Almost Nonlinear Contractions of Pant Type Employing Locally Finitely Transitive Relations with an Application to Nonlinear Integral Equations

Abstract

In this research, a few metrical fixed-point outcomes consisting of an almost nonlinear Pant-type contraction employing a locally finitely transitive relation have been established. The findings of our research extrapolate, unify, develop, and improve a number of previously mentioned results. In the present investigation, we formulate a fixed-point finding for almost nonlinear Pant-type contractions in abstract metric space. To assist our study, we formulate numerous examples to illustrate our outcomes. Using our findings, we describe the existence and uniqueness of solutions to a nonlinear Fredholm integral equation.

1. Introduction

Nonlinear functional analysis relies extensively on metric fixed-point theory, which has advantages arising from its uses in many various disciplines. The field of metric fixed-point theory formed in 1922 with investigation of classical BCP. Indeed, the BCP is one of the most notable outcomes on fixed points, which even motivates researchers studying metric fixed-point theory presently.

The relational variation in BCP was designed by Alam and Imdad [1]. As relational contractions are connected by a BR, they are actually significantly more general than usual contractions. A major feature of relational contractions lies in the fact that just comparative elements should satisfy the contraction criterion instead of all element pairings. In recognition of this fact, multiple kinds of boundary value problems, nonlinear matrix equations, and nonlinear integral equations can be resolved through the findings involving relational contractions, whereas the findings on fixed points of abstract MS cannot be applied. Consequently, numerous findings are established in this regard, e.g., [2,3,4,5,6,7,8,9,10].

In 2004, Berinde [11] proposed an inventive extension of the BCP, which is often referred to as an “almost contraction.”

Definition 1

([11]). A map $\mathcal{P}$ from an MS $(\mathbf{M},\mu )$ into itself is termed an almost contraction if $\exists 0<a<1$ and $l\in {\mathbb{R}}^{+}$ that verify

$$\mu (\mathcal{P}u,\mathcal{P}w)\le a·\mu (u,w)+l·\mu (u,\mathcal{P}w),\forall u,w\in \mathbf{M}.$$

The symmetric property of $\mu$ enables the foregoing contraction inequality condition to be identical to the following one.

$$\mu (\mathcal{P}u,\mathcal{P}w)\le a·\mu (u,w)+l·\mu (w,\mathcal{P}u),\forall u,w\in \mathbf{M}.$$

Theorem 1

([11]). An almost contraction on a CMS owns a fixed point.

Although an almost contraction $\mathcal{P}$ generally is not continuous, it is on the set $Fix\left(\mathcal{P}\right)$ (c.f. [12]). Apart from the usual contraction, numerous well-known generalized contractions are also extended by almost contractions. For further generalizations of almost contractions, we refer to [13,14,15,16,17]. Khan [18], Khan et al. [19], and Algehyne and Khan [20] extensively established some outcomes on fixed points over almost contractions in setup of relational MS.

The below-mentioned class of almost contractions has been suggested by Babu et al. [21] to strengthen the uniqueness theorem.

Definition 2

([21]). A map $\mathcal{P}$ from an MS $(\mathbf{M},\mu )$ into itself is termed a strict almost contraction if $\exists 0<a<1$ and $l\in {\mathbb{R}}^{+}$ such that

$$\mu (\mathcal{P}u,\mathcal{P}w)\le a·\mu (u,w)+l·min\left\{\mu \right(u,\mathcal{P}u),\mu (w,\mathcal{P}w),\mu (u,\mathcal{P}w),\mu (w,\mathcal{P}u\left)\right\},\forall u,w\in \mathbf{M}.$$

It is nevertheless evident that any strict almost contraction is almost contraction. Typically, opposite situation is not valid, as shown by Example $2.6$ [21].

Theorem 2

([21]). Every strict almost contraction on a CMS owns a unique fixed point.

Turinici [22] (eventually, Alfuraidan et al. [23]) formed a nonlinear simulation of an almost contraction by inserting an auxiliary function $\varrho$ for the Lipschitz constant l. In actuality, the following class of functions was considered by Alfuraidan et al. [23].

$$\Omega =\{\varrho :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}:\underset{\mathsf{p}\to 0^{+}}{lim}\varrho \left(\mathsf{p}\right)=\theta \left(0\right)=0\}.$$

A while ago, the next non-unique fixed-point outcome was established by Pant [24].

Theorem 3.

Let $\mathcal{P}$ be a map from a CMS $(\mathbf{M},\mu )$ into itself such that $\exists 0\le a<1$ such that

$$\begin{array}{c}\hfill \mu (\mathcal{P}u,\mathcal{P}w)\le a·\mu (u,w),\forall u,w\in \mathbf{V}with[u\ne \mathcal{P}(u)orw\ne \mathcal{P}(w\left)\right].\end{array}$$

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

Afterwards, Pant [25] improved Theorem 3 over $\mathrm{\Phi }$-contraction.

A few findings on fixed points of a relational nonlinear contraction map were investigated by Alam et al. [4], employing the locally finitely $\mathcal{P}$-transitive BR. They adopted the following class of control functions:

$$\mathrm{\Psi }=\{\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}:\mathsf{p}>0\Rightarrow \psi \left(\mathsf{p}\right)<\mathsf{p}\mathrm{and}\underset{r\to \mathsf{p}}{lim\; sup}\psi \left(\mathsf{r}\right)<\mathsf{p}\}.$$

Quiet recently, Alshaban et al. [26] proved fixed-point outcomes over almost nonlinear contractions of Pant type under an arbitrary BR. The outcomes of Alshaban et al. [26] are further improved by Filali et al. [27] and Filali and Khan [28] for expanded contraction conditions under the restricted class of BRs (namely, locally $\mathcal{P}$-transitive BRs).

The foremost intention of the present analysis is to expand the class of nonlinear contractions for investigating certain novel analogs of the findings of Alshaban et al. [26]. In the hypotheses of our existing findings, the underlying BR is required to be locally $\mathcal{P}$-transitive and $\mathcal{P}$-closed. Meanwhile, to obtain the uniqueness result, an additional presumption ($\mathbf{R}$-directedness property) is required. We provide a number of scenarios that demonstrate the effectiveness of our outcomes. Using our findings, we derive some classical outcomes on fixed points, especially those of Alam et al. [4], Berinde [11], Khan et al. [19], Babu et al. [21], Turinici [22], Pant [25], and similar others. Our findings allow us to pick a unique solution to a specific (nonlinear) Fredholm integral equation.

2. Preliminaries

Recall that a subset of ${\mathbf{M}}^2$ is named a BR on the set $\mathbf{M}$. As outlined below, let $\mathbf{M}$ be a set, $\mu$ a metric on $\mathbf{M}$, $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ a map, and $\mathbf{R}$ a BR on $\mathbf{M}$. We say the following.

Definition 3

([1]). Two points $u,w\in \mathbf{M}$ are $\mathit{R}$-comparative if $(u,w)\in \mathit{R}$ or $(w,u)\in \mathit{R}$. Such a pair is indicated by $[u,w]\in \mathit{R}$.

Definition 4

([29]). The BR ${\mathit{R}}^{-1}:=\{(u,w)\in {\mathbf{M}}^2:(w,u)\in \mathit{R}\}$ is inverse of $\mathit{R}$. Also, the BR ${\mathit{R}}^s:=\mathit{R}\cup {\mathit{R}}^{-1}$ is a symmetric closure of $\mathit{R}$.

Remark 1

([1]). $(u,w)\in {\mathit{R}}^s⟺[u,w]\in \mathit{R}.$

Definition 5

([1]). $\mathit{R}$ is $\mathcal{P}$-closed BR if for all $(u,w)\in \mathit{R}$,

$$(\mathcal{P}u,\mathcal{P}w)\in \mathit{R}.$$

Proposition 1

([3]). If $\mathit{R}$ is $\mathcal{P}$-closed, then $\mathit{R}$ is ${\mathcal{P}}^n$-closed for all $n\in {\mathbb{N}}_0$.

