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

Abstract

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

1. Introduction

One of the powerful and fundamental results of metric fixed point theory is the Banach contraction principle (abbreviation: BCP). Indeed, BCP guarantees a unique fixed point for a self-contraction on a complete metric space. This result also offers an iterative scheme to compute the unique fixed point. In the last century, this result has been extended by various researchers. In this direction, several authors enlarged the usual contraction to be a $\psi$-contraction by governing the contraction condition via suitable auxiliary function $\psi :[0,\infty )\to [0,\infty )$. By varying $\psi$ suitably, various generalizations were obtained and this theme now has a considerable literature. A noted class of $\psi$-contraction is essentially due to Boyd and Wong [1] wherein the author improved the contraction condition by replacing Lipschitz constant $c\in (0,1)$ with a control function belonging to the following family:

$$\mathsf{\Omega }=\{\psi :[0,\infty )\to [0,\infty ):\psi \left(s\right)<s,\forall s>0\mathrm{and}\underset{t\to s^{+}}{lim\; sup}\psi \left(t\right)<s,\forall s>0\}.$$

Theorem 1

([1]). Assume that a self-map $\mathcal{L}$ on a complete metric space $(\mathbf{Z},\rho )$ satisfies for some $\psi \in \mathsf{\Omega }$ that

$$\rho (\mathcal{L}v,\mathcal{L}z)\le \psi \left(\rho \right(v,z\left)\right),\forall v,z\in \mathbf{Z}.$$

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

The above contractivity condition is called nonlinear contraction or $\psi$-contraction. Under the restriction $\psi \left(s\right)=cs,0<c<1$, $\psi$-contraction reduces to usual contraction and Theorem 1 reduces to the BCP.

In 2004, Berinde [2] introduced yet a new generalization of BCP, often called “almost contraction”.

Definition 1

([2,3]). A self-map $\mathcal{L}$ on a metric space $(\mathbf{Z},\rho )$ is termed as almost contraction if $\exists c\in (0,1)$ and $\exists K\in [0,\infty )$, satisfying

$$\rho (\mathcal{L}v,\mathcal{L}z)\le c\rho (v,z)+K\rho (v,\mathcal{L}z),\forall v,z\in \mathbf{Z}.$$

By symmetric property of $\rho$, the above condition is equivalent to the following one:

$$\rho (\mathcal{L}v,\mathcal{L}z)\le c\rho (v,z)+K\rho (z,\mathcal{L}v),\forall v,z\in \mathbf{Z}.$$

Theorem 2

([2]). An almost contraction self-map on a complete metric space owns a fixed point.

The notion of almost contraction has been developed by various researchers, e.g., see [4,5,6,7,8,9]. Any almost contraction hasn’t always a unique fixed point, but a sequence of Picard iterations remains convergent to a fixed point of the underlying mapping. To obtain a uniqueness theorem, Babu et al. [4] defined a slightly stronger class of almost contraction conditions.

Definition 2

([4]). A self-map $\mathcal{L}$ on a metric space $(\mathbf{Z},\rho )$ is termed as strict almost contraction if $\exists c\in (0,1)$ and $\exists K\in [0,\infty )$, satisfying

$$\rho (\mathcal{L}v,\mathcal{L}z)\le c\rho (v,z)+Kmin\left\{\rho \right(v,\mathcal{L}v),\rho (z,\mathcal{L}z),\rho (v,\mathcal{L}z),\rho (z,\mathcal{L}v\left)\right\},\forall v,z\in \mathbf{Z}.$$

Clearly, a strict almost contraction is an almost contraction. However, the converse is not generally true, see; Example $2.6$ [4].

Theorem 3

([4]). A strict almost contraction on a complete metric space owns a unique fixed point.

A novel extension of BCP in relational metric space was investigated by Alam and Imdad [10]. Since then, various fixed point theorems have been established employing different contractivity conditions in this context, e.g., [11,12,13,14,15,16,17,18,19]. In such results, the contraction map is verified only for comparative pairs. Consequently, the relation-theoretic contractions remain weaker than usual contractions. The fixed point results obtained in the relation-theoretic setting are applicable into specific periodic BVPs (i.e., boundary value problems).

The aim of the present manuscript is to subsume two contractivity conditions, as mentioned earlier (i.e., $\psi$-contraction and strict almost contraction), and utilize this newly obtained contraction to establish relevant fixed-point theorems in a metric space with a locally $\mathcal{L}$-transitive relation. We illustrate our results by adopting some examples. To validate our results, we adopt an application to a BVP, satisfying certain additional hypotheses.

2. Preliminaries

In the aftermath, the sets of natural, whole and real numbers will be denoted by $\mathbb{N}$, ${\mathbb{N}}_0$ and $\mathbb{R}$, respectively. Recall that a subset of ${\mathbf{Z}}^2$ is a binary relation (or, a relation) on the set $\mathbf{Z}$.

Let us assume that $\mathbf{Z}$ is the given set, $\mathcal{L}:\mathbf{Z}\to \mathbf{Z}$ is a mapping, $\mathcal{S}$ is a relation on $\mathbf{Z}$ and $\rho$ remains a metric on $\mathbf{Z}$.

Definition 3

([10]). The points $v,z\in \mathbf{Z}$ are called $\mathcal{S}$-comparative if $(v,z)\in \mathcal{S}$ or $(z,v)\in \mathcal{S}$. We denote such a pair by $[v,z]\in \mathcal{S}$.

Definition 4

([20]). The relation ${\mathcal{S}}^{-1}:=\{(v,z)\in {\mathbf{Z}}^2:(z,v)\in \mathcal{S}\}$ is called inverse of $\mathcal{S}$. Also, ${\mathcal{S}}^s:=\mathcal{S}\cup {\mathcal{S}}^{-1}$ defines a symmetric relation on $\mathbf{Z}$, often called symmetric closure of $\mathcal{S}$.

Remark 1

([10]). $(v,z)\in {\mathcal{S}}^s⟺[v,z]\in \mathcal{S}.$

Definition 5

([21]). For a subset $\mathbf{Q}\subseteq \mathbf{Z}$, the set

$${\mathcal{S}|}_{\mathbf{Q}}:=\mathcal{S}\cap {\mathbf{Q}}^2,$$

a relation on $\mathbf{Q}$, is named as the restriction of $\mathcal{S}$ on $\mathbf{Q}$.

