Solving Integral Equations via Hybrid Interpolative ℛℐ-Type Contractions in

Abstract

Karapinar et al. established a more general class of contractions, namely, hybrid interpolative $\mathcal{R}$iech $\mathcal{I}$strǎstescue-type contractions, and presented some results on the platform of metric spaces. This research uses the domain of $\mathfrak{b}$-metric spaces to modify this class proficiently. Several interesting fixed-point results are presented by using this new class defined on $\mathfrak{b}$-metric space, where the symmetric condition is preserved in this study. Examples are provided for the authentication of proved results. Eventually, an application is also provided in order to comprehend our extensive effort in a better way.

1. Introduction and Preliminaries

Today, fixed-point theory is a rapidly expanding and intriguing topic of mathematics with significant applications in a variety of disciplines. One of the powerful applications is in the context of integral equations, where recent numerical approaches have achieved significant scientific advancements. The objectives of this study are to propose some useful methods for predicting the solution of a family of integral equations by using certain modified results from fixed-point theory.

In 1922, Banach [1] established the famous Banach contraction principle (BCP), which used contraction mapping on complete metric spaces. It was later considered an effective technique for finding unique fixed points. It was also useful in nonlinear analysis, which resulted in a slew of developments in all directions. There are several generalizations of the Banach contraction principle in the literature on metric fixed-point theory. Three ingredients are very crucial for establishing fixed-point results. These are the underlying space, the completeness property, and an equality that is strong enough to guarantee the existence of a fixed point. It is suggested that readers look at some of the latest extensions established by [2,3,4,5,6,7]. In addition, during the past several decades, fixed-point theory has played a key role in solving many problems arising in nonlinear analysis and optimization [8,9].

In this context, Czerwik [10] and Bakhtin [11] proposed an interesting notion of metric spaces, termed $\mathfrak{b}$-metric space in $1993,$ by keeping the symmetric condition and changing the triangular inequality of the metric spaces. Furthermore, they presented some valuable fixed-point results to demonstrate the validity of this extension. This space was used by several authors for establishing new directions to prove the existence of fixed-points [12,13,14,15,16,17,18,19,20,21,22,23,24]. Recently, Karapinar et al. [25] presented a novel family of contraction mappings that combined some linear and non-linear contractions in metric spaces. These results are the generalization of the theorems of [26,27,28].

This research provided some interesting results in the context of $\mathfrak{b}$MS by using the contraction condition introduced by Karapinar et al. [25]. The fundamental advantage of these new conditions is that they allow us to include contractivity conditions involving a large number of terms, and we can also use either addition or multiplication at the same time. We suggested some significant convergence criteria and obtained fixed-point results in this regard. Finally, one of our results is applied to establish a novel existence condition for the solution of a class of the integral equations. Our results extend many existing results, including those given in [25].

Throughout the article, $\mathcal{M}$ refers to a non-empty set, $\mathbb{N}$ represents the set of natural numbers, $\mathbb{R}$ corresponds to the collection of real numbers, and $Fi{x}_S\left(\mathcal{M}\right)$ denotes set of all fixed points of a mapping S on $\mathcal{M}$.

Let us have a look at some core concepts that will be helpful for the proof of our main results.

Definition 1.

Let $\mathcal{M}\ne \varphi$ and $\mathfrak{b}\ge 1$ be any real number, a map $d_{\mathfrak{b}}:\mathcal{M}\times \mathcal{M}\to {\mathbb{R}}^{+}$ satisfying the following properties on $\mathcal{M}$ is called a $\mathfrak{b}$ metric on $\mathcal{M}$:

$\acute{d}b\left(1\right)$: $d_{\mathfrak{b}}(m,n)=0⟺m=n$;

$\acute{d}b\left(2\right)$: $d_{\mathfrak{b}}(m,n)=d_{\mathfrak{b}}(n,m)$;

$\acute{d}b\left(3\right)$: $d_{\mathfrak{b}}(m,n)\le \mathfrak{b}\{d_{\mathfrak{b}}(m,k)+d_{\mathfrak{b}}(k,n)\},$

for all $m,n,k\in \mathcal{M}$.

The pair $(\mathcal{M},d_{\mathfrak{b}})$ is said to be a $\mathfrak{b}$-metric space ($\mathfrak{b}$MS).

Example 1.

Let $\mathcal{M}={\mathbb{R}}^{+},$ define $d_{\mathfrak{b}}:\mathcal{M}\times \mathcal{M}\to \mathcal{M}$ as $d_{\mathfrak{b}}(m,n)=\mid m-n{\mid }^q$ for all $m,n\in \mathcal{M},$ $q>1$ be any constant; then, $(\mathcal{M},d_{\mathfrak{b}})$ is a $\mathfrak{b}$MS with $\mathfrak{b}=2^{q-1}$.

Definition 2.

For a mapping $S:\mathcal{M}\to \mathcal{M}$ and $m\in \mathcal{M}$, the orbit of m with respect to S is defined as the following sequence of points:

$$O_S\left(m\right)=\{m,Sm,S^{2}m,...\}.$$

Definition 3.

Let $(\mathcal{M},d_{\mathfrak{b}})$ be a $\mathfrak{b}$MS. A sequence $\left\{m_k\right\}$ defined by $m_k=S^k{m}_0$ for all $k\in \mathbb{N}$ is called the Picard sequence of S based on $m_0$, where $S^k=S\circ S\circ ...{.}^{\left(k\right)}\circ S:\mathcal{M}\to \mathcal{M}$ is the k-th iterate of S and the mapping S is called a Picard operator if each Picard sequence of such an operator converges to one of its fixed points.

Definition 4.