Definition 6

([1]). A sequence $\left\{u_n\right\}\subset \mathbf{M}$ with the property $(u_n,u_{n+1})\in \mathit{R}$, $\mathrm{\forall }n\in \mathbb{N}$ is $\mathit{R}$-preserving.

Definition 7

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

$${\mathit{R}|}_{\mathbf{S}}:=\mathit{R}\cap {\mathbf{S}}^2,$$

(on $\mathbf{S}$) is a restriction of $\mathit{R}$ in $\mathbf{S}$.

Definition 8

([3]). $\mathit{R}$ is locally $\mathcal{P}$-transitive if for any $\mathit{R}$-preserving sequence $\left\{w_n\right\}\subset \mathcal{P}\left(\mathbf{M}\right)$, ${\mathit{R}|}_{\mathcal{Z}}$ remains transitive, where $\mathcal{Z}=\{w_n:n\in \mathbb{N}\}$.

Definition 9

([31]). Given $\nu \in \mathbb{N}-\left\{1\right\}$, $\mathit{R}$ is ν-transitive if for any $u_0,u_1,\dots ,u_{\nu }\in \mathbf{M}$,

$$(u_{i-1},u_i)\in \mathit{R}\mathrm{for}\mathrm{each}i(1\le i\le \nu )\Rightarrow (u_0,u_{\nu })\in \mathit{R}.$$

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

Definition 10

([32]). $\mathit{R}$ is finitely transitive if we can determine $\nu \in \mathbb{N}-\left\{1\right\}$, whereas $\mathit{R}$ is ν-transitive.

Definition 11

([4]). $\mathit{R}$ is locally finitely $\mathcal{P}$-transitive if for any $\mathit{R}$-preserving sequence $\left\{w_n\right\}\subset \mathcal{P}\left(\mathbf{M}\right)$ with range $\mathcal{Z}=\{w_n:n\in \mathbb{N}\}$, ${\mathit{R}|}_{\mathcal{Z}}$ remains finitely transitive.

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

Definition 12

([1]). $\mathit{R}$ is μ-self-closed if each $\mathit{R}$-preserving convergent sequence in $\mathbf{M}$ contains a subsequence with terms that are $\mathit{R}$-comparative with the convergence limit.

Definition 13

([2]). The MS $(\mathbf{M},\mu )$ is $\mathit{R}$-complete if any $\mathit{R}$-preserving Cauchy sequence in $\mathbf{M}$ is convergent.

A CMS is obviously $\mathbf{R}$-complete MS. In particular, for $\mathbf{R}={\mathbf{M}}^2$ both concepts coincide.

Definition 14

([2]). The map $\mathcal{P}$ is $\mathit{R}$-continuous if for every $u\in \mathbf{M}$ and for any $\mathit{R}$-preserving sequence $\left\{u_n\right\}\subset \mathbf{M}$ with $u_n\stackrel{\mu }{⟶}u$,

$$\mathcal{P}\left(u_n\right)\stackrel{\mu }{⟶}\mathcal{P}\left(u\right).$$

Clearly, each continuous map is $\mathbf{R}$-continuous. In particular, for $\mathbf{R}={\mathbf{M}}^2$ both concepts coincide.

Definition 15

([33]). A subset $\mathbf{S}\subseteq \mathbf{M}$ is $\mathit{R}$-directed if for all $u,w\in \mathbf{S}$, $\exists v\in \mathbf{M}$ verifying $(u,v)\in \mathit{R}$ and $(w,v)\in \mathit{R}$.

Definition 16

([22]). A sequence $\left\{u_n\right\}$ in an MS $(\mathbf{M},\mu )$ is semi-Cauchy if

$$\underset{n\to \infty }{lim}\mu (u_n,u_{n+1})=0.$$

Any Cauchy sequence is actually semi-Cauchy.

Lemma 1

([31]). Let $\left\{u_n\right\}$ be a non-Cauchy sequence in an MS $(\mathbf{M},\mu )$; then $\exists {ϵ}_0>0$ and subsequences $\left\{u_{n_i}\right\}$ and $\left\{u_{m_i}\right\}$ of $\left\{u_n\right\}$ with

(i)

  $i\le m_i<n_{i}foralli\in \mathbb{N}$;

(ii)

 $\mu (u_{m_i},u_{n_i})\ge {ϵ}_{0}foralli\in \mathbb{N}$;

(iii)

$\mu (u_{m_i},u_{{\nu }_i})<{ϵ}_0$ for all ${\nu }_i\in \{m_i+1,m_i+2,\dots ,n_i-2,n_i-1\}$.

Moreover, if $\underset{n\to \infty }{lim}\mu (u_n,u_{n+1})=0$, then

$$\underset{i\to \infty }{lim}\mu (u_{m_i},u_{n_i+\eta })={ϵ}_0,\forall \eta \in {\mathbb{N}}_0.$$

Lemma 2

([32]). Let $\mathbf{M}$ be a set composed with a BR $\mathit{R}$. Assume that $\left\{u_n\right\}\subset \mathbf{M}$ is an $\mathbf{R}$-preserving sequence and $\mathit{R}$ is an ν-transitive on $\mathcal{Z}=\{u_n:n\in {\mathbb{N}}_0\}$; then

$$(u_n,u_{n+1+\lambda (\nu -1)})\in \mathit{R},\forall n,\lambda \in {\mathbb{N}}_0.$$

Proposition 2.

Given $\psi \in \Psi$ and $\varrho \in \Omega$, (A) and (B) are equivalent:

(A)

$\mu (\mathcal{P}u,\mathcal{P}w)\le \psi \left(\mu \right(u,w\left)\right)+min\left\{\varrho \right(\mu (u,\mathcal{P}u)),\varrho (\mu (w,\mathcal{P}w)),\varrho (\mu (u,\mathcal{P}w)),\varrho (\mu (w,\mathcal{P}u)\left)\right\},$

$\forall u,w\in \mathbf{M}with(u,w)\in \mathbf{R}.$

(B)

$\mu (\mathcal{P}u,\mathcal{P}w)\le \psi \left(\mu \right(u,w\left)\right)+min\left\{\varrho \right(\mu (u,\mathcal{P}u)),\varrho (\mu (w,\mathcal{P}w)),\varrho (\mu (u,\mathcal{P}w)),\varrho (\mu (w,\mathcal{P}u)\left)\right\},$

$\forall u,w\in \mathbf{M}with[u,w]\in \mathbf{R}.$

Proof. 

The conclusion (B)⇒(A) is straightforward. On the contrary, assume that (A) is valid. Suppose that $u,w\in \mathbf{M}$ with $[u,w]\in \mathbf{R}$. Then, in case $(u,w)\in \mathbf{R}$, (A) yields (B). Otherwise, we have $(w,u)\in \mathbf{R}$. In this case, using symmetric property of metric $\mu$ and (A), we conclude that

$$\begin{array}{ccc}\hfill \mu (\mathcal{P}u,\mathcal{P}w)& =& \mu (\mathcal{P}w,\mathcal{P}u)\hfill \\ & \le & \psi \left(\mu \right(w,u\left)\right)+min\left\{\varrho \right(\mu (w,\mathcal{P}w)),\varrho (\mu (u,\mathcal{P}u)),\varrho (\mu (w,\mathcal{P}u)),\varrho (\mu (u,\mathcal{P}w)\left)\right\}\hfill \\ & =& \psi \left(\mu \right(u,w\left)\right)+min\left\{\varrho \right(\mu (u,\mathcal{P}u)),\varrho (\mu (w,\mathcal{P}w)),\varrho (\mu (u,\mathcal{P}w)),\varrho (\mu (w,\mathcal{P}u)\left)\right\}.\hfill \end{array}$$

It follows that (A)⇒(B). □

3. Main Results

We furnish the following outcomes on fixed points for a relational almost nonlinear contraction of the Pant type.

Theorem 4.

