Extensions of Orthogonal p-Contraction on Orthogonal Metric Spaces

Abstract

Fixed-point theory and symmetry are major and vigorous tools to working nonlinear analysis and applications, specially nonlinear operator theory and applications. The subject of examining the presence and inimitableness of fixed points of self-mappings defined on orthogonal metric spaces has become very popular in the latest decade. As a result, many researchers reached more relevant conclusions. In this study, the notion of $\varphi$-Kannan orthogonal p-contractive conditions in orthogonal complete metric spaces is presented. W-distance mappings do not need to satisfy the symmetry condition, that is, such mappings can be symmetrical or asymmetrical. Self-distance does not need to be zero in w-distance mappings. The intent of this study is to enhance the recent development of fixed-point theory in orthogonal metric spaces and related nonlinear problems by using w-distance. On this basis, some fixed-point results are debated. Some explanatory examples are shown that indicate the currency of the hypotheses and grade of benefit of the suggested conclusions. Lastly, sufficient cases for the presence of a solution to nonlinear Fredholm integral equations are investigated through the main results in this study.

1. Introduction and Preliminaries

Fixed-point theorems were used to indicate the presence and inimitableness of the solution of differential equation

$$y^{\prime }\left(x\right)=F(x;y);y\left(x_0\right)=y_0,$$

(1)

where F is a continuously differentiable function. In 1877, this proof was simplified by R. Lipschitz. In 1890, G. Peano generalized the idea of Cauchy by supposing only the continuity of $F(x;y).$ In 1922, Banach [1] stated his famous theorem, the Banach contraction principle:

Every self-mapping h on complete metric spaces $(\mathsf{\Omega },d)$ satisfying case

$$d\left(h\right(k),h(l\left)\right)\le \theta d(k,l),forallk,l\in \mathsf{\Omega },\theta \in [0,1)$$

(2)

has a unique fixed point.

This theorem has been used to show the presence and inimitableness of solutions to (1) after that time. This result is more powerful since it not only guarantees the presence and inimitableness of fixed points of exact self-maps of metric spaces, but it also ensures a constitutive technique to discover those fixed points. After Banach, many authors worked on fixed-point theory and gave generalizations of the Banach contraction principle on complete metrics.

In 1968, for example, Kannan proved the next theorem by adding a new contraction condition on h mapping:

Theorem 1

([2]). Let $(\mathsf{\Omega },d)$ be a complete metric space, and $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be a given mapping. Assume there exists $\theta \in [0,\frac{1}{2})$ such that, for all $k,l\in \mathsf{\Omega }$,

$$d\left(h\right(k),h(l\left)\right)\le \theta \left(d\right(h\left(k\right),k)+d(h\left(l\right),l\left)\right).$$

(3)

In this case, h has a unique fixed point.

A comparison of the Kannan and Banach fixed-point theorems shows that, although Inequality (2) guarantees the continuity of the given transformation, Inequality (3) does not guarantee the continuity of the given transformation. So, Inequalities (2) and (3) are independent conditions.

Kada, Suzuki, and Takahashi [3] introduced the concept of w-distance on metric spaces, and appointed different famous conclusions using this field in 1996. Then, Alber and Guerre-Delabriere [4] presented fixed-point theorems for weakly contractive maps in Hilbert spaces in 1997. Results of [4] were shown to be valuable in complete metric spaces by Rhoades [5]. In 2019, Lakzian, Rakocevic, and Aydi [6], inspired by [2,4,5], introduced and studied fixed-point results for $\varphi$-Kannan contractions on metric spaces by w-distance.

Notions of orthogonal sets and orthogonal metric spaces were otherwise described by Gordji et al. [7]. Then, Gordji and Habibi [8] proved the presence and inimitableness theorem of fixed points for mappings on a generalized orthogonal metric space. They also applied this method on first-order differential equations. Moreover, Ramezani [9] established generalized convex contractions on orthogonal metric spaces that might be called their definitive versions. Baghani et al. [10] proved some fixed-point theorems on orthogonal spaces. These theorems improved the consequence of the paper by Eshaghi Gordji et al. [7]. Then, Ramezani and Baghani [11] presented the concept of strongly orthogonal sets, and obtained an actual generalization of Banach’s fixed-point theorem. They also obtained an illustrative example that highlights the importance of their main theorem.

In 2018, Senapati et al. [12], inspired by [3,7], defined orthogonal lower semicontinuity and offered the notion w-distance on an orthogonal metric space. They also presented the version of Banach’s fixed-point theorem on orthogonal metric spaces via w-distance.

Then, the researchers acquired generalized fixed-point theorems in this field. Indeed, the concept of an orthogonal F contraction mapping is defined, and many fixed-point results were produced for this type of contraction mapping on orthogonal metric spaces by Sawangsup et al. [13]. Very recently, in 2021, Uddin et al. [14] presented the notion of orthogonal m-metric space and gave fixed-point theorems on orthogonal m-metric space. Uddin et al. [15] generalized the notion of control fuzzy metric spaces via presentation orthogonal control fuzzy metric spaces. Then, Beg et al. [16] presented the notion of generalized orthogonal F-Suzuki contraction mapping and obtained fixed-point theorems on orthogonal b-metric spaces. Furthermore, notions of generalized orthogonal F-contraction and orthogonal F-Suzuki contraction mappings were presented by Mani et al. [17]. Thus, many results that were very common in the literature were generalized.

In this study, the notion of $\varphi$-Kannan orthogonal p-contractive conditions in orthogonal complete metric spaces is presented. W-distance mappings do not need to satisfy the symmetry condition, that is, such mappings can be symmetrical or asymmetrical. Self-distance also does not need to be zero in w-distance mappings. The intent of this study is to enhance the new improvement of fixed-point theory on orthogonal metric spaces and related nonlinear problems via the notion of w-distance. On this basis, some fixed-point results are debated. Explanatory examples are delivered that indicate the currency of the hypotheses and grade of benefit of the suggested conclusions. Lastly, sufficient cases for the presence of a solution to nonlinear Fredholm integral equations are explored through the main results in this paper.

Throughout this study, $\mathbb{R}and\mathbb{Z}$ denote real numbers and integers, respectively.

Definition 1

([7]). Let $\mathsf{\Omega }\ne \varnothing$ and $\perp \subseteq \mathsf{\Omega }\times \mathsf{\Omega }$ be a binary relation. If ⊥ satisfies condition

$$\exists k_0\in \mathsf{\Omega };(\forall k\in \mathsf{\Omega },k\perp k_0)\vee (\forall k\in \mathsf{\Omega },k_0\perp k),$$

(4)

then $(\mathsf{\Omega },\perp )$ is called an orthogonal set. In this case, $k_0$ is called an orthogonal element.

Example 1

([8]). Let $\mathsf{\Omega }=\mathbb{Z}$. Determine $k\perp l$ if there exists $a\in \mathbb{Z}$, such that $k=al$. One can see that $0\perp l$ for all $l\in \mathbb{Z}$. Hence, $(\mathsf{\Omega },\perp )$ is an orthogonal set.

Definition 2

([7]). Let $(\mathsf{\Omega },\perp )$ be an orthogonal set, and d be a metric on Ω. Then, $(\mathsf{\Omega },\perp ,d)$ is called an orthogonal metric space.

