Matkowski-Type Functional Contractions under Locally Transitive Binary Relations and Applications to Singular Fractional Differential Equations

Abstract

This article presents a few fixed-point results under Matkowski-type functional contractive mapping using locally $\mathbf{J}$-transitive binary relations. Our results strengthen, enhance, and consolidate numerous existent fixed-point results. To argue for the efficacy of our results, several illustrated examples are supplied. With the help of our findings, we deal with the existence and uniqueness theorems pertaining to the solution of a variety of singular fractional differential equations.

1. Introduction

A crucial and vital outcome of metric fixed-point theory is the Banach contraction principle (BCP for short). In reality, the BCP assesses whether a unique fixed point for a self-contraction on a whole metric space occurs. Furthermore, this result proposes an iterative method for estimating this fixed point. Over the past century, numerous researchers have broadened and refined this core result. One of the natural generalizations of the BCP is determined by expanding the usual contraction to nonlinear contraction through an appropriate auxiliary function $\theta :[0,\infty )\to [0,\infty )$. Numerous generalizations have been obtained by appropriately modifying $\theta$, so there is now an extensive collection of studies on this topic. A notable class of nonlinear contractions was launched by Matkowski [1], wherein the author initiated the idea of comparison functions. In recent years, Matkowski’s fixed-point theorem has been further extended to the frameworks of more generalized distance spaces, such as Menger space [2], partial metric space [3], fuzzy normed linear space [4], $\nu$-generalized metric space [5], and several others. Order-theoretic variants of the Matkowski fixed point theorem were studied by Agarwal et al. [6] and O’Regan and Petruşel [7], which were then further enriched by Aydi et al. [8] to prove certain tripled coincidence point theorems. On the other hand, Khantwal and Gairola [9] presented a fixed-point theorem for a system of mappings on the finite product of metric spaces employing the concept of the comparison function and utilized the same to solve a system of functional equations. Very recently, Barcz [10] provided a new proof of the Matkowski fixed-point theorem using Cantor’s intersection theorem instead of Picard iterations.

Another variety of generalizations of the BCP encompass wider contractivity conditions on a given metric space $(\bigvee ,\sigma )$ comprising the distance $\sigma (v,w)$ on R.H.S., whereas $v,w\in \bigvee$. Nonetheless, some contraction conditions comprise $\sigma (v,w)$ with the displacements of $v,w\in \bigvee$ for the mapping $\mathbf{J}$: $\sigma (v,\mathbf{J}v),\sigma (w,\mathbf{J}w),\sigma (v,\mathbf{J}w),\sigma (w,\mathbf{J}v)$. Functional contractions remain a broad variety of such contractions, represented as follows:

$$\sigma (\mathbf{J}v,\mathbf{J}w)\le \mathcal{G}\left(\sigma \right(v,w),\sigma (v,\mathbf{J}v),\sigma (w,\mathbf{J}w),\sigma (v,\mathbf{J}w),\sigma (w,\mathbf{J}v\left)\right),\forall v,w\in \bigvee ,$$

where the proper choice of the function $\mathcal{G}:{[0,\infty )}^5\to [0,\infty )$ is made.

Alam and Imdad [11] developed the BCP in the framework of relational metric space, which is refined by Alam et al. [12]. Relational contractions are, in fact, much broader than Banach contractions, as they are applied for contractions that are linked by a binary relation. Following the initial unveiling of this groundbreaking discovery, the results in relational metric spaces have drawn a lot of attention over the last eight years. Subsequently, Alam and Imdad [13] investigated the coincidence and common fixed-point theorems in the framework of relational metric space. As it turns out, multiple works have already been published in the field of relational metric space by enlarging the class of relational contractions, e.g., relational Boyd–Wong contraction [14,15], relational F-contraction [16], relational multivalued Suzuki-type $\theta$-contraction [17], relational functional contraction [18], relational Matkowski contractions [19], relational $(\psi ,\varphi )$-contraction [20], relational L-contraction [21], relational almost $\varphi$-contraction [22], and relational Geraghty contraction [23].

In the recent past, fractional differential equations (FDEs for short) were discussed with regard to the impassable development and applicability of the area of fractional calculus. In recent years, some typical boundary value problems (BVPs for short) of FDEs have been solved using the fixed-point theorems proven in ordered metric space by Liang and Zhang [24] and Cabrera et al. [25], relational metric space by Saleh et al. [26] and Alamer et al. [27], and orthogonal metric space by Abdou [28].

The intention of this manuscript is to prove a few fixed-point theorems in a metric space possessed with a local class of transitive relation subjects to a relation-preserving functional contraction of the form:

$$\sigma (\mathbf{J}v,\mathbf{J}w)\le \theta \left(\sigma \right(v,w\left)\right)+\zeta \left(\sigma \right(v,\mathbf{J}v),\sigma (w,\mathbf{J}w),\sigma (v,\mathbf{J}w),\sigma (w,\mathbf{J}v\left)\right).$$

We also construct a variety of illustrative examples demonstrating the main results. Furthermore, using our results, we obtain the positive solutions of certain BVP associated with a singular FDE.

2. Preliminaries

In the following, the set of real numbers will be symbolized by $\mathbb{R}$, and the set of natural numbers by $\mathbb{N}$. In the context of a set ⋁, a binary relation (a relation) refers to the subset $\mathfrak{B}\subseteq {\bigvee }^2$. In the following definitions, ⋁ is a set, $\sigma$ is a metric on ⋁, $\mathfrak{B}$ is a relation on ⋁ and $\mathbf{J}:\bigvee \to \bigvee$ is a mapping.

Definition 1 

([11]). A pair $v,w\in \bigvee$ is described as $\mathfrak{B}$-comparative if $(v,w)\in \mathfrak{B}$ or $(w,v)\in \mathfrak{B}$. We denote it by $[v,w]\in \mathfrak{B}$.

Definition 2 

([29]). ${\mathfrak{B}}^{-1}:=\{(v,w)\in {\bigvee }^2:(w,v)\in \mathfrak{B}\}$ is described as transpose of $\mathfrak{B}$.

Definition 3 

([29]). By the symmetric closure of $\mathfrak{B}$, one means the relation ${\mathfrak{B}}^s:=\mathfrak{B}\cup {\mathfrak{B}}^{-1}$.

Remark 1 

([11]). $(v,w)\in {\mathfrak{B}}^s⟺[v,w]\in \mathfrak{B}.$

Definition 4 

([11]). A sequence $\left\{v_n\right\}\subset \bigvee$ verifying $(v_n,v_{n+1})\in \mathfrak{B}$, ∀$n\in \mathbb{N}$, is described as $\mathfrak{B}$-preserving.

Definition 5 

([11]). $\mathfrak{B}$ is described as $\mathbf{J}$-closed if, for any $v,w\in \bigvee$ verifying $(v,w)\in \mathfrak{B}$, we have

$$(\mathbf{J}v,\mathbf{J}w)\in \mathfrak{B}.$$

Definition 6 

([13]). $(\bigvee ,\sigma )$ is described as $\mathfrak{B}$-complete if each $\mathfrak{B}$-preserving Cauchy sequence in ⋁ remains convergent.

Definition 7 

([13]). $\mathbf{J}$ is described as $\mathfrak{B}$-continuous at $v\in \bigvee$ if for every $\mathfrak{B}$-preserving sequence $\left\{v_n\right\}\subset \bigvee$ with $v_n\stackrel{\sigma }{⟶}v$,

$$\mathbf{J}\left(v_n\right)\stackrel{\sigma }{⟶}\mathbf{J}\left(v\right).$$

Furthermore, any $\mathfrak{B}$-continuous function at each point is termed as $\mathfrak{B}$-continuous.

Definition 8 

([11]). $\mathfrak{B}$ is described as σ-self-closed if every $\mathfrak{B}$-preserving convergent sequence in ⋁ contains a subsequence for which each term remains $\mathfrak{B}$-comparative with the convergence limit.

Definition 9 

([30]). A subset $\mho \subseteq \bigvee$ is described as $\mathfrak{B}$-directed if for every pair $v,w\in \mho$, ∃$u\in \bigvee$ satisfying $(v,u)\in \mathfrak{B}$ and $(w,u)\in \mathfrak{B}$.