Let $(\mathbf{M},\mu )$ be an MS, $\mathit{R}$ a BR on $\mathbf{M}$, and $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ a map. Also,

(a) 

$(\mathbf{M},\mu )$ is $\mathit{R}$-complete;

(b) 

$(u_0,\mathcal{P}{u}_0)\in \mathit{R}$, for some $u_0\in \mathbf{M}$;

(c) 

$\mathit{R}$ remains locally finitely $\mathcal{P}$-transitive and $\mathcal{P}$-closed;

(d) 

$\mathbf{M}$ remains $\mathit{R}$-continuous, or $\mathit{R}$ is μ-self-closed;

(e) 

$\exists \psi \in \mathrm{\Psi }$ and $\varrho \in \Omega$ holds.

$$\begin{array}{c}\hfill \mu (\mathcal{P}u,\mathcal{P}w)\le \psi \left(\mu \right(u,w\left)\right)+min\left\{\varrho \right(\mu (u,\mathcal{P}u)),\varrho (\mu (w,\mathcal{P}w)),\varrho (\mu (u,\mathcal{P}w)),\varrho (\mu (w,\mathcal{P}u)\left)\right\},\\ \hfill \forall (u,w)\in \mathit{R}with[u\ne \mathcal{P}(u)orw\ne \mathcal{P}(w\left)\right].\end{array}$$

Then, $\mathcal{P}$ possess a fixed point.

Proof. 

We intend to accomplish the proof in six step.

• Step I. Consider a sequence $\left\{u_{\mathrm{n}}\right\}\subset \mathbf{M}$ of Picard iteration starting with $u_0\in \mathbf{M}$; i.e.,

  $$u_{\mathrm{n}}:={\mathcal{P}}^{\mathrm{n}}\left(u_0\right)=\mathcal{P}\left(u_{\mathrm{n}-1}\right),\forall \mathrm{n}\in \mathbb{N}.$$

  (1)
• Step II. We demonstrate that the sequence $\left\{u_{\mathrm{n}}\right\}$ is $\mathbf{R}$-preserving. Owing to supposition $\left(b\right)$, the $\mathcal{P}$-closedness of $\mathbf{R}$, and Proposition 1, we attain

  $$({\mathcal{P}}^{\mathrm{n}}{u}_0,{\mathcal{P}}^{\mathrm{n}+1}{u}_0)\in \mathbf{R},$$

  which, using (1), becomes

  $$(u_{\mathrm{n}},u_{\mathrm{n}+1})\in \mathbf{R},\forall \mathrm{n}\in {\mathbb{N}}_0.$$

  (2)

• Step III. Define ${\mu }_{\mathrm{n}}:=\mu (u_{\mathrm{n}},u_{\mathrm{n}+1})$. If $\exists {\mathrm{n}}_0\in {\mathbb{N}}_0$ for which ${\mu }_{{\mathrm{n}}_0}=0$, then from (1) we obtain $u_{{\mathrm{n}}_0}=u_{{\mathrm{n}}_0+1}=\mathcal{P}\left(u_{{\mathrm{n}}_0}\right)$; therefore $u_{n_0}$ remains a fixed point of $\mathcal{P}$ and hence we have finished. Otherwise, we attain ${\mu }_{\mathrm{n}}>0$, ∀$\mathrm{n}\in {\mathbb{N}}_0$ so that we continue to Step 4.
• Step IV. We demonstrate that $\left\{u_{\mathrm{n}}\right\}$ is a semi-Cauchy sequence; i.e., $\underset{\mathrm{n}\to \infty }{lim}\mu (u_{\mathrm{n}},u_{\mathrm{n}+1})=0$. From assumption $\left(e\right)$, (1), and (2), we get

  $$\begin{array}{ccc}\hfill {\mu }_{\mathrm{n}}& =& \mu (u_{\mathrm{n}},u_{\mathrm{n}+1})=\mu (\mathcal{P}{u}_{\mathrm{n}-1},\mathcal{P}{u}_{\mathrm{n}})\hfill \\ & \le & \psi \left(\mu (u_{\mathrm{n}-1},u_{\mathrm{n}})\right)+min\{\varrho \left(\mu (u_{\mathrm{n}-1},u_{\mathrm{n}})\right),\varrho \left(\mu (u_{\mathrm{n}},u_{\mathrm{n}+1})\right),\varrho \left(\mu (u_{\mathrm{n}-1},u_{\mathrm{n}+1})\right),\varrho \left(0\right)\},\hfill \end{array}$$

  which, using the property of $\varrho$, reduces to

  $$\begin{array}{c}\hfill {\mu }_{\mathrm{n}}\le \psi \left({\mu }_{\mathrm{n}-1}\right),\forall \mathrm{n}\in {\mathbb{N}}_0.\end{array}$$

  (3)

By virtue of $\psi$ in (3), we find

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

Hence, the sequence $\left\{{\mu }_{\mathrm{n}}\right\}$ remains decreasing in ${\mathbb{R}}^{+}$. Now, $\left\{{\mu }_{\mathrm{n}}\right\}$ being bounded below confirms the existence of $\overline{\mu }\ge 0$, verifying

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

(4)

Now, we show that $\overline{\mu }=0.$ If $\overline{\mu }>0$ then using the limit superior in (3) and by (4) and virtue of $\psi$, we get

$$\overline{\mu }=\underset{\mathrm{n}\to \infty }{lim\; sup}{\mu }_{\mathrm{n}}\le \underset{\mathrm{n}\to \infty }{lim\; sup}\psi \left({\mu }_{\mathrm{n}-1}\right)=\underset{{\mu }_{\mathrm{n}}\to l^{+}}{lim\; sup}\psi \left({\mu }_{\mathrm{n}-1}\right)<\overline{\mu },$$

which produces a contradiction. Therefore, we infer $\overline{\mu }=0$ so that

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

(5)

• Step V. We demonstrate that $\left\{u_{\mathrm{n}}\right\}$ is Cauchy. In contrast, let $\left\{u_{\mathrm{n}}\right\}$ be not Cauchy. By Lemma 1, $\exists {ϵ}_0>0$ and subsequences $\left\{u_{{\mathrm{n}}_{\mathrm{i}}}\right\}$ and $\left\{u_{{\mathrm{m}}_{\mathrm{i}}}\right\}$ of $\left\{u_{\mathrm{n}}\right\}$ that satisfy

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

Using (5) and Lemma 1, we conclude that

$$\begin{array}{c}\hfill \underset{\mathrm{i}\to \infty }{lim}\mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{n}}_{\mathrm{i}}+\eta })={ϵ}_0,\mathrm{for}\mathrm{all}\eta \in {\mathbb{N}}_0.\end{array}$$

(6)

From (1), we conclude that $\mathcal{Z}:=\{u_{\mathrm{n}}:\mathrm{n}\in {\mathbb{N}}_0\}\subset \mathcal{P}\left(\mathbf{M}\right)$. Owing to the locally finitely $\mathcal{P}$-transitiveness of $\mathbf{R}$, $\exists \nu \in \{2,3,\cdots \}$, verifying that ${\mathbf{R}|}_{\mathcal{Z}}$ is $\nu$-transitive.