Definition 3

([7]). Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal metric space. A sequence $\left(k_n\right)\subseteq \mathsf{\Omega }$ is called orthogonal if

$$(\forall n\in \mathbb{N};k_n\perp k_{n+1})\vee (\forall n\in \mathbb{N};k_{n+1}\perp k_n).$$

(5)

In the same way, a Cauchy sequence $\left(k_n\right)\subseteq \mathsf{\Omega }$ is orthogonal if

$$(\forall n\in \mathbb{N};k_n\perp k_{n+1})\vee (\forall n\in \mathbb{N};k_{n+1}\perp k_n).$$

(6)

Definition 4

([7]). Orthogonal metric space $(\mathsf{\Omega },\perp ,d)$ is an orthogonal complete metric space if every orthogonal Cauchy sequence converges in Ω.

Definition 5

([7]). Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal metric space. Function $h:\mathsf{\Omega }\to \mathsf{\Omega }$ is orthogonal continuous at k if for each orthogonal sequence $\left(k_n\right)$ converging to k implies $h\left(k_n\right)\to h\left(k\right)$ as $n\to \infty$. h is orthogonal continuous on Ω if h is orthogonal continuous in each $k\in \mathsf{\Omega }$.

Definition 6

([7]). Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal metric space, and $\theta \in \mathbb{R}$, $0<\theta <1$. A function $h:\mathsf{\Omega }\to \mathsf{\Omega }$ is an orthogonal contraction with Lipschitz constant θ if

$$d\left(h\right(k),h(l\left)\right)\le \theta d(k,l)$$

(7)

for all $k,l\in \mathsf{\Omega }$ whenever $k\perp l$.

Definition 7

([7]). Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal metric space. A function $h:\mathsf{\Omega }\to \mathsf{\Omega }$ is orthogonal preserving if $h\left(k\right)\perp h\left(l\right)$ whenever $k\perp l$.

Theorem 2

([7]). Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal complete metric space and $0<\theta <1$. Let $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be an orthogonal preserving mapping that is orthogonal continuous and orthogonal contraction mapping with Lipschitz constant θ. Then, h is a Picard operator, that is, h has a unique fixed point, and every Picard iteration in Ω converges to this fixed point.

Lemma 1

([12]). Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal metric space and $p:\mathsf{\Omega }\times \mathsf{\Omega }\to [0,\infty )$ be a w-distance. Assume that $\left(k_n\right)$ and $\left(l_n\right)$ are two orthogonal sequences in Ω and $k,l,m\in \mathsf{\Omega }$. Let $\left(u_n\right)$ and $\left(v_n\right)$ be sequences of positive real numbers converging to 0. Then, we have the following:

(i) 

If $p(k_n,l)\le u_n$ and $p(k_n,m)\le v_n$ then $l=m$. Moreover, if $p(k,l)=0$ and $p(k,m)=0$, then $l=m$.

(ii) 

If $p(k_n,l_n)\le u_n$ and $p(k_n,m)\le v_n$, then $l_n\to m$ as $n\to \infty$.

(iii) 

If $p(k_n,k_m)\le u_n$ for all $m>n$, then $\left(k_n\right)$ is a orthogonal Cauchy sequence in Ω.

(iv) 

If $p(k_n,l)\le u_n$, then $\left(k_n\right)$ is a orthogonal Cauchy sequence in Ω.

Definition 8

([12]). Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal metric space, and $p:\mathsf{\Omega }\times \mathsf{\Omega }\to [0,\infty )$ be a w-distance. Mapping $h:\mathsf{\Omega }\to \mathsf{\Omega }$ is an orthogonal p-contraction if there exists $\theta \in [0,1)$, such that

$$p\left(h\right(k),h(l\left)\right)\le \theta p(k,l),$$

(8)

for all $k,l\in \mathsf{\Omega }$ with $k\perp l$.

Theorem 3

([12]). Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal complete metric space with a w-distance p. If h is an orthogonal p-contractive, orthogonal preserving and orthogonal continuous self-mapping, then

(a) 

h has a unique fixed point $k^{*}\in \mathsf{\Omega }$;

(b) 

Picard sequence $h^n\left(k\right)$ converges to $k^{*}\in \mathsf{\Omega }$ for every $k\in \mathsf{\Omega }$.

2. Main Results

Definition 9.

Let $(\mathsf{\Omega },\perp )$ be an orthogonal set. Any two elements $k,l\in \mathsf{\Omega }$ are orthogonally related if $k\perp l$ or $l\perp k$.

Theorem 4.

Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal complete metric space with a w-distance p, $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be a self-map, $\varphi :[0,\infty )\times [0,\infty )\to [0,\infty )$ be continuous function, and $\varphi (a,b)=0\iff a=b=0$. Assume that h is orthogonal preserving self-mapping satisfying inequality

$$p(h\left(k\right),h\left(l\right)))\le \frac{1}{2}(p(h\left(k\right),k)+p(h\left(l\right),l))-\varphi (p(h\left(k\right),k),p(h\left(l\right),l))$$

(9)

for all orthogonally related $k,l\in \mathsf{\Omega }$. Then,

$\left(i\right)$ there exists a point $k^{*}\in \mathsf{\Omega }$, such that, for any orthogonal element $k_0\in \mathsf{\Omega }$, iteration sequence $\left(h^n\left(k_0\right)\right)$ converges to this point.

$\left(ii\right)$ either

$\left(a\right)$ h is orthogonal continuous at $k^{*}\in \mathsf{\Omega }$

or

$\left(b\right)$ if an orthogonal sequence $\left(k_n\right)\subseteq \mathsf{\Omega }$ converges to $k\in \mathsf{\Omega }$ then $k\perp k_n$ or $k_n\perp k$ for all n,

in this case, $k^{*}\in \mathsf{\Omega }$ is the unique fixed point of h.

$\left(iii\right)$$p(k^{*},k^{*})=0$.

Proof. 

Because $(\mathsf{\Omega },\perp )$ is an orthogonal set,

$$\exists k_0\in \mathsf{\Omega };(\forall x\in \mathsf{\Omega },x\perp k_0)\vee (\forall x\in \mathsf{\Omega },k_0\perp x).$$

(10)

If for any orthogonal element $k_0\in \mathsf{\Omega }$, since h is self-mapping on $\mathsf{\Omega }$, $k_1\in \mathsf{\Omega }$ can be chosen to be $k_1=h\left(k_0\right)$. Thus,

$$\begin{array}{cc}\hfill & k_0\perp h\left(k_0\right)\vee h\left(k_0\right)\perp k_0\hfill \\ \hfill & \Rightarrow k_0\perp k_1\vee k_1\perp k_0.\hfill \end{array}$$

(11)

Then, if we continue in the same way,

$$k_1=h\left(k_0\right),k_2=h\left(k_1\right)=h^2\left(k_0\right),...,k_n=h\left(k_{n-1}\right)=h^n\left(k_0\right)$$