A function $\pi :\mathcal{M}\times \mathcal{M}\to [0,\infty )$ in a $\mathfrak{b}$MS is called π-orbital admissible if for $m\in \mathcal{M}$ it satisfies

$$\pi (m,Sm)\ge 1⟹\pi (Sm,S^{2}m)\ge 1.$$

Definition 5.

Let $(\mathcal{M},d_{\mathfrak{b}})$ be a $\mathfrak{b}$MS and $\pi :\mathcal{M}\times \mathcal{M}\to [0,\infty )$ be a function. A mapping $S:\mathcal{M}\to \mathcal{M}$ is called a hybrid interpolative $\mathcal{RI}$-type contraction if there exists a $w\in [0,1)$ such that

$$\pi (m,n)d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w\mathbb{M}(m,n),$$

here

$$\mathbb{M}(m,n)=\left\{\begin{array}{c}[c_1{d}_{\mathfrak{b}}{(m,n)}^p+c_2{d}_{\mathfrak{b}}{(m,Sm)}^p+c_3{d}_{\mathfrak{b}}{(n,Sn)}^p\hfill \\ +c_4{d}_{\mathfrak{b}}{(Sm,Sn)}^p+c_5{d}_{\mathfrak{b}}{(Sm,S^{2}m)}^p+q{d}_{\mathfrak{b}}{(Sn,S^{2}n)}^p{]}^{\frac{1}{p}}\hfill & \sum _{i=1}^5{c}_i+q\le 1\hfill \\ & ifp>0\hfill \\ d_{\mathfrak{b}}{(m,n)}^{c_1}.d_{\mathfrak{b}}{(m,Sm)}^{c_2}.d_{\mathfrak{b}}{(n,Sn)}^{c_3}.d_{\mathfrak{b}}{(Sm,Sn)}^{c_4}\hfill \\ .d_{\mathfrak{b}}{(Sm,S^{2}m)}^{c_5}.d_{\mathfrak{b}}{(Sn,S^{2}n)}^q\hfill & \sum _{i=1}^5{c}_i+q=1\hfill \\ & ifp=0.\hfill \end{array}\right.$$

(1)

with $\{c_i:i=1,2,..5\ge 0\},p\in \mathbb{R}$ and $q>0$.

2. Main Results

In this section, we develop some fixed-point results for $\mathcal{RI}$-type contractions in the context of $\mathfrak{b}$MS. Furthermore, an example and an application are also presented for a deeper understanding of our results.

Proposition 1.

Given $w\in [0,1)$ and $\left\{m_k\right\}\subset {\mathbb{R}}^{+}$ be any sequence such that

$$m_{k+2}\le wmax\{m_k,m_{k+1}\}\mathit{for}\mathit{all}k\in \mathbb{N}\cup 0$$

(2)

then,

$$m_{2k}\le w^k\mathsf{\Lambda },m_{2k+1}\le w^k\mathsf{\Lambda }\forall k\ge 1,$$

(3)

where $\mathsf{\Lambda }=max\{m_0,m_1\}$.

Proof. 

Setting $k=0$ in (2), we obtain

$$m_2\le wmax\{m_0,m_1\}=w\mathsf{\Lambda },$$

for $k=1$ we obtain

$$\begin{array}{cc}\hfill m_3& \le wmax\{m_1,m_2\}\hfill \\ \hfill & \le wmax\{m_1,wmax\{m_0,m_1\}\}\hfill \\ \hfill & \le wmax\{m_1,w\mathsf{\Lambda }\}\hfill \\ \hfill & \le w\mathsf{\Lambda }.\hfill \end{array}$$

Suppose that (3) holds for some $k\in \mathbb{N}$; then,

$$\begin{array}{cc}\hfill m_{2k+2}& \le wmax\{m_{2k},m_{2k+1}\}\hfill \\ \hfill & \le wmax\{w^k\mathsf{\Lambda },w^k\mathsf{\Lambda }\}\hfill \\ \hfill & \le w^{k+1}\mathsf{\Lambda },\hfill \end{array}$$

on the same lines we will obtain $m_{2k+3}\le w^{k+1}\mathsf{\Lambda }$.

This completes the proof by induction. □

Lemma 1.

Let $\left\{r_k\right\}$ be a sequence on a $\mathfrak{b}$MS, and there exists $w\in [0,1)$ such that

$$d_{\mathfrak{b}}(r_{k+2},r_{k+3})\le wmax\{d_{\mathfrak{b}}(r_k,r_{k+1}),d_{\mathfrak{b}}(r_{k+1},r_{k+2})\},\forall k\in \mathbb{N},$$

(4)

then, $\left\{r_k\right\}$ is a Cauchy sequence in $(\mathcal{M},d_{\mathfrak{b}})$.

Proof. 

Let us consider a sequence $\left\{m_k\right\}$ in $\mathcal{M}$ defined as

$$m_k=d_{\mathfrak{b}}(r_k,r_{k+1})$$

for all $k\in \mathbb{N}$, this sequence guarantees the condition (3); then, for all $k\ge \mathbb{N}$ we have

$$d_{\mathfrak{b}}(r_{2k},r_{2k+1})=m_{2k}\le w^k\mathsf{\Lambda },$$

also,

$$d_{\mathfrak{b}}(r_{2k+1},r_{2k+2})=m_{2k+1}\le w^k\mathsf{\Lambda }.$$

Adding both vertically

$$d_{\mathfrak{b}}(r_{2k},r_{2k+1})+d_{\mathfrak{b}}(r_{2k+1},r_{2k+2})\le 2w^k\mathsf{\Lambda }.$$

Notice that if $w=0$ or $\mathsf{\Lambda }=0$, we will have a constant sequence which is Cauchy. So, let $w>0$ and $\mathsf{\Lambda }>0$; let $ϵ>0$ be any arbitrary real number, so $\frac{ϵ}{2\mathsf{\Lambda }{\mathfrak{b}}^{n^{*}}}>0$; and for $w\in (0,1)$, there is a natural number $n^{*}>k_0\ge 1$ such that

$$\sum _{k=k_0}^{+\infty }{w}^k<\frac{ϵ}{2\mathsf{\Lambda }{\mathfrak{b}}^{n^{*}}},$$

in particular,

$$2{\mathfrak{b}}^{n^{*}}\mathsf{\Lambda }\sum _{k=k_0}^n{w}^k<2{\mathfrak{b}}^{n^{*}}\mathsf{\Lambda }\sum _{k=k_0}^{+\infty }{w}^k<ϵ,$$

for all $n\in \mathbb{N}$ such that $n\ge k$.