Definition 10 

([31]). A relation on a subset $\mho \subseteq \bigvee$ defined by

$${\mathfrak{B}|}_{\mho }:=\mathfrak{B}\cap {\mho }^2$$

is described as the restriction of $\mathfrak{B}$ on ℧.

Definition 11 

([14]). $\mathfrak{B}$ is described as locally $\mathbf{J}$-transitive if for each (effectively) $\mathfrak{B}$-preserving sequence $\left\{v_n\right\}\subset \mathbf{J}(\bigvee )$ (with range $\mho =\{v_n:n\in \mathbb{N}\})$, ${\mathfrak{B}|}_{\mho }$ is transitive.

Proposition 1 

([14]). $\mathfrak{B}$ remains ${\mathbf{J}}^n$-closed relation $\forall n\in \mathbb{N}$, whenever $\mathfrak{B}$ is $\mathbf{J}$-closed.

We will implement the following notations going forward:

•

$Fix\left(\mathbf{J}\right)$:=the set of fixed points of $\mathbf{J}$;

•

$\bigvee (\mathbf{J},\mathfrak{B}):=\{v\in \bigvee :(v,\mathbf{J}v)\in \mathfrak{B}\}$.

Definition 12 

([32]). A function $\theta :[0,\infty )\to [0,\infty )$ is termed a comparison function if

(i)

θ is increasing;

(ii)

$\forall t>0$, $\underset{n\to \infty }{lim}{\theta }^n\left(t\right)=0$.

Finally, one indicates the following auxiliary results.

Proposition 2 

([14]). If $\mathfrak{B}$ is $\mathbf{J}$-closed, then for each $n\in {\mathbb{N}}_0$, $\mathfrak{B}$ is also ${\mathbf{J}}^n$-closed.

Proposition 3 

([32]). If θ is a comparison function, then

$$\theta \left(0\right)=0and\theta \left(t\right)<t,whenevert>0.$$

The family of all continuous functions $\zeta :{[0,\infty )}^4\to [0,\infty )$ verifying $\zeta (t_1,t_2,t_3,t_4)=0$$\iff t_i=0$ for at least one $i\in \{1,2,3,4\}$ will be denoted by $\mathrm{\Lambda }$. This family is proposed by Jleli et al. [33].

Proposition 4. 

If θ is a comparison function and $\zeta \in \mathrm{\Lambda }$, then the following are equivalent:

(i)

$$\begin{gathered} \sigma (\mathbf{J}v,\mathbf{J}w)\le \theta \left(\sigma \right(v,w\left)\right)+\zeta \left(\sigma \right(v,\mathbf{J}v),\sigma (w,\mathbf{J}w),\sigma (v,\mathbf{J}w),\sigma (w,\mathbf{J}v\left)\right), \\ \forall v,w\in \bigvee with(v,w)\in \mathfrak{B}, \end{gathered}$$

(ii)

$$\begin{gathered} \sigma (\mathbf{J}v,\mathbf{J}w)\le \theta \left(\sigma \right(v,w\left)\right)+\zeta \left(\sigma \right(v,\mathbf{J}v),\sigma (w,\mathbf{J}w),\sigma (v,\mathbf{J}w),\sigma (w,\mathbf{J}v\left)\right), \\ \forall v,w\in \bigvee with[v,w]\in \mathfrak{B}. \end{gathered}$$

Proof. 

The conclusion is immediate by using the symmetric property of $\sigma$. □

3. Main Results

We propose the following results relating the uniqueness and existence of fixed points for mappings based on relational nonlinear contractions.

Theorem 1. 

Assume that $(\bigvee ,\sigma )$ remains a metric space endowed with a relation $\mathfrak{B}$, while $\mathbf{J}:\bigvee \to \bigvee$ is a map. Also assume the following:

(a)

$\bigvee (\mathbf{J},\mathfrak{B})\ne \varnothing$;

(b)

$\mathfrak{B}$ is locally $\mathbf{J}$-transitive and $\mathbf{J}$-closed;

(c)

$(\bigvee ,\sigma )$ is $\mathfrak{B}$-complete;

(d)

$\mathbf{J}$ remains $\mathfrak{B}$-continuous or $\mathfrak{B}$ remains σ-self-closed;

(e)

there exists a comparison function θ and $\zeta \in \mathrm{\Lambda }$ verifying

$$\begin{array}{ccc}\hfill \sigma (\mathbf{J}v,\mathbf{J}w)& \le & \theta \left(\sigma \right(v,w\left)\right)+\zeta \left(\sigma \right(v,\mathbf{J}v),\sigma (w,\mathbf{J}w),\sigma (v,\mathbf{J}w),\sigma (w,\mathbf{J}v\left)\right),\hfill \\ & & \forall v,w\in \bigvee with(v,w)\in \mathfrak{B}.\hfill \end{array}$$

Then, $\mathbf{J}$ admits a fixed point.

Proof. 

By (a), choose $v_0\in \bigvee (\mathbf{J},\mathfrak{B})$. Define the sequence $\left\{v_n\right\}\subset \bigvee$ verifying

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

(1)

By assumption (a), we have $(v_0,\mathbf{J}{v}_0)\in \mathfrak{B}$. Employing (b) and Proposition 2, we obtain

$$({\mathbf{J}}^n{v}_0,{\mathbf{J}}^{n+1}{v}_0)\in \mathfrak{B}$$

so that

$$(v_n,v_{n+1})\in \mathfrak{B},\forall n\in \mathbb{N}.$$

(2)

Therefore, $\left\{v_n\right\}$ is an $\mathfrak{B}$-preserving sequence.

Let us denote ${\sigma }_n:=\sigma (v_n,v_{n+1})$. If ${\sigma }_{n_0}=0$ for some $n_0\in {\mathbb{N}}_0$, then in lieu of (1), one has $\mathbf{J}\left(v_{n_0}\right)=v_{n_0}$. Thus, $v_{n_0}$ is a fixed point of $\mathbf{J}$, and hence we are done.

In case ${\sigma }_n>0,\forall n\in {\mathbb{N}}_0$, one can use assumption (e), (1) and (2) to obtain

$$\begin{array}{ccc}\hfill \sigma (v_n,v_{n+1})& =& \sigma (\mathbf{J}{v}_{n-1},\mathbf{J}{v}_n)\le \theta \left(\sigma (v_{n-1},v_n)\right)\hfill \\ & +& \zeta (\sigma (\mathbf{J}{v}_n,v_n),\sigma (\mathbf{J}{v}_{n-1},v_{n-1}),\sigma (\mathbf{J}{v}_{n-1},v_n),\sigma (\mathbf{J}{v}_n,v_{n-1})),\forall n\in \mathbb{N}\hfill \end{array}$$

so that

$$\begin{array}{c}\hfill \sigma (v_n,v_{n+1})\le \theta \left(\sigma (v_{n-1},v_n)\right)+\zeta (\sigma (v_{n+1},v_n),\sigma (v_n,v_{n-1}),0,\sigma (v_{n+1},v_{n-1})),\forall n\in \mathbb{N}\end{array}$$

which, using the definition of $\mathrm{\Lambda }$ and the monotonicity of $\theta$, becomes

$$\sigma (v_n,v_{n+1})\le {\theta }^n\sigma (v_0,\mathbf{J}{v}_0),\forall n\in \mathbb{N},$$

or,

$$\begin{array}{c}\hfill {\sigma }_n\le {\theta }^n\left({\sigma }_0\right),\forall n\in \mathbb{N}.\end{array}$$

(3)

Using $n\to \infty$ in (3) and making use of property of $\theta$, one obtains

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

(4)

Fix $\epsilon >0.$ Then, due to availability of (4), one can find $n\in {\mathbb{N}}_0$ verifying

$$\begin{array}{c}\hfill {\sigma }_n<\epsilon -\theta \left(\epsilon \right).\end{array}$$

(5)