(12)

so $\left(h^n\left(k_0\right)\right)$ is an iteration sequence. Since h is orthogonal preserving, $\left(h^n\left(k_0\right)\right)$ is an orthogonal sequence. If $k_{n+1}=k_n$ for some n, then $k_n$ is a fixed point of h. So, we assume that $k_{n+1}\ne k_n$ for each n. By using (9) with $k=k_n,l=k_{n-1}$, we obtain

$$\begin{array}{cc}\hfill p(k_{n+1},k_n)& =p(h\left(k_n\right),h\left(k_{n-1}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(h\left(k_n\right),k_n)+p(h\left(k_{n-1}\right),k_{n-1}))-\varphi (p(h\left(k_n\right),k_n),p(h\left(k_{n-1}\right),k_{n-1}))\hfill \\ \hfill & =\frac{1}{2}(p(k_{n+1},k_n)+p(k_n,k_{n-1}))-\varphi (p(k_{n+1},k_n),p(k_n,k_{n-1}))\hfill \end{array}$$

(13)

So, $p(k_{n+1},k_n)\le p(k_n,k_{n-1})$ for all $n\ge 1$. Thus, real sequence $\left(p(k_{n+1},k_n)\right)$ is monotone nonincreasing and bounded below; so, there exists $s\ge 0$, such that ${lim}_{n\to \infty }p(k_{n+1},k_n)=s$. Letting $n\to \infty$ in (13) and using the continuity of $\varphi$, we obtain $\varphi (s,s)=0$, and so $s=0$ by a feature of $\varphi$. Thus, ${lim}_{n\to \infty }p(k_{n+1},k_n)=0$ is obtained.

On the other hand, by using (9) with $k=k_n,l=k_{n+1}$, we obtain

$$\begin{array}{cc}\hfill p(k_n,k_{n+1})& =p(h\left(k_{n-1}\right),h\left(k_n\right))\hfill \\ \hfill & \le \frac{1}{2}(p(k_n,k_{n-1})+p(k_{n+1},k_n))-\varphi (p(k_n,k_{n-1}),p(k_{n+1},k_n)).\hfill \end{array}$$

(14)

So $p(k_n,k_{n+1})\le \frac{1}{2}(p(k_n,k_{n-1})+p(k_{n+1},k_n))$ and letting $n\to \infty$ in the last inequality, ${lim}_{n\to \infty }p(k_n,k_{n+1})=0$ is obtained.

Next, we prove that $\left(k_n\right)$ is an orthogonal Cauchy sequence. If $\left(k_n\right)$ is not an orthogonal Cauchy sequence, by using Lemma 1$\left(iii\right)$, there exists a sequence $\left(u_n\right)$ of positive real numbers converging to 0, and corresponding subsequences $\left(m\right(n\left)\right)$ and $\left(t\right(n\left)\right)$ of $\mathbb{N}$ satisfying $t\left(n\right)>m\left(n\right)$, for which

$$p(k_{m\left(n\right)},k_{t\left(n\right)})>u_{m\left(n\right)}.$$

(15)

Thus, there exists $\epsilon >0$ that satisfies

$$p(k_{m\left(n\right)},k_{t\left(n\right)})>u_{m\left(n\right)}\ge \epsilon .$$

(16)

If $t\left(n\right)$ is chosen as the smallest integer satisfying (16), that is,

$$p(k_{m\left(n\right)},k_{t\left(n\right)-1})<\epsilon .$$

(17)

By the triangular inequality of p and (16), we easily derive that

$$\epsilon \le p(k_{m\left(n\right)},k_{t\left(n\right)})\le p(k_{m\left(n\right)},k_{m\left(n\right)+1})+p(k_{m\left(n\right)+1},k_{m\left(n\right)+2})+...+p(k_{t\left(n\right)-1},k_{t\left(n\right)}).$$

(18)

Letting $n\to \infty$ in the last inequality, since ${lim}_{n\to \infty }p(k_n,k_{n+1})=0$, a contradiction is obtained. So, $\left(k_n\right)$ is an orthogonal Cauchy sequence. By the orthogonal completeness of $\mathsf{\Omega }$, there exists $k^{*}\in \mathsf{\Omega }$ such that $\left(k_n\right)=\left(h^n\left(k_0\right)\right)$ converges to this point.

Suppose that h is orthogonal continuous mapping. In that case,

$$d(k^{*},h\left(k^{*}\right))=\underset{n\to \infty }{lim}d(k_{n+1},h\left(k^{*}\right))=\underset{n\to \infty }{lim}d(h\left(k_n\right),h\left(k^{*}\right))=d(h\left(k^{*}\right),h\left(k^{*}\right))=0,$$

(19)

and so $k^{*}$ is a fixed point of h.

Suppose that, if a sequence $\left(k_n\right)\subseteq \mathsf{\Omega }$ converges to $k\in \mathsf{\Omega }$, then $k\perp k_n$ or $k_n\perp k$ for all n. In that case, by using the existence of $k^{*}\in \mathsf{\Omega }$, such that $\left(k_n\right)=\left(h^n\left(k_0\right)\right)$ converges to this point, then $k^{*}\perp k_n$ or $k_n\perp k^{*}$ for all n. From p being orthogonal lower semicontinuous, we obtain

$$\begin{array}{cc}\hfill p(h\left(k^{*}\right),k^{*})& \le \underset{n\to \infty }{lim\; inf}p(h\left(k^{*}\right),k_{n+1})\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; inf}\frac{1}{2}(p(h\left(k^{*}\right),k^{*})+p(k_{n+1},k_n))-\varphi (p(h\left(k^{*}\right),k^{*}),p(k_{n+1},k_n))\hfill \end{array}$$

(20)

and so $\frac{1}{2}p(h\left(k^{*}\right),k^{*})\le -\varphi (p(h\left(k^{*}\right),k^{*}),0)\Rightarrow p(h\left(k^{*}\right),k^{*})=0$ is obtained, and

$$\begin{array}{cc}\hfill p(h\left(k^{*}\right),h\left(k^{*}\right))& \le p(h\left(k^{*}\right),k_{n+1})+p(k_{n+1},h\left(k^{*}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(h\left(k^{*}\right),k^{*})+p(k_{n+1},k_n))-\varphi (p(h\left(k^{*}\right),k^{*}),p(k_{n+1},k_n))\hfill \\ \hfill & +\frac{1}{2}(p(k_{n+1},k_n)+p(h\left(k^{*}\right),k^{*}))-\varphi (p(k_{n+1},k_n),p(h\left(k^{*}\right),k^{*})).\hfill \end{array}$$

(21)

Letting $n\to \infty$ in the last inequality, since ${lim}_{n\to \infty }p(k_{n+1},k_n)=0$ and $p(h\left(k^{*}\right),k^{*})=0$, $p(h\left(k^{*}\right),h\left(k^{*}\right))=0$ is obtained. Hence, by using Lemma 1$\left(i\right)$, $k^{*}=h\left(k^{*}\right)$.