Let $n^{*},k,l\in \mathbb{N}$ such that $n^{*}>l>k\ge 2k_0$ and $n\ge k_0+1$ and $2n\ge l$; thus, we have

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}(r_k,r_l)& \le \mathfrak{b}[d_{\mathfrak{b}}(r_k,r_{k+1})+d_{\mathfrak{b}}(r_{k+1},r_l)]\hfill \\ \hfill & =\mathfrak{b}{d}_{\mathfrak{b}}(r_k,r_{k+1})+\mathfrak{b}{d}_{\mathfrak{b}}(r_{k+1},r_l)\hfill \\ \hfill & \le \mathfrak{b}{d}_{\mathfrak{b}}(r_k,r_{k+1})+{\mathfrak{b}}^2[d_{\mathfrak{b}}(r_{k+1},r_{k+2})+d_{\mathfrak{b}}(r_{k+2},r_l)]\hfill \\ \hfill & \le \mathfrak{b}{d}_{\mathfrak{b}}(r_k,r_{k+1})+{\mathfrak{b}}^2{d}_{\mathfrak{b}}(r_{k+1},r_{k+2})+{\mathfrak{b}}^3{d}_{\mathfrak{b}}(r_{k+2},r_{k+3})+...\hfill \\ \hfill & ..+{\mathfrak{b}}^{l-k}{d}_{\mathfrak{b}}(r_{l-1},r_l)\hfill \\ \hfill & =\sum _{j=k}^{l-1}{\mathfrak{b}}^{j-k+1}{d}_{\mathfrak{b}}(r_j,r_{j+1})\hfill \\ \hfill & \le \sum _{j=k}^{l-1}{\mathfrak{b}}^{n^{*}}{d}_{\mathfrak{b}}(r_j,r_{j+1})\hfill \\ \hfill & \le \sum _{j=2k_0}^{2n-1}{\mathfrak{b}}^{n^{*}}{d}_{\mathfrak{b}}(r_j,r_{j+1})\hfill \\ \hfill & \le \sum _{k=k_0}^{2n-1}{\mathfrak{b}}^{n^{*}}\{d_{\mathfrak{b}}(r_{2k},r_{2k+1})+d_{\mathfrak{b}}(r_{2k+1},r_{2k+2})\}\hfill \\ \hfill & \le \sum _{k=k_0}^{n-1}{\mathfrak{b}}^{n^{*}}2w^k\mathsf{\Lambda }\le \sum _{k=k_0}^n{\mathfrak{b}}^{n^{*}}2w^k\mathsf{\Lambda }\hfill \\ \hfill & \le \sum _{k=k_0}^{+\infty }{\mathfrak{b}}^{n^{*}}2\mathsf{\Lambda }{w}^k\hfill \\ \hfill & <ϵ,\hfill \end{array}$$

showing that $\left\{r_k\right\}$ is a Cauchy sequence in $\mathfrak{b}$MS. □

Corollary 1.

Let $\left\{r_k\right\}$ be a sequence in $\mathfrak{b}$MS, and $w\in [0,1)$ exists such that

$$d_{\mathfrak{b}}(r_{k+2},r_{k+3})\le w[d_{\mathfrak{b}}{(r_k,r_{k+1})}^{{\beta }_1}.d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^{{\beta }_2}],\forall k\in \mathbb{N},$$

(5)

then, $\left\{r_k\right\}$ is a Cauchy sequence in $(\mathcal{M},d_{\mathfrak{b}}),$ where ${\beta }_1,{\beta }_2\in [0,1]$ satisfy ${\beta }_1+{\beta }_2=1$.

Proof. 

Consider

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}(r_{k+2},r_{k+3})& \le w[d_{\mathfrak{b}}{(r_k,r_{k+1})}^{{\beta }_1}.d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^{{\beta }_2}]\hfill \\ \hfill & \le w[max{\{d_{\mathfrak{b}}(r_k,r_{k+1}),d_{\mathfrak{b}}(r_{k+1},r_{k+2})\}}^{{\beta }_1}\hfill \\ \hfill & .max{\{d_{\mathfrak{b}}(r_k,r_{k+1}),d_{\mathfrak{b}}(r_{k+1},r_{k+2})\}}^{{\beta }_2}]\hfill \\ \hfill & =w{[max\{d_{\mathfrak{b}}(r_k,r_{k+1}),d_{\mathfrak{b}}(r_{k+1},r_{k+2})\}]}^{{\beta }_1+{\beta }_2}\hfill \\ \hfill & =w[max\{d_{\mathfrak{b}}(r_k,r_{k+1}),d_{\mathfrak{b}}(r_{k+1},r_{k+2})\}],\hfill \end{array}$$

then, the result follows form Lemma (1). □

Theorem 1.