Definition 6

([10]). $\mathcal{S}$ is referred to as $\mathcal{L}$-closed if, for every pair $v,z\in \mathbf{Z}$ verifying $(v,z)\in \mathcal{S}$, one has

$$(\mathcal{L}v,\mathcal{L}z)\in \mathcal{S}.$$

Definition 7

([10]). A sequence $\left\{v_n\right\}\subset \mathbf{Z}$, satisfying $(v_n,v_{n+1})\in \mathcal{S}$, $\forall n\in \mathbb{N}$, is termed as $\mathcal{S}$-preserving.

Definition 8

([22]). A subset $\mathbf{Q}\subseteq \mathbf{Z}$ is named as $\mathcal{S}$-directed if, for every pair $v,z\in \mathbf{Q}$, $\exists \omega \in \mathbf{Z}$ verifying $(v,\omega )\in \mathcal{S}$ and $(z,\omega )\in \mathcal{S}$.

For each fixed $v_0\in P$, the set ${\mathcal{O}}_{\mathcal{L}}\left(v_0\right):=\{{\mathcal{L}}^n{v}_0:n\in \mathbb{N}\}$ is referred to as the orbit of $v_0$. If $\mathcal{L}$ is understood, then we write $\mathcal{O}\left(v_0\right)$ instead of ${\mathcal{O}}_{\mathcal{L}}\left(v_0\right)$. A sequence is termed a $\mathcal{L}$-orbital sequence if its range remains $\mathcal{O}\left(v\right)$ for some $v\in \mathbf{Z}$ (c.f. [16]).

Definition 9

([16]). $(\mathbf{Z},\rho )$ is termed as ($\mathcal{O},\mathcal{S}$)-complete metric space if every $\mathcal{L}$-orbital $\mathcal{S}$-preserving Cauchy sequence in $\mathbf{Z}$ is convergent.

Definition 10

([16]). $\mathcal{L}$ is termed as ($\mathcal{O},\mathcal{S}$)-continuous map at a point $v\in \mathbf{Z}$ if, for any $\mathcal{L}$-orbital $\mathcal{S}$-preserving sequence $\left\{v_n\right\}\subset \mathbf{Z}$ satisfying $v_n\stackrel{\rho }{\to }v$, we have $\mathcal{L}\left(v_n\right)\stackrel{\rho }{\to }\mathcal{L}\left(v\right)$.

Definition 11

([16]). $\mathcal{S}$ is referred to as ($\mathcal{O},\rho$)-self-closed if, for each $\mathcal{L}$-orbital $\mathcal{S}$-preserving sequence $\left\{v_n\right\}\subset \mathbf{Z}$ converging to $v\in \mathbf{Z}$, $\exists$ a subsequence $\left\{v_{n_k}\right\}$ such that $[v_{n_k},v]\in \mathcal{S}$, $\forall k\in \mathbb{N}$.

If we ignore the orbital properties in Definitions 9–11, we obtain the notions of ‘$\mathcal{S}$-complete metric space’, ‘$\mathcal{S}$-continuous map’ and ‘$\rho$-self-closed relation’, respectively (cf. [11]).

Definition 12

([12]). $\mathcal{S}$ is referred to as locally $\mathcal{L}$-transitive if, for each $\mathcal{S}$-preserving sequence $\left\{v_n\right\}\subset \mathcal{L}\left(\mathbf{Z}\right)$ possessing the range $\mathbf{E}=\{v_n:n\in \mathbb{N}\}$, ${\mathcal{S}|}_{\mathbf{E}}$ remains transitive.

Proposition 1

([12]). For each $n\in {\mathbb{N}}_0$, $\mathcal{S}$ is ${\mathcal{L}}^n$-closed whenever $\mathcal{S}$ is $\mathcal{L}$-closed.

Lemma 1

([23]). Assume that $\left\{v_n\right\}$ remains a sequence in a metric space $(\mathbf{Z},\rho )$. If $\left\{v_n\right\}$ is not Cauchy, then $\exists$ subsequences $\left\{v_{n_k}\right\}$ and $\left\{v_{l_k}\right\}$ of $\left\{v_n\right\}$ and $\exists {ϵ}_0>0$ verifying

(i)

$k\le l_k<n_k\forall k\in \mathbb{N}$,

(ii)

$\rho (v_{l_k},v_{n_k})>{ϵ}_0\forall k\in \mathbb{N}$,

(iii)

$\rho (v_{l_k},v_{n_{k-1}})\le {ϵ}_0\forall k\in \mathbb{N}$.

Moreover, if $\underset{n\to \infty }{lim}\rho (v_n,v_{n+1})=0$, then

(iv)

$\underset{k\to \infty }{lim}\rho (v_{l_k},v_{n_k})={ϵ}_0,$

(v)

$\underset{k\to \infty }{lim}\rho (v_{l_k},v_{n_k+1})={ϵ}_0,$

(vi)

$\underset{k\to \infty }{lim}\rho (v_{l_k+1},v_{n_k})={ϵ}_0,$

(vii)

$\underset{k\to \infty }{lim}\rho (v_{l_k+1},v_{n_k+1})={ϵ}_0.$

Making use of the symmetric property of metric $\rho$, we get the following:

Proposition 2.

For a given control function $\psi \in \mathsf{\Omega }$ and $K\ge 0$, the following conditions are equivalent:

(I)

$\rho (\mathcal{L}v,\mathcal{L}z)\le \psi \left(\rho \right(v,z\left)\right)+Kmin\left\{\rho \right(v,\mathcal{L}v),\rho (z,\mathcal{L}z),\rho (v,\mathcal{L}z),\rho (z,\mathcal{L}v\left)\right\}$,

$\hspace{1em}\hspace{0.5em}\forall v,z\in \mathbf{Z}with(v,z)\in \mathcal{S}.$

(II)

$\rho (\mathcal{L}v,\mathcal{L}z)\le \psi \left(\rho \right(v,z\left)\right)+Kmin\left\{\rho \right(v,\mathcal{L}v),\rho (z,\mathcal{L}z),\rho (v,\mathcal{L}z),\rho (z,\mathcal{L}v\left)\right\}$,

$\hspace{1em}\hspace{0.5em}\forall v,z\in \mathbf{Z}with[v,z]\in \mathcal{S}.$

3. Main Results

We are going to prove the results about existence and uniqueness of fixed points for relational strict almost $\psi$-contractions.