We will have to prove that $\left\{v_n\right\}$ is Cauchy. By triangular inequality, monotonicity of $\theta$, (3) and (5), one obtains

$$\begin{array}{ccc}\hfill \sigma (v_n,v_{n+2})& \le & \sigma (v_n,v_{n+1})+\sigma (v_{n+1},v_{n+2})={\sigma }_n+{\sigma }_{n+1}\hfill \\ & \le & {\sigma }_n+\theta \left({\sigma }_n\right)\hfill \\ & <& \epsilon -\theta \left(\epsilon \right)+\theta [\epsilon -\theta (\epsilon \left)\right]\le \epsilon -\theta \left(\epsilon \right)+\theta \left(\epsilon \right)\hfill \\ & =& \epsilon .\hfill \end{array}$$

Using locally $\mathbf{J}$-transitivity of $\mathfrak{B}$, assumption (e), monotonicity of $\theta$, (1) and (5), one gets

$$\begin{array}{ccc}\hfill \sigma (v_n,v_{n+3})& \le & \sigma (v_n,v_{n+1})+\sigma (v_{n+1},v_{n+3})\hfill \\ & =& {\sigma }_n+\sigma (\mathbf{J}{v}_n,\mathbf{J}{v}_{n+2})\hfill \\ & <& \epsilon -\theta \left(\epsilon \right)+\theta \left(\sigma (v_n,v_{n+2})\right)\hfill \\ & \le & \epsilon -\theta \left(\epsilon \right)+\theta \left(\epsilon \right)\hfill \\ & =& \epsilon .\hfill \end{array}$$

Thus, by induction, one has

$$\sigma (v_n,v_{n+v})<\epsilon ,\forall v\in \mathbb{N}.$$

We shall apply (d) to conclude that $v\in Fix\left(\mathbf{J}\right)$. Let $\mathbf{J}$ be $\mathfrak{B}$-continuous. Since $\left\{v_n\right\}$ is $\mathfrak{B}$-preserving, verifying $v_n\stackrel{\sigma }{⟶}v$, therefore, by $\mathfrak{B}$-continuity of $\mathbf{J}$, we have $v_{n+1}=\mathbf{J}\left(v_n\right)\stackrel{\sigma }{⟶}\mathbf{J}\left(v\right)$ so that $\mathbf{J}\left(v\right)=v$. If $\mathfrak{B}$ is $\sigma$-self closed then $\left\{v_n\right\}$, being $\mathfrak{B}$-preserving along with $v_n\stackrel{\sigma }{⟶}v$, induces a subsequence $\left\{v_{n_k}\right\}$ verifying $[v_{n_k},v]\in \mathfrak{B}$, $\forall k\in \mathbb{N}.$ Using contractivity condition (e), Proposition 4 and $[v_{n_k},v]\in \mathfrak{B}$, we have

$$\begin{array}{ccc}\hfill \sigma (v_{n_k+1},\mathbf{J}v)& =& \sigma (\mathbf{J}{v}_{n_k},\mathbf{J}v)\hfill \\ & \le & \theta \left(\sigma (v_{n_k},v)\right)+\zeta (\sigma (v,\mathbf{J}v),\sigma (v_{n_k},v_{n_k+1}),\sigma (v,v_{n_k+1}),\sigma (v_{n_k},\mathbf{J}v)).\hfill \end{array}$$

Based on Remark 3 (whether $\sigma (v_{n_k},v)$ is non-zero or zero), the above inequality gives rise to

$$\begin{array}{c}\hfill \sigma (v_{n_k+1},\mathbf{J}v)\le \sigma (v_{n_k},v)+\zeta (\sigma (v,\mathbf{J}v),\sigma (v_{n_k},v_{n_k+1}),\sigma (v,v_{n_k+1}),\sigma (v_{n_k},\mathbf{J}v)).\end{array}$$

(6)

Letting $k\to \infty$ in (6) and using $v_{n_k}\stackrel{\sigma }{⟶}v$ and the definition of $\mathrm{\Lambda }$, we have

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}\sigma (v_{n_k+1},\mathbf{J}v)=0.\end{array}$$

(7)

thereby implying $v_{n_k+1}\stackrel{\sigma }{⟶}\mathbf{J}\left(v\right)$. Due to the uniqueness of limit, we obtain $\mathbf{J}\left(v\right)=v$. □

Theorem 2. 

Under the assumptions(a)–(d) of Theorem 1, if $\mathbf{J}(\bigvee )$ is ${\mathfrak{B}}^s$-directed, then $\mathbf{J}$ admits a unique fixed point.

Proof. 

Take $v_1,v_2\in Fix\left(\mathbf{J}\right)$, we have

$${\mathbf{J}}^n\left(v_1\right)=v_1\mathrm{and}{\mathbf{J}}^n\left(v_2\right)=v_2,\forall n\in \mathbb{N}.$$

(8)

As $\mathbf{J}(\bigvee )$ is ${\mathfrak{B}}^s$-directed and contains $v_1,v_2\in \mathbf{J}(\bigvee )$, $\exists \omega \in \bigvee$, verifying

$$[v_1,\omega ]\in \mathfrak{B}\mathrm{and}[v_2,\omega ]\in \mathfrak{B}.$$

(9)

By assumption (b) and Proposition 2, we have

$$[v_1,{\mathbf{J}}^n\omega ]\in \mathfrak{B}\mathrm{and}[v_2,{\mathbf{J}}^n\omega ]\in \mathfrak{B}.$$

(10)

Using (10), assumption (e) and Proposition 4, we obtain

$$\begin{array}{ccc}& & \sigma (v_1,{\mathbf{J}}^{n+1}\omega )=\sigma (\mathbf{J}{v}_1,\mathbf{J}\left({\mathbf{J}}^n\omega \right))\le \theta (\sigma (v_1,{\mathbf{J}}^n\omega )\hfill \\ & & +\zeta (\sigma (v_1,v_1),\sigma ({\mathbf{J}}^n\omega ,{\mathbf{J}}^{n+1}\omega ),\sigma (v_1,{\mathbf{J}}^{n+1}\omega ),\sigma ({\mathbf{J}}^n\omega ,v_1))\hfill \end{array}$$

so that

$$\sigma (v_1,{\mathbf{J}}^{n+1}\omega )\le \theta \left(\sigma (v_1,{\mathbf{J}}^n\omega )\right).$$

(11)

Set ${\varsigma }_n:=\sigma (v_1,{\mathbf{J}}^n\omega )$. We assert that

$${\displaystyle \underset{n\to \infty }{lim}{\varsigma }_n=0.}$$

(12)

Consider the following cases:

• Case-I: If ${\varsigma }_{n_0}=0$ for some $n_0\in \mathbb{N}$, then $v_1={\mathbf{J}}^{n_0}\left(\omega \right)$. This yields that $v_1=\mathbf{J}\left(v_1\right)={\mathbf{J}}^{n_0+1}\left(\omega \right)$ so that ${\varsigma }_{n_0+1}=0$. Hence by induction, we obtain ${\varsigma }_n=0,\forall n\ge n_0,$ so that $\underset{n\to \infty }{lim}{\varsigma }_n=0$.
• Case-II: If ${\varsigma }_n>0$, for all $n\in \mathbb{N}$, then using induction on n in (11) and monotonicity of $\theta$, we get

$$\begin{array}{c}\hfill {\varsigma }_{n+1}\le \theta \left({\varsigma }_n\right)\le {\theta }^2\left({\varsigma }_{n-1}\right)\le \cdots \end{array}$$

so that

$$\begin{array}{c}\hfill {\varsigma }_{n+1}\le {\theta }^n\left({\varsigma }_1\right),n\in \mathbb{N}.\end{array}$$

(13)

Letting $n\to \infty$ in (13) and using the definition of comparison function, we have

$${\displaystyle \underset{n\to \infty }{lim}{\varsigma }_{n+1}\le \underset{n\to \infty }{lim}{\theta }^n\left({\varsigma }_1\right)=0.}$$

Thus, in both the cases, (12) is verified. Employing (8), (9), (12) and triangular inequality, we have

$$\sigma (v_1,v_2)=\sigma (v_1,{\mathbf{J}}^n\omega )+\sigma ({\mathbf{J}}^n\omega ,v_2)\to 0,\mathrm{as}n\to \infty$$

thereby yielding $v_1=v_2$. Therefore, $\mathbf{J}$ admits a unique fixed point. □

Remark 2. 

In particular, for $\zeta =0$, Theorems 1 and 2 reduce to the corresponding results of Arif et al. [19].