Since ${\mathrm{m}}_{\mathrm{i}}<{\mathrm{n}}_{\mathrm{i}}$ and $\nu -1>0$, the division algorithm yields

$${\mathrm{n}}_{\mathrm{i}}-{\mathrm{m}}_{\mathrm{i}}=(\nu -1)({\alpha }_{\mathrm{i}}-1)+(\nu -{\beta }_{\mathrm{i}})$$

$${\alpha }_{\mathrm{i}}-1\ge 0,0\le \nu -{\beta }_{\mathrm{i}}<\nu -1$$

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

As ${\beta }_{\mathrm{i}}\in (1,\nu ]$, the subsequences $\left\{u_{{\mathrm{n}}_{\mathrm{i}}}\right\}$ and $\left\{u_{{\mathrm{m}}_{\mathrm{i}}}\right\}$ of $\left\{u_{\nu }\right\}$ (verifying (6)) can therefore be selected in a manner that ensures ${\beta }_{\mathrm{i}}=\beta$ as a constant. Hence, we get

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

(7)

Using (6) and (7), we find

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

(8)

Using the triangular inequality, we find

$$\mu (u_{{\mathrm{m}}_{\mathrm{i}}+1},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1})\le \mu (u_{{\mathrm{m}}_{\mathrm{i}}+1},u_{{\mathrm{m}}_{\mathrm{i}}})+\mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})+\mu (u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1})$$

and

$$\mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})\le \mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}+1})+\mu (u_{{\mu }_{\mathrm{i}+1}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1})+\mu (u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }}).$$

Therefore, we conclude that

$$\begin{array}{c}\mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})-\mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}+1})-\mu (u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})\le \mu (u_{{\mu }_{\mathrm{i}+1}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1})\hfill \\ \hfill \le \mu (u_{{\mathrm{m}}_{\mathrm{i}}+1},u_{{\mathrm{m}}_{\mathrm{i}}})+\mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})+\mu (u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1}).\end{array}$$

Letting $\mathrm{i}\to \infty$ and using (5) and (13)), the last inequality reduces to

$$\begin{array}{c}\hfill \underset{\mathrm{i}\to \infty }{lim}\mu (u_{{\mathrm{m}}_{\mathrm{i}}+1},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1})={ϵ}_0.\end{array}$$

(9)

Using (7) and Lemma 1, we obtain $(u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})\in \mathbf{R}.$

Define ${\delta }_{\mathrm{i}}:=\mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})$. Applying contraction condition $\left(e\right)$, we get

$$\begin{array}{ccc}\hfill \mu (u_{{\mathrm{m}}_{\mathrm{i}}+1},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1})& =& \mu (\mathcal{P}{u}_{{\mathrm{m}}_{\mathrm{i}}},\mathcal{P}{u}_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})\le \psi \left(\mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})\right)\hfill \\ & & +min\{\varrho \left(\mu (u_{{\mathrm{m}}_{\mathrm{i}}},\mathcal{P}{u}_{{\mathrm{m}}_{\mathrm{i}}})\right),\varrho \left(\mu (u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }},\mathcal{P}{u}_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})\right),\varrho \left(\mu (u_{{\mathrm{m}}_{\mathrm{i}}},\mathcal{P}{u}_{{\mathrm{m}}_{\mathrm{i}}^{\prime }})\right),\varrho \left(\mu (u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }},\mathcal{P}{u}_{{\mathrm{m}}_{\mathrm{i}}})\right)\}\hfill \end{array}$$

so that

$$\mu (u_{{\mathrm{m}}_{\mathrm{i}}+1},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1})\le \psi \left({\delta }_{\mathrm{i}}\right)+min\{\varrho \left({\mu }_{{\mathrm{m}}_{\mathrm{i}}}\right),\varrho \left({\mu }_{{\mathrm{m}}_{\mathrm{i}}^{\prime }}\right),\varrho \left(\mu (u_{{\mathrm{m}}_{\mathrm{i}}},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1})\right),\varrho \left(\mu (u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }},u_{{\mathrm{m}}_{\mathrm{i}}+1})\right)\}.$$

(10)

From (5) and the feature of $\psi$, we find

$${\displaystyle \underset{\mathrm{i}\to \infty }{lim}\varrho \left({\delta }_{l_{\mathrm{i}}}\right)=\underset{\mathrm{i}\to \infty }{lim}\varrho \left({\mu }_{{\mathrm{n}}_{\mathrm{i}}}\right)=\underset{\mathsf{p}\to 0^{+}}{lim}\varrho \left(\mathsf{p}\right)=0.}$$

(11)

Taking upper limit in (7) and utilizing (11) and by virtue of $\psi$, we get

$${\displaystyle {ϵ}_0=\underset{\mathrm{i}\to \infty }{lim\; sup}\mu (u_{{\mathrm{m}}_{\mathrm{i}}+1},u_{{\mathrm{m}}_{\mathrm{i}}^{\prime }+1})\le \underset{\mathrm{i}\to \infty }{lim\; sup}\psi \left({\delta }_{\mathrm{i}}\right)+0=\underset{s\to {ϵ}_0^{+}}{lim\; sup}\psi \left(s\right)<{ϵ}_0,}$$

which contradicts the axiom of $\psi$. Thus, $\left\{u_{\mathrm{n}}\right\}$ is $\mathbf{R}$-preserving and Cauchy. Now, as $\mathbf{M}$ is $\mathbf{R}$-complete, $\exists \overline{u}\in \mathbf{M}$ with $u_{\mathrm{n}}\stackrel{\mu }{⟶}\overline{u}$.

• Step VI. We show that $\overline{u}\in F\mathrm{i}x\left(\mathcal{P}\right)$ using the assumption $\left(d\right)$. Assuming that $\mathcal{P}$ is $\mathbf{R}$-continuous, then $u_{\mathrm{n}+1}=\mathcal{P}\left(u_{\mathrm{n}}\right)\stackrel{\mu }{⟶}\mathcal{P}\left(\overline{u}\right)$. Thus, we conclude $\mathcal{P}\left(\overline{u}\right)=\overline{u}$.

Now, assuming that $\mathbf{R}$ is $\mu$-self-closed, then $\exists$ a subsequence $\left\{u_{{\mathrm{n}}_{\mathrm{i}}}\right\}$ of $\left\{u_{\mathrm{n}}\right\}$ that satisfies $[u_{{\mathrm{n}}_{\mathrm{i}}},\overline{u}]\in \mathbf{R},\forall \mathrm{i}\in \mathbb{N}.$ Incorporating the supposition $\left(e\right)$, Proposition 2, $[u_{{\mathrm{n}}_{\mathrm{i}}},\overline{u}]\in \mathbf{R}$, and the property of $\varrho$, we conclude that

$$\begin{array}{ccc}\hfill \mu (u_{{\mathrm{n}}_{\mathrm{i}}+1},\mathcal{P}\overline{u})& =& \mu (\mathcal{P}{u}_{{\mathrm{n}}_{\mathrm{i}}},\mathcal{P}\overline{u})\hfill \\ & \le & \psi \left(\mu (u_{{\mathrm{n}}_{\mathrm{i}}},\overline{u})\right)+min\{\varrho \left(\mu (u_{{\mathrm{n}}_{\mathrm{i}}},u_{{\mathrm{n}}_{\mathrm{i}}+1})\right),\varrho \left(0\right),\varrho \left(\mu (u_{{\mathrm{n}}_{\mathrm{i}}},\overline{u})\right),\varrho \left(\mu (\overline{u},u_{{\mathrm{n}}_{\mathrm{i}}+1})\right)\}\hfill \\ & =& \psi \left(\mu (u_{{\mathrm{n}}_{\mathrm{i}}},\overline{u})\right).\hfill \end{array}$$

We shall show that

$$\mu (u_{{\mathrm{n}}_{\mathrm{i}}+1},\mathcal{P}\overline{u})\le \mu (u_{{\mathrm{n}}_{\mathrm{i}}},\overline{u}),\forall \mathrm{i}\in \mathbb{N}.$$

(12)

If $\mu (u_{{\mathrm{n}}_{{\mathrm{i}}_0}},\overline{u})=0$ for some ${\mathrm{i}}_0\in \mathbb{N}$, then we get $\mu (\mathcal{P}{u}_{{\mathrm{n}}_{{\mathrm{i}}_0}},\mathcal{P}\overline{u})=0$, so $\mu (u_{{\mathrm{n}}_{{\mathrm{i}}_0}+1},\mathcal{P}\overline{u})=0$ and (12) pertains to these ${\mathrm{i}}_0\in \mathbb{N}$. In either situation, we conclude that $\mu (u_{{\mathrm{n}}_{\mathrm{i}}},\overline{u})>0,\forall \mathrm{i}\in \mathbb{N}.$ By the property of $\psi$, we conclude that $\mu (u_{{\mathrm{n}}_{\mathrm{i}}+1},\mathcal{P}\overline{u})\le \psi \left(\mu (u_{{\mathrm{n}}_{\mathrm{i}}},\overline{u})\right)<\mu (u_{{\mathrm{n}}_{\mathrm{i}}},\overline{u}),\forall \mathrm{i}\in \mathbb{N}$. Thus, (12) pertains for all $\mathrm{i}\in \mathbb{N}.$ Taking the limit in (12) and using $u_{{\mathrm{n}}_{\mathrm{i}}}\stackrel{\mu }{⟶}\overline{u}$, we conclude that $u_{{\mathrm{n}}_{\mathrm{i}}+1}\stackrel{\mu }{⟶}\mathcal{P}\left(\overline{u}\right)$, so $\mathcal{P}\left(\overline{u}\right)=\overline{u}$. Thus, $\overline{u}$ is a fixed point of $\mathcal{P}$. □

Theorem 5.

