0>\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\nu }_{\mathsf{ȷ}}}),\forall \mathsf{ȷ}\in \mathbb{N},{\nu }_{\mathsf{ȷ}}\in \{{\mathrm{m}}_{\mathsf{ȷ}}+1,{\mathrm{m}}_{\mathsf{ȷ}}+2,\dots ,{\mathrm{n}}_{\mathsf{ȷ}}-2,{\mathrm{n}}_{\mathsf{ȷ}}-1\}.$$

  Making use of (5) and Lemma 1, we attain

  $$\begin{array}{c}\hfill \underset{\mathsf{ȷ}\to \infty }{lim}\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{n}}_{\mathsf{ȷ}}+\gamma })={ϵ}_0,\forall \gamma \in {\mathbb{N}}_0.\end{array}$$

  (6)

Employing (1), we have $\mathcal{Z}:=\{z_{\mathrm{n}}:\mathrm{n}\in {\mathbb{N}}_0\}\subset \mathcal{L}\left(\mathbf{S}\right)$. From the locally finitely $\mathcal{L}$-transitivity of $\zeta$, ∃$\nu \in \mathbb{N}\backslash \left\{0\right\}$ for which ${\zeta |}_{\mathcal{Z}}$ remains $\nu$-transitive.

Due to the fact that ${\mathrm{m}}_{\mathsf{ȷ}}<{\mathrm{n}}_{\mathsf{ȷ}}$ and $\nu -1>0$, the division algorithm shows that

$${\mathrm{n}}_{\mathsf{ȷ}}-{\mathrm{m}}_{\mathsf{ȷ}}=(\nu -1)(p_{\mathsf{ȷ}}-1)+(\nu -q_{\mathsf{ȷ}})$$

$$p_{\mathsf{ȷ}}-1\ge 0,0\le \nu -q_{\mathsf{ȷ}}<\nu -1$$

$$⟺\left\{\begin{array}{c}{\mathrm{n}}_{\mathsf{ȷ}}+q_{\mathsf{ȷ}}={\mathrm{m}}_{\mathsf{ȷ}}+1+(\nu -1)p_{\mathsf{ȷ}}\hfill \\ p_{\mathsf{ȷ}}\ge 1,1<q_{\mathsf{ȷ}}\le \nu .\hfill \end{array}\right.$$

Owing to $q_{\mathsf{ȷ}}\in (1,\nu ]$, the subsequences $\left\{z_{{\mathrm{n}}_{\mathsf{ȷ}}}\right\}$ and $\left\{z_{{\mathrm{m}}_{\mathsf{ȷ}}}\right\}$ of $\left\{z_{\mathrm{n}}\right\}$ (verifying (6)) may be assumed such that $q_{\mathsf{ȷ}}=\beta$. Thus, we arrive at

$$\begin{array}{c}\hfill {\mathrm{m}}_{\mathsf{ȷ}}^{\prime }={\mathrm{n}}_{\mathsf{ȷ}}+\beta ={\mathrm{m}}_{\mathsf{ȷ}}+1+(\nu -1)p_{\mathsf{ȷ}}.\end{array}$$

(7)

Making use of (6) and (7), we obtain

$$\begin{array}{c}\hfill \underset{\mathsf{ȷ}\to \infty }{lim}\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }})=\underset{\mathsf{ȷ}\to \infty }{lim}\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{n}}_{\mathsf{ȷ}}+\beta })={ϵ}_0.\end{array}$$

(8)

By the triangle inequality, we arrive at

$$\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}+1},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }+1})\le \sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}+1},z_{{\mathrm{m}}_{\mathsf{ȷ}}})+\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }})+\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }+1})$$

and

$$\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }})\le \sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{m}}_{\mathsf{ȷ}}+1})+\sigma (z_{{\sigma }_{\mathsf{ȷ}+1}},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }+1})+\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }+1},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }}).$$

Thus, we find

$$\begin{array}{c}\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }})-\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{m}}_{\mathsf{ȷ}}+1})-\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }+1},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }})\le \sigma (z_{{\sigma }_{\mathsf{ȷ}+1}},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }+1})\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}+1},z_{{\mathrm{m}}_{\mathsf{ȷ}}})+\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }})+\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }+1}).\end{array}$$

Taking $\mathsf{ȷ}\to \infty$ and employing (5) and (11), the above inequality becomes

$$\begin{array}{c}\hfill \underset{\mathsf{ȷ}\to \infty }{lim}\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}+1},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }+1})={ϵ}_0.\end{array}$$

(9)

Making use of (7) and Lemma 1, we get $(z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }})\in \zeta .$ Denote ${\delta }_{\mathsf{ȷ}}:=\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{m}}_{\mathsf{ȷ}}^{\prime }})$. Using contraction condition $\left(e\right)$, we find

$$\varphi (z_{{\mathrm{m}}_{\mathsf{ȷ}}+1},z_{{\mathrm{n}}_{\mathsf{ȷ}}+1}))=\varphi \left(\sigma (\mathcal{L}{z}_{{\mathrm{m}}_{\mathsf{ȷ}}},\mathcal{L}{z}_{{\mathrm{n}}_{\mathsf{ȷ}}})\right)\le \varphi \left(\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{n}}_{\mathsf{ȷ}}})\right)-\psi \left(\sigma (z_{{\mathrm{m}}_{\mathsf{ȷ}}},z_{{\mathrm{n}}_{\mathsf{ȷ}}})\right),\forall \mathsf{ȷ}\in {\mathbb{N}}_0.$$