4. Illustrative Examples

To highlight our results proven in the previous section, we adopt the following examples.

Example 1. 

Consider $\bigvee =[1,3]$ with usual metric σ and a relation $\mathfrak{B}=\left\{\right(1,1),(1,2),(2,1),(2,2),(1,3\left)\right\}$. Define a map $\mathbf{J}:\bigvee \to \bigvee$ by

$$\mathbf{J}\left(v\right)=\left\{\begin{array}{c}1\mathrm{if}1\le v\le 2\hfill \\ 2\mathrm{if}2<v\le 3.\hfill \end{array}\right.$$

Then, $(\bigvee ,\sigma )$ is a $\mathfrak{B}$-complete metric space and $\mathfrak{B}$ is $\mathbf{J}$-closed. Define the auxiliary functions $\theta \left(t\right)=t/3$ and $\zeta (t_1,t_2,t_3,t_4)=t_1{t}_2{t}_3{t}_4$. Then, θ is a comparison function and $\zeta \in \mathrm{\Lambda }$. It can be easily shown that the contractivity condition (e) of Theorem 1 holds under these auxiliary functions.

Suppose that $\left\{v_n\right\}\subset \bigvee$ is $\mathfrak{B}$-preserving sequence satisfying $v_n\stackrel{\sigma }{⟶}w$ so that $(v_n,v_{n+1})\in \mathfrak{B},$$\forall n\in \mathbb{N}$. Here, $(v_n,v_{n+1})\notin \left\{(1,3)\right\}$, so that $(v_n,v_{n+1})\in \{(1,1),(1,2),(2,1),(2,2)\}$, $\forall n\in \mathbb{N}$, and hence $\left\{v_n\right\}\subset \{1,2\}$. Since $\{1,2\}$ is closed, we therefore have $[v_n,w]\in \mathfrak{B}$. Hence, $\mathfrak{B}$ is σ-self-closed. The remaining requirements of Theorems 1 and 2 are likewise satisfied, and hence $\mathbf{J}$ has a unique fixed point $v=1$.

Example 2. 

Consider $\bigvee =(0,1]\cup (2,3)$ with usual metric $\sigma (v,w)=|v-w|$. On ⋁, define a relation $\mathfrak{B}:=\{(v,w)\in {\bigvee }^2:v\ge w\}$. Then, the metric space $(\bigvee ,\sigma )$ is $\mathfrak{B}$-complete; however, it is not complete. Define a map $\mathbf{J}:\bigvee \to \bigvee$ by

$$\mathbf{J}\left(v\right)=\left\{\begin{array}{cc}0,\hfill & ifv=1\hfill \\ 1/7,\hfill & otherwise.\hfill \end{array}\right.$$

Then, $\mathfrak{B}$ is $\mathbf{J}$-closed. Also, $\mathbf{J}$ is $\mathfrak{B}$-continuous. Define an auxiliary function $\theta \left(t\right)=t/7$ and $\zeta (t_1,t_2,t_3,t_4)=t_4/7$. Clearly, θ remains a comparison function and $\zeta \in \mathrm{\Lambda }$. Now, for all $v,w\in \bigvee$, verifying $(v,w)\in \mathfrak{B}$, one has

$$\sigma (\mathbf{J}v,\mathbf{J}w)\le \theta \left(\sigma \right(v,w\left)\right)+\zeta \left(\sigma \right(v,\mathbf{J}v),\sigma (w,\mathbf{J}w),\sigma (v,\mathbf{J}w),\sigma (w,\mathbf{J}v\left)\right).$$

Therefore, all the hypotheses of Theorem 1, and similarly those of Theorem 2, hold. Consequently, $\mathbf{J}$ possesses a unique fixed point, namely: $v=1/7$.

5. Applications to Fractional Differential Equations

Let us consider the following singular fractional three-point BVP:

$$\left\{\begin{array}{c}{\mathfrak{D}}_{0^{+}}^{\kappa }v\left(s\right)+h(s,v\left(s\right))=0,\forall s\in (0,1),\hfill \\ v\left(0\right)=v^{\prime }\left(0\right)=v^{″}\left(0\right)=0,v^{″}\left(1\right)=\lambda v^{″}\left(\gamma \right),\hfill \end{array}\right.$$

(14)

where $3<\kappa \le 4,0<\gamma <1$, $0<\lambda {\gamma }^{\kappa -3}<1$ and $h:[0,1]\times [0,\infty )\to [0,\infty )$ remains continuous.

As usual, the classical gamma function and the classical beta function will be represented by $\mathrm{\Gamma }(·)$ and $\beta (·,·)$, respectively. Motivated by [24,25], we shall compute the unique positive solution of (14) under the assumption that h is singular at $s=0$ (i.e., $\underset{s\to 0^{+}}{lim}h(s,·)=\infty$).

Definition 13 

([34]). Given a real valued function ϕ on $(0,\infty )$, the Riemann–Liouville fractional integral of ϕ of order $\kappa >0$ is

$${\mathfrak{I}}_{0^{+}}^{\kappa }\varphi \left(s\right)=\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^s{(s-\xi )}^{\kappa -1}\varphi \left(\xi \right)d\xi ,$$

provided that R.H.S. is pointwise defined on $(0,\infty )$.

Definition 14 

([35]). Given a real valued function ϕ on $(0,\infty )$, the Riemann–Liouville fractional derivative of ϕ of order $\kappa >0$ is

$${\mathfrak{D}}_{0^{+}}^{\kappa }\varphi \left(s\right)=\frac{1}{\mathrm{\Gamma }(n-\kappa )}{\left(\frac{d}{ds}\right)}^n{\int }_0^s\frac{\varphi \left(\xi \right)}{{(s-\xi )}^{\kappa -n+1}}d\xi .$$

Here, $n=\left[\kappa \right]+1$ and $\left[\kappa \right]$ represents the integer part of κ.

Lemma 1 

([24]). The BVP

$${\mathfrak{D}}_{0+}^{\kappa }v\left(s\right)+f\left(s\right)=0,0<s<1,$$

$$v\left(0\right)=v^{\prime }\left(0\right)=v^{″}\left(0\right),v^{″}\left(1\right)=\lambda v^{″}\left(\gamma \right),$$

(15)

where $3<\kappa \le 4,0<\gamma <1,0<\lambda {\gamma }^{\kappa -3}<1$ and $f:[0,1]\to [0,\infty )$ remains continuous, admits a unique solution of the form:

$$v\left(s\right)={\int }_0^1\mathcal{G}(s,\xi )h\left(\xi \right)d\xi +\frac{\lambda s^{\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi )h\left(\xi \right)d\xi ,$$

where

$$\mathcal{G}(s,\xi )=\left\{\begin{array}{cc}\frac{s^{\kappa -1}{(1-\xi )}^{\kappa -3}-{(s-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)},\hfill & 0\le \xi \le s\le 1,\hfill \\ \frac{s^{\kappa -1}{(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)},\hfill & 0\le s\le \xi \le 1\hfill \end{array}\right.$$

and

$$\mathcal{H}(s,\xi )=\frac{{\partial }^2\mathcal{G}(s,\xi )}{\partial s^2}=\left\{\begin{array}{cc}\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}\left[s^{\kappa -3}{(1-\xi )}^{\kappa -3}-{(s-\xi )}^{\kappa -3}\right],\hfill & 0\le \xi \le s\le 1,\hfill \\ \frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{s}^{\kappa -3}{(1-\xi )}^{\kappa -3},\hfill & 0\le s\le \xi \le 1.\hfill \end{array}\right.$$

Remark 3. 

Under the hypotheses of Lemma 1, $\mathcal{G}$ and $\mathcal{H}$ admit the following properties:

• $\mathcal{G}$ remains continuous on $[0,1]\times [0,1]$;
• $\mathcal{G}(s,\xi )\ge 0$;
• $\mathcal{G}(s,1)=0$;
• $\underset{0\le s\le 1}{sup}{\int }_0^1\mathcal{G}(s,\xi )d\xi =\frac{2}{(\kappa -2)\mathrm{\Gamma }(\kappa +1)}$;
• ${\int }_0^1\mathcal{H}(\gamma ,\xi )d\xi =\frac{{\gamma }^{\kappa -3}(\kappa -1)(1-\gamma )}{\mathrm{\Gamma }\left(\kappa \right)}$.