The uniqueness of the fixed point is shown as follows:

Assume that $l^{*}\in \mathsf{\Omega }$ is another fixed point of h, such that, for any orthogonal element $k_0\in \mathsf{\Omega }$, iteration sequence $\left(h^n\left(k_0\right)\right)$ converges to this point. Using the triangular inequality of p and Inequality (9),

$$\begin{array}{cc}\hfill p(k^{*},k^{*})=& p(h\left(k^{*}\right),h\left(k^{*}\right))\hfill \\ \hfill & \le p(h\left(k^{*}\right),k_{n+1})+p(k_{n+1},h\left(k^{*}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(h\left(k^{*}\right),k^{*})+p(k_{n+1},k_n))-\varphi (p(h\left(k^{*}\right),k^{*}),p(k_{n+1},k_n))\hfill \\ \hfill & +\frac{1}{2}(p(k_{n+1},k_n)+p(h\left(k^{*}\right),k^{*}))-\varphi (p(k_{n+1},k_n),p(h\left(k^{*}\right),k^{*})).\hfill \end{array}$$

(22)

Letting $n\to \infty$ in the last inequality, since ${lim}_{n\to \infty }p(k_{n+1},k_n)=0$ and $p(h\left(k^{*}\right),k^{*})=0$, $p(k^{*},k^{*})=0$ is obtained. In this case, we obtain

$$\begin{array}{cc}\hfill p(k^{*},l^{*})=& p(h\left(k^{*}\right),h\left(l^{*}\right))\hfill \\ \hfill & \le p(h\left(k^{*}\right),k_{n+1})+p(k_{n+1},h\left(l^{*}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(h\left(k^{*}\right),k^{*})+p(k_{n+1},k_n))-\varphi (p(h\left(k^{*}\right),k^{*}),p(k_{n+1},k_n))\hfill \\ \hfill & +\frac{1}{2}(p(k_{n+1},k_n)+p(h\left(l^{*}\right),l^{*}))-\varphi (p(k_{n+1},k_n),p(h\left(l^{*}\right),l^{*})).\hfill \end{array}$$

(23)

Letting $n\to \infty$ in the last inequality, since ${lim}_{n\to \infty }p(k_{n+1},k_n)=0$, $p(h\left(k^{*}\right),k^{*})=0$ and $p(h\left(l^{*}\right),l^{*})=0$, $p(k^{*},l^{*})=0$ is obtained. Hence, by using Lemma 1$\left(i\right)$, $k^{*}=l^{*}$. □

Now, we can give an example for Theorem 4. The space given in this example is an orthogonal complete metric space, but not a complete metric space. Therefore, this example cannot be applied to Kannan or orthogonal weak Kannan-type contraction theorems given in complete metric spaces, so the concept of orthogonal metric space is very important in theory.

Example 2.

Let $\mathsf{\Omega }=[0,1)$ be a set, and define $d:\mathsf{\Omega }\times \mathsf{\Omega }\to \mathbb{R}$, such that $d(k,l)=\mid k-l\mid$. Let binary relation ⊥ on Ω, such that $k\perp l\iff kl\le max\{\frac{k}{3},\frac{l}{3}\}$. Then, $(\mathsf{\Omega },\perp )$ is an orthogonal set, and d is a metric on Ω. So, $(\mathsf{\Omega },\perp ,d)$ is an orthogonal metric space. In this space, any orthogonal Cauchy sequence is convergent. Indeed, if $\left(k_n\right)$ is an arbitrary orthogonal Cauchy sequence in Ω, then there exists a subsequence $\left(k_{n_m}\right)$ of $\left(k_n\right)$, for all $n\in \mathbb{N}$ $k_{n_m}=0$ or a subsequence $\left(k_{n_m}\right)$ of $\left(k_n\right)$ for all $n\in \mathbb{N}$, $k_{n_m}\le \frac{1}{3}$. So, this subsequence is convergent in Ω. Every Cauchy sequence with a convergent subsequence is convergent, so $\left(k_n\right)$ is convergent in Ω. So, $(\mathsf{\Omega },\perp ,d)$ is an orthogonal complete metric space. Consider $p:\mathsf{\Omega }\times \mathsf{\Omega }\to [0,\infty )$, $p(k,l)=l$, which is a w-distance on Ω. Let $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be defined as

$$h\left(k\right)=\left\{\begin{array}{cc}\hfill \frac{k}{3}& ,0\le k\le \frac{1}{3},\hfill \\ \hfill 0& ,\frac{1}{3}<k<1.\hfill \end{array}\right.$$

(24)

In this case, h is orthogonal preserving mapping. Indeed, suppose that $k\perp l$. Without loss of generality, $kl\le \frac{k}{3}$ can be chosen. So, $k\ge 0$ and $0\le l\le \frac{1}{3}$. Thus, two cases are obtained:

Case (I): $0\le k\le \frac{1}{3}$ and $l\le \frac{1}{3}$; then, $h\left(k\right)=\frac{k}{3},h\left(l\right)=\frac{l}{3}$.

Case (II): $1>k>\frac{1}{3}$ and $l\le \frac{1}{3}$; then, $h\left(k\right)=0,h\left(l\right)=\frac{l}{3}$.

These cases imply that $h\left(k\right)\perp h\left(l\right)$.

Consider $\varphi :[0,\infty )\times [0,\infty )\to [0,\infty )$, $\varphi (k,l)=\frac{k}{2}+\frac{l}{8}$ for all $k,l\ge 0$. ϕ is a continuous function, and $\varphi (a,b)=0\iff a=b=0$. h also satisfies Inequality (9). Indeed, for any orthogonally related $k,l\in X$, $kl\le max\{\frac{k}{3},\frac{l}{3}\}$ is obtained. Then, there are two cases:

Case (I): Suppose that $\frac{k}{3}\ge \frac{l}{3}$, and so $kl\le \frac{k}{3}$. Then, ($l\le \frac{1}{3})\wedge (k\le \frac{1}{3})$ or $(l\le \frac{1}{3})\wedge (k>\frac{1}{3})$; in both cases, Inequality (9) is satisfied.

Case (II): Suppose that $\frac{k}{3}<\frac{l}{3}$, and so $kl\le \frac{l}{3}$. Then, ($k\le \frac{1}{3})\wedge (l\le \frac{1}{3})$ or $(k\le \frac{1}{3})\wedge (l>\frac{1}{3})$; in both cases, Inequality (9) is satisfied.

Therefore, all hypotheses of Theorem 4 are satisfied. For any orthogonal element $k_0\in \mathsf{\Omega }$, iteration sequence $\left(h^n\left(k_0\right)\right)$ converges to $k^{*}=0\in \mathsf{\Omega }$. h is orthogonal continuous at $k^{*}=0$, so this point is the unique fixed point of h and $p(k^{*},k^{*})=p(0,0)=0$.

Theorem 5.

Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal complete metric space with a w-distance p, $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be a self-map, $\varphi :[0,\infty )\times [0,\infty )\to [0,\infty )$ be continuous function, and $\varphi (a,b)=0\iff a=b=0$. Assume that f is orthogonal preserving self-mapping satisfying inequality

$$p(h\left(k\right),h\left(l\right)))\le \frac{1}{2}(p(h\left(k\right),k)+p(l,h\left(l\right)))-\varphi (p(h\left(k\right),k),p(l,h\left(l\right)))$$

(25)

for all orthogonally related $k,l\in \mathsf{\Omega }$. Then,

$\left(i\right)$ there exists a point $k^{*}\in \mathsf{\Omega }$, such that, for any orthogonal element $k_0\in \mathsf{\Omega }$, iteration sequence $\left(h^n\left(k_0\right)\right)$ converges to this point.