Theorem 4.

Suppose that $(\mathbf{Z},\rho )$ is metric space endued with a relation $\mathcal{S}$ and $\mathcal{L}:\mathbf{Z}\to \mathbf{Z}$ is a map. Moreover,

(i)

$(\mathbf{Z},\rho )$ is ($\mathcal{O},\mathcal{S}$)-complete,

(ii)

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

(iii)

$\exists v_0\in \mathbf{Z}$ satisfying $(v_0,\mathcal{L}{v}_0)\in \mathcal{S}$,

(iv)

$\mathcal{L}$ remains ($\mathcal{O},\mathcal{S}$)-continuous, or $\mathcal{S}$ remains ($\mathcal{O},\rho$)-self-closed,

(v)

$\exists \psi \in \mathsf{\Omega }$ and $K\ge 0$ verifying

$$\begin{array}{ccc}& & \rho (\mathcal{L}v,\mathcal{L}z)\le \psi \left(\rho \right(v,z\left)\right)+Kmin\left\{\rho \right(v,\mathcal{L}v),\rho (z,\mathcal{L}z),\rho (v,\mathcal{L}z),\rho (z,\mathcal{L}v\left)\right\},\hfill \\ & & \forall v,z\in \mathbf{Z}with(v,z)\in \mathcal{S}.\hfill \end{array}$$

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

Proof. 

Given $v_0\in \mathbf{Z}$ (by (iii)). Construct a sequence $\left\{v_n\right\}\subset \mathbf{Z}$, as follows:

$$v_n:={\mathcal{L}}^n\left(v_0\right)=\mathcal{L}\left(v_{n-1}\right),\forall n\in \mathbb{N}.$$

(1)

By assumption (iii), $\mathcal{L}$-closedness of $\mathcal{S}$ and Proposition 1, we get

$$({\mathcal{L}}^n{v}_0,{\mathcal{L}}^{n+1}{v}_0)\in \mathcal{S},$$

which, due to availability of (1), reduces to

$$(v_n,v_{n+1})\in \mathcal{S},\forall n\in {\mathbb{N}}_0.$$

(2)

Hence, $\left\{v_n\right\}$ is a $\mathcal{S}$-preserving sequence.

Let us denote ${\rho }_n:=\rho (v_n,v_{n+1})$. If ${\rho }_{n_0}=\rho (v_{n_0},v_{n_0+1})=0$ for some $n_0\in {\mathbb{N}}_0$, then in lieu of (1), one has $\mathcal{L}\left(v_{n_0}\right)=v_{n_0}$. Thus, $v_{n_0}$ is a fixed point of $\mathcal{L}$; hence, we have completed the solution.

In case ${\rho }_n>0,\forall n\in {\mathbb{N}}_0$, employing assumption (v), (1) and (2), we get

$$\begin{array}{ccc}\hfill {\rho }_n& =& \rho (v_n,v_{n+1})=\rho (\mathcal{L}{v}_{n-1},\mathcal{L}{v}_n)\hfill \\ & \le & \psi \left(\rho (v_{n-1},v_n)\right)+Kmin\{\rho (v_{n-1},v_n),\rho (v_n,v_{n+1}),\rho (v_{n-1},v_{n+1}),0\},\hfill \end{array}$$

so that

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

(3)

Employing the property of $\psi$ in (3), we have

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

This embraces that $\left\{{\rho }_n\right\}$ is a monotonically decreasing sequence of positive reals. Further, $\left\{{\rho }_n\right\}$ remains bounded below by ‘0’. Consequently, $\exists \delta \ge 0$ such that

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\rho }_n=\delta .\end{array}$$

(4)

We assert $\delta =0.$ Assuming, to contrary, that $\delta >0$. Invoking to limit superior in (3), employing (4) and the property of $\mathsf{\Omega }$, one finds

$$\delta =\underset{n\to \infty }{lim\; sup}{\rho }_n\le \underset{n\to \infty }{lim\; sup}\psi \left({\rho }_{n-1}\right)=\underset{{\rho }_n\to {\delta }^{+}}{lim\; sup}\psi \left({\rho }_{n-1}\right)<\delta .$$

This contradiction implies that $\delta =0$. Thus, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{\rho }_n=0.\end{array}$$

(5)

Now, we assert that $\left\{v_n\right\}$ is Cauchy. Assuming, to contrary, that $\left\{v_n\right\}$ is not Cauchy. Then, by Lemma 1, $\exists$ subsequences $\left\{v_{n_k}\right\}$ and $\left\{v_{l_k}\right\}$ of $\left\{v_n\right\}$ and $\exists {ϵ}_0>0$, satisfying

$$k\le l_k<n_k,\rho (v_{l_k},v_{n_k})>{ϵ}_0\ge \rho (v_{l_k},v_{n_{k-1}}),\forall k\in \mathbb{N}.$$

Denote ${\alpha }_k:=\rho (v_{l_k},v_{n_k})$. As $\left\{v_n\right\}$ is $\mathcal{S}$-preserving (due to (2)) and $\left\{v_n\right\}\subset \mathcal{L}\left(\mathbf{Z}\right)$ (due to (1)), using locally $\mathcal{L}$-transitivity of $\mathcal{S}$, we find $(v_{l_k},v_{n_k})\in \mathcal{S}$. Therefore, by using the contractivity condition (v), we obtain

$$\begin{array}{ccc}\hfill \rho (v_{l_k+1},v_{n_k+1})& =& \rho (\mathcal{L}{v}_{l_k},\mathcal{L}{v}_{n_k})\hfill \\ & \le & \psi \left(\rho (v_{l_k},v_{n_k})\right)+Kmin\{\rho (v_{l_k},\mathcal{L}{v}_{l_k}),\rho (v_{n_k},\mathcal{L}{v}_{n_k}),\rho (v_{l_k},\mathcal{L}{v}_{n_k}),\rho (v_{n_k},\mathcal{L}{v}_{l_k})\}\hfill \end{array}$$

so that

$$\rho (v_{l_k+1},v_{n_k+1})\le \psi \left({\alpha }_k\right)+Kmin\{{\rho }_{l_k},{\rho }_{n_k},\rho (v_{l_k},v_{n_k+1}),\rho (v_{n_k},v_{l_k+1})\}.$$