In combination with the assumptions of Theorem 4, if $\mathcal{P}\left(\mathbf{M}\right)$ is $\mathit{R}$-directed, then $\mathcal{P}$ admits a unique fixed point.

Proof. 

Using Theorem 4, $\mathcal{P}$ possesses at least one fixed point. Let $\overline{u},\overline{w}\in \mathrm{Fix}\left(\mathcal{P}\right)$; i.e.,

$$\mathcal{P}\left(\overline{u}\right)=\overline{u}\mathrm{and}\mathcal{P}\left(\overline{w}\right)=\overline{w}.$$

(13)

Since $\overline{u},\overline{w}\in \mathcal{P}\left(\mathbf{M}\right)$, due to our assumption, $\exists v\in \mathbf{M}$ verifying

$$(\overline{u},v)\in \mathbf{R}a\mathrm{n}d(\overline{w},v)\in \mathbf{R}.$$

(14)

Denote $D_{\mathrm{n}}:=\mu (\overline{u},{\mathcal{P}}^{\mathrm{n}}v)$. From (13), (14), and assumption $\left(e\right)$, we obtain

$$\begin{array}{ccc}\hfill D_{\mathrm{n}}=\mu (\overline{u},{\mathcal{P}}^{\mathrm{n}}v)& =& \mu (\mathcal{P}\overline{u},\mathcal{P}\left({\mathcal{P}}^{\mathrm{n}-1}v\right))\le \psi \left(\mu (\overline{u},{\mathcal{P}}^{\mathrm{n}-1}v)\right)\hfill \\ & & +min\{\varrho \left(0\right),\varrho \left(\mu ({\mathcal{P}}^{\mathrm{n}-1}v,{\mathcal{P}}^{\mathrm{n}}v)\right),\varrho \left(\mu (\overline{u},{\mathcal{P}}^{\mathrm{n}}v)\right),\varrho \left(\mu ({\mathcal{P}}^{\mathrm{n}-1}v,\overline{u})\right)\}\hfill \\ & =& \psi (D_{\mathrm{n}-1})\hfill \end{array}$$

so that

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

(15)

If $\exists {\mathrm{n}}_0\in \mathbb{N}$ for which $D_{{\mathrm{n}}_0}=0$, then we conclude that $D_{{\mathrm{n}}_0}\le D_{{\mathrm{n}}_0-1}$. Nevertheless, we have $D_{\mathrm{n}}>0,forall\mathrm{n}\in \mathbb{N}$, and applying the property of $\psi$, (15) yields $D_{\mathrm{n}}<D_{\mathrm{n}-1}$. Thereby, in each scenario, we obtain

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

Using arguments similar to those utilized earlier in Theorem 4, the last inequality yields

$${\displaystyle \underset{\mathrm{n}\to \infty }{lim}{D}_{\mathrm{n}}=\underset{\mathrm{n}\to \infty }{lim}\mu (\overline{u},{\mathcal{P}}^{\mathrm{n}}v)=0.}$$

(16)

Similarly, we can get

$${\displaystyle \underset{\mathrm{n}\to \infty }{lim}\mu (\overline{u},{\mathcal{P}}^{\mathrm{n}}v)=0.}$$

(17)

Using (16), (17), and the triangle inequality, we find

$$\mu (\overline{u},\overline{w})=\mu (\overline{u},{\mathcal{P}}^{\mathrm{n}}v)+\mu ({\mathcal{P}}^{\mathrm{n}}v,\overline{w})\to 0,as\mathrm{n}\to \infty .$$

Thus, $\overline{u}=\overline{w}$, so $\mathcal{P}$ admits a unique fixed point. □

4. Illustrative Examples

To describe Theorems 4 and 5, we deal with the following scenarios.

Example 1.

Take the MS $\mathbf{M}={\mathbb{R}}^{+}$ with standard metric μ and the BR $\mathit{R}:=\{(u,w)\in {\mathbf{M}}^2:u-w>0\}$. Draw a mapping $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ by $\mathcal{P}\left(u\right)={\displaystyle \frac{u}{u+1}}$. Then, $\mathit{R}$ is a locally $\mathcal{P}$-transitive and $\mathcal{P}$-closed BR. Also, $(\mathbf{M},\mu )$ is an $\mathit{R}$-complete MS and $\mathcal{P}$ is an $\mathit{R}$-continuous map. Define the auxiliary functions $\psi \left(p\right)={\displaystyle \frac{p}{p+1}}$ and $\varrho \in \Omega$ as arbitrary. Then for all $(u,w)\in \mathit{R}$, we attain

$$\begin{array}{ccc}\hfill \mu (\mathcal{P}u,\mathcal{P}w)& =& |{\displaystyle \frac{w}{u+1}}-{\displaystyle \frac{w}{w+1}}|=\left|{\displaystyle \frac{u-w}{1+u+w+pq}}\right|\hfill \\ & \le & {\displaystyle \frac{u-w}{1+(u-w)}}={\displaystyle \frac{\mu (u,w)}{1+\mu (u,w)}}\hfill \\ & \le & \psi \left(\mu \right(u,w\left)\right)+min\left\{\varrho \right(\mu (u,\mathcal{P}u)),\varrho (\mu (w,\mathcal{P}w)),\varrho (\mu (u,\mathcal{P}w)),\varrho (\mu (w,\mathcal{P}u)\left)\right\}.\hfill \end{array}$$

It follows that the contraction condition $\left(e\right)$ of Theorem 4 holds. Similarly, the rest of the suppositions of Theorems 4 and 5 can be fulfilled. Consequently, $\mathcal{P}$ possesses a unique fixed point ($\overline{u}=0$).

Example 2.

Take the MS $\mathbf{M}=[0,1]$ with standard metric μ and the BR $\mathit{R}:=\le$. Draw the mapping $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ by

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

Then $(0,\mathcal{P}0)\in \mathit{R}$. Clearly, $\mathit{R}$ is a locally finitely $\mathcal{P}$-transitive and $\mathcal{P}$-closed BR. Also, $(\mathbf{M},\mu )$ is an $\mathit{R}$-complete MS.

Let $\left\{u_n\right\}\subset \mathbf{M}$ be an $\mathit{R}$-preserving convergent sequence with $u_n\stackrel{\mu }{⟶}u$. Then $\left\{u_n\right\}$ as an increasing convergent sequence verifies $u_n\le u$ so that $(u_n,u)\in \mathit{R}$ for every $n\in \mathbb{N}$. This yields that $\mathit{R}$ is μ-self-closed.

Define the auxiliary functions $\psi \left(p\right)=p/2$ and $\varrho \left(p\right)=p$. Then, the contraction condition $\left(e\right)$ of Theorem 4 can be easily confirmed. Herein, the set $\mathcal{P}\left(\mathbf{M}\right)$ is ${\mathit{R}}^s$-directed as for any pair $u,w\in \mathcal{P}\left(\mathbf{M}\right)$, $w:=max\{u,w\}$ satisfies $(u,w)\in \mathit{R}$ and $(w,w)\in \mathit{R}$. Lastly, it is intuitive to examine all the suppositions of Theorems 4 and 5. Consequently, $\mathcal{P}$ possesses a unique fixed point, $\overline{u}=0$.

Example 3.