$\left(ii\right)$ either

$\left(a\right)$ h is orthogonal continuous at $k^{*}\in \mathsf{\Omega }$

or

$\left(b\right)$ if an orthogonal sequence $\left(k_n\right)\subseteq \mathsf{\Omega }$ converges to $k\in \mathsf{\Omega }$, then $k\perp k_n$ or $k_n\perp k$ for all n and also $k,hk$ are orthogonally related elements; in this case, $k^{*}\in \mathsf{\Omega }$ is the unique fixed point of h.

$\left(iii\right)$$p(k^{*},k^{*})=0$.

Proof. 

Because $(\mathsf{\Omega },\perp )$ is an orthogonal set,

$$\exists k_0\in \mathsf{\Omega };(\forall x\in \mathsf{\Omega },x\perp k_0)\vee (\forall x\in \mathsf{\Omega },k_0\perp x).$$

(26)

If for any orthogonal element $k_0\in \mathsf{\Omega }$, since h is a self-mapping on $\mathsf{\Omega }$, $k_1\in \mathsf{\Omega }$ can be chosen as $k_1=h\left(k_0\right)$. Thus,

$$\begin{array}{cc}\hfill & k_0\perp h\left(k_0\right)\vee h\left(k_0\right)\perp k_0\hfill \\ \hfill & \Rightarrow k_0\perp k_1\vee k_1\perp k_0.\hfill \end{array}$$

(27)

Then, if we continue in the same way,

$$k_1=h\left(k_0\right),k_2=h\left(k_1\right)=h^2\left(k_0\right),...,k_n=h\left(k_{n-1}\right)=h^n\left(k_0\right);$$

(28)

so, $\left(h^n\left(k_0\right)\right)$ is an iteration sequence. Since h is orthogonal preserving, $\left(h^n\left(k_0\right)\right)$ is an orthogonal sequence. If $k_{n+1}=k_n$ for some n, then $k_n$ is a fixed point of h. So, we assume that $k_{n+1}\ne k_n$ for each n. By using (25) with $k=k_n,l=k_{n-1}$, we obtain

$$\begin{array}{cc}\hfill p(k_{n+1},k_n)& =p(h\left(k_n\right),h\left(k_{n-1}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(h\left(k_n\right),k_n)+p(k_{n-1},h\left(k_{n-1}\right)))-\varphi (p(h\left(k_n\right),k_n),p(k_{n-1},h\left(k_{n-1}\right)))\hfill \\ \hfill & =\frac{1}{2}(p(k_{n+1},k_n)+p(k_{n-1},k_n))-\varphi (p(k_{n+1},k_n),p(k_{n-1},k_n))\hfill \end{array}$$

(29)

and so $p(k_{n+1},k_n)\le p(k_{n-1},k_n)$ for all $n\ge 1$. Furthermore,

$$\begin{array}{cc}\hfill p(k_n,k_{n+1})& =p(h\left(k_{n-1}\right),h\left(k_n\right))\hfill \\ \hfill & \le \frac{1}{2}(p(h\left(k_{n-1}\right),k_{n-1})+p(k_n,h\left(k_n\right)))-\varphi (p(h\left(k_{n-1}\right),k_{n-1}),p(k_n,h\left(k_n\right)))\hfill \\ \hfill & =\frac{1}{2}(p(k_n,k_{n-1})+p(k_n,k_{n+1}))-\varphi (p(k_n,k_{n-1}),p(k_n,k_{n+1}));\hfill \end{array}$$

(30)

so, $p(k_n,k_{n+1})\le p(k_n,k_{n-1})$ for all $n\ge 1$. Thus,

$$p(k_{n+3},k_{n+2})\le p(k_{n+1},k_{n+2})\le p(k_{n+1},k_n)\le p(k_{n-1},k_n)\mathrm{for}\mathrm{all}n\ge 1.$$

(31)

Similarly

$$p(k_{n+1},k_n)\le p(k_{n-1},k_n)\le p(k_{n-1},k_{n-2})\le p(k_{n-3},k_{n-2})\mathrm{for}\mathrm{all}n\ge 1.$$

(32)

Now, (31) and (32) imply

$$\underset{n\to \infty }{lim}p(k_{2n+1},k_{2n})=\underset{n\to \infty }{lim}p(k_{2n-1},k_{2n})=\underset{n\to \infty }{lim}p(k_{2n+1},k_{2n+2})=s\ge 0.$$

(33)

By using (25),

$$\begin{array}{cc}\hfill p(k_{2n+1},k_{2n+2})& =p(h\left(k_{2n}\right),h\left(k_{2n+1}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(k_{2n+1},k_{2n})+p(k_{2n+1},k_{2n+2}))-\varphi (p(k_{2n+1},k_{2n}),p(k_{2n+1},k_{2n+2})).\hfill \end{array}$$

(34)

Letting $n\to \infty$ in (34) and using the continuity of $\varphi$, we obtain $\varphi (s,s)=0$; so, $s=0$ by the property of $\varphi$.

On the other hand, by using (25),

$$\begin{array}{cc}\hfill p(k_{2n},k_{2n-1})& =p(h\left(k_{2n-1}\right),h\left(k_{2n-2}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(k_{2n-1},k_{2n})+p(k_{2n-2},k_{2n-1}))-\varphi (p(k_{2n-1},k_{2n}),p(k_{2n-2},k_{2n-1}));\hfill \end{array}$$

(35)

so, ${lim}_{n\to \infty }p(k_{2n},k_{2n-1})=0$ is obtained. Similarly, ${lim}_{n\to \infty }p(k_{2n+2},k_{2n+1})=0$ and ${lim}_{n\to \infty }p(k_{2n},k_{2n+1})=0$ are obtained by using (25).

Now, using the way in the proof of Theorem 4, we conclude that $\left(k_n\right)$ is an orthogonal Cauchy sequence. By the orthogonal completeness of $\mathsf{\Omega }$, there exists $k^{*}\in \mathsf{\Omega }$, such that $\left(k_n\right)=\left(h^n\left(k_0\right)\right)$ converges to this point.

Suppose that h is orthogonal continuous mapping. In that case,

$$d(k^{*},h\left(k^{*}\right))=\underset{n\to \infty }{lim}d(k_{n+1},h\left(k^{*}\right))=\underset{n\to \infty }{lim}d(h\left(k_n\right),h\left(k^{*}\right))=d(h\left(k^{*}\right),h\left(k^{*}\right))=0,$$

(36)

so $k^{*}$ is a fixed point of h.