(6)

Letting the upper limit in (6) and making use of Lemma (1) and definition of $\mathsf{\Omega }$, one finds

$${\displaystyle {ϵ}_0=\underset{k\to \infty }{lim\; sup}\rho (v_{l_k+1},v_{n_k+1})\le \underset{k\to \infty }{lim\; sup}\psi \left({\alpha }_k\right)+Kmin\{0,0,{ϵ}_0,{ϵ}_0\}=\underset{s\to {ϵ}_0^{+}}{lim\; sup}\psi \left(s\right)<{ϵ}_0,}$$

which gives rise to a contradiction. Thus, $\left\{v_n\right\}$ remains Cauchy. Since $\left\{v_n\right\}$ is also $\mathcal{L}$-orbital and $\mathcal{S}$-preserving, therefore, by $(\mathcal{O},\mathcal{S})$-completeness of $\mathbf{Z}$, $\exists \overline{v}\in \mathbf{Z}$ verifying $v_n\stackrel{\rho }{⟶}\overline{v}$.

Finally, we conclude the proof using the assumption (iv). Suppose that the mapping $\mathcal{L}$ is $(\mathcal{O},\mathcal{S})$-continuous. As $\left\{v_n\right\}$ remains $\mathcal{L}$-orbital and $\mathcal{S}$-preserving, verifying $v_n\stackrel{\rho }{⟶}\overline{v}$, by $(\mathcal{O},\mathcal{S})$-continuity of $\mathcal{L}$, we obtain $v_{n+1}=\mathcal{L}\left(v_n\right)\stackrel{\rho }{⟶}\mathcal{L}\left(\overline{v}\right)$. By uniqueness property of convergence limit, we get $\mathcal{L}\left(\overline{v}\right)=\overline{v}$.

If $\mathcal{S}$ is $(\mathcal{O},\rho )$-self closed, then as $\left\{v_n\right\}$ remains is $\mathcal{L}$-orbital and $\mathcal{S}$-preserving, verifying $v_n\stackrel{\rho }{⟶}\overline{v}$, $\exists$ a subsequence $\left\{v_{n_k}\right\}$ of $\left\{v_n\right\}$ satisfying $[v_{n_k},\overline{v}]\in \mathcal{S},\forall k\in \mathbb{N}.$ Using assumption (v), Proposition 2 and $[v_{n_k},\overline{v}]\in \mathcal{S}$, we obtain

$$\begin{array}{ccc}\hfill \rho (v_{n_k+1},\mathcal{L}\overline{v})& =& \rho (\mathcal{L}{v}_{n_k},\mathcal{L}\overline{v})\hfill \\ & \le & \psi \left(\rho (v_{n_k},\overline{v})\right)+Kmin\{\rho (v_{n_k},v_{n_k+1}),0,\rho (v_{n_k},\overline{v}),\rho (\overline{v},v_{n_k+1})\}\hfill \\ & =& \psi \left(\rho (v_{n_k},\overline{v})\right).\hfill \end{array}$$

We claim that

$$\rho (v_{n_k+1},\mathcal{L}\overline{v})\le \rho (v_{n_k},\overline{v}),\forall k\in \mathbb{N}.$$

(7)

If $\rho (v_{n_{k_0}},\overline{v})=0$ for some $k_0\in \mathbb{N}$, then we find $\rho (\mathcal{L}{v}_{n_{k_0}},\mathcal{L}\overline{v})=0$ so that $\rho (v_{n_{k_0}+1},\mathcal{L}\overline{v})=0$; hence, (7) holds for such $k_0\in \mathbb{N}$. In either case, we have $\rho (v_{n_k},\overline{v})>0,\forall k\in \mathbb{N}.$ By the definition of $\mathsf{\Omega }$, we get $\rho (v_{n_k+1},\mathcal{L}\overline{v})\le \psi \left(\rho (v_{n_k},\overline{v})\right)<\rho (v_{n_k},\overline{v}),\forall k\in \mathbb{N}$. Thus, (7) holds for all $k\in \mathbb{N}.$ On letting the limit of (7) and employing $v_{n_k}\stackrel{\rho }{⟶}\overline{v}$, we get $v_{n_k+1}\stackrel{\rho }{⟶}\mathcal{L}\left(\overline{v}\right)$. By uniqueness property of limit, we find $\mathcal{L}\left(\overline{v}\right)=\overline{v}$, so that $\overline{v}$ remains a fixed point of $\mathcal{L}$. □

Theorem 5.

Along with the hypotheses of Theorem 4, if $\mathcal{L}\left(\mathbf{Z}\right)$ is $\mathcal{S}$-directed, then $\mathcal{L}$ possesses a unique fixed point.

Proof. 