Lemma 2. 

Assume that $0<\rho <1,3<\kappa \le 4$ and $F:(0,1]\to \mathbb{R}$ remains continuous such that $\underset{s\to 0^{+}}{lim}F(s,·)=\infty .$ If the function $s^{\rho }F\left(s\right)$ is continuous on $[0,1]$, then

$$\mathcal{L}\left(s\right)={\int }_0^1\mathcal{G}(s,\xi )F\left(\xi \right)d\xi$$

(16)

is also a continuous function on $[0,1]$, whereas $\mathcal{G}(s,\xi )$ remains the Green’s function given in Lemma 1.

Proof. 

Our proof will be distinguished into three cases.

• Case-I: When $s_0=0$. Obviously, we have $\mathcal{L}\left(0\right)=0$. As $s^{\rho }F\left(s\right)$ is continuous, ∃ a constant $M>0$ verifying

$$|s^{\rho }F\left(s\right)|\le M,\forall s\in [0,1].$$

Thus, we have

$$\begin{array}{cc}\hfill & |\mathcal{L}\left(s\right)-\mathcal{L}\left(0\right)|=\left|\mathcal{L}\left(s\right)\right|=\left|{\int }_0^1\mathcal{G}(s,\xi )F\left(\xi \right)d\xi \right|=\left|{\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right|\hfill \\ \hfill & =\left|{\int }_0^1\frac{s^{\kappa -1}{(1-\xi )}^{\kappa -3}-{(s-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi +{\int }_s^1\frac{s^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right|\hfill \\ \hfill & =\left|{\int }_0^1\frac{s^{\kappa -1}{(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi -{\int }_0^s\frac{{(s-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le \left|{\int }_0^1\frac{s^{\kappa -1}{(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right|+\left|{\int }_0^s\frac{{(s-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le \frac{M{s}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^1{(1-\xi )}^{\kappa -3}{\xi }^{-\kappa }d\xi +\frac{M}{\mathrm{\Gamma }\left(\kappa \right){\int }_0^s}{(s-\xi )}^{\kappa -1}{s}^{-\kappa }d\xi \hfill \\ \hfill & =\frac{M{s}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^1{(1-\xi )}^{\kappa -3}d\xi +\frac{M{s}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^1{\left(1-\frac{\xi }{s}\right)}^{\kappa -1}{\xi }^{-\kappa }d\xi \hfill \\ \hfill & =\frac{M{s}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}\beta (1-\rho ,\kappa -2)+\frac{M{s}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^1{\left(1-\frac{\xi }{s}\right)}^{\kappa -1}{\xi }^{-\kappa }d\xi .\hfill \end{array}$$

(17)

Making use of change of variables $v=\xi /s$ in integral ${\int }_0^1{\left(1-(\xi /s)\right)}^{\kappa -1}{\xi }^{-\rho }$, we get

$${\int }_0^s{\left(1-\frac{\xi }{s}\right)}^{\kappa -1}{\xi }^{-\rho }d\xi =s^{s-\rho }{\int }_0^1{(1-v)}^{\kappa -1}{v}^{-\rho }du=s^{1-\rho }\beta (1-\rho ,\kappa ).$$

(18)

On combining (17) and (18), we obtain

$$\left|\mathcal{L}\left(s\right)\right|\le \frac{M{s}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}\beta (1-\rho ,\kappa -2)+\frac{M{s}^{\kappa -\rho }}{\mathrm{\Gamma }\left(\kappa \right)}\beta (1-\rho ,\kappa ),$$

which upon letting the limit $s\to 0$ gives rise to $\left|\mathcal{L}\right(s\left)\right|\to 0$. This concludes that $\mathcal{L}$ is continuous at $s_0=0$.

• Case-II: When $s_0\in (0,1)$. Assuming without loss of generality that $s_n>s_0$, as the same argument will be hold in case $s_n<s_0$, we shall now have to verify that $\mathcal{L}\left(s_n\right)\to \mathcal{L}\left(s_0\right)$. Indeed, we have

$$\begin{array}{cc}\hfill |\mathcal{L}\left(s_n\right)-\mathcal{L}\left(s_0\right)|& =\left|{\int }_0^{s_n}\frac{s_n^{\kappa -3}{(s_n-\xi )}^{\kappa }-1}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\kappa }{\xi }^{\rho }F\left(\xi \right)d\xi \right.\hfill \\ \hfill & +{\int }_{s_n}^1\frac{s_n^{\kappa -1}{(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \hfill \\ \hfill & -{\int }_0^{s_0}\frac{s_0^{\kappa -1}{(1-\xi )}^{\kappa -3}-{(s_0-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \hfill \\ \hfill & \left.-{\int }_{s_0}^1\frac{s_0^{\kappa -1}{(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right|\hfill \\ \hfill & =\left|{\int }_0^1\frac{s_n^{\kappa -1}{(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi -{\int }_0^{s_n}\frac{{(s_n-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{S}^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right.\hfill \\ \hfill & \left.-{\int }_0^1\frac{s_0^{\kappa -1}{(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi +{\int }_0^{s_0}\frac{{(s_0-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right|\hfill \\ \hfill & =\left|{\int }_0^1\frac{(s_n^{\kappa -1}-s_0^{\kappa -1}){(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right.\hfill \\ \hfill & -{\int }_0^{s_0}\frac{{(s_n-\xi )}^{\kappa -1}-{(s_0-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \hfill \\ \hfill & \left.-{\int }_{s_0}^{s_n}\frac{{(s_n-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le \frac{M(s_n^{\kappa -1}-s_0^{\kappa -1})}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^1{(1-\xi )}^{\kappa -3}{\xi }^{-\rho }d\xi \hfill \\ \hfill & +\frac{M}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{s_0}(\left(s_n{-\xi )}^{\kappa -1}-{(s_0-\xi )}^{\kappa -1}\right){\xi }^{-\rho }d\xi \hfill \\ \hfill & +\frac{M}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{s_n}{(s_n-\xi )}^{\kappa -1}{\xi }^{-\rho }d\xi \hfill \\ \hfill & \le \frac{M(s_n^{\kappa -1}-s_0^{\kappa -1})}{\mathrm{\Gamma }\left(\kappa \right)}\beta (1-\rho ,\kappa -2)+\frac{M}{\mathrm{\Gamma }\left(\kappa \right)}{\mathfrak{I}}_n^1+\frac{M}{\mathrm{\Gamma }\left(\kappa \right)}{\mathfrak{I}}_n^2,\hfill \end{array}$$

(19)

where

$${\mathfrak{I}}_n^1={\int }_0^{s_0}\left({(s_n-\xi )}^{\kappa -1}-(s_0-\xi )\right){\xi }^{-\rho }d\xi$$

and

$${\mathfrak{I}}_n^2={\int }_{s_0}^{s^n}{(s_n-\xi )}^{\kappa -1}{\xi }^{-\rho }d\xi .$$

Firstly, we shall show that ${\mathfrak{I}}_n^1\to 0$, whenever $n\to \infty$. Using the fact that $s_n\to s_0$, we have

$$\left[{(s_n-\xi )}^{\kappa -1}-{(s_0-\xi )}^{\kappa -1}\right]{\xi }^{-\rho }\to 0 \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} n\to \infty$$

(20)

On the other hand, we have

$$\left|\left({(s_n-\xi )}^{\kappa -1}-{(s_0-\xi )}^{\kappa -1}\right){\xi }^{-\rho }\right|\le \left(|s_n{-\xi |}^{\kappa -1}+{|s_0-\xi |}^{\kappa -1}\right){\xi }^{-\rho }\le 2{\xi }^{-\rho }.$$

(21)

Moreover,

$${\int }_0^{1}2{\xi }^{-\rho }d\xi =2\left[\frac{{\xi }^{-\rho +1}}{-\rho +1}\right]=\frac{2}{1-\rho }<\infty .$$

(22)