Let $(\mathcal{M},d_{\mathfrak{b}})$ be a complete $\mathfrak{b}$MS, $d_{\mathfrak{b}}$ be continuous, and S be a hybrid interpolative $\mathcal{RI}$-type contraction mapping. Assume that

(1) 

S is continuous;

(2) 

S is a π orbital-admissible mapping;

(3) 

$r_0\in \mathcal{M}$ exists such that $\pi (r_0,S{r}_0)\ge 1$.

then S has a fixed point.

Proof. 

For $r_0\in \mathcal{M}$ by assumption, $\pi (r_0,S{r}_0)\ge 1$. Since S is an π orbital-admissible, we may write

$$\pi (S{r}_0,S^2{r}_0)\ge 1.$$

Continuing in the same manner, for $r_0$ we will obtain $\pi (S^k{r}_0,S^{k+1}{r}_0)\ge 1$ for any $k\in \mathbb{N}$, and for such a sequence $\{r_{k\}}$ we can write

$$r_1=S{r}_0,r_2=S{r}_1=S^2{r}_0,...,r_k=S{r}_{k-1}=S^k{r}_0.$$

If $r_k\in S{r}_{k+1}$ this means that $r_k$ is the fixed point of S trivially. Suppose $r_k\ne S{r}_{k+1}$ for all $k\in \mathbb{N}.$ Now consider

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}(r_{k+2},r_{k+3})& \le \pi (r_{k+2},r_{k+3})d_{\mathfrak{b}}(r_{k+2},r_{k+3})\hfill \\ \hfill & \le \pi (r_{k+2},r_{k+3})d_{\mathfrak{b}}(S^2{r}_k,S^2{r}_{k+1})\hfill \\ \hfill & \le w\mathbb{M}(r_k,r_{k+1})\hfill \end{array}$$

(6)

Now, we will discuss both possible cases of choice of p.

Case-I:- If $p>0$,

$$\begin{array}{cc}\hfill \mathbb{M}(r_k,r_{k+1})& =[c_1{d}_{\mathfrak{b}}{(r_k,r_{k+1})}^p+c_2{d}_{\mathfrak{b}}{(r_k,S{r}_k)}^p+c_3{d}_{\mathfrak{b}}{(r_{k+1},S{r}_{k+1})}^p\hfill \\ \hfill & +c_4{d}_{\mathfrak{b}}{(S{r}_k,S{r}_{k+1})}^p+c_5{d}_{\mathfrak{b}}{(S{r}_k,S^2{r}_k)}^p+q{d}_{\mathfrak{b}}{(S{r}_{k+1},S^2{r}_{k+1})}^p{]}^{\frac{1}{p}}\hfill \\ \hfill & =[c_1{d}_{\mathfrak{b}}{(r_k,r_{k+1})}^p+c_2{d}_{\mathfrak{b}}{(r_k,r_{k+1})}^p+c_3{d}_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^p\hfill \\ \hfill & +c_4{d}_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^p+c_5{d}_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^p+q{d}_{\mathfrak{b}}(r_{k+2},r_{k+3}){]}^{\frac{1}{p}}\hfill \\ \hfill & =[(c_1+c_2)d_{\mathfrak{b}}{(r_k,r_{k+1})}^p+(c_3+c_4+c_5)d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^p\hfill \\ \hfill & +q{d}_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^p{]}^{\frac{1}{p}}\hfill \\ \hfill & \le [(c_1+c_2+c_3+c_4+c_5)max\{d_{\mathfrak{b}}{(r_k,r_{k+1})}^p,d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^p\}\hfill \\ \hfill & +q{d}_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^p{]}^{\frac{1}{p}},\hfill \end{array}$$

(7)

taking power of p on both sides of (6), we have

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^p\le & w^p(c_1+c_2+c_3+c_4+c_5)max\{d_{\mathfrak{b}}{(r_k,r_{k+1})}^p,d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^p\}\hfill \\ \hfill & +w^{p}q{d}_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^p.\hfill \end{array}$$

(8)

Therefore,

$$\begin{array}{cc}\hfill (1-w^{p}q)d_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^p\le & w^p(c_1+c_2+c_3+c_4+c_5)max\{d_{\mathfrak{b}}{(r_k,r_{k+1})}^p,d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^p\}\hfill \\ \hfill & \le w^p(1-q)max{\{d_{\mathfrak{b}}(r_k,r_{k+1}),d_{\mathfrak{b}}(r_{k+1},r_{k+2})\}}^p.\hfill \end{array}$$

(9)

So for all $k\in \mathbb{N}$,

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^p& \le \frac{w^p(1-q)}{1-w^{p}q}max{\{d_{\mathfrak{b}}(r_k,r_{k+1}),d_{\mathfrak{b}}(r_{k+1},r_{k+2})\}}^p,\hfill \end{array}$$

(10)

or equally

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}(r_{k+2},r_{k+3})& ={\left(\frac{w^p(1-q)}{1-w^{p}q}\right)}^{\frac{1}{p}}max\{d_{\mathfrak{b}}(r_k,r_{k+1}),d_{\mathfrak{b}}(r_{k+1},r_{k+2})\}\hfill \\ \hfill & =wmax\{d_{\mathfrak{b}}(r_k,r_{k+1}),d_{\mathfrak{b}}(r_{k+1},r_{k+2})\}.\hfill \end{array}$$

(11)

Here, we used

$$w={\left(\frac{w^p(1-q)}{1-w^{p}q}\right)}^{\frac{1}{p}}$$

and it is easy to verify that $w\in (0,1)$, so Lemma (1) concludes that $\left\{r_k\right\}$ is a Cauchy sequence.