Take the MS $\mathbf{M}=[0,1]$ with standard metric μ and the BR $\mathit{R}:=[0,1]\times (\mathbb{M}\cap [0,1\left]\right)$. Assume that $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ is the identity mapping. Then, $\mathit{R}$ is a locally $\mathcal{P}$-transitive and $\mathcal{P}$-closed BR. Also, $(\mathbf{M},\mu )$ forms an $\mathit{R}$-complete MS and $\mathcal{P}$ remains an $\mathit{R}$-continuous map. Fix $\mu \in [0,1)$ and consider the functions $\psi \left(p\right)=\mu ·p$ and $\varrho \left(p\right)=p-\mu ·p$. Then, the condition $\left(e\right)$ of Theorem 4 can be easily verified. Similarly, the rest of the suppositions of Theorem 4 are readily verifiable.

Since $\mathcal{P}\left(\mathbf{M}\right)$ is not $\mathit{R}$-directed in the present scenario, Theorem 5 is no longer valid. Indeed, $\mathrm{Fix}\left(\mathcal{P}\right)=\mathbf{M}$.

5. Consequences

Our findings are utilized to infer some documented fixed-point outcomes. Under the BR $\mathbf{R}={\mathbf{M}}^2$, Theorem 5 determines the following outcome on fixed points of a nonlinear almost $\psi$-contraction of Pant type.

Corollary 1.

Let $(\mathbf{M},\mu )$ be a CMS and $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ be a map. If $\exists \psi \in \mathrm{\Phi }$ and $\varrho \in \Omega$ with

$$\begin{array}{ccc}& & \mu (\mathcal{P}u,\mathcal{P}w)\le \psi \left(\mu \right(u,w\left)\right)+min\left\{\varrho \right(\mu (u,\mathcal{P}u)),\varrho (\mu (w,\mathcal{P}w)),\varrho (\mu (u,\mathcal{P}w)),\varrho (\mu (w,\mathcal{P}u)\left)\right\},\hfill \\ & & forallu,w\in \mathbf{M}with[u\ne \mathcal{P}(u)orw\ne \mathcal{P}(w\left)\right],\hfill \end{array}$$

then $\mathcal{P}$ owns a unique fixed point.

If we look at $\psi \left(\mathsf{p}\right)=a·\mathsf{p}$ ($0<a<1$) in Corollary 1, then we achieve the below-mentioned finding of Turinici [22].

Corollary 2

([22]). Let $(\mathbf{M},\mu )$ be a CMS and $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ be a map. If $\exists a\in (0,1)$ and $\varrho \in \Omega$ with

$$\begin{array}{ccc}& & \mu (\mathcal{P}u,\mathcal{P}w)\le a·\mu (u,w)+min\left\{\varrho \right(\mu (u,\mathcal{P}u)),\varrho (\mu (w,\mathcal{P}w)),\varrho (\mu (u,\mathcal{P}w)),\varrho (\mu (w,\mathcal{P}u)\left)\right\},\hfill \\ & & forallu,w\in \mathbf{M},\hfill \end{array}$$

then $\mathcal{P}$ owns a unique fixed point.

For $\psi \left(\mathsf{p}\right)=a·\mathsf{p}$ ($0<a<1$) and $\varrho \left(\mathsf{p}\right)=l·\mathsf{p}$ ($l\in {\mathbb{R}}^{+}$), the finding of Babu et al. [21] (also established by Berinde [11]) is determined from Corollary 1.

Corollary 3

([11,21]). Let $(\mathbf{M},\mu )$ be a CMS and $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ be a map. If $\exists a\in (0,1)$ and $l\in {\mathbb{R}}^{+}$ with

$$\begin{array}{c}\hfill \mu (\mathcal{P}u,\mathcal{P}w)\le a·\mu (u,w)+l·min\left\{\mu \right(u,\mathcal{P}u),\mu (w,\mathcal{P}w),\mu (u,\mathcal{P}w),\mu (w,\mathcal{P}u\left)\right\},forallu,w\in \mathbf{M},\end{array}$$

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

If we choose $\varrho \left(\mathsf{p}\right)=0$, then Corollary 1 yields the following result of Pant [25].

Corollary 4

([25]). Let $(\mathbf{M},\mu )$ be a CMS and $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ be a map. If $\exists \psi \in \mathrm{\Phi }$ with

$$\begin{array}{ccc}& & \mu (\mathcal{P}u,\mathcal{P}w)\le \psi \left(\mu \right(u,w\left)\right),forallu,w\in \mathbf{M}with[u\ne \mathcal{P}(u)orw\ne \mathcal{P}(w\left)\right],\hfill \end{array}$$

then $\mathcal{P}$ owns a unique fixed point.

By substitution $\varrho \left(\mathsf{p}\right)=0$, Theorem 5 provides the following finding of Alam et al. [4].

Corollary 5

([4]). Let $(\mathbf{M},\mu )$ be an MS, $\mathit{R}$ be BR on $\mathbf{M}$, and $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ be a mapping. Also, the following hold:

(a) 

$(\mathbf{M},\mu )$ is $\mathit{R}$-complete;

(b) 

$\exists u_0\in \mathbf{M}$ with $(u_0,\mathcal{P}{u}_0)\in \mathit{R}$;

(c) 

$\mathit{R}$ remains locally finitely $\mathcal{P}$-transitive and $\mathcal{P}$-closed;

(d) 

$\mathbf{M}$ remains $\mathit{R}$-continuous, or $\mathit{R}$ is μ-self-closed;

(e) 

$\exists \psi \in \mathrm{\Phi }$ with

$$\begin{array}{ccc}& & \mu (\mathcal{P}u,\mathcal{P}w)\le \psi \left(\mu \right(u,w\left)\right)forall(u,w)\in \mathit{R}.\hfill \end{array}$$

Then, $\mathcal{P}$ owns a fixed point. Moreover, if $\mathcal{P}\left(\mathbf{M}\right)$ is $\mathit{R}$-directed, then $\mathcal{P}$ admits a unique fixed point.

Particularly, for $\varrho \left(\mathsf{p}\right)=l·\mathsf{p}$ (where $l\in {\mathbb{R}}^{+}$), Theorem 5 is transformed into the following outcome of Khan et al. [19].

Corollary 6

([19]). Let $(\mathbf{M},\mu )$ be an MS, $\mathit{R}$ be a BR on $\mathbf{M}$, and $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ be a mapping. Also, the following hold:

(a) 

$(\mathbf{M},\mu )$ is $\mathit{R}$-complete;

(b) 

$\exists u_0\in \mathbf{M}$ with $(u_0,\mathcal{P}{u}_0)\in \mathit{R}$;

(c) 

$\mathit{R}$ remains locally finitely $\mathcal{P}$-transitive and $\mathcal{P}$-closed;

(d) 

$\mathbf{M}$ remains $\mathit{R}$-continuous, or $\mathit{R}$ is μ-self-closed;

(e) 

$\exists \psi \in \mathrm{\Phi }$ and $l\in {\mathbb{R}}^{+}$ with

$$\begin{array}{ccc}\hfill \mu (\mathcal{P}u,\mathcal{P}w)& \le & \psi \left(\mu \right(u,w\left)\right)+l·min\left\{\mu \right(u,\mathcal{P}u),\mu (w,\mathcal{P}w),\mu (u,\mathcal{P}w),\mu (w,\mathcal{P}u\left)\right\},\hfill \\ \hfill forall(u,w)\in \mathit{R}.& \end{array}$$

Then, $\mathcal{P}$ owns a fixed point. Moreover, if $\mathcal{P}\left(\mathbf{M}\right)$ is $\mathit{R}$-directed, then $\mathcal{P}$ admits a unique fixed point.