In view of Theorem 4, if $\exists \overline{v},\overline{z}\in \mathbf{Z}$ verifying

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

(8)

As $\overline{v},\overline{z}\in \mathcal{L}\left(\mathbf{Z}\right)$, by our hypothesis, $\exists \omega \in \mathbf{Z}$, satisfying

$$(\overline{v},\omega )\in \mathcal{S}\mathrm{and}(\overline{z},\omega )\in \mathcal{S}.$$

(9)

Denote ${\varrho }_n:=\rho (\overline{v},{\mathcal{L}}^n\omega )$. Using (8) and (9) and assumption (v), one obtains

$$\begin{array}{ccc}\hfill {\varrho }_n=\rho (\overline{v},{\mathcal{L}}^n\omega )& =& \rho (\mathcal{L}\overline{v},\mathcal{L}\left({\mathcal{L}}^{n-1}\omega \right))\hfill \\ & \le & \psi \left(\rho (\overline{v},{\mathcal{L}}^{n-1}\omega )\right)+Lmin\{0,\rho ({\mathcal{L}}^{n-1}\omega ,{\mathcal{L}}^n\omega ),\rho (\overline{v},{\mathcal{L}}^n\omega ),\rho ({\mathcal{L}}^{n-1}\omega ,\overline{v})\}\hfill \\ & =& \psi \left({\varrho }_{n-1}\right)\hfill \end{array}$$

so that

$${\varrho }_n\le \psi \left({\varrho }_{n-1}\right).$$

(10)

If $\exists n_0\in \mathbb{N}$ such that ${\varrho }_{n_0}=0$, then we have ${\varrho }_{n_0}\le {\varrho }_{n_0-1}$. Otherwise, in case ${\varrho }_n>0,\forall n\in \mathbb{N}$, using the definition of $\mathsf{\Omega }$, (10) reduces to ${\varrho }_n<{\varrho }_{n-1}$. Hence, in both cases, we have

$${\varrho }_n\le {\varrho }_{n-1}.$$

Using the arguments similar to Theorem 4, the above inequality gives rise to

$${\displaystyle \underset{n\to \infty }{lim}{\varrho }_n=\underset{n\to \infty }{lim}\rho (\overline{v},{\mathcal{L}}^n\omega )=0.}$$

(11)

Similarly, one can find

$${\displaystyle \underset{n\to \infty }{lim}\rho (\overline{z},{\mathcal{L}}^n\omega )=0.}$$

(12)

By using (11) and (12) and the triangular inequality, one has

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

thereby implying $\overline{v}=\overline{z}$. Therefore, $\mathcal{L}$ possesses a unique fixed point. □

4. Consequences

Outlined, by making use of our findings, we shall obtain few familiar fixed-point theorems from the literature. In particular, for the universal relation, $\mathcal{S}={\mathbf{Z}}^2$, Theorem 5 reduces to the following corollary:

Corollary 1.

If $(\mathbf{Z},\rho )$ is a complete metric space, $\mathcal{L}:\mathbf{Z}\to \mathbf{Z}$ is a map and $\exists \psi \in \mathsf{\Omega }$ and $K\ge 0$, enjoying

$$\begin{array}{c}\hfill \rho (\mathcal{L}v,\mathcal{L}z)\le \psi \left(\rho \right(v,z\left)\right)+Kmin\left\{\rho \right(v,\mathcal{L}v),\rho (z,\mathcal{L}z),\rho (v,\mathcal{L}z),\rho (z,\mathcal{L}v\left)\right\},\forall v,z\in \mathbf{Z},\end{array}$$

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

For $K=0$, Theorem 4 reduces to an enhanced variant of the fixed-point theorem of Alam and Imdad [12], given below.

Corollary 2.

Suppose that $(\mathbf{Z},\rho )$ is metric space endued with a relation $\mathcal{S}$ and $\mathcal{L}:\mathbf{Z}\to \mathbf{Z}$ is a map. Moreover,

(i)

$(\mathbf{Z},\rho )$ is ($\mathcal{O},\mathcal{S}$)-complete,

(ii)

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

(iii)

$\exists v_0\in \mathbf{Z}$ satisfying $(v_0,\mathcal{L}{v}_0)\in \mathcal{S}$,

(iv)

$\mathcal{L}$ remains ($\mathcal{O},\mathcal{S}$)-continuous, or $\mathcal{S}$ remains ($\mathcal{O},\rho$)-self-closed,

(v)

$\exists \psi \in \mathsf{\Omega }$ verifying

$$\begin{array}{ccc}& & \rho (\mathcal{L}v,\mathcal{L}z)\le \psi \left(\rho \right(v,z\left)\right)\forall v,z\in \mathbf{Z}with(v,z)\in \mathcal{S}.\hfill \end{array}$$

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

5. Examples

Intending to illustrate Theorems 4 and 5, we undertake some examples.

Example 1.

Consider $\mathbf{Z}=[0,\infty )$ along with metric $\rho (v,z)=|v-z|$. Let $\mathcal{L}:\mathbf{Z}\to \mathbf{Z}$ be a map defined by $\mathcal{L}\left(v\right)=\frac{v}{v+1}$. Take a relation $\mathcal{S}:=\{(v,z)\in {\mathbf{Z}}^2:v<z\}$. Then $(\mathbf{Z},\rho )$ is $(\mathcal{O},\mathcal{S})$-complete and $\mathcal{L}$ is $(\mathcal{O},\mathcal{S})$-continuous. Also, $\mathcal{S}$ is locally $\mathcal{L}$-transitive and $\mathcal{L}$-closed relation on $\mathbf{Z}$. Define the auxiliary function $\psi :[0,\infty )\to [0,\infty )$ by $\psi \left(s\right)=\frac{s}{s+1}$ and choose $K\ge 0$ arbitrarily. Then, for all $(v,z)\in \mathcal{S}$, we have

$$\begin{array}{ccc}\hfill \rho (\mathcal{L}v,\mathcal{L}z)& =& |\frac{v}{v+1}-\frac{z}{z+1}|=\left|\frac{v-z}{1+v+z+vz}\right|\hfill \\ & \le & \frac{v-z}{1+(v-z)}=\frac{\rho (v,z)}{1+\rho (v,z)}\hfill \\ & \le & \psi \left(\rho \right(v,z\left)\right)+Kmin\left\{\rho \right(v,\mathcal{L}v),\rho (z,\mathcal{L}z),\rho (v,\mathcal{L}z),\rho (z,\mathcal{L}v\left)\right\}.\hfill \end{array}$$

Thus, the map $\mathcal{L}$ satisfies the condition (v) of Theorem 4. Similarly, rest assumptions of Theorems 4 and 5 can be verified. In turn, $\mathcal{L}$ owns a unique fixed point, namely, $\overline{v}=0$.

Example 2.