Taking the upper limit in the above and by (8) and (9), along with the properties of $\varphi$ and $\psi$, we attain

$$\begin{array}{c}\hfill \varphi \left({ϵ}_0\right)\le \varphi \left({ϵ}_0\right)-\underset{\mathrm{t}\to {ϵ}_0}{lim\; inf}\psi \left(\mathrm{t}\right)<\varphi \left({ϵ}_0\right),\end{array}$$

implying thereby

$$\begin{array}{c}\hfill \underset{\mathrm{t}\to {ϵ}_0}{lim\; inf}\psi \left(\mathrm{t}\right)\le 0,\end{array}$$

which contradicts ${\Psi }_2$. Therefore, $\left\{z_{\mathrm{n}}\right\}$ is a $\zeta$-preserving Cauchy sequence. By the $\zeta$-completeness of $\mathbf{S}$, ∃$\overline{z}\in \mathbf{S}$ with $z_{\mathrm{n}}\stackrel{\sigma }{⟶}\overline{z}$.

• Step–VI. We prove that $\overline{z}\in Fix\left(\mathcal{L}\right)$ through hypothesis $\left(d\right)$. Suppose that $\mathcal{L}$ is $\zeta$-continuous; then $z_{\mathrm{n}+1}=\mathcal{L}\left(z_{\mathrm{n}}\right)\stackrel{\sigma }{⟶}\mathcal{L}\left(\overline{z}\right)$. Therefore, we attain $\mathcal{L}\left(\overline{z}\right)=\overline{z}$.

Now, let $\zeta$ be $\sigma$-self-closed; then ∃ a subsequence $\left\{z_{{\mathrm{n}}_{\mathsf{ȷ}}}\right\}$ of $\left\{z_{\mathrm{n}}\right\}$ that verifies $[z_{{\mathrm{n}}_{\mathsf{ȷ}}},\overline{z}]\in \zeta ,\forall \mathsf{ȷ}\in \mathbb{N}$. We assert that

$$\sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}+1},\mathcal{L}\overline{z})\le \sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}},\overline{z}),\forall \mathsf{ȷ}\in \mathbb{N}.$$

(10)

Consider a partition $\{{\mathbb{N}}^0,{\mathbb{N}}^{+}\}$ of $\mathbb{N}$; i.e., ${\mathbb{N}}^0\cup {\mathbb{N}}^{+}=\mathbb{N}$ and ${\mathbb{N}}^0\cap {\mathbb{N}}^{+}=\varnothing$, which verify

(i)

$\sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}},\overline{z})=0,\forall \mathsf{ȷ}\in {\mathbb{N}}^0$;

(ii)

$\sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}},\overline{z})>0,\forall \mathsf{ȷ}\in {\mathbb{N}}^{+}.$

In case (i), we conclude that $\sigma (\mathcal{L}{z}_{{\mathrm{n}}_{\mathsf{ȷ}}},\mathcal{L}\overline{z})=0,\forall \mathsf{ȷ}\in {\mathbb{N}}^0$, which yields $\sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}+1},\mathcal{L}\overline{z}),\forall \mathsf{ȷ}\in {\mathbb{N}}^0$ and so (10) holds for all $\mathsf{ȷ}\in {\mathbb{N}}^0.$ In case (ii), employing $\left(e\right)$, Proposition 3, and $[z_{{\mathrm{n}}_{\mathsf{ȷ}}},\overline{z}]\in \zeta$, we attain

$$\begin{array}{c}\hfill \varphi \left(\sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}+1},\mathcal{L}\overline{z})\right)=\varphi \left(\sigma (\mathcal{L}{z}_{{\mathrm{n}}_{\mathsf{ȷ}}},\mathcal{L}\overline{z})\right)\le \varphi \left(\sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}},\overline{z})\right)-\psi \left(\sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}},\overline{z})\right),\forall \mathsf{ȷ}\in {\mathbb{N}}^{+}.\end{array}$$

Utilizing Proposition 2, we arrive at $\sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}+1},\mathcal{L}\overline{z})<\sigma (z_{{\mathrm{n}}_{\mathsf{ȷ}}},\overline{z}),\forall \mathsf{ȷ}\in {\mathbb{N}}^{+}$. For all $\mathsf{ȷ}\in \mathbb{N}$, (10) is thus accurate. Taking the limit as $\mathsf{ȷ}\to \infty$ in (10), we attain $z_{{\mathrm{n}}_{\mathsf{ȷ}}+1}\stackrel{\sigma }{\to }\mathcal{L}\left(\overline{z}\right)$ so that $\mathcal{L}\left(\overline{z}\right)=\overline{z}$. Thus, $\overline{z}\in Fix\left(\mathcal{L}\right)$. □

Theorem 3.

Turning to the conclusions of Theorem 2, when $\mathcal{L}\left(\mathbf{S}\right)$ is ${\zeta }^s$-directed, then $\mathcal{L}$ possesses a unique fixed point.

Proof. 