Case-II:- If $p=0$.

$$\begin{array}{cc}\hfill \mathbb{M}(r_k,r_{k+1})& =d_{\mathfrak{b}}{(r_k,r_{k+1})}^{c_1}.d_{\mathfrak{b}}{(r_k,S{r}_k)}^{c_2}.d_{\mathfrak{b}}{(r_{k+1},S{r}_{k+1})}^{c_3}\hfill \\ \hfill & .d_{\mathfrak{b}}{(S{r}_k,S{r}_{k+1})}^{c_4}.d_{\mathfrak{b}}{(S{r}_k,S^2{r}_k)}^{c_5}.d_{\mathfrak{b}}{(S{r}_{k+1},S^2{r}_{k+1})}^q\hfill \\ \hfill & =d_{\mathfrak{b}}{(r_k,r_{k+1})}^{c_1}.d_{\mathfrak{b}}{(r_k,r_{k+1})}^{c_2}.d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^{c_3}\hfill \\ \hfill & .d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^{c_4}.d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^{c_5}.d_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^q\hfill \\ \hfill & =d_{\mathfrak{b}}{(r_k,r_{k+1})}^{c_1+c_2}.d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^{c_3+c_4+c_5}\hfill \\ \hfill & .d_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^q\hfill \end{array}$$

(12)

(6) implies that

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}(r_{k+2},r_{k+3})& \le w{d}_{\mathfrak{b}}{(r_k,r_{k+1})}^{c_1+c_2}.d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^{c_3+c_4+c_5}\hfill \\ \hfill & .d_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^q.\hfill \end{array}$$

(13)

As by assumption

$$\sum _{i=1}^5{c}_i+q=1,$$

so notice that if we set $q=1$, then (13) will be a contradiction. Then, necessarily $q<1$, so

$$\sum _{i=1}^5{c}_i=1-q>0.$$

Let us consider,

$${\beta }_1=\frac{c_1+c_2}{1-q},{\beta }_2=\frac{c_3+c_4+c_5}{1-q},$$

satisfying ${\beta }_1+{\beta }_2=1$. Now, setting these in (13),

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}{(r_{k+2},r_{k+3})}^{1-q}& \le w{d}_{\mathfrak{b}}{(r_k,r_{k+1})}^{c_1+c_2}.d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^{c_3+c_4+c_5}\hfill \\ \hfill ⟹d_{\mathfrak{b}}(r_{k+2},r_{k+3})& \le w^{\frac{1}{1-q}}{d}_{\mathfrak{b}}{(r_k,r_{k+1})}^{{\beta }_1}.d_{\mathfrak{b}}{(r_{k+1},r_{k+2})}^{{\beta }_2}.\hfill \end{array}$$

As $w\in (0,1)$; therefore,

$$0<1-q\le 1⟹1\le \frac{1}{1-q}⟹w^{\frac{1}{1-q}}\le w<1,$$

so $\left\{r_k\right\}$ is a Cauchy sequence by Corollary (1). In both cases, we have shown that this Picard sequence is a Cauchy sequence. As $\mathcal{M}$ is complete, so there exists a point $r^{*}\in \mathcal{M}$ such that

$$d_{\mathfrak{b}}(r^{*},S{r}^{*})=\underset{k\to \infty }{lim}{d}_{\mathfrak{b}}(r_{k+1},S{r}^{*})=\underset{k\to \infty }{lim}{d}_{\mathfrak{b}}(S{r}_k,S{r}^{*})=0,$$

so $S{r}^{*}=r^{*}$, that is, $r^{*}$ is the fixed point of S. □

Corollary 2.

Let $(\mathcal{M},d_{\mathfrak{b}})$ be a complete $\mathfrak{b}$MS and $d_{\mathfrak{b}}$ be continuous; also, $S:\mathcal{M}\to \mathcal{M}$ is a continuous map. Suppose that $c_1,c_2\in (0,1)$ exists satisfying $c_1+c_2<1$ such that

$$d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le c_1{d}_{\mathfrak{b}}(m,n)+c_1{d}_{\mathfrak{b}}(Sm,Sn),$$

then S has a fixed point.

Theorem 2.

Let $(\mathcal{M},d_{\mathfrak{b}})$ be a complete $\mathfrak{b}$MS, $d_{\mathfrak{b}}$ be continuous and S be a hybrid interpolative $\mathcal{RI}$-type contraction mapping. Assume that

(1) 

$S^2$ is continuous;

(2) 

S is a π orbital-admissible mapping;

(3) 

$m_0\in \mathcal{M}$ exists such that $\pi (m_0,S{m}_0)\ge 1$,

(4) 

$\pi (m,Sm)\ge 1$ for all $m\in Fi{x}_{S^2}\left(\mathcal{M}\right)$

then S has a fixed point.

Proof. 

Let $\left\{r_k\right\}$ be the Picard sequence of S based on $m_0$ defined by $m_k=S^k{m}_0$. By completeness of $\mathcal{M}$, we have $m^{*}$ such that

$$d_{\mathfrak{b}}(m^{*},S^2{m}^{*})=\underset{k\to \infty }{lim}{d}_{\mathfrak{b}}(m_{k+1},S^2{m}^{*})=\underset{k\to \infty }{lim}{d}_{\mathfrak{b}}(S^2{m}_k,S^2{m}^{*})=0,$$

showing that $m^{*}$ is fixed point of $S^2.$ This shows that the set $Fi{x}_{S^2}\left(\mathcal{M}\right)$ is non empty. Next, we verify that $m^{*}$ is also a fixed point of S. On the contrary, assume that $S{m}^{*}\ne m^{*}$ then $m^{*}\notin Fi{x}_S\left(\mathcal{M}\right).$

$$\begin{array}{cc}\hfill 0\le d_{\mathfrak{b}}(m^{*},S{m}^{*})& \le \pi (m^{*},S{m}^{*})d_{\mathfrak{b}}(m^{*},S{m}^{*})\hfill \\ \hfill & \le \pi (m^{*},S{m}^{*})d_{\mathfrak{b}}(S^2{m}^{*},S^2{m}^{*})\hfill \\ \hfill & \le w\mathbb{M}(m^{*},S{m}^{*})\hfill \end{array}$$

(14)

Now, we will discuss both possible cases of choice of p.