Consider $\mathbf{Z}=[0,1)$ along with the metric $\rho (v,z)=|v-z|$. Let $\mathcal{L}:\mathbf{Z}\to \mathbf{Z}$ be a map defined by

$$\mathcal{L}\left(v\right)=\left\{\begin{array}{c}{v}^2,\mathrm{if}0\le v<1/3\hfill \\ 0,\mathrm{if}1/3\le v<1.\hfill \end{array}\right.$$

Take a relation $\mathcal{S}:=\{(v,z)\in {\mathbf{Z}}^2:v\ge z\}$. Clearly, $(\mathbf{Z},\rho )$ is $(\mathcal{O},\mathcal{S})$-complete. Also, $\mathcal{S}$ is locally $\mathcal{L}$-transitive and $\mathcal{L}$-closed binary relation on $\mathbf{Z}$. Here, $\mathcal{L}$ is not $(\mathcal{O},\mathcal{S})$-continuous. However, $\mathcal{S}$ is $(\mathcal{O},\rho )$-self-closed. Also, $\mathcal{L}$ satisfies the contractivity condition (v) for the auxiliary function $\psi \left(s\right)=2s/3$ and for the constant $K=1$. Similarly, rest assumptions of Theorems 4 and 5 can be verified. In turn, $\mathcal{L}$ owns a unique fixed point, namely, $\overline{v}=0$.

6. Applications to Boundary Value Problems

Considering the BVP:

$$\left\{\begin{array}{c}{w}^{\prime }\left({\displaystyle \ell }\right)={\displaystyle Ϝ}({\displaystyle \ell },w\left({\displaystyle \ell }\right)),{\displaystyle \ell }\in [0,l]\hfill \\ w\left(0\right)=w\left(l\right)\hfill \end{array}\right.$$

(13)

where ${\displaystyle Ϝ}:[0,l]\times \mathbb{R}\to \mathbb{R}$ remains a continuous function. By $\mathsf{\Phi }$, we denote the family of continuous and monotonic increasing functions $\varphi :[0,\infty )\to [0,\infty )$, enjoying $\varphi \left(s\right)<s,\forall s>0$. Obviously, $\mathsf{\Phi }\subset \mathsf{\Omega }$.

As usual, by $\mathcal{C}[0,l]$ and ${\mathcal{C}}^{\prime }[0,l]$, we denote respectively the class of: continuous functions and continuously differentiable functions from the interval $[0,l]$ to $\mathbb{R}$.

Definition 13

([24]). $\underline{w}\in {\mathcal{C}}^{\prime }[0,l]$ is called a lower solution of (13) if

$$\left\{\begin{array}{c}{\underline{w}}^{\prime }\left({\displaystyle \ell }\right)\le {\displaystyle Ϝ}({\displaystyle \ell },\underline{w}\left({\displaystyle \ell }\right)),{\displaystyle \ell }\in [0,l]\hfill \\ \underline{w}\left(0\right)\le \underline{w}\left(l\right).\hfill \end{array}\right.$$

Definition 14

([24]). $\overline{w}\in {\mathcal{C}}^{\prime }[0,l]$ is called an upper solution of (13) if

$$\left\{\begin{array}{c}{\overline{w}}^{\prime }\left({\displaystyle \ell }\right)\le {\displaystyle Ϝ}({\displaystyle \ell },\overline{w}\left({\displaystyle \ell }\right)),{\displaystyle \ell }\in [0,l]\hfill \\ \overline{w}\left(0\right)\le \overline{w}\left(l\right).\hfill \end{array}\right.$$

Our main result of this section runs as under:

Theorem 6.

Along with the Problem (13), assume that $\exists r>0$ and $\exists \varphi \in \mathsf{\Phi }$, verifying

$$0\le \left[{\displaystyle Ϝ}\right({\displaystyle \ell },q)+rq]-\left[{\displaystyle Ϝ}\right({\displaystyle \ell },p)+rp]\le r\varphi (q-p),\forall {\displaystyle \ell }\in [0,l]and\forall p,q\in \mathbb{R}withp\le q.$$

(14)

If Problem (13) possesses a lower or an upper solution, then it admits a unique solution.

Proof. 

Clearly, (13) can be written as

$$\left\{\begin{array}{c}{w}^{\prime }\left({\displaystyle \ell }\right)+{\displaystyle \ell }w\left({\displaystyle \ell }\right)={\displaystyle Ϝ}({\displaystyle \ell },w\left({\displaystyle \ell }\right))+{\displaystyle \ell }w\left({\displaystyle \ell }\right),\forall {\displaystyle \ell }\in [0,l]\hfill \\ w\left(0\right)=w\left(l\right)\hfill \end{array}\right.$$

which remains equivalent to the Fredholm integral equation:

$$w\left({\displaystyle \ell }\right)={\int }_0^{l}N({\displaystyle \ell },\xi )[{\displaystyle Ϝ}(\xi ,w\left(\xi \right))+{\displaystyle \ell }w\left(\xi \right)]d\xi .$$

(15)

Here, $N({\displaystyle \ell },\xi )$ is the Green function, defined by

$$N({\displaystyle \ell },\xi )=\left\{\begin{array}{c}\frac{e^{r(l+\xi -{\displaystyle \ell })}}{e^{rl}-1},0\le \xi <{\displaystyle \ell }\le l\hfill \\ \frac{e^{r(\xi -{\displaystyle \ell })}}{e^{rl}-1},0\le {\displaystyle \ell }<\xi \le l.\hfill \end{array}\right.$$

Denote $\mathbf{Z}:=\mathcal{C}[0,l]$. Consider the function $\mathcal{L}:\mathbf{Z}\to \mathbf{Z}$ defined by

$$\left(\mathcal{L}w\right)\left({\displaystyle \ell }\right)={\int }_0^{l}N({\displaystyle \ell },\xi )[{\displaystyle Ϝ}(\xi ,w\left(\xi \right))+rw\left(\xi \right)]d\xi ,\forall {\displaystyle \ell }\in [0,l].$$

(16)

Define the two relations on $\mathbf{Z}$ as follows:

$$\underline{\mathcal{S}}=\{(w,v)\in {\mathbf{Z}}^2:w\left({\displaystyle \ell }\right)\le v\left({\displaystyle \ell }\right),\forall {\displaystyle \ell }\in [0,l]\}$$