Take $\overline{z},\overline{w}\in \mathrm{Fix}\left(\mathcal{L}\right)$; i.e.,

$$\mathcal{L}\left(\overline{z}\right)=\overline{z}\mathrm{and}\mathcal{L}\left(\overline{w}\right)=\overline{w}.$$

(11)

As $\overline{z},\overline{w}\in \mathcal{L}\left(\mathbf{S}\right)$ and $\mathcal{L}\left(\mathbf{S}\right)$ is ${\zeta }^s$-directed, ∃$v\in \mathbf{S}$ verifying

$$(\overline{z},v)\in \zeta \mathrm{a}\mathrm{n}\mathrm{d}(\overline{w},v)\in \zeta .$$

(12)

Denote $d_{\mathrm{n}}:=\sigma (\overline{z},{\mathcal{L}}^{\mathrm{n}}v)$. From (11), (12) and $\left(e\right)$, we arrive at

$$\varphi \left(\sigma (\overline{z},{\mathcal{L}}^{\mathrm{n}}v)\right)=\varphi \left(\sigma (\mathcal{L}\overline{z},\mathcal{L}\left({\mathcal{L}}^{\mathrm{n}-1}v\right))\right)\le \varphi \left(\sigma (\overline{z},{\mathcal{L}}^{\mathrm{n}-1}v)\right)-\psi \left(\sigma (\overline{z},{\mathcal{L}}^{\mathrm{n}-1}v)\right)$$

so that

$$\varphi \left(d_{\mathrm{n}}\right)\le \varphi \left(d_{\mathrm{n}-1}\right)-\psi \left(d_{\mathrm{n}-1}\right).$$

(13)

If ∃${\mathrm{n}}_0\in \mathbb{N}$ verifying $d_{{\mathrm{n}}_0}=0$, then we conclude that $d_{{\mathrm{n}}_0}\le d_{{\mathrm{n}}_0-1}$, unless we attain $d_{\mathrm{n}}>0,\forall \mathrm{n}\in \mathbb{N}$. Utilizing Proposition 2, we find $d_{\mathrm{n}}<d_{\mathrm{n}-1}$. Thus, in each case, we arrive at

$$d_{\mathrm{n}}\le d_{\mathrm{n}-1}.$$

Utilizing the same reasoning as previously in Theorem 2, the last inequality determines

$${\displaystyle \underset{\mathrm{n}\to \infty }{lim}{d}_{\mathrm{n}}=\underset{\mathrm{n}\to \infty }{lim}\sigma (\overline{z},{\mathcal{L}}^{\mathrm{n}}v)=0.}$$

(14)

Similarly, we can derive

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

(15)

From (14), (15) and the triangle inequality, we conclude that

$$\sigma (\overline{z},\overline{w})=\sigma (\overline{z},{\mathcal{L}}^{\mathrm{n}}v)+\sigma ({\mathcal{L}}^{\mathrm{n}}v,\overline{w})\to 0,\mathrm{as}\mathrm{n}\to \infty .$$

Thereby, $\overline{z}=\overline{w}$, so $\mathcal{L}$ possesses a unique fixed point. □

4. Illustrative Examples

The following scenarios are taken into consideration to clarify Theorems 2 and 3.

Example 1.

Let $\mathbf{S}=[0,1)\cup \mathbb{N}$ with the following metric:

$$\sigma (z,w)=\left\{\begin{array}{cc}\mid z-w\mid ,\hfill & ifz,w\in [0,1]andz\ne w,\hfill \\ z+w,\hfill & ifat\mathit{least}\mathit{one}\mathit{of}z\mathit{or}w\mathit{does}\mathit{not}\mathit{belong}\mathit{to}[0,1]\mathit{and}z\ne w,\hfill \\ 0,\hfill & ifz=w.\hfill \end{array}\right.$$

Define a BR ζ on $\mathbf{S}$ by

$$\zeta =\{(z,w)\in {\mathbf{S}}^2:z>w\mathit{and}z\in \{3,4,5,..\},w\ne 2\}.$$

Then, $\mathbf{S}$ remains a ζ-complete MS.

Consider a map $\mathcal{L}:\mathbf{S}\to \mathbf{S}$ as

$$\mathcal{L}\left(z\right)=\left\{\begin{array}{cc}z-{\displaystyle \frac{z^3}{4}},\hfill & \mathit{if}0\le z\le 1,\hfill \\ z-1,\hfill & \mathit{if}z\in \{2,3,4,\dots \}.\hfill \end{array}\right.$$

Then, ζ remains a locally finitely $\mathcal{L}$-transitive and $\mathcal{L}$-closed BR.

Define

$$\varphi \left(t\right)=t^2$$

and

$$\psi \left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{t^2}{4}},\hfill & \mathit{if}0\le t\le 1,\hfill \\ {\displaystyle \frac{1}{5}},\hfill & \mathrm{if}t>1.\hfill \end{array}\right.$$

Then $\varphi \in \Phi$ and $\psi \in \Psi$.

We will confirm the contraction condition $\left(e\right)$. Take $(z,w)\in \zeta$ verifying $z\ne \mathcal{L}\left(z\right)$ or $w\ne \mathcal{L}\left(w\right)$. If $z\in \{3,4,\dots .\}$, then w can be picked in one of two ways. We begin by taking $w\in [0,1]$. Then, we conclude that

$$\begin{array}{ccc}\hfill \sigma (\mathcal{L}z,\mathcal{L}w)& =& \sigma \left(z-1,w-{\displaystyle \frac{1}{4}}{w}^3\right)=z-1+w-{\displaystyle \frac{w^3}{4}}\hfill \\ & \le & z+w-1.\hfill \end{array}$$

Second, we take $w\in \{3,4,\dots \}$. Then, we attain

$$\begin{array}{ccc}\hfill \sigma (\mathcal{L}z,\mathcal{L}w)& =& \sigma (z-1,w-1)=z+w-2\hfill \\ & <& z+w-1.\hfill \end{array}$$

In both the cases, we thereby conclude that

$$\begin{array}{ccc}\hfill \varphi \left(\sigma \right(\mathcal{L}z,\mathcal{L}w\left)\right)& =& {\left(\sigma (\mathcal{L}z,\mathcal{L}w)\right)}^2<{(z+w-1)}^2\hfill \\ & <& (z+w-1)(z+w+1)={(z+w)}^2-1\hfill \\ & <& {(z+w)}^2-{\displaystyle \frac{1}{5}}\hfill \\ & =& \varphi \left(\sigma \right(z,w\left)\right)-\psi \left(\sigma \right(z,w\left)\right).\hfill \end{array}$$

The contraction inequality $\left(e\right)$ is thus justified. Hence, by Theorem 2, $\mathcal{L}$ admits a fixed point. Also, here $\mathcal{L}\left(\mathbf{S}\right)$ remains ${\zeta }^s$-directed; consequently by Theorem 3, $\mathcal{L}$ possesses a unique fixed point: $\overline{z}=0$.