Suppose that, if a sequence $\left(k_n\right)\subseteq \mathsf{\Omega }$ converges to $k\in \mathsf{\Omega }$, then $k\perp k_n$ or $k_n\perp k$ for all n. In that case, by using the existence $k^{*}\in X$, such that $\left(k_n\right)=\left(h^n\left(k_0\right)\right)$ converges to this point, then $k^{*}\perp k_n$ or $k_n\perp k^{*}$ for all n, and $k^{*},h\left(k^{*}\right)$ are orthogonally related elements. From p being orthogonal lower semicontinuous, we obtain

$$\begin{array}{cc}\hfill p(h\left(k^{*}\right),k^{*})& \le \underset{n\to \infty }{lim\; inf}p(h\left(k^{*}\right),k_{2n+1})\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; inf}\frac{1}{2}(p(h\left(k^{*}\right),k^{*})+p(k_{2n},k_{2n+1}))-\varphi (p(h\left(k^{*}\right),k^{*}),p(k_{2n},k_{2n+1}));\hfill \end{array}$$

(37)

so, $\frac{1}{2}p(h\left(k^{*}\right),k^{*})\le -\varphi (p(h\left(k^{*}\right),k^{*}),0)\Rightarrow p(h\left(k^{*}\right),k^{*})=0$ is obtained, and

$$\begin{array}{c}\hfill p(h\left(k^{*}\right),h^2\left(k^{*}\right))\le \frac{1}{2}(p(h\left(k^{*}\right),k^{*})+p(h\left(k^{*}\right),h^2\left(k^{*}\right)))-\varphi (p(h\left(k^{*}\right),k^{*}),p(h\left(k^{*}\right),h^2\left(k^{*}\right))).\end{array}$$

(38)

Thus, $p(h\left(k^{*}\right),h^2\left(k^{*}\right))=0$ is obtained. Hence, by using Lemma 1$\left(i\right)$, $h^2\left(k^{*}\right)=k^{*}$, and

$$\begin{array}{cc}\hfill p(k^{*},h\left(k^{*}\right))& =p(h^2\left(k^{*}\right),h\left(k^{*}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(h^2\left(k^{*}\right),h\left(k^{*}\right))+p(k^{*},h\left(k^{*}\right)))-\varphi (p(h^2\left(k^{*}\right),h\left(k^{*}\right)),p(k^{*},h\left(k^{*}\right))),\hfill \end{array}$$

(39)

$p(k^{*},h\left(k^{*}\right))=0$. Thus, $p(h\left(k^{*}\right),h\left(k^{*}\right))\le p(h\left(k^{*}\right),k^{*})+p(k^{*},h\left(k^{*}\right))=0$; so, by using Lemma 1, $\left(i\right)$, $k^{*}=h\left(k^{*}\right)$.

The uniqueness of the fixed point is shown in the following:

Assume that $l^{*}\in \mathsf{\Omega }$ is another fixed point of h, such that, for any orthogonal element $k_0\in \mathsf{\Omega }$, iteration sequence $\left(h^n\left(k_0\right)\right)$ converges to this point. Using the triangular inequality of p and (25),

$$\begin{array}{cc}\hfill p(k^{*},k^{*})=& p(h\left(k^{*}\right),h\left(k^{*}\right))\hfill \\ \hfill & \le p(h\left(k^{*}\right),k_{2n+1})+p(k_{2n+1},h\left(k^{*}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(h\left(k^{*}\right),k^{*})+p(k_{2n},k_{2n+1}))-\varphi (p(h\left(k^{*}\right),k^{*}),p(k_{2n},k_{2n+1}))\hfill \\ \hfill & +\frac{1}{2}(p(k_{2n+1},k_{2n})+p(k^{*},h\left(k^{*}\right)))-\varphi (p(k_{2n+1},k_{2n}),p(k^{*},h\left(k^{*}\right))).\hfill \end{array}$$

(40)

Letting $n\to \infty$ in the last inequality, since ${lim}_{n\to \infty }p(k_{2n+1},k_{2n})={lim}_{n\to \infty }p(k_{2n},k_{2n+1})$ $=0$ and $p(h\left(k^{*}\right),k^{*})=p(k^{*},h\left(k^{*}\right))=0$, $p(k^{*},k^{*})=0$ is obtained. Similarly, using the triangular inequality of p and (25),

$$\begin{array}{cc}\hfill p(k^{*},l^{*})=& p(h\left(k^{*}\right),h\left(l^{*}\right))\hfill \\ \hfill & \le p(h\left(k^{*}\right),k_{2n+1})+p(k_{2n+1},h\left(l^{*}\right))\hfill \\ \hfill & \le \frac{1}{2}(p(h\left(k^{*}\right),k^{*})+p(k_{2n},k_{2n+1}))-\varphi (p(h\left(k^{*}\right),k^{*}),p(k_{2n},k_{2n+1}))\hfill \\ \hfill & +\frac{1}{2}(p(k_{2n+1},k_{2n})+p(l^{*},h\left(l^{*}\right)))-\varphi (p(k_{2n+1},k_{2n}),p(l^{*},h\left(l^{*}\right))).\hfill \end{array}$$

(41)

Letting $n\to \infty$ in the last inequality, since ${lim}_{n\to \infty }p(k_{2n+1},k_{2n})={lim}_{n\to \infty }p(k_{2n},k_{2n+1})$ $=0$, $p(h\left(k^{*}\right),k^{*})=p(k^{*},h\left(k^{*}\right))=0$ and $p(h\left(l^{*}\right),l^{*})=p(l^{*},h\left(l^{*}\right))=0$, $p(k^{*},l^{*})=0$ is obtained. Hence, by using Lemma 1$\left(i\right)$, $k^{*}=l^{*}$. □

Now, we give an example for Theorem 5. The space given in this example is an orthogonal complete metric space, but not a complete metric space. Therefore, the classical Kannan theorem is not applicable to the following example. In this example, h is not a Kannan contraction with the w-distance p for $k=\frac{1}{7}$. Both cases are examined in detail at the end of the example.

Example 3.

Let $\mathsf{\Omega }=[0,2)$ be a set, and define the discrete metric $d:\mathsf{\Omega }\times \mathsf{\Omega }\to \mathbb{R}$ such that

$$d(k,l)=\left\{\begin{array}{cc}\hfill 0& ,k=l,\hfill \\ \hfill 1& ,k\ne l.\hfill \end{array}\right.$$

(42)

Let binary relation ⊥ on Ω, such that $k\perp l\iff (kl\ge k)\vee (kl\ge l)$. Then, $(\mathsf{\Omega },\perp )$ is an orthogonal set, and d is a metric on Ω. So, $(\mathsf{\Omega },\perp ,d)$ is an orthogonal metric space. In this space, any orthogonal Cauchy sequence is convergent. Indeed, if $\left(k_n\right)$ is an arbitrary orthogonal Cauchy sequence in Ω. Then, there exists a subsequence $\left(k_{n_m}\right)$ of $\left(k_n\right)$, for all $n\in \mathbb{N}$, $k_{n_m}=c$ where c is a constant element of Ω. So, this subsequence is convergent in Ω. Every Cauchy sequence with a convergent subsequence is convergent, so $\left(k_n\right)$ is convergent in Ω. So, $(\mathsf{\Omega },\perp ,d)$ is an orthogonal complete metric space. Consider $p:\mathsf{\Omega }\times \mathsf{\Omega }\to [0,\infty )$, $p(k,l)=l$ which is a w-distance on Ω. Let $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be defined as

$$h\left(k\right)=\left\{\begin{array}{cc}\hfill \frac{k}{6}& ,0\le k<1,\hfill \\ \hfill 0& ,1\le k<2.\hfill \end{array}\right.$$

(43)

In this case, h is orthogonal preserving mapping. Without loss of generality, $kl\ge k$ can be chosen. So, $k\ge 0$ and $l\ge 1$. Thus, two cases are obtained:

Case (I): $0\le k<1$ and $1\le l<2$, then $h\left(k\right)=\frac{k}{6},h\left(l\right)=0$.