Case-I:- If $p>0$,

$$\begin{array}{cc}\hfill \mathbb{M}(m^{*},S{m}^{*})& =[c_1{d}_{\mathfrak{b}}{(m^{*},S{m}^{*})}^p+c_2{d}_{\mathfrak{b}}{(m^{*},S{m}^{*})}^p+c_3{d}_{\mathfrak{b}}{(S{m}^{*},S^2{m}^{*})}^p\hfill \\ \hfill & +c_4{d}_{\mathfrak{b}}{(S{m}^{*},S^2{m}^{*})}^p+c_5{d}_{\mathfrak{b}}{(S{m}^{*},S^2{m}^{*})}^p+q{d}_{\mathfrak{b}}{(S^2{m}^{*},S^3{m}^{*})}^p{]}^{\frac{1}{p}}\hfill \\ \hfill & =[c_1{d}_{\mathfrak{b}}{(m^{*},S{m}^{*})}^p+c_2{d}_{\mathfrak{b}}{(m^{*},S{m}^{*})}^p+c_3{d}_{\mathfrak{b}}{(S{m}^{*},m^{*})}^p\hfill \\ \hfill & +c_4{d}_{\mathfrak{b}}{(S{m}^{*},m^{*})}^p+c_5{d}_{\mathfrak{b}}{(S{m}^{*},m^{*})}^p+q{d}_{\mathfrak{b}}{(m^{*},S{m}^{*})}^p{]}^{\frac{1}{p}}\hfill \\ \hfill & ={\left[(c_1+c_2+c_3+c_4+c_5+q)d_{\mathfrak{b}}{(m^{*},S{m}^{*})}^p\right]}^{\frac{1}{p}}\hfill \\ \hfill & \le {\left[d_{\mathfrak{b}}{(m^{*},S{m}^{*})}^p\right]}^{\frac{1}{p}}\hfill \\ \hfill & =d_{\mathfrak{b}},(m^{*},S{m}^{*}),\hfill \end{array}$$

which contradicts (14).

Case-II:- If $p=0$.

$$\begin{array}{cc}\hfill \mathbb{M}(m^{*},S{m}^{*})& =d_{\mathfrak{b}}{(m^{*},S{m}^{*})}^{c_1}.d_{\mathfrak{b}}{(m^{*},S{m}^{*})}^{c_2}.d_{\mathfrak{b}}{(S{m}^{*},S^2{m}^{*})}^{c_3}\hfill \\ \hfill & .d_{\mathfrak{b}}{(S{m}^{*},S^2{m}^{*})}^{c_4}.d_{\mathfrak{b}}{(S{m}^{*},S^2{m}^{*})}^{c_5}.d_{\mathfrak{b}}{(S^2{m}^{*},S^3{m}^{*})}^q\hfill \\ \hfill & =d_{\mathfrak{b}}{(m^{*},S{m}^{*})}^{c_1}.d_{\mathfrak{b}}{(m^{*},S{m}^{*})}^{c_2}.d_{\mathfrak{b}}{(S{m}^{*},m^{*})}^{c_3}\hfill \\ \hfill & .d_{\mathfrak{b}}{(S{m}^{*},m^{*})}^{c_4}.d_{\mathfrak{b}}{(S{m}^{*},m^{*})}^{c_5}.d_{\mathfrak{b}}{(S{m}^{*},m^{*})}^q\hfill \\ \hfill & =d_{\mathfrak{b}}{(m^{*},m^{*})}^{c_1+c_2+c_3+c_4+c_5+q}\hfill \\ \hfill & =d_{\mathfrak{b}}(m^{*},S{m}^{*})\hfill \end{array}$$

which is again a contradiction to (14), thus $m^{*}$ is fixed point of S. □

Theorem 3.

Let $(\mathcal{M},d_{\mathfrak{b}})$ be a complete $\mathfrak{b}$MS, $d_{\mathfrak{b}}$ be continuous. Let $\pi :\mathcal{M}\times \mathcal{M}\to [0,\infty )$ be a function, and let $S:\mathcal{M}\to \mathcal{M}$ be a mapping such that

(1) 

Either S is $S^2$ is continuous;

(2) 

S is a π orbital-admissible mapping;

(3) 

$m_0\in \mathcal{M}$ exists such that $\pi (m_0,S{m}_0)\ge 1$,

(4) 

A constant $w\in [0,1)$ exists such that at least one of the following conditions is fulfilled for all distinct $m,n\in \mathcal{M}\setminus Fi{x}_S\left(\mathcal{M}\right)$:

(a) 

$\pi (m,n)d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(m,n)$

(b) 

$\pi (m,n)d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(m,Sm)$

(c) 

$\pi (m,n)d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(n,Sn)$

(d) 

$\pi (m,n)d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(Sm,Sn)$

(e) 

$\pi (m,n)d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(Sm,S^{2}m)$

(f) 

$\pi (m,n)d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(Sn,S^{2}n).$

then S has a fixed point.

Proof. 