From (20)–(22), the sequence $[{(s_n-\xi )}^{\kappa -1}-{(s_0-\xi )}^{\kappa -1}]{\xi }^{-\rho }$ converges pointwise to $\mathbf{0}$ (zero function) and $|\left[{(s_n-\xi )}^{\kappa -1}{(s_0-\xi )}^{\kappa -1}\right]{\xi }^{-\rho }|$ is bounded above by a function belonging to $L^1[0,1]$. Theretofore, using Lebesgue’s dominated convergence theorem, we have

$${\mathfrak{I}}_n^1\to 0 \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} n\to \infty .$$

(23)

Next, we shall show that $\mathfrak{I}-n^2\to 0$, whenever $n\to \infty$. Indeed, we have

$${\mathfrak{I}}_n^2={\int }_{s_0}^{s_n}{(s_n-\xi )}^{\kappa -1}{\xi }^{-\rho }d\xi \le {\int }_{s_0}^{s_n}{\xi }^{-\rho }d\xi =\frac{1}{1-\rho }\left(s_n^{1-\rho }-s_0^{1-\rho }\right)$$

Letting $n\to \infty$ and using the fact $s_n\to s_0$, we obtain

$${\mathfrak{I}}_n^2\to 0 \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} n\to \infty .$$

(24)

Finally, making use of (23), and (24), (19) gives rise to

$$|\mathcal{L}\left(s_n\right)-\mathcal{L}\left(s_0\right)|\to \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n} n\to \infty .$$

(25)

It follows that $\mathcal{L}$ is continuous at $s_0$.

• Case-III: When $s_0=1$. It can be easily shown that $\mathcal{L}\left(1\right)=0$. Proceeding the lines of Case I, we can verify the continuity of $\mathcal{L}$ at $s_0=1$. This completes the proof. □

Lemma 3. 

Assume that $0<\rho <1,3<\kappa \le 4,0<\lambda {\gamma }^{\kappa -3}<1,$ and F remains a continuous real function defined on $(0,1]$ such that $\underset{s\to 0^{+}}{lim}F\left(s\right)=\infty$. If $s^{\rho }F\left(s\right)$ remains a continuous function on $[0,1]$, then

$$\mathcal{N}\left(s\right)=\frac{\lambda s^{\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi )F\left(\xi \right)d\xi$$

(26)

is also a continuous function on $[0,1]$, whereas $\mathcal{H}(s,\xi )$ is the function given in Lemma 1.

Proof. 

As $s^{\rho }F\left(s\right)$ remains continuous, ∃$M>0$ verifying

$$|s^{\rho }F\left(s\right)|\le M,\forall s\in [0,1].$$

(27)

Also, one has

$$\left|\mathcal{H}(s,\xi )\right|\le \frac{2(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}.$$

(28)

From (27) and (28), we obtain

$$\begin{array}{cc}\hfill \left|{\int }_0^1\mathcal{H}(\gamma ,\xi )F\left(\xi \right)d\xi \right|& =\left|{\int }_0^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }{\xi }^{\rho }F\left(\xi \right)d\xi \right|\hfill \\ \hfill & \le \frac{2M(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^1{\xi }^{-\rho }d\xi \hfill \\ \hfill & =\frac{2M(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)(1-\kappa )}<\infty .\hfill \end{array}$$

This yields that $\mathcal{N}$ is continuous. □

Remark 4. 

The function $\mathcal{H}(s,\xi )$ (given in Lemma 1) defined by

$$\mathcal{H}(s,\xi )=\left\{\begin{array}{cc}\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}\left[s^{\kappa -3}{(1-\xi )}^{\kappa -3}-{(s-\xi )}^{\kappa -3}\right],\hfill & 0\le \xi \le s\le 1,\hfill \\ \frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{s}^{\kappa -3}{(1-\xi )}^{\kappa -3},\hfill & 0\le s\le \xi \le 1,\hfill \end{array}\right.$$

(29)

is continuous on ${[0,1]}^2$. Also, we have $\mathcal{H}(s,\xi )\ge 0$.

In case $0\le s\le \xi \le 1$, obviously, we have $\mathcal{H}(s,\xi )\ge 0$. Also in case $0\le \xi \le s\le 1$, we have

$$\begin{array}{cc}\hfill \mathcal{H}(s,\xi )& =\frac{(\kappa -1)\kappa -2}{\mathrm{\Gamma }\left(\kappa \right)}\left[s^{\kappa -3}{(1-\xi )}^{\kappa -3}-{(s-\xi )}^{\kappa -3}\right]\hfill \\ \hfill & =\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}\left[{(s-ts)}^{\kappa -3}-{(s-\xi )}^{\kappa -3}\right]\ge 0.\hfill \end{array}$$

This shows the non-negative character of $\mathcal{H}$ on ${[0,1]}^2$.

Lemma 4. 

If $0<\rho <1$, then

$$\underset{0\le s\le 1}{sup}{\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }d\xi =\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}(\beta (1-\rho ,\kappa -2)-\beta (1-\rho ,\kappa )),$$

(30)

whereas $\mathcal{G}(s,\xi )$ remains the Green’s function given in Lemma 1

Proof. 

By definition of $\mathcal{G}$, we have

$$\begin{array}{cc}\hfill {\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }d\xi & ={\int }_0^s\mathcal{G}(s,\xi ){\xi }^{-\rho }d\xi +{\int }_s^1\mathcal{G}(s,\xi ){\xi }^{-\rho }d\xi \hfill \\ & ={\int }_0^s\frac{s^{\kappa -1}{(1-\xi )}^{\kappa -3}-{(s-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }d\xi +{\int }_s^1\frac{s^{\kappa -1}{(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }d\xi \hfill \\ & ={\int }_0^1\frac{s^{\kappa -1}{(1-\xi )}^{\kappa -3}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\rho }d\xi -{\int }_0^s\frac{{(s-\xi )}^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\xi }^{-\kappa }d\xi \hfill \\ \hfill & =\frac{s^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^1{(1-\xi )}^{\kappa -3}{\xi }^{-\rho }d\xi -\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^s{(s-\xi )}^{\kappa -1}{\xi }^{-\rho }d\xi .\hfill \end{array}$$

(31)

As shown earlier (Case I of Lemma 2), we have

$${\int }_0^1{(s-\xi )}^{\kappa -1}{\xi }^{-\rho }d\xi =\frac{s^{\kappa -\rho }}{\mathrm{\Gamma }\left(\kappa \right)}\beta (1-\rho ,\kappa ).$$

(32)

Using (31) and (32), we get

$${\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }d\xi =\frac{s^{\kappa -1}}{\mathrm{\Gamma }\left(\kappa \right)}\beta (1-\rho ,\kappa -2)-\frac{s^{\kappa -\rho }}{\mathrm{\Gamma }\left(\kappa \right)}\beta (1-\rho ,\kappa ).$$

Employing the elemental calculus, it can be easily verified that

$$\phi \left(s\right)=\frac{\beta (1-\rho ,\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{s}^{\kappa -1}-\frac{\beta (1-\rho ,\kappa )}{\mathrm{\Gamma }\left(\kappa \right)}{s}^{\kappa -\rho }$$

is a monotonically increasing function on $[0,1]$. Consequently, we obtain

$$\underset{0\le s\le 1}{sup}{\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }d\xi =\underset{0\le s\le 1}{sup}\phi \left(s\right)=\phi \left(1\right)=\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}[\beta (1-\rho ,\kappa -2)-\beta (1-\rho ,\kappa )].$$

This concludes the proof. □

Lemma 5. 

If $0<\rho <1$, then

$${\int }_0^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }d\xi =\frac{(\kappa -1(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}\left({\gamma }^{\kappa -3}-{\gamma }^{\kappa -\rho -2}\beta (1-\rho ,\kappa -2)\right),$$

(33)

whereas $\mathcal{H}(s,\xi )$ remains the function given in Lemma 1

Proof. 