Example 2.

Let $\mathbf{S}=[0,1]$ with Euclidean metric σ. Define BR $\zeta :=\le$ on $\mathbf{S}$. Let $\mathcal{L}:\mathbf{S}\to \mathbf{S}$ be a map defined by

$$\mathcal{L}\left(z\right)=\left\{\begin{array}{c}{z}^2,\mathrm{if}0\le z<1/4\hfill \\ 0,\mathrm{if}1/4\le z\le 1.\hfill \end{array}\right.$$

Then $(0,\mathcal{L}0)\in \zeta$. Also, ζ is a locally finitely $\mathcal{L}$-transitive and $\mathcal{L}$-closed BR. Further, $(\mathbf{S},\sigma )$ remains a ζ-complete MS.

If $\left\{z_n\right\}\subset \mathbf{S}$ is a ζ-preserving convergent sequence verifying $z_n\stackrel{\sigma }{⟶}z$, then $\left\{z_n\right\}$ remains an increasing and convergent sequence that must fulfill $z_n\le z$, implying thereby $(z_n,z)\in \zeta$ for every $n\in \mathbb{N}$. Thus ζ is σ-self-closed.

Consider the auxiliary functions $\varphi \left(t\right)=t/2$ and $\psi \left(\right(t)=t/4$. Then, the contraction inequality $\left(e\right)$ of Theorem 2 would be met. Furthermore, $\mathcal{L}\left(\mathbf{S}\right)$ remains a ${\zeta }^s$-directed set due to the fact that each pair $z,w\in \mathcal{L}\left(\mathbf{S}\right)$ verifies $(z,w)\in \zeta$ and $(w,w)\in \zeta$ for $w:=max\{z,w\}$. Ultimately, each speculation outlined in Theorems 2 and 3 is proven; thereby, $\mathcal{L}$ possesses a unique fixed point, $\overline{z}=0$.

5. Consequences

In this part, a few existing fixed-point outcomes are inferred from our findings. Under the BR $\zeta ={\mathbf{S}}^2$, Theorem 3 derives the following outcome on fixed points of extended $(\varphi ,\psi )$-contractions in ordinary MS.

Corollary 1.

Let $(\mathbf{S},\sigma )$ be a complete MS and $\mathcal{L}:\mathbf{S}\to \mathbf{S}$ a map. If ∃$\varphi \in \Phi$ and $\psi \in \Psi$ with

$$\begin{array}{c}\hfill \varphi \left(\sigma \right(\mathcal{L}z,\mathcal{L}w\left)\right)\le \varphi \left(\sigma \right(z,w\left)\right)-\psi \left(\sigma \right(z,w\left)\right),\forall z,w\in \mathbf{S},\end{array}$$

then $\mathcal{L}$ possesses a unique fixed point.

Clearly, Corollary 1 reduces to the main outcome of Pant [24] if we choose $\varphi \left(\mathrm{t}\right)=\mathrm{t}$ and $\psi \left(\mathrm{t}\right)=\mathrm{t}-\phi \left(\mathrm{t}\right)$, whereas $\phi \left(\mathrm{t}\right)<\mathrm{t}$ for every $\mathrm{r}>0$ and $\underset{\mathrm{t}\to \mathrm{r}}{lim\; sup}\phi \left(\mathrm{t}\right)<\mathrm{r}$ for every $\mathrm{r}>0$.

If we remove the restriction $z\ne \mathcal{L}\left(z\right)$ or $w\ne \mathcal{L}\left(w\right)$ from contraction inequality of Theorem 2, then we determine the following outcome of Sk et al. [14].

Corollary 2

([14]). Let $(\mathbf{S},\sigma )$ be an MS equipped with a BR ζ and $\mathcal{L}:\mathbf{S}\to \mathbf{S}$ a map. Also,

(a) 

$(\mathbf{S},\sigma )$ remains ζ-complete;

(b) 

∃$z_0\in \mathbf{S}$ with $(z_0,\mathcal{L}{z}_0)\in \zeta$;

(c) 

ζ remains locally finitely $\mathcal{L}$-transitive and $\mathcal{L}$-closed;

(d) 

$\mathbf{S}$ remains ζ-continuous, or ζ remains σ-self-closed;

(e) 

∃$\varphi \in \Phi$ and $\psi \in \Psi$ with

$$\begin{array}{c}\hfill \varphi \left(\sigma \right(\mathcal{L}z,\mathcal{L}w\left)\right)\le \varphi \left(\sigma \right(z,w\left)\right)-\psi \left(\sigma \right(z,w\left)\right),\forall (z,w)\in \zeta .\end{array}$$

Then, $\mathcal{L}$ possesses a fixed point.

On setting $\varphi \left(\mathrm{t}\right)=\mathrm{t}$ and $\psi \left(\mathrm{t}\right)=\mathrm{t}-\phi \left(\mathrm{t}\right)$ in Theorem 2, we get the main finding of Alam et al. [13].