6. An Application

In this part, we apply our findings to find the unique solution of an innovative (nonlinear) Fredholm integral equation of the form

$$\vartheta \left(s\right)=\mathbf{F}\left(s\right)+{\int }_a^b\mathbf{L}(s,\zeta )\psi (\zeta ,\vartheta \left(\zeta \right))d\zeta ,\forall s\in [a,b],$$

(18)

where $\mathbf{F}:\mathcal{I}\to \mathbb{R}$, $\mathbf{L}:{\mathcal{I}}^2\to \mathbb{R}$, and $\psi :\mathcal{I}\times \mathbb{R}\to \mathbb{R}$ are functions, where $\mathcal{I}:=[a,b]$.

Definition 17

([34]). $\underline{\vartheta }\in \mathcal{C}\left(\mathcal{I}\right)$ is termed a lower solution of (18) if $\forall s\in \mathcal{I}$

$$\underline{\vartheta }\left(s\right)\le \mathit{F}\left(s\right)+{\int }_a^b\mathit{L}(s,\zeta )\psi (\zeta ,\underline{\vartheta }\left(\zeta \right))d\zeta .$$

Definition 18

([34]). $\overline{\vartheta }\in \mathcal{C}\left(\mathcal{I}\right)$ is termed an upper solution of (18) if $\forall s\in \mathcal{I}$

$$\overline{\vartheta }\left(s\right)\ge \mathit{F}\left(s\right)+{\int }_a^b\mathit{L}(s,\zeta )\psi (\zeta ,\overline{\vartheta }\left(\zeta \right))d\zeta .$$

Define a subfamily of $\mathrm{\Psi }$ given by

$$\mathrm{\Phi }=\{\psi \in \mathrm{\Psi }:\varphi ismonotoneincearsing\}.$$

We right away investigate the principal findings of this part.

Theorem 6.

In combination with Problem (18), assume that the following presumptions hold:

(i)

  $\mathit{F}$, ℏ, and $\mathit{L}$ remain continuous.

(ii)

 $\mathit{L}(s,\zeta )>0,\forall s,\zeta \in \mathcal{I}$.

(iii)

$\exists 0<\eta \le 1$ and $\exists \psi \in \mathrm{\Phi }$ obeying

$$0\le \hslash (s,x)-\hslash (s,y)\le {\displaystyle \frac{1}{\eta }}\varphi (x-y),\forall s\in \mathcal{I}and\forall x,y\in \mathbb{R}withx\ge y,$$

(iv)

$\underset{s\in \mathcal{I}}{sup}{\int }_a^b\mathit{L}(s,\zeta )d\zeta \le \eta$.

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

Proof. 

Let $\mathbf{M}:=\mathcal{C}\left(\mathcal{I}\right)$. On $\mathbf{M}$, define the following metric:

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

(19)

Consider a BR $\mathbf{R}$ on $\mathbf{M}$ as

$$\mathbf{R}=\{(\vartheta ,\omega )\in {\mathbf{M}}^2:\vartheta \left(s\right)\le \omega \left(s\right),\forall s\in \mathcal{I}\}.$$

(20)

Let $\mathcal{P}:\mathbf{M}\to \mathbf{M}$ be a map defined by

$$\left(\mathcal{P}\vartheta \right)\left(s\right)=\mathbf{F}\left(s\right)+{\int }_a^b\mathbf{L}(s,\zeta )\hslash (\zeta ,\vartheta \left(\zeta \right))d\zeta ,\forall s\in \mathbf{M}.$$

(21)

Trivially, $\vartheta \in \mathbf{M}$ solves (18) if $\vartheta$ is a fixed point of $\mathcal{P}$.

We will verify all the premises of Theorems 4 and 5.

(a) 

$(\mathbf{M},\mu )$ as a CMS is an $\mathbf{R}$-complete MS.

(b) 

If $\underline{\vartheta }\in \mathbf{M}$ is the lower solution of (18), then we have

$$\underline{\vartheta }\left(s\right)\le \mathbf{F}\left(s\right)+{\int }_a^b\mathbf{L}(s,\zeta )\hslash (\zeta ,\underline{\vartheta }\left(\zeta \right))d\zeta =\left(\mathcal{P}\underline{\vartheta }\right)\left(s\right)$$

thereby yielding $(\underline{\vartheta },\mathcal{P}\underline{\vartheta })\in \mathbf{R}$.

(c) 

Pick $\vartheta ,\omega \in \mathbf{M}$ such that $(\vartheta ,\omega )\in \mathbf{R}$. From (iii), we attain

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

(22)

From (21), (22), and (ii), it follows that

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

so that $\left(\mathcal{P}\vartheta \right)\left(s\right)\le \left(\mathcal{P}\omega \right)\left(s\right)$. Due to (20), we find that $(\mathcal{P}\vartheta ,\mathcal{P}\omega )\in \mathbf{R}$. Thus, $\mathbf{R}$ is $\mathcal{P}$-closed.

(d) 

If $\left\{{\vartheta }_n\right\}\subset \mathbf{M}$ is an $\mathbf{R}$-preserving sequence and ${\vartheta }_n\to \omega \in \mathbf{M}$, then for every $s\in \mathcal{I}$, $\left\{{\vartheta }_n\left(s\right)\right\}\subset \mathbb{R}$ increases and converges to $\omega \left(s\right)$. This yields ${\vartheta }_n\left(s\right)\le \omega \left(s\right),\forall n\in \mathbb{N}$, and $\forall s\in \mathcal{I}$. Owing to (20), we find $({\vartheta }_n,\omega )\in \mathbf{R},\forall n\in \mathbb{N}$. Hence, $\mathbf{R}$ is $\mu$-self-closed.

(e) 

Pick $\vartheta ,\omega \in \mathbf{M}$ such that $(\vartheta ,\omega )\in \mathbf{R}$. Making use of (iii) along with (19) and (21), we get

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

(23)

Monotonicity of $\psi$ and the fact that $0\le \omega \left(\zeta \right)-\vartheta \left(\zeta \right)\le \mu (\vartheta ,\omega )$ provides $\psi \left(\omega \right(\zeta )-\vartheta (\zeta \left)\right)\le \psi \left(\mu \right(\vartheta ,\omega \left)\right)$. So, (23) determines

$$\begin{array}{ccc}\hfill \mu (\mathcal{P}\vartheta ,\mathcal{P}\omega )& \le & {\displaystyle \frac{1}{\eta }}\psi \left(\mu (\vartheta ,\omega )\right)\underset{s\in \mathcal{I}}{sup}{\int }_a^b\mathbf{L}(s,\zeta )d\zeta \hfill \\ & \le & {\displaystyle \frac{1}{\eta }}\psi \left(\mu (\vartheta ,\omega )\right).\eta \hfill \\ & =& \psi \left(\mu \right(\vartheta ,\omega \left)\right)\hfill \end{array}$$

or

$$\begin{array}{ccc}\hfill \mu (\mathcal{P}\vartheta ,\mathcal{P}\omega )& \le & \psi \left(\mu \right(\vartheta ,\omega \left)\right)+min\left\{\varrho \right(\mu (\vartheta ,\mathcal{P}\vartheta )),\varrho (\mu (\omega ,\mathcal{P}\omega )),\varrho (\mu (\vartheta ,\mathcal{P}\omega )),\varrho (\mu (\omega ,\mathcal{P}\vartheta )\left)\right\},\hfill \\ & \forall & (\vartheta ,\omega )\in \mathbf{R},with[\vartheta \ne \mathcal{P}(\vartheta )or\omega \ne \mathcal{P}(\omega \left)\right].\hfill \end{array}$$

where $\varrho \in \Omega$ is chosen arbitrarily.

Choose an arbitrary pair $\vartheta ,\omega \in \mathbf{M}$. Set $\vartheta :=max\{\mathcal{P}\vartheta ,\mathcal{P}\omega \}\in \mathbf{M}$. Then, we have $(\mathcal{P}\vartheta ,\vartheta )\in \mathbf{R}$ and $(\mathcal{P}\omega ,\vartheta )\in \mathbf{R}$. Therefore, $\mathcal{P}\left(\mathbf{M}\right)$ is ${\mathbf{R}}^s$-directed. Consequently, in lieu of Theorem 5, $\mathcal{P}$ possesses a unique fixed point, which is a unique solution of (18). □

Theorem 7.