(17)

and

$$\overline{\mathcal{S}}=\{(w,v)\in {\mathbf{Z}}^2:w\left({\displaystyle \ell }\right)\ge v\left({\displaystyle \ell }\right),\forall {\displaystyle \ell }\in [0,l]\}.$$

In lieu of one of the hypotheses, let $\underline{w}\in {\mathcal{C}}^{\prime }[0,l]$ be a lower solution of (13). Now, we shall show that $(\underline{w},\mathcal{L}\underline{w})\in \underline{\mathcal{S}}$. One has

$${\underline{w}}^{\prime }\left({\displaystyle \ell }\right)+r\underline{w}\left({\displaystyle \ell }\right)\le {\displaystyle Ϝ}({\displaystyle \ell },\underline{w}\left({\displaystyle \ell }\right))+r\underline{w}\left({\displaystyle \ell }\right),\forall {\displaystyle \ell }\in [0,l].$$

By multiplying to both of the sides with $e^{r{\displaystyle \ell }}$, we find

$${\left(\underline{w}\left({\displaystyle \ell }\right)e^{r{\displaystyle \ell }}\right)}^{\prime }\le [{\displaystyle Ϝ}({\displaystyle \ell },\underline{w}\left({\displaystyle \ell }\right))+r\underline{w}\left({\displaystyle \ell }\right)]e^{r{\displaystyle \ell }},\forall {\displaystyle \ell }\in [0,l]$$

thereby yielding

$$\underline{w}\left({\displaystyle \ell }\right)e^{r{\displaystyle \ell }}\le \underline{w}\left(0\right)+{\int }_0^{{\displaystyle \ell }}[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]e^{r\xi }d\xi ,\forall {\displaystyle \ell }\in [0,l].$$

(18)

Using the fact $\underline{w}\left(0\right)\le \underline{w}\left(l\right)$, we find

$$\underline{w}\left(0\right)e^{rl}\le \underline{w}\left(l\right)e^{rl}\le \underline{w}\left(0\right)+{\int }_0^l[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]e^{r\xi }d\xi$$

so that

$$\underline{w}\left(0\right)\le {\int }_0^l\frac{e^{r\xi }}{e^{rl}-1}[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]d\xi .$$

(19)

Employing (18) and (19), we find

$$\begin{array}{ccc}\hfill \underline{w}\left({\displaystyle \ell }\right)e^{r{\displaystyle \ell }}& \le & {\int }_0^l\frac{e^{r\xi }}{e^{rl}-1}[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]d\xi +{\int }_0^{{\displaystyle \ell }}{e}^{r\xi }[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]d\xi \hfill \\ & =& {\int }_0^{{\displaystyle \ell }}\frac{e^{r(l+\xi )}}{e^{rl}-1}[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]d\xi +{\int }_{{\displaystyle \ell }}^l\frac{e^{r\xi }}{e^{rl}-1}[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]d\xi \hfill \end{array}$$

thereby implying

$$\begin{array}{ccc}\hfill \underline{w}\left({\displaystyle \ell }\right)& \le & {\int }_0^{{\displaystyle \ell }}\frac{e^{r(l+\xi -{\displaystyle \ell })}}{e^{rl}-1}[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]d\xi +{\int }_{{\displaystyle \ell }}^l\frac{e^{r(\xi -{\displaystyle \ell })}}{e^{rl}-1}[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]d\xi \hfill \\ & =& {\int }_0^{l}N({\displaystyle \ell },\xi )[{\displaystyle Ϝ}(\xi ,\underline{w}\left(\xi \right))+r\underline{w}\left(\xi \right)]d\xi \hfill \\ & =& \left(\mathcal{L}\underline{w}\right)\left({\displaystyle \ell }\right),\forall {\displaystyle \ell }\in [0,l]\hfill \end{array}$$

so that $(\underline{w},\mathcal{L}\underline{w})\in \underline{\mathcal{S}}$. Similarly, if $\overline{w}\in {\mathcal{C}}^{\prime }[0,l]$ is an upper solution of (13), then we can prove that $(\overline{w},\mathcal{L}\overline{w})\in \overline{\mathcal{S}}$.

Next, we shall verify that $\underline{\mathcal{S}}$ is $\mathcal{L}$-closed. Choose $w,v\in \mathbf{Z}$ such that $(w,v)\in \underline{\mathcal{S}}$. Making use of (14), we find

$${\displaystyle Ϝ}({\displaystyle \ell },w({\displaystyle \ell }\left)\right)+rw\left({\displaystyle \ell }\right)\le {\displaystyle Ϝ}({\displaystyle \ell },v({\displaystyle \ell }\left)\right)+rv\left({\displaystyle \ell }\right),\forall {\displaystyle \ell }\in [0,l].$$

(20)

By (16) and (20) and $N({\displaystyle \ell },\xi )>0$, $\forall {\displaystyle \ell },\xi \in [0,l]$, we obtain

$$\begin{array}{ccc}\hfill \left(\mathcal{L}w\right)\left({\displaystyle \ell }\right)& =& {\int }_0^{l}N({\displaystyle \ell },\xi )[{\displaystyle Ϝ}(\xi ,w\left(\xi \right))+{\displaystyle \ell }w\left(\xi \right)]d\xi \hfill \\ & \le & {\int }_0^{l}N({\displaystyle \ell },\xi )[{\displaystyle Ϝ}(\xi ,v\left(\xi \right))+{\displaystyle \ell }v\left(\xi \right)]d\xi \hfill \\ & =& \left(\mathcal{L}v\right)\left({\displaystyle \ell }\right),\forall {\displaystyle \ell }\in [0,l],\hfill \end{array}$$

which in view of (6) yields that $(\mathcal{L}w,\mathcal{L}v)\in \underline{\mathcal{S}}$; hence, the conclusion is immediate. Similarly, we can verify that $\overline{\mathcal{S}}$ is also $\mathcal{L}$-closed.