Corollary 3

([13]). Let $(\mathbf{S},\sigma )$ be a MS equipped with a BR ζ and $\mathcal{L}:\mathbf{S}\to \mathbf{S}$ a map. Also,

(a) 

$(\mathbf{S},\sigma )$ remains ζ-complete,

(b) 

∃$z_0\in \mathbf{S}$ with $(z_0,\mathcal{L}{z}_0)\in \zeta$,

(c) 

ζ remains locally finitely $\mathcal{L}$-transitive and $\mathcal{L}$-closed,

(d) 

$\mathbf{S}$ remains ζ-continuous, or ζ remains σ-self-closed,

(e) 

∃$\phi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying $\phi \left(t\right)<t$, for every $r>0$ and $\underset{t\to r}{lim\; sup}\phi \left(t\right)<r$, for every $r>0$ with

$$\begin{array}{c}\hfill \sigma (\mathcal{L}z,\mathcal{L}w)\le \phi \left(\sigma \right(z,w\left)\right),\forall (z,w)\in \zeta .\end{array}$$

Then, $\mathcal{L}$ possesses a fixed point.

In particular, if $\varphi$ is an identity function, then Theorem 2 reduces the following finding of Hossain et al. [15].

Corollary 4

([15]). Let $(\mathbf{S},\sigma )$ be an MS equipped with a BR ζ and $\mathcal{L}:\mathbf{S}\to \mathbf{S}$ a map. Also,

(a) 

$(\mathbf{S},\sigma )$ remains ζ-complete;

(b) 

∃$z_0\in \mathbf{S}$ with $(z_0,\mathcal{L}{z}_0)\in \zeta$;

(c) 

ζ remains locally finitely $\mathcal{L}$-transitive and $\mathcal{L}$-closed;

(d) 

$\mathbf{S}$ remains ζ-continuous, or ζ remains σ-self-closed;

(e) 

∃$\psi \in \Psi$ with

$$\begin{array}{c}\hfill \sigma (\mathcal{L}z,\mathcal{L}w)\le \sigma (z,w)-\psi \left(\sigma \right(z,w\left)\right),\forall (z,w)\in \zeta .\end{array}$$

Then, $\mathcal{L}$ possesses a fixed point.

6. An Application to a Nonlinear FIE

This portion consists of determining the unique solution to the following (nonlinear) FIE:

$$\mu \left(\mathrm{s}\right)=\mathsf{ϝ}\left(\mathrm{s}\right)+{\int }_a^b\pounds (\mathrm{s},\vartheta )\mho (\vartheta ,\mu \left(\vartheta \right))d\vartheta ,\forall \mathrm{s}\in \mathcal{I}:=[a,b],$$

(16)

where $\mathsf{ϝ}:\mathcal{I}\to \mathbb{R}$, $\mho :\mathcal{I}\times \mathbb{R}\to \mathbb{R}$, and $\pounds :{\mathcal{I}}^2\to \mathbb{R}$ are functions.

Definition 15.

$\underline{\mu }\in \mathcal{C}\left(\mathcal{I}\right)$ is named as a lower solution of (16) when

$$\underline{\mu }\left(s\right)\le \mathsf{ϝ}\left(s\right)+{\int }_a^b\pounds (s,\vartheta )\mho (\vartheta ,\underline{\mu }\left(\vartheta \right))d\vartheta ,\forall s\in \mathcal{I}.$$

Definition 16.

$\overline{\mu }\in \mathcal{C}\left(\mathcal{I}\right)$ is named as an upper solution of (16) when

$$\overline{\mu }\left(s\right)\ge \mathsf{ϝ}\left(s\right)+{\int }_a^b\pounds (s,\vartheta )\mho (\vartheta ,\overline{\mu }\left(\vartheta \right))d\vartheta ,\forall s\in \mathcal{I}.$$

Let $\Theta$ denote the family of functions $\theta :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ verifying the following axioms:

• $\theta$ remains increasing;
• $\theta \left(\mathrm{t}\right)<\mathrm{t},\forall \mathrm{r}>0$;
• $\underset{\mathrm{t}\to \mathrm{r}}{lim\; sup}\theta \left(\mathrm{t}\right)<\mathrm{r},\forall \mathrm{r}>0$.

We immediately explore the main insights of this portion.

Theorem 4.

In collaboration with Problem (16), suppose that the aforementioned circumstances are valid:

(i)

ϝ, £, and ℏ remain continuous.

(ii)

$\pounds (s,\vartheta )>0,\forall s,\vartheta \in \mathcal{I}$.

(iii)

∃$0<\gamma \le 1$ and ∃$\theta \in \Theta$ that satisfy

$$0\le \hslash (s,x)-\hslash (s,y)\le {\displaystyle \frac{1}{\gamma }}\theta (x-y),\forall s\in \mathcal{I}a\mathrm{n}d\forall x,y\in \mathbb{R}withx\ge y,$$

(iv)

$\underset{s\in \mathcal{I}}{sup}{\int }_a^b\pounds (s,\vartheta )d\vartheta \le \gamma$.

• Then the problem admits a unique solution provided it possesses a lower solution.

Proof. 