We have

$$\begin{array}{cc}& {\int }_0^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }d\xi ={\int }_0^{\gamma }\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }d\xi +{\int }_{\gamma }^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }d\xi \hfill \\ & ={\int }_0^{\gamma }\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}\left[{\gamma }^{\kappa -3}{(1-\xi )}^{\kappa -3}-{(\gamma -\xi )}^{\kappa -3}\right]{\xi }^{-\rho }d\xi \hfill \\ \hfill & +{\int }_{\gamma }^1\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{\gamma }^{\kappa -3}{(1-\xi )}^{\kappa -3}{\xi }^{-\rho }d\xi \hfill \\ \hfill & ={\int }_0^1\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{\gamma }^{\kappa -3}{(1-\xi )}^{\kappa -3}{\xi }^{-\rho }d\xi \hfill \\ \hfill & -{\int }_0^{\gamma }\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{(\gamma -\xi )}^{\kappa -3}{\xi }^{-\rho }d\xi \hfill \\ \hfill & =\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{\gamma }^{\kappa -3}{\int }_0^1{(1-\xi )}^{\kappa -1}{\xi }^{-\rho }d\xi \hfill \\ \hfill & -\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{\gamma }{(\gamma -\xi )}^{\kappa -3}{\xi }^{-\rho }d\xi \hfill \\ & =\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{\gamma }^{\kappa -3}\beta (1-\rho ,\kappa -2)-\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{\gamma }{(\gamma -\xi )}^{\kappa -3}{\xi }^{-\rho }d\xi \hfill \end{array}$$

As shown earlier (Case I of Lemma 2), we have

$$\begin{array}{cc}\hfill {\int }_0^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }d\xi & =\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{\gamma }^{\kappa -3}\beta (1-\rho ,\kappa -2)\hfill \\ \hfill & -\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}{\gamma }^{\kappa -\rho -2}\beta (1-\rho ,\kappa -2)\hfill \\ \hfill & =\frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}\left({\gamma }^{\kappa -3}-{\gamma }^{\kappa -\rho -2}\right)\beta (1-\rho ,\kappa -2).\hfill \end{array}$$

This completes the proof. □

Remark 5. 

Denote

$$K:=\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}\left[\left(1+\frac{\beta ({\gamma }^{\kappa -3}-{\gamma }^{\kappa -\rho -2})}{1-\beta {\gamma }^{\kappa -3}}\right)\beta (1-\rho ,\kappa -2)-\beta (1-\rho ,\kappa )\right].$$

The main result of this section runs as follows:

Theorem 3. 

Let $0<\rho <1,3<\kappa \le 4,0<\gamma <1,0<\lambda {\gamma }^{\kappa -3}<1$. Also, suppose that $h:(0,1]\times [0,\infty )\to [0,\infty )$ remains a continuous function verifying $\underset{s\to 0^{+}}{lim}h(s,·)=\infty$ and that $s^{\rho }h(s,y)$ is continuous on $[0,1]\times [0,\infty )$. If ∃$0<\mu \le 1/K$ and a comparison function θ such that $\forall x,y\in [0,\infty )$ verifying $x\ge y$ and $\forall s\in [0,1]$, one has

$$0\le s^{\rho }[h(s,x)-h(s,y)]\le \mu \theta (x-y),$$

(34)

then BVP (14) has a unique positive solution.

Proof. 

Consider the Banach space $\mathcal{C}[0,1]$ with the metric defined by

$$\sigma (v,w)=\underset{0\le s\le 1}{sup}|v\left(s\right)-w\left(s\right)|.$$

Define the cone

$$\bigvee =\{v\in C[0,1]:v(s)\ge 0\}.$$

On ⋁, consider the relation

$$\mathfrak{B}=\{(v,w)\in {\bigvee }^2:v\left(s\right)\le w\left(s\right),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}s\in [0,1]\}.$$

Now, define the map $\mathcal{T}:\bigvee \to \bigvee$ by

$$\begin{array}{cc}\hfill \left(\mathbf{J}v\right)\left(s\right)=& {\int }_0^1\mathcal{G}(s,\xi )h(\xi ,v\left(\xi \right))d\xi +\frac{\lambda s^{\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi )h(\xi ,v\left(\xi \right))d\xi .\hfill \end{array}$$

Now, we are going to investigate each of the assumptions in Theorems 1 and 2.

(a)

Clearly, ⋁ being a closed set of $\mathcal{C}[0,1]$ is a complete metric space with regard to the metric $\sigma$ and hence is a $\mathfrak{B}$-complete metric space.

(b)

Take $(v,w)\in \mathfrak{B}$ implying thereby $v\left(s\right)\le w\left(s\right)$, for each $s\in [0,1]$. Consequently, we have

$$\begin{array}{cc}\hfill \left(\mathbf{J}v\right)\left(s\right)& ={\int }_0^1\mathcal{G}(s,\xi )h(\xi ,v\left(\xi \right))d\xi +\frac{\lambda s^{\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi )h(\xi ,v\left(\xi \right))d\xi .\hfill \\ \hfill & ={\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }{\xi }^{\rho }h(x,v\left(\xi \right))d\xi \hfill \\ \hfill & +\frac{\lambda s^{\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }{\xi }^{\rho }h(\xi ,v\left(\xi \right))d\xi \hfill \\ \hfill & \le {\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }{\xi }^{\rho }h(\xi ,w\left(\xi \right))d\xi \hfill \\ \hfill & +\frac{\lambda s^{\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }{\xi }^{\rho }h(\xi ,w\left(\xi \right))d\xi \hfill \\ \hfill & ={\int }_0^1\mathcal{G}(s,\xi )h(\xi ,w\left(\xi \right))d\xi +\frac{{\lambda }^{\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi )h(\xi ,w\left(\xi \right))d\xi \hfill \\ \hfill & =\left(\mathbf{J}w\right)\left(s\right)\hfill \end{array}$$

yielding $(\mathbf{J}v,\mathbf{J}w)\in \mathfrak{B}$. Therefore, $\mathfrak{B}$ is $\mathbf{J}$-closed. $\mathfrak{B}$ is also a locally $\mathbf{J}$-transitive binary relation.

(c)

Let $\mathbf{0}\in \bigvee$ be a zero function. Then, for each $s\in [0,1]$, we have $\mathbf{0}\left(s\right)\le \left(\mathbf{J}\mathbf{0}\right)\left(s\right)$, thereby yielding $(\mathbf{0},\mathbf{J}\mathbf{0})\in \mathfrak{B}$.

(d)

Let $\left\{v_n\right\}\subset \bigvee$ be a $\mathfrak{B}$-preserving sequence converging to $v\in \bigvee$. Then, $\forall s\in [0,1]$, $\left\{v_n\left(s\right)\right\}$ is an increasing sequence of reals converging to $v\left(s\right)$. This implies that $\forall n\in \mathbb{N}$ and $\forall s\in [0,1]$, $v_n\left(s\right)\le v\left(s\right)$ so that $(v_n,v)\in \mathfrak{B},\forall n\in \mathbb{N}$. Consequently, $\mathfrak{B}$ remains $\sigma$-self-closed.

(e)

Take $(v,w)\in \mathfrak{B}$ implying thereby $v\left(s\right)\le w\left(s\right)$, for each $s\in [0,1]$. Thus,

$$\begin{array}{cc}\hfill \sigma (\mathbf{J}v,\mathbf{J}w)& =\underset{0\le s\le 1}{sup}|\left(\mathbf{J}v\right)\left(s\right)-\left(\mathbf{J}w\right)\left(s\right)|=\underset{0\le s\le 1}{sup}[\left(\mathbf{J}w\right)\left(s\right)-\left(\mathbf{J}v\right)\left(s\right)]\hfill \\ \hfill & =\underset{0\le s\le 1}{sup}\left[{\int }_0^1\mathcal{G}(s,\xi )(h(\xi ,w\left(\xi \right))-h(\xi ,v\left(\xi \right)))\right.d\xi \hfill \\ \hfill & \left.+\frac{\lambda s^{\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi )(h(\xi ,w\left(\xi \right))-h(\xi ,v)\left(\xi \right))d\xi \right]\hfill \\ \hfill & \le \underset{0\le s\le 1}{sup}{\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }{\xi }^{\rho }[h(\xi ,w\left(\xi \right))-h(\xi ,v\left(\xi \right))]d\xi \hfill \\ \hfill & +\frac{\lambda }{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }{\xi }^{\rho }[h(\xi ,w\left(\xi \right))-h(\xi ,v)\left(\xi \right)]d\xi \hfill \\ \hfill & \le sup{\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }\mu \theta (w\left(\xi \right)-v\left(\xi \right))d\xi \hfill \\ \hfill & +\frac{\lambda }{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_{{}^0}^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }\mu \theta \left(w\left(\xi \right)\right)-v\left(\xi \right)d\xi .\hfill \end{array}$$