In combination with assumptions (i)–(iv) of Theorem 6, the problem (18) possesses a unique solution if it admits an upper solution.

Proof. 

Take a metric $\mu$ on $\mathbf{M}:=\mathcal{C}\left(\mathcal{I}\right)$ and a map $\mathcal{P}:\mathbf{M}\to \mathbf{M}$, as defined in the proof of Theorem 6. Define a BR ${\mathbf{R}}^{\prime }$ on $\mathbf{M}$ as follows:

$${\mathbf{R}}^{\prime }=\{(\vartheta ,\omega )\in {\mathbf{M}}^2:\vartheta \left(s\right)\ge \omega \left(s\right),\forall s\in \mathcal{I}\}.$$

(24)

We will verify all the premises of Theorems 4 and 5.

(a) 

$(\mathbf{M},\mu )$ as a CMS is an ${\mathbf{R}}^{\prime }$-complete MS.

(b) 

If $\overline{\vartheta }\in \mathbf{M}$ is the upper solution of (18), then we conclude that

$$\overline{\vartheta }\left(s\right)\ge \mathbf{F}\left(s\right)+{\int }_a^b\mathbf{L}(s,\zeta )\hslash (\zeta ,\overline{\vartheta }\left(\zeta \right))d\zeta =\left(\mathcal{P}\overline{\vartheta }\right)\left(s\right)$$

so that $(\overline{\vartheta },\mathcal{P}\overline{\vartheta })\in {\mathbf{R}}^{\prime }$.

(c) 

Pick $\vartheta ,\omega \in \mathbf{M}$ such that $(\vartheta ,\omega )\in {\mathbf{R}}^{\prime }$. Using (iii), we find

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

(25)

From (21), (22), and item (ii), we attain

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

thereby yielding $\left(\mathcal{P}\vartheta \right)\left(s\right)\ge \left(\mathcal{P}\omega \right)\left(s\right)$. Thus, in lieu of (24), we conclude that $(\mathcal{P}\vartheta ,\mathcal{P}\omega )\in {\mathbf{R}}^{\prime }$. Therefore, ${\mathbf{R}}^{\prime }$ is $\mathcal{P}$-closed.

(d) 

If $\left\{{\vartheta }_n\right\}\subset \mathbf{M}$ is an ${\mathbf{R}}^{\prime }$-preserving sequence and ${\vartheta }_n\to \omega \in \mathbf{M}$, then for every $s\in \mathcal{I}$, $\left\{{\vartheta }_n\left(s\right)\right\}\subset \mathbb{R}$ decreases and converges to $\omega \left(s\right)$. This yields ${\vartheta }_n\left(s\right)\ge \omega \left(s\right),\forall n\in \mathbb{N}$, and $\forall s\in \mathcal{I}$. Owing to (20), we find $({\vartheta }_n,\omega )\in {\mathbf{R}}^{\prime },\forall n\in \mathbb{N}$. Therefore, ${\mathbf{R}}^{\prime }$ is $\mu$-self-closed.

(e) 

Pick $\vartheta ,\omega \in \mathbf{M}$ such that $(\vartheta ,\omega )\in {\mathbf{R}}^{\prime }$. Utilization of (iii) along with (19) and (21) provides us with

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

(26)

Employing the monotone property of $\psi$ and $0\le \vartheta \left(\zeta \right)-\omega \left(\zeta \right)\le \mu (\vartheta ,\omega )$, we conclude that $\psi \left(\vartheta \right(\zeta )-\omega (\zeta \left)\right)\le \psi \left(\mu \right(\vartheta ,\omega \left)\right)$. Therefore (26) becomes

$$\begin{array}{ccc}\hfill \mu (\mathcal{P}\vartheta ,\mathcal{P}\omega )& \le & {\displaystyle \frac{1}{\eta }}\psi \left(\mu (\vartheta ,\omega )\right)\underset{s\in \mathcal{I}}{sup}{\int }_a^b\mathbf{L}(s,\zeta )d\zeta \hfill \\ & \le & {\displaystyle \frac{1}{\eta }}\psi \left(\mu (\vartheta ,\omega )\right).\eta \hfill \\ & =& \psi \left(\mu \right(\vartheta ,\omega \left)\right)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \mu (\mathcal{P}\vartheta ,\mathcal{P}\omega )& \le & \psi \left(\mu \right(\vartheta ,\omega \left)\right)+min\left\{\varrho \right(\mu (\vartheta ,\mathcal{P}\vartheta )),\varrho (\mu (\omega ,\mathcal{P}\omega )),\varrho (\mu (\vartheta ,\mathcal{P}\omega )),\varrho (\mu (\omega ,\mathcal{P}\vartheta )\left)\right\},\hfill \\ & \forall & (\vartheta ,\omega )\in \mathbf{R},with[\vartheta \ne \mathcal{P}(\vartheta )or\omega \ne \mathcal{P}(\omega \left)\right].\hfill \end{array}$$

where $\varrho \in \Omega$ is chosen arbitrarily.

Let $\vartheta ,\omega \in \mathbf{M}$ be arbitrary. Define $\vartheta :=max\{\mathcal{P}\vartheta ,\mathcal{P}\omega \}\in \mathbf{M}$. Then $(\vartheta ,\mathcal{P}\vartheta )\in {\mathbf{R}}^{\prime }$ and $(\vartheta ,\mathcal{P}\omega )\in {\mathbf{R}}^{\prime }$. Thus, $\mathcal{P}\left(\mathbf{M}\right)$ is ${\mathbf{R}}^{\prime s}$-directed. Finally, using Theorem 5, $\mathcal{P}$ admits a unique fixed point, which is a unique solution of (18). □

We explore the following instance to convey Theorem 6.

Example 4.

Let (18) be an (nonlinear) Fredholm integral equation with $\mathit{F}\left(s\right)=2(1-2s^2)$, $\mathit{L}(s,\zeta )=2s\zeta$, and $\hslash (\zeta ,\zeta )={\displaystyle \frac{1}{3}}\zeta$. Fix $\eta =1$. Define the function $\psi \left(t\right)={\displaystyle \frac{2}{3}}t$. Then, we have $\psi \in \mathrm{\Phi }$. Clearly, the assumptions $\left(\mathrm{I}\right)$–$\left(\mathrm{IV}\right)$ of Theorem 6 hold and $\underline{\vartheta }=0$ forms a lower solution. Thus, Theorem 6 is applicable for this integral equation. Observe that $\vartheta \left(s\right)=2(1-2s^2)$ is the unique solution of the given equation.

7. Conclusions

Quiet recently, Filali et al. [27] demonstrated the fixed-point theorems for relational almost Matkowski $\psi$-contractions, while Filali and Khan [28] investigated the results for relational almost Boyd–Wong $\psi$-contractions. Both types of results involved a locally $\mathcal{P}$-transitive BR. In our outcomes, the contraction condition is slightly different; however we utilized a locally finitely $\mathcal{P}$-transitive BR, which is wider than a locally $\mathcal{P}$-transitive BR. Our findings enriched, sharpened, and unified a few existing outcomes on fixed points, notably those of Alam et al. [4], Berinde [11], Khan et al. [19], Babu et al. [21], Turinici [22] and Pant [25]. In our outcomes, the contraction inequality is solely incorporated into the comparison elements. To illustrate our outcomes, we presented a few scenarios. We additionally implemented our findings to certain nonlinear Fredholm integral equations to bring out the relevance of the theory and wide variety of our findings.

Regarding potential future research, our outcomes can be generalized in the following aspects:

1.

To expand our findings over semimetric space, metric-like space, quasimetric space, $G$-metric space, b-metric space, etc., endowed with a BR;

2.

To improve the axioms of auxiliary functions $\psi$ and $\varrho$;

3.

To extend our outcomes for two maps by demonstrating outcomes on coincidence points and common fixed points;

4.

To apply our results in fractional differential equations along the lines of Filali et al. [27], Baleanu and Shiri [35,36], Shiri et. al [37], and similar others.