Consider $\mathbf{S}:=\mathcal{C}\left(\mathcal{I}\right)$ with the metric

$$\sigma (\mu ,\omega )=\underset{\mathrm{s}\in \mathcal{I}}{sup}|\mu \left(\mathrm{s}\right)-\omega \left(\mathrm{s}\right)|,\forall \mu ,\omega \in \mathbf{S}.$$

(17)

Consider the following BR $\zeta$ on $\mathbf{S}$:

$$\zeta =\{(\mu ,\omega )\in {\mathbf{S}}^2:\mu \left(\mathrm{s}\right)\le \omega \left(\mathrm{s}\right),\forall \mathrm{s}\in \mathcal{I}\}.$$

(18)

Define the map $\mathcal{L}:\mathbf{S}\to \mathbf{S}$ as follows:

$$\left(\mathcal{L}\mu \right)\left(\mathrm{s}\right)=\mathsf{ϝ}\left(\mathrm{s}\right)+{\int }_a^b\pounds (\mathrm{s},\vartheta )\hslash (\vartheta ,\mu \left(\vartheta \right))d\vartheta ,\forall \mathrm{s}\in \mathbf{S}.$$

(19)

Trivially, $\mu \in \mathbf{S}$ solves (16) iff $\mu$ is a fixed point of $\mathcal{L}$.

• We will confirm all presumptions of Theorems 2 and 3.

(a)

$(\mathbf{S},\sigma )$, being a complete MS, is $\zeta$-complete.

(b)

If $\underline{\mu }\in \mathbf{S}$ serves as lower solution of (16), then

$$\underline{\mu }\left(\mathrm{s}\right)\le \mathsf{ϝ}\left(\mathrm{s}\right)+{\int }_a^b\pounds (\mathrm{s},\vartheta )\hslash (\vartheta ,\underline{\mu }\left(\vartheta \right))d\vartheta =\left(\mathcal{L}\underline{\mu }\right)\left(\mathrm{s}\right)$$

yielding thereby $(\underline{\mu },\mathcal{L}\underline{\mu })\in \zeta$.

(c)

Take $\mu ,\omega \in \mathbf{S}$ verifying $(\mu ,\omega )\in \zeta$. From (iii), we find

$$\hslash (\mathrm{s},\mu (\vartheta \left)\right)-\hslash (\mathrm{s},\omega (\vartheta \left)\right)\le 0,\forall \mathrm{s},\vartheta \in \mathcal{I}.$$

(20)

By (19), (20) and (ii), we conclude that

$$\begin{array}{c}\hfill \left(\mathcal{L}\mu \right)\left(\mathrm{s}\right)-\left(\mathcal{L}\omega \right)\left(\mathrm{s}\right)={\int }_a^b\pounds (\mathrm{s},\vartheta )[\hslash (\vartheta ,\mu \left(\vartheta \right))-\hslash (\vartheta ,\omega \left(\vartheta \right))]d\vartheta \le 0,\end{array}$$

i.e., $\left(\mathcal{L}\mu \right)\left(\mathrm{s}\right)\le \left(\mathcal{L}\omega \right)\left(\mathrm{s}\right)$. By (18), we find that $(\mathcal{L}\mu ,\mathcal{L}\omega )\in \zeta$. Thus, $\zeta$ is $\mathcal{L}$-closed.

(d)

Let $\left\{{\mu }_{\mathrm{n}}\right\}\subset \mathbf{S}$ be a $\zeta$-preserving sequence and ${\mu }_{\mathrm{n}}\to \omega \in \mathbf{S}$. Then for each $\mathrm{s}\in \mathcal{I}$, $\left\{{\mu }_{\mathrm{n}}\left(\mathrm{s}\right)\right\}\subset \mathbb{R}$ is increasing and converging to $\omega \left(\mathrm{s}\right)$ so that ${\mu }_{\mathrm{n}}\left(\mathrm{s}\right)\le \omega \left(\mathrm{s}\right),\forall \mathrm{n}\in \mathbb{N}$ and $\forall \mathrm{s}\in \mathcal{I}$. From (18), we attain $({\mu }_{\mathrm{n}},\omega )\in \zeta ,\forall \mathrm{n}\in \mathbb{N}$. Therefore, $\zeta$ remains $\sigma$-self-closed.

(e)

Take $(\mu ,\omega )\in \zeta$ verifying either $\mu \ne \mathcal{L}\left(\mu \right)$, or $\omega \ne \mathcal{L}\left(\omega \right)$. Employing (iii), (17) and (19), we arrive at

$$\begin{array}{ccc}\hfill \sigma (\mathcal{L}\mu ,\mathcal{L}\omega )& =& \underset{\mathrm{s}\in \mathcal{I}}{sup}|\left(\mathcal{L}\mu \right)\left(\mathrm{s}\right)-\left(\mathcal{L}\omega \right)\left(\mathrm{s}\right)|=\underset{\mathrm{s}\in \mathcal{I}}{sup}[\left(\mathcal{L}\omega \right)\left(\mathrm{s}\right)-\left(\mathcal{L}\mu \right)\left(\mathrm{s}\right)]\hfill \\ & =& \underset{\mathrm{s}\in \mathcal{I}}{sup}{\int }_a^b\pounds (\mathrm{s},\vartheta )[\hslash (\vartheta ,\omega \left(\vartheta \right))-\hslash (\vartheta ,\mu \left(\vartheta \right))]d\vartheta \hfill \\ & \le & \underset{\mathrm{s}\in \mathcal{I}}{sup}{\int }_a^b\pounds (\mathrm{s},\vartheta ){\displaystyle \frac{1}{\gamma }}\theta (\omega \left(\vartheta \right)-\mu \left(\vartheta \right))d\vartheta .\hfill \end{array}$$

(21)