Using the increasing property of $\theta$, the above inequality reduces to

$$\begin{array}{cc}\hfill \sigma (\mathbf{J}v,\mathbf{J}w)& \le \mu \theta \left(\sigma (v,w)\right)\underset{0\le s\le 0}{sup}{\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }d\xi \hfill \\ \hfill & +\frac{\lambda }{(\kappa -1)(\kappa -2)(1-\lambda \gamma \kappa -3)}\mu \theta \left(\sigma (w,v)\right){\int }_0^1\mathcal{H}(\gamma ,\xi ){\xi }^{\rho }d\xi \hfill \\ \hfill & =\mu \theta \left(\sigma (v,w)\right)\left[\underset{0\le s\le 0}{sup}{\int }_0^1\mathcal{G}(s,\xi ){\xi }^{-\rho }d\xi \right.\hfill \\ \hfill & \left.+\frac{\lambda }{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi ){\xi }^{-\rho }d\xi \right].\hfill \end{array}$$

(35)

Using Lemmas 4 and 5, (35) reduces to

$$\begin{array}{cc}\hfill \sigma (\mathbf{J}v,\mathbf{J}w)\le & \mu \theta \left(\sigma (v,w)\right)\left[\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}(\beta (1-\rho ,\kappa -2)-\beta (1-\rho \kappa ))+\frac{\lambda }{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}\right.\hfill \\ \hfill & \left.\times \frac{(\kappa -1)(\kappa -2)}{\mathrm{\Gamma }\left(\kappa \right)}\left({\gamma }^{\kappa -3}-{\gamma }^{\kappa -\rho -2}\right)\right]\hfill \\ \hfill & =\mu \theta \left(\sigma (v,w)\right)\left[\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}(\beta (1-\rho ,\kappa -2)-\beta (1-\rho ,\kappa ))\right.\hfill \\ \hfill & \left.+\frac{\lambda ({\gamma }^{\kappa -3}-{\gamma }^{\kappa -\rho -2})}{(1-\lambda {\gamma }^{\kappa -3})\mathrm{\Gamma }\left(\kappa \right)}\beta (1-\rho ,\kappa -2)\right]\hfill \\ \hfill & =\mu \theta \left(\sigma (v,w)\right)\left[\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}\left[\left(1+\frac{\lambda ({\gamma }^{\kappa -3}-{\gamma }^{\kappa -\rho -2})}{1-\lambda -\lambda {\gamma }^{\kappa -3}}\right)\beta (1-\rho ,\kappa -2)-\beta (1-\rho ,\kappa )\right]\right]\hfill \\ \hfill & =\mu \theta \left(\sigma \right(v,w\left)\right)K.\hfill \end{array}$$

As $0<\mu \le 1/K$, the last inequality becomes

$$\sigma (\mathbf{J}v,\mathbf{J}w)\le \mu \theta \left(\sigma \right(v,w\left)\right)K\le \theta \left(\sigma \right(v,w\left)\right),$$

If $v\ne w,$ then the above inequality can be written as

$$\sigma (\mathbf{J}v,\mathbf{J}w)\le \frac{\theta \left(\sigma \right(v,w\left)\right)}{\sigma (v,w)}\sigma (v,w)$$

so that

$$\sigma (\mathbf{J}v,\mathbf{J}w)\le \alpha \left(\sigma \right(v,w\left)\right)\sigma (v,w),\forall v,w\in \bigvee \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}(v,w)\in \mathfrak{B}.$$

In case $v=w$, the above inequality is obviously satisfied.

Thus all premises of Theorem 1 are met and hence $\mathbf{J}$ enjoys a fixed point. Choose $v,w\in \mathbf{J}(\bigvee )$. Define $\omega :=max\{v,w\}\in \bigvee$. Then, $\{v,\omega ,w\}$ forms a path in ${\mathfrak{B}}^s$ from v to w. It follows that $\mathbf{J}(\bigvee )$ is ${\mathfrak{B}}^s$-connected. Consequently, in view of Theorem 2, the fixed point of $\mathbf{J}$, say $\overline{w}\in \bigvee$, remains unique, which indeed forms a unique non-negative solution to BVP (14).

Finally, we shall have to prove that $\overline{w}$ is a positive solution. On the contrary, suppose that $\exists 0<s^{*}<1$, verifying $\overline{w}\left(s^{*}\right)=0$. As the non-negative solution $\overline{w}$ of (14) remains fixed point of $\mathbf{J}$, we have

$$\overline{w}\left(s\right)={\int }_0^1\mathcal{G}(s,\xi )h(\xi ,x\left(\xi \right))d\xi +\frac{\lambda s^{\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi )h(\xi ,\overline{w}\left(\xi \right))d\xi .$$

In particular, we have

$$\overline{w}\left(s^{*}\right)={\int }_0^1\mathcal{G}(s^{*},\xi )h(\xi ,\overline{w}\left(\xi \right))d\xi +\frac{\lambda s^{*\kappa -1}}{(\kappa -1)(\kappa -2)(1-\lambda {\gamma }^{\kappa -3})}{\int }_0^1\mathcal{H}(\gamma ,\xi )h(\xi ,x\left(\xi \right))d\xi =0.$$

As both summands in RHS are non-negative, by Remarks 3 and 4, we have

$$\begin{array}{cc}\hfill & {\int }_0^1\mathcal{G}(s^{*},\xi )h(\xi ,\overline{w}\left(\xi \right))d\xi =0,\hfill \\ \hfill & {\int }_0^1\mathcal{H}(\gamma ,\xi )h(\xi ,\xi \left(\xi \right))d\xi =0.\hfill \end{array}$$

Due to the non-negative character of $\mathcal{G}(s,\xi )$, $\mathcal{H}(\gamma ,\xi )$ and $h(\xi ,\overline{w})$, we have

$$\left\{\begin{array}{c}\mathcal{G}(s^{*},\xi )h(\xi ,\overline{w}\left(\xi \right))=0,a.e.\left(\xi \right),\hfill \\ \mathcal{H}(\gamma ,\xi )h(\xi ,\overline{w}\left(\xi \right))=0,a.e\left(\xi \right).\hfill \end{array}\right.$$

(36)

Using the fact $\underset{s\to 0^{+}}{lim}h(s,0)=\infty$, it follows that for given $M>0$, $\exists \delta >0$ such that $h(\xi ,0)>M$, for every $\xi \in [0,1]\cap (0,\delta )$. Note that

$$[0,1]\cap (0,\delta )\subset \{\xi \in [0,1]:h(\xi ,x\left(\xi \right))>M\}$$

and

$$m\left(\right[0,1]\cap (0,\delta \left)\right)>0,$$

where m remains the Lebesque measure. Therefore, (36) gives rise to

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathcal{G}(s^{*},\xi )=0,a.e.\left(\xi \right),\hfill \\ \mathcal{H}(\gamma ,\xi )=0,a.e.\left(\xi \right)\hfill \end{array}\right.\end{array}$$

which raises a contradiction, as $\mathcal{G}(s^{*},\xi )$ and $\mathcal{H}(\gamma ,\xi )$ are rational functions of $\xi$. Consequently, we have $\overline{w}\left(s\right)>0$, $\forall s\in (0,1)$. □

6. Conclusions

Recently, Ansari et al. [18] investigated the fixed-point results via an amorphous relation under functional contraction containing a (c)-comparison function. In this work, we utilized a functional contraction containing the control function of Matkowski [1]. The underlying relation in our results being locally $\mathbf{J}$-transitive is restrictive, but the class of functional contractions is weakened. To illustrate the results, numerous examples are performed. Our results are applied to determine a unique positive solution for certain singular nonlinear FDEs prescribed by three-point boundary conditions, which shows the efficiency of our findings. Inspired by the novelty of the relation metric space, in future, researchers can prove the same results for different types of contractions or in the frameworks of generalized metrical structures (e.g., dislocated space, symmetric space, etc). Also, our results can be extended to a pair of self-maps by investigating the common fixed-point results.