The proof of this result follows from the case of $p>0$ from of theorem (1) and theorem (2), and it can be observed easily that the contractions condition (1) is satisfied by the following suitable choice of parameters

$$\begin{array}{cc}\hfill & c_1=1,c_2=0,c_3=0,c4=0,c_5=0,q=0\hfill \\ \hfill & c_1=0,c_2=1,c_3=0,c4=0,c_5=0,q=0\hfill \\ \hfill & c_1=0,c_2=0,c_3=1,c4=0,c_5=0,q=0\hfill \\ \hfill & c_1=0,c_2=0,c_3=0,c4=1,c_5=0,q=0\hfill \\ \hfill & c_1=0,c_2=0,c_3=0,c4=0,c_5=1,q=0\hfill \\ \hfill & c_1=0,c_2=0,c_3=0,c4=0,c_5=0,q=1\hfill \end{array}$$

(15)

□

Corollary 3.

Let $(\mathcal{M},d_{\mathfrak{b}})$ be a complete $\mathfrak{b}$MS, $d_{\mathfrak{b}}$ be continuous, and $S:\mathcal{M}\to \mathcal{M}$ be a mapping such that either S is $S^2$ or is continuous. Suppose that there exists a constant $w\in (0,1)$ such that at least one of the following conditions is fulfilled for all distinct $m,n\in \mathcal{M}Fi{x}_S\left(\mathcal{M}\right)$:

1.

$d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(m,n)$

2.

$d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(m,Sm)$

3.

$d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(n,Sn)$

4.

$d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(Sm,Sn)$

5.

$d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(Sm,S^{2}m)$

6.

$d_{\mathfrak{b}}(S^{2}m,S^{2}n)\le w{d}_{\mathfrak{b}}(Sn,S^{2}n)$

then S has a fixed point.

In particular, if we drop the condition of continuity of either S or $S^2$, then we have a special case of existence of the fixed point provided that $\delta =0.$ For this, observe the following example.

Example 2.

Let $(\mathcal{M},d_{\mathfrak{b}})$ be a complete metric space where $\mathcal{M}=[-1,1]$, and $d_{\mathfrak{b}}:\mathcal{M}\times \mathcal{M}\to (0,\infty )$ is defined as $d_{\mathfrak{b}}(m,n)={|m-n|}^2.$ Let $S:\mathcal{M}\to \mathcal{M}$ be defined by

$$Sm=\left\{\begin{array}{cc}\sqrt{1-m^2}\hfill & if-1\le m\le 0\hfill \\ \\ \frac{m^2}{2}\hfill & if 0\le m\le 1,\hfill \end{array}\right.$$

then

$$S^{2}m=\left\{\begin{array}{cc}\frac{1-m^2}{2}\hfill & if-1\le m\le 0\hfill \\ \\ \frac{m^4}{8}\hfill & if 0\le m\le 1.\hfill \end{array}\right.$$

Next, define $\pi :\mathcal{M}\times \mathcal{M}\to [0,\infty )$, which is defined as

$$\pi (m,n)=\left\{\begin{array}{cc}\frac{3}{2}\hfill & if 0\le m\le 1\hfill \\ 1\hfill & if n=1,m=-1\hfill \\ 0\hfill & if \mathrm{otherwise}\hfill \end{array}\right.$$

Now to check the contractivity condition, for $0\le m\le 1$

$$\begin{array}{cc}\hfill \pi (m,n)d_{\mathfrak{b}}(S^{2}m,S^{2}n)& =\frac{3}{\left(2\right)\left(8\right)}|m^4-n^4|\hfill \\ \hfill & =\frac{3}{\left(2\right)\left(8\right)}\left|(m^2-n^2)(m^2+n^2)\right|\hfill \\ \hfill & \le \frac{3}{8}|m^2-n^2|\hfill \\ \hfill & =\frac{3}{8}|\sqrt{|m^2-n^2|}\sqrt{|m^2-n^2|}\hfill \\ \hfill & =\frac{3}{8}\sqrt{|m-n|.|m+n|}\sqrt{2\frac{|m^2-n^2|}{2}}\hfill \\ \hfill & \le \frac{3}{4}{d}_{\mathfrak{b}}{(m,n)}^{\frac{1}{4}}{d}_{\mathfrak{b}}{(Sm,Sn)}^{\frac{1}{4}}.\hfill \end{array}$$

For $m=-1,n=1$, we have

$$\begin{array}{cc}\hfill \pi (m,n)d_{\mathfrak{b}}(S^{2}m,S^{2}n)& =\frac{1}{8}\hfill \\ \hfill & <\frac{3}{4}\hfill \\ \hfill & =\frac{3}{4}{d}_{\mathfrak{b}}{(m,n)}^{\frac{1}{4}}{d}_{\mathfrak{b}}{(Sm,Sn)}^{\frac{1}{4}}.\hfill \end{array}$$

All of the conditions of theorem (1) except the continuity of S are satisfied with $\delta =0$. Hence, S has a fixed point.

3. Application

Now, we apply our main result to find a solution to an integral equation of the Fredholm type.

Suppose $I=[0,1]$ and $\mathcal{M}=\mathcal{C}(I,{\mathbb{R}}^2)$ are the space of all continuous functions defined from I to ${\mathbb{R}}^2$, endowed with the usual sup-norm.

We define a $\mathfrak{b}$ metric on $\mathcal{M}$ as

$$d_{\mathfrak{b}}(\varphi ,\psi )={\Vert \varphi -\psi \Vert }_{\infty }=\underset{m\in I}{sup}\left\{{\left|\varphi \right(m)-\psi (m\left)\right|}^{q^{*}}\right\}{q}^{*}>1,$$

for all $\varphi ,\psi \in \mathcal{M}.$ It is easy to verify that $(\mathcal{M},d_{\mathfrak{b}})$ is a complete $\mathfrak{b}$MS.