As $\theta$ remains increasing and $0\le \omega \left(\vartheta \right)-\mu \left(\vartheta \right)\le \sigma (\mu ,\omega )$, we find $\theta \left(\omega \right(\vartheta )-\mu (\vartheta \left)\right)\le \theta \left(\sigma \right(\mu ,\omega \left)\right)$. Thereby, (21) derives

$$\begin{array}{ccc}\hfill \sigma (\mathcal{L}\mu ,\mathcal{L}\omega )& \le & {\displaystyle \frac{1}{\gamma }}\theta \left(\sigma (\mu ,\omega )\right)\underset{\mathrm{s}\in \mathcal{I}}{sup}{\int }_a^b\pounds (\mathrm{s},\vartheta )d\vartheta \hfill \\ & \le & {\displaystyle \frac{1}{\gamma }}\theta \left(\sigma (\mu ,\omega )\right).\gamma \hfill \\ & =& \theta \left(\sigma \right(\mu ,\omega \left)\right)\hfill \end{array}$$

i.e.,

$$\sigma (\mathcal{L}\mu ,\mathcal{L}\omega )\le \sigma (\mu ,\omega )-\left[\sigma \right(\mu ,\omega )-\theta (\sigma (\mu ,\omega )\left)\right].$$

(22)

Define $\varphi \in \Phi$ and $\psi \in \Psi$ by

$$\varphi \left(\mathrm{t}\right)=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\psi \left(\mathrm{t}\right)=\mathrm{t}-\theta \left(\mathrm{t}\right).$$

Henceforth, (22) becomes

$$\varphi \left(\sigma \right(\mathcal{L}\mu ,\mathcal{L}\omega \left)\right)\le \varphi \left(\sigma \right(\mu ,\omega \left)\right)-\psi \left(\sigma \right(\mu ,\omega \left)\right).$$

Take any arbitrary $\mu ,\omega \in \mathbf{S}$. Denote $\nu :=max\{\mathcal{L}\mu ,\mathcal{L}\omega \}\in \mathbf{S}$. Then, we conclude that $(\mathcal{L}\mu ,\nu )\in \zeta$ and $(\mathcal{L}\omega ,\nu )\in \zeta$. Hence, $\mathcal{L}\left(\mathbf{S}\right)$ remains ${\zeta }^s$-directed. It turns out that employing Theorem 3, $\mathcal{L}$ admits a unique fixed point that serves as a unique solution of (16). □

Theorem 5.

In collaboration with circumstances (i)–(iv) of Theorem 4, problem (16) admits a unique solution provided it possesses an upper solution.

Proof. 

On $\mathbf{S}:=\mathcal{C}\left(\mathcal{I}\right)$, let us define a metric $\sigma$ and a self-map $\mathcal{L}$ as defined in the proof of Theorem 4. Consider the following BR ${\zeta }^{\prime }$ on $\mathbf{S}$:

$${\zeta }^{\prime }=\{(\mu ,\omega )\in {\mathbf{S}}^2:\mu \left(\mathrm{s}\right)\ge \omega \left(\mathrm{s}\right),\forall \mathrm{s}\in \mathcal{I}\}.$$

(23)

We will confirm all presumptions of Theorems 2 and 3.

(a)

$(\mathbf{S},\sigma )$, being a complete MS, is ${\zeta }^{\prime }$-complete.

(b)

If $\overline{\mu }\in \mathbf{S}$ serves as upper solution of (16), then

$$\overline{\mu }\left(\mathrm{s}\right)\ge \mathsf{ϝ}\left(\mathrm{s}\right)+{\int }_a^b\pounds (\mathrm{s},\vartheta )\hslash (\vartheta ,\overline{\mu }\left(\vartheta \right))d\vartheta =\left(\mathcal{L}\overline{\mu }\right)\left(\mathrm{s}\right)$$

yielding thereby $(\overline{\mu },\mathcal{L}\overline{\mu })\in {\zeta }^{\prime }$.

(c)

Take $\mu ,\omega \in \mathbf{S}$ verifying $(\mu ,\omega )\in {\zeta }^{\prime }$. From (iii), we find

$$\hslash (\mathrm{s},\mu (\vartheta \left)\right)-\hslash (\mathrm{s},\omega (\vartheta \left)\right)\ge 0,\forall \mathrm{s},\vartheta \in \mathcal{I}.$$

(24)

By (19), (20) and (ii), we conclude that

$$\begin{array}{c}\hfill \left(\mathcal{L}\mu \right)\left(\mathrm{s}\right)-\left(\mathcal{L}\omega \right)\left(\mathrm{s}\right)={\int }_a^b\pounds (\mathrm{s},\vartheta )[\hslash (\vartheta ,\mu \left(\vartheta \right))-\hslash (\vartheta ,\omega \left(\vartheta \right))]d\vartheta \ge 0,\end{array}$$

i.e., $\left(\mathcal{L}\mu \right)\left(\mathrm{s}\right)\ge \left(\mathcal{L}\omega \right)\left(\mathrm{s}\right)$. By (23), we find that $(\mathcal{L}\mu ,\mathcal{L}\omega )\in {\zeta }^{\prime }$. Thus, ${\zeta }^{\prime }$ is $\mathcal{L}$-closed.