Case (II): $1\le k<2$ and $1\le l<2$, then $h\left(k\right)=h\left(l\right)=0.$

These cases imply that $h\left(k\right)\perp h\left(l\right)$.

Consider $\varphi :[0,\infty )\times [0,\infty )\to [0,\infty )$, $\varphi (k,l)=\frac{k}{12}+\frac{l}{2}$ for all $k,l\ge 0$. ϕ is a continuous function, and $\varphi (a,b)=0\iff a=b=0$. h also satisfies Inequality (25). Indeed, for any orthogonally related $k,l\in \mathsf{\Omega }$, $(kl\ge k)\vee (kl\ge l)$ is obtained. Then, there are two cases:

Case (I): Suppose that $kl\ge k$, and so $k\ge 0$ and $l\ge 1$. Then, $(0\le k<1)\wedge (1\le l<2)$ or $(1\le k<2)\wedge (1\le l<2)$ and in both cases, Inequality (25) is satisfied.

Case (II): Suppose that $kl\ge l$ and so $l\ge 0$ and $k\ge 1$. Then, $(0\le l<1)\wedge (1\le k<2)$ or $(1\le l<2)\wedge (1\le k<2)$ and in both cases, Inequality (25) is satisfied.

Therefore, all conditions of Theorem 5 are satisfied. Careful examination shows that, for any orthogonal element $k_0\in \mathsf{\Omega }$, theiteration sequence $\left(h^n\left(k_0\right)\right)$ converges to $k^{*}=0\in \mathsf{\Omega }$. h is orthogonal continuous at $k^{*}=0$, so this point is the unique fixed point of h, and $p(k^{*},k^{*})=p(0,0)=0$.

Otherwise, h is not a Kannan contraction connected to metric d, so the classical Kannan theorem is not practicable to the metric d:

$$d(h\left(0\right),h\left(\frac{1}{2}\right))=1>\frac{1}{2}=\frac{1}{2}(d(h\left(0\right),0)+d(h\left(\frac{1}{2}\right),\frac{1}{2}).$$

(44)

h is not a Kannan contraction with the w-distance p for $k=\frac{1}{7}$. Actually,

$$\begin{array}{cc}\hfill p(h\left(0\right),h\left(\frac{5}{6}\right))=\frac{5}{36}>\frac{5}{42}& =\frac{1}{7}(p(h\left(0\right),0)+p(h\left(\frac{5}{6}\right),\frac{5}{6})\hfill \\ \hfill & =k(p(h\left(0\right),0)+p(h\left(\frac{5}{6}\right),\frac{5}{6}).\hfill \end{array}$$

(45)

Corollary 1.

Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal complete metric space with a w-distance p, $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be a self-map, $\psi :[0,\infty )\to [0,\infty )$ be a continuous function, and ${\psi }^{-1}\left(\left\{0\right\}\right)=0$. Suppose that h is orthogonal preserving self-mapping satisfying inequalities

$$p(h\left(k\right),h\left(l\right)))\le \frac{1}{2}(p(h\left(k\right),k)+p(h\left(l\right),l))-\psi \left(\frac{1}{2}(p(h\left(k\right),k)+p(h\left(l\right),l))\right)$$

(46)

for all orthogonally related $k,l\in \mathsf{\Omega }$. Then,

$\left(i\right)$ there exists a point $k^{*}\in \mathsf{\Omega }$, such that, for any orthogonal element $k_0\in \mathsf{\Omega }$, iteration sequence $\left(h^n\left(k_0\right)\right)$ converges to this point.

$\left(ii\right)$ either

$\left(a\right)$h is ⊥-continuous at $k^{*}\in \mathsf{\Omega }$

or

$\left(b\right)$ if an orthogonal sequence $\left(k_n\right)\subseteq \mathsf{\Omega }$ converges to $k\in \mathsf{\Omega }$, then $k\perp k_n$ or $k_n\perp k$ for all n;

in this case, $k^{*}\in \mathsf{\Omega }$ is the unique fixed point of h.

$\left(iii\right)$$p(k^{*},k^{*})=0$.

Proof. 

In Theorem 4, it is sufficient to choose $\varphi (a,b)=\psi \left(\frac{1}{2}(a+b)\right).$ □

Corollary 2.

Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal complete metric space with a w-distance p, $h:\mathsf{\Omega }\to \mathsf{\Omega }$ be a self-map, $\psi :[0,\infty )\to [0,\infty )$ be a continuous function, and ${\psi }^{-1}\left(\left\{0\right\}\right)=0$. Suppose that h is orthogonal preserving self-mapping satisfying inequalities

$$p(h\left(k\right),h\left(\right)l))\le \frac{1}{2}(p(h\left(k\right),k)+p(l,h\left(l\right)))-\psi \left(\frac{1}{2}(p(h\left(k\right),k)+p(l,h\left(l\right)))\right)$$

(47)

for all orthogonally related $k,l\in \mathsf{\Omega }$. Then,

$\left(i\right)$ there exists a point $k^{*}\in \mathsf{\Omega }$, such that, for any orthogonal element $k_0\in \mathsf{\Omega }$, iteration sequence $\left(h^n\left(k_0\right)\right)$ converges to this point.

$\left(ii\right)$ either

$\left(a\right)$ h is orthogonal continuous at $k^{*}\in \mathsf{\Omega }$

or

$\left(b\right)$ if an orthogonal sequence $\left(k_n\right)\subseteq \mathsf{\Omega }$ converges to $k\in \mathsf{\Omega }$, then $k\perp k_n$ or $k_n\perp k$ for all n, and $k,hk$ are orthogonally related elements.

In this case, $k^{*}\in \mathsf{\Omega }$ is the unique fixed point of h.

$\left(iii\right)$$p(k^{*},k^{*})=0$.

Proof. 

In Theorem 5, it is sufficient to choose $\varphi (a,b)=\psi \left(\frac{1}{2}(a+b)\right).$ □

In the following, another consequence of Theorem 4 corresponds to the contraction, which we can call the orthogonal p-Kannan or orthogonal weak Kannan-type contraction.

Corollary 3.

Let $(\mathsf{\Omega },\perp ,d)$ be an orthogonal complete metric space with a w-distance p. Suppose that $h:\mathsf{\Omega }\to \mathsf{\Omega }$ is orthogonal preserving self-mapping, there exists $\theta \in [0,\frac{1}{2})$, such that

$$p\left(h\right(k),h(l\left)\right))\le \theta (p\left(h\right(k),k)+p\left(h\right(l),l))$$

(48)

for all orthogonally related $k,l\in \mathsf{\Omega }$. Then,

$\left(i\right)$ there exists a point $k^{*}\in \mathsf{\Omega }$ such that, for any orthogonal element $k_0\in X$, iteration sequence $\left(h^n\left(k_0\right)\right)$ converges to this point.

$\left(ii\right)$ either

$\left(a\right)$ h is orthogonal continuous at $k^{*}\in \mathsf{\Omega }$

or

$\left(b\right)$ if an orthogonal sequence $\left(k_n\right)\subseteq \mathsf{\Omega }$ converges to $k\in \mathsf{\Omega }$, then $k\perp k_n$ or $k_n\perp k$ for all n.

In this case, $k^{*}\in \mathsf{\Omega }$ is the unique fixed point of h.