Consider a Fredholm integral equation

$$\varphi \left(\zeta \right)=f\left(\zeta \right)+{\int }_0^1{k}_{\varphi }(\zeta ,x^{*},\varphi \left(x^{*}\right))d{x}^{*}.$$

(16)

Define a mapping $S:\mathcal{M}\to \mathcal{M}$, as

$$S\left({\varphi }^{*}\left(\zeta \right)\right)=f\left(\zeta \right)+{\int }_0^1{k}_{\varphi }(\zeta ,x^{*},\varphi \left(x^{*}\right))d{x}^{*}$$

(17)

Theorem 4.

Suppose that the following conditions hold:

(1) 

Let $k_{\varphi }:I\times I\times {\mathbb{R}}^2\to {\mathbb{R}}^2$ and $f:I\to {\mathbb{R}}^2$ be continuous;

(2) 

${\varphi }_0\in \mathcal{M}$ exists such that ${\varphi }_k=S{\varphi }_{k-1}$;

(3) 

A continuous function $\mathfrak{f}:I\times I\to I$ exists such that

$$|k_{\varphi }(m,m^{*},\varphi \left(m^{*}\right))-k_{\psi }(m,m^{*},\psi \left(m^{*}\right)){|}^{q^{*}}\le |\mathfrak{f}(\varphi \left(m^{*}\right),\psi \left(m^{*}\right))\left|\right|\varphi \left(m^{*}\right)-\psi \left(m^{*}\right){|}^{q^{*}},$$

for each $m,m^{*}\in I$ and $|\mathfrak{f}(\varphi \left(m^{*}\right),\psi \left(m^{*}\right))|\le \frac{1}{\nu }$ where $\nu >0$.

Then, the integral Equation (16) has a solution.

Proof. 

Let $(\mathcal{M},d_{\mathfrak{b}})$ be a complete $\mathfrak{b}$MS and $\pi (\varphi ,\psi )=1$. Let $p^{*}>1$ such that $\frac{1}{p^{*}}+\frac{1}{q^{*}}=1$; then, for ${\varphi }^{*}\in S\left(\varphi \right)$ we have

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}(S{\varphi }^{*}\left(m\right),S\left({\psi }^{*}\left(m\right)\right))& =\underset{m\in I}{sup}{|S{\varphi }^{*}\left(m\right)-S{\psi }^{*}\left(m\right)|}^{q^{*}}\hfill \\ \hfill & =\underset{m\in I}{sup}|{\int }_0^1{k}_{\varphi }(m,m^{*},\varphi \left(m^{*}\right))-k_{\psi }(m,m^{*},\psi \left(m^{*}\right)){|}^{q^{*}}d{m}^{*}\hfill \\ \hfill & \le \underset{m\in I}{sup}[{\left({\int }_0^1{\left|1\right|}^{p^{*}}d{m}^{*}\right)}^{\frac{1}{p^{*}}}\hfill \\ \hfill & {\int }_0^1{\left(|k_{\varphi }(m,m^{*},\varphi \left(m^{*}\right))-k_{\psi }(m,m^{*},\psi \left(m^{*}\right)){|}^{q^{*}}\right)}^{\frac{1}{q^{*}}}{]}^{q^{*}}d{m}^{*}\hfill \\ \hfill & =\underset{m\in I}{sup}{\int }_0^1{|k_{\varphi }(m,m^{*},\varphi \left(m^{*}\right))-k_{\psi }(m,m^{*},\psi \left(m^{*}\right))|}^{q^{*}}d{m}^{*}\hfill \\ \hfill & \le \underset{m\in I}{sup}{\int }_0^1\left|\mathfrak{f}(\varphi \left(m^{*}\right),\psi \left(m^{*}\right))\right||\varphi \left(m^{*}\right)-\psi \left(m^{*}\right){|}^{q^{*}}d{m}^{*}\hfill \\ \hfill & \le \left(\frac{1}{\nu }\right){\Vert \varphi \left(m^{*}\right)-\psi \left(m^{*}\right)\Vert }_{\infty }\hfill \\ \hfill & =\left(\frac{1}{\nu }\right)d_{\mathfrak{b}}(\varphi \left(m^{*}\right),\psi \left(m^{*}\right))\hfill \end{array}$$

Similarly, one can easily obtain

$$\begin{array}{cc}\hfill d_{\mathfrak{b}}(S^2{\varphi }^{*}\left(m\right),S^2{\psi }^{*}\left(m\right))& =\underset{m\in I}{sup}{|S^2{\varphi }^{*}\left(m\right)-S^2{\psi }^{*}\left(m\right)|}^{q^{*}}\hfill \\ \hfill & \le {\left(\frac{1}{\nu }\right)}^2{d}_{\mathfrak{b}}(\varphi \left(m^{*}\right),\psi \left(m^{*}\right))\hfill \\ \hfill & ={\left(\frac{1}{\nu }\right)}^2\mathbb{M}{d}_{\mathfrak{b}}(\varphi \left(m^{*}\right),\psi \left(m^{*}\right)).\hfill \end{array}$$

All of the conditions of the theorem (1) are satisfied by choice of $w={\left(\frac{1}{\nu }\right)}^2\in (0,1)$ and

$$c_1=1,c_2=c_3=c_4=c_5=\delta =0,$$

hence the integral Equation (16) has a solution. □

4. Conclusions and Future Work

Several results from the literature may be considered as a special cases of our developed extension, which illustrates the degree of the validity of our results. If $\mathfrak{b}=1$ then the main results of Karapinar et al. [25] are just a subcase of our results. The proposed idea also yields some consequences of work done by Istrăţescu [26,27].

Further work should be considered on how to redevelop the contractivity condition in order to formulate these results in more general abstract spaces, for example, controlled, as well as double-controlled, metric spaces; fuzzy metric spaces; fuzzy $\mathfrak{b}$-metric spaces; and generalized fuzzy metric spaces.