(d)

Let $\left\{{\mu }_{\mathrm{n}}\right\}\subset \mathbf{S}$ be a ${\zeta }^{\prime }$-preserving sequence and ${\mu }_{\mathrm{n}}\to \omega \in \mathbf{S}$. Then for each $\mathrm{s}\in \mathcal{I}$, $\left\{{\mu }_{\mathrm{n}}\left(\mathrm{s}\right)\right\}\subset \mathbb{R}$ is decreasing and converging to $\omega \left(\mathrm{s}\right)$ so that ${\mu }_{\mathrm{n}}\left(\mathrm{s}\right)\ge \omega \left(\mathrm{s}\right),\forall \mathrm{n}\in \mathbb{N}$ and $\forall \mathrm{s}\in \mathcal{I}$. From (18), we attain $({\mu }_{\mathrm{n}},\omega )\in {\zeta }^{\prime },\forall \mathrm{n}\in \mathbb{N}$. Therefore, ${\zeta }^{\prime }$ remains $\sigma$-self-closed.

(e)

Take $(\mu ,\omega )\in {\zeta }^{\prime }$ verifying either $\mu \ne \mathcal{L}\left(\mu \right)$, or $\omega \ne \mathcal{L}\left(\omega \right)$. Employing (iii), (17) and (19), we arrive at

$$\begin{array}{ccc}\hfill \sigma (\mathcal{L}\mu ,\mathcal{L}\omega )& =& \underset{\mathrm{s}\in \mathcal{I}}{sup}|\left(\mathcal{L}\mu \right)\left(\mathrm{s}\right)-\left(\mathcal{L}\omega \right)\left(\mathrm{s}\right)|=\underset{\mathrm{s}\in \mathcal{I}}{sup}[\left(\mathcal{L}\mu \right)\left(\mathrm{s}\right)-\left(\mathcal{L}\omega \right)\left(\mathrm{s}\right)]\hfill \\ & =& \underset{\mathrm{s}\in \mathcal{I}}{sup}{\int }_a^b\pounds (\mathrm{s},\vartheta )[\hslash (\vartheta ,\mu \left(\vartheta \right))-\hslash (\vartheta ,\omega \left(\vartheta \right))]d\vartheta \hfill \\ & \le & \underset{\mathrm{s}\in \mathcal{I}}{sup}{\int }_a^b\pounds (\mathrm{s},\vartheta ){\displaystyle \frac{1}{\gamma }}\theta (\mu \left(\vartheta \right)-\omega \left(\vartheta \right))d\vartheta .\hfill \end{array}$$

(25)

As $\theta$ remains increasing and $0\le \mu \left(\vartheta \right)-\omega \left(\vartheta \right)\le \sigma (\mu ,\omega )$, we find that $\theta \left(\mu \right(\vartheta )-\omega (\vartheta \left)\right)\le \theta \left(\sigma \right(\mu ,\omega \left)\right)$. Thereby, (25) derives

$$\begin{array}{ccc}\hfill \sigma (\mathcal{L}\mu ,\mathcal{L}\omega )& \le & {\displaystyle \frac{1}{\gamma }}\theta \left(\sigma (\mu ,\omega )\right)\underset{\mathrm{s}\in \mathcal{I}}{sup}{\int }_a^b\pounds (\mathrm{s},\vartheta )d\vartheta \hfill \\ & \le & {\displaystyle \frac{1}{\gamma }}\theta \left(\sigma (\mu ,\omega )\right).\gamma \hfill \\ & =& \theta \left(\sigma \right(\mu ,\omega \left)\right)\hfill \end{array}$$

so that

$$\sigma (\mathcal{L}\mu ,\mathcal{L}\omega )\le \sigma (\mu ,\omega )-\left[\sigma \right(\mu ,\omega )-\theta (\sigma (\mu ,\omega )\left)\right].$$

(26)

Define $\varphi \in \Phi$ and $\psi \in \Psi$ by

$$\varphi \left(\mathrm{t}\right)=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\psi \left(\mathrm{t}\right)=\mathrm{t}-\theta \left(\mathrm{t}\right).$$

Henceforth, (26) becomes

$$\varphi \left(\sigma \right(\mathcal{L}\mu ,\mathcal{L}\omega \left)\right)\le \varphi \left(\sigma \right(\mu ,\omega \left)\right)-\psi \left(\sigma \right(\mu ,\omega \left)\right).$$

Take any arbitrary $\mu ,\omega \in \mathbf{S}$. Denote $\nu :=max\{\mathcal{L}\mu ,\mathcal{L}\omega \}\in \mathbf{S}$. Then, we conclude that $(\nu ,\mathcal{L}\mu )\in {\zeta }^{\prime }$ and $(\nu ,\mathcal{L}\omega )\in {\zeta }^{\prime }$. Hence, $\mathcal{L}\left(\mathbf{S}\right)$ remains ${\zeta }^{\prime s}$-directed. It turns out that employing Theorem 3, $\mathcal{L}$ admits a unique fixed point that serves a unique solution of (16). □

7. Conclusions

We have evidenced the accuracy of fixed points and their uniqueness for expanded $(\varphi ,\psi )$-contractions in an MS employing a locally finitely $\mathcal{L}$-transitive BR. The study we conducted broadened, refined, and combined a number of outcomes on fixed points, notably those by Alam et al. [13], Sk et al. [14], Hossain et al. [15] and Pant [24]. In our findings, the contraction inequality merely holds for the comparison elements. For evidence of the findings of our study, we offered a few instances. We additionally adapted our outcomes to a specific nonlinear FIE to draw attention to the generality of the theory and the breadth of our discoveries.

Our findings can be extrapolated to possible future studies in the following ways:

• Generalizing our outcomes to dislocated space, symmetric space, quasi-metric space, fuzzy MS, etc., with a BR;
• Strengthening the properties of functions $\varphi$ and $\psi$;
• Extending these findings for two maps by proving results on common fixed points and coincidence points;
• Implementing our findings in nonlinear matrix equations or periodic boundary value problems.