Endow the following metric $\rho$ on $\mathbf{Z}$:

$$\rho (w,v)=\underset{{\displaystyle \ell }\in [0,l]}{sup}|w\left({\displaystyle \ell }\right)-v\left({\displaystyle \ell }\right)|,\forall w,v\in \mathbf{Z}.$$

(21)

Clearly, the metric space $(\mathbf{Z},\rho )$ is ($\mathcal{O},\underline{\mathcal{S}}$)-complete as well as ($\mathcal{O},\overline{\mathcal{S}}$)-complete. To verify the contraction condition, take $w,v\in \mathbf{Z}$ such that $(w,v)\in \underline{\mathcal{S}}$. Making use of (14), (16) and (21), we find

$$\begin{array}{ccc}\hfill \rho (\mathcal{L}w,\mathcal{L}v)& =& \underset{{\displaystyle \ell }\in [0,l]}{sup}|\left(\mathcal{L}w\right)\left({\displaystyle \ell }\right)-\left(\mathcal{L}v\right)\left({\displaystyle \ell }\right)|=\underset{{\displaystyle \ell }\in [0,l]}{sup}\left(\left(\mathcal{L}v\right)\left({\displaystyle \ell }\right)-\left(\mathcal{L}w\right)\left({\displaystyle \ell }\right)\right)\hfill \\ \hfill & \le & \underset{{\displaystyle \ell }\in [0,l]}{sup}{\int }_0^{l}N({\displaystyle \ell },\xi )[{\displaystyle Ϝ}(\xi ,v\left(\xi \right))+rv\left(\xi \right)-{\displaystyle Ϝ}(\xi ,w\left(\xi \right))-rw\left(\xi \right)]d\xi \hfill \\ & \le & \underset{{\displaystyle \ell }\in [0,l]}{sup}{\int }_0^{l}N({\displaystyle \ell },\xi )r\varphi (v\left(\xi \right)-w\left(\xi \right))d\xi .\hfill \end{array}$$

(22)

Since, we have $0\le v\left(\xi \right)-w\left(\xi \right)\le \rho (w,v)$, therefore, monotonicity of $\varphi$ provides that

$$\varphi \left(v\right(\xi )-w(\xi \left)\right)\le \varphi \left(\rho \right(w,v\left)\right).$$

Using the above inequality, (22) reduces to

$$\begin{array}{ccc}\hfill \rho (\mathcal{L}w,\mathcal{L}v)& \le & r\varphi \left(\rho (w,v)\right)\underset{{\displaystyle \ell }\in [0,l]}{sup}{\int }_0^{l}N({\displaystyle \ell },\xi )d\xi \hfill \\ & =& r\varphi \left(\rho (w,v)\right)\underset{{\displaystyle \ell }\in [0,l]}{sup}\frac{1}{e^{rl}-1}\left[\frac{1}{r}{e}^{r(l+\xi -{\displaystyle \ell })}{|}_0^{{\displaystyle \ell }}+\frac{1}{r}{e}^{r(\xi -{\displaystyle \ell })}{|}_{{\displaystyle \ell }}^l\right]\hfill \\ & =& r\varphi \left(\rho (w,v)\right)\frac{1}{r(e^{rl}-1)}(e^{rl}-1)\hfill \\ & =& \varphi \left(\rho \right(w,v\left)\right)\hfill \end{array}$$

thereby implying

$$\begin{array}{ccc}& & \rho (\mathcal{L}w,\mathcal{L}v)\le \varphi \left(\rho \right(w,v\left)\right)+Kmin\left\{\rho \right(w,\mathcal{L}w),\rho (v,\mathcal{L}v),\rho (w,\mathcal{L}v),\rho (v,\mathcal{L}w\left)\right\},\hfill \\ & & \forall w,v\in \mathbf{Z},\mathrm{satisfying}(w,v)\in \underline{\mathcal{S}}\hfill \end{array}$$

where $K\ge 0$ is arbitrary. A similar contraction condition can be verified analogously for the relation $\overline{\mathcal{S}}$.

Let $\left\{w_n\right\}\subset \mathbf{Z}$ be an $\mathcal{L}$-orbital $\underline{\mathcal{S}}$-preserving sequence converging to $\tilde{w}\in \mathbf{Z}$. Then, we have $w_n\left({\displaystyle \ell }\right)\le \tilde{w}\left({\displaystyle \ell }\right)$, $\forall n\in \mathbb{N}$ and $\forall {\displaystyle \ell }\in [0,l]$. By (6), we have $(w_n,\tilde{w})\in \underline{\mathcal{S}},\forall n\in \mathbb{N}$. In turn, $\underline{\mathcal{S}}$ is ($\mathcal{O},\rho$)-self-closed. Similarly, we can verify that $\overline{\mathcal{S}}$ is also ($\mathcal{O},\rho$)-self-closed.

Thus conditions (i)–(v) of Theorem 4 are verified for the relational metric spaces $(\mathbf{Z},\rho ,\underline{\mathcal{S}})$ and $(\mathbf{Z},\rho ,\overline{\mathcal{S}})$. Consequently, $\mathcal{L}$ has a fixed point.

Take arbitrary $w,v\in \mathbf{Z}$ so that $\mathcal{L}\left(w\right),\mathcal{L}\left(v\right)\in \mathcal{L}\left(\mathbf{Z}\right)$. Set $u:=max\{\mathcal{L}w,\mathcal{L}v\}$, thereby implying $(\mathcal{L}w,u)\in \underline{\mathcal{S}}$ and $(\mathcal{L}v,u)\in \underline{\mathcal{S}}$. This shows that the set $\mathcal{L}\left(\mathbf{Z}\right)$ is $\underline{\mathcal{S}}$-directed. Similarly, $\mathcal{L}\left(\mathbf{Z}\right)$ is also $\overline{\mathcal{S}}$-directed. Thus, using Theorem 5, $\mathcal{L}$ ows a unique fixed point, which in turns remains the unique solution of (13). □

7. Conclusions

In this manuscript, some fixed point results employing a locally $\mathcal{L}$-transitive relation under strict almost $\psi$-contraction in the sense of Boyd and Wong [1] have been investigated. As a future work, one can prove the analogues of Theorems 4 and 5 for locally finitely $\mathcal{L}$-transitive relation under strict almost $\psi$-contraction following the results of Alam et al. [13]. Besides BVP, one can also apply our results to certain types of matrix equations or integral equations.