$\left(iii\right)$$p(k^{*},k^{*})=0$.

Proof. 

In Theorem 4, it is sufficient to choose $\varphi (a,b)=\frac{1-\theta }{2}(a+b).$ □

Taking $p=d$ in Corollary 3, orthogonal Kannan fixed-point theorem is obtained, which is a generalization of Theorem 1 on orthogonal metric spaces.

3. Application on Nonlinear Fredholm Integral Equations

In this part, we employ our main result to show the presence and inimitableness of a solution for the following nonlinear Fredholm integral equation:

$$k\left(t\right)=\psi \left(t\right)+{\int }_0^{1}F(t,s,k\left(s\right))ds$$

(49)

where $k\in C\left(\right[0,1],\mathbb{R})$ (the set of all continuous functions from $[0,1]$ into $\mathbb{R}$), such that $\psi :[0,1]\to [0,\infty )$ and $F:{[0,1]}^2\times [0,\infty )\to [0,\infty )$ are given continuous mappings.

Theorem 6.

Suppose that there exists $\alpha \in (0,\frac{1}{2})$, such that

$$|F(t,s,k\left(s\right))\mid +\mid F(t,s,l\left(s\right))\mid \le \alpha \left(\underset{s\in {[0,1]}^{}}{sup}\mid k\left(s\right)\mid +\underset{s\in [0,1]}{\overset{}{sup}}\mid l\left(s\right)\mid \right)-2\mid \psi \left(t\right)|$$

(50)

for all $k,l\in C\left(\right[0,1],\mathbb{R})$ and for all $t,s\in [0,1]$. Then, nonlinear integral Equation (49) has a unique solution.

Proof. 

Now, we take into account the following orthogonality relation on $\mathsf{\Omega }=C\left(\right[0,1],\mathbb{R})$:

$$k\perp l\iff k\left(t\right)l\left(t\right)\ge 0,\mathrm{for}\mathrm{all}t\in [0,1].$$

(51)

Obviously, $(\mathsf{\Omega },\perp )$ is an orthogonal set since, for every $k\in \mathsf{\Omega }$, there exist $l\left(t\right)=0,$ for all $t\in [0,1]$ such that $k\left(t\right)l\left(t\right)=0.$ We consider

$$d(k,l)=\underset{t\in [0,1]}{\overset{}{sup}}\mid k\left(t\right)-l\left(t\right)\mid$$

(52)

for all $k,l\in \mathsf{\Omega }$. So $(\mathsf{\Omega },\perp ,d)$ is an orthogonal complete metric space. We define a mapping $H:\mathsf{\Omega }\to \mathsf{\Omega }$ by

$$\left(Hk\right)\left(t\right)=\psi \left(t\right)+{\int }_0^{1}F(t,s,k\left(s\right))ds$$

(53)

for all $k\in \mathsf{\Omega }$ and $t\in [0,1]$. Consider on $\mathsf{\Omega }$ the w-distance $p:\mathsf{\Omega }\times \mathsf{\Omega }\to [0,\infty )$ given by

$$p(k,l)=\underset{t\in [0,1]}{\overset{}{sup}}\mid k\left(t\right)\mid +\underset{t\in [0,1]}{\overset{}{sup}}\mid l\left(t\right)\mid$$

(54)

for all $k,l\in \mathsf{\Omega }.$ To demonstrate the presence of a fixed point of H, we show that H is orthogonal preserving self-mapping that satisfies Inequality (9) by choosing $\varphi (t,s)=\left(\frac{1}{2}-\alpha \right)(t+s)$ for all $t,s\ge 0,k\in (0,\frac{1}{2}).$

First, we show that H is orthogonal preserving. Let $k\perp l$ for $k,l\in \mathsf{\Omega }$. Then, $k\left(t\right)l\left(t\right)\ge 0$ for all $t\in [0,1].$ Now, we have

$$\left(Hk\right)\left(t\right)=\psi \left(t\right)+{\int }_0^{1}F(t,s,k\left(s\right))ds\ge 0$$

(55)

and

$$\left(Hl\right)\left(t\right)=\psi \left(t\right)+{\int }_0^{1}F(t,s,l\left(s\right))ds\ge 0,$$

(56)

which implies that $H\left(k\right)\perp H\left(l\right)$, i.e., H is orthogonal preserving.

We now show that H satisfies Iinequality (9) by choosing $\varphi (t,s)=\left(\frac{1}{2}-\alpha \right)(t+s)$ for all $t,s\ge 0,k\in (0,\frac{1}{2}).$ Consider $k,l\in \mathsf{\Omega }$ and $t\in [0,1].$ We have

$$\begin{array}{cc}\hfill \mid \left(Hk\right)\left(t\right)\mid +\mid \left(Hl\right)\left(t\right)\mid & =\mid \psi \left(t\right)+{\int }_0^{1}F(t,s,k\left(s\right))ds\mid +\mid \psi \left(t\right)+{\int }_0^{1}F(t,s,l\left(s\right))ds\mid \hfill \\ \hfill & \le \mid \psi \left(t\right)\mid +\mid {\int }_0^{1}F(t,s,k\left(s\right))ds\mid +\mid \psi \left(t\right)\mid +\mid {\int }_0^{1}F(t,s,l\left(s\right))ds\mid \hfill \\ \hfill & \le 2\mid \psi \left(t\right)\mid +{\int }_0^1\mid F(t,s,k\left(s\right))\mid ds+{\int }_0^1\mid F(t,s,l\left(s\right))\mid ds\hfill \\ \hfill & =2\mid \psi \left(t\right)\mid +{\int }_0^1\left(\mid F(t,s,k(s\left)\right)\mid +\mid F(t,s,l(s\left)\right)\mid \right)ds\hfill \\ \hfill & \le 2\mid \psi \left(t\right)\mid +{\int }_0^1[\alpha \left(\underset{s\in [0,1]}{\overset{}{sup}}\mid k\left(s\right)\mid +\underset{s\in [0,1]}{\overset{}{sup}}\mid l\left(s\right)\mid \right)\hfill \\ \hfill & -2\mid \psi \left(t\right)\mid ]ds\hfill \\ \hfill & \le 2\mid \psi \left(t\right)\mid +{\int }_0^1\left[\alpha \right(\underset{s\in [0,1]}{\overset{}{sup}}\mid \left(Hk\right)\left(s\right)\mid +\underset{s\in [0,1]}{\overset{}{sup}}\mid k\left(s\right)\mid \hfill \\ \hfill & +\underset{s\in [0,1]}{\overset{}{sup}}\mid \left(Hl\right)\left(s\right)\mid +\underset{s\in [0,1]}{\overset{}{sup}}\mid l\left(s\right)\mid )-2\mid \psi \left(t\right)\mid ]ds\hfill \\ \hfill & =2\mid \psi \left(t\right)\mid +{\int }_0^1\left[\alpha \left(p\left(H\right(k),k)+p\left(H\right(l),l)\right)-2\mid \psi \left(t\right)\mid \right]ds\hfill \\ \hfill & =\alpha \left(p\left(H\right(k),k)+p\left(H\right(l),l)\right)\hfill \\ \hfill & =\frac{1}{2}\left(p\left(H\right(k),k)+p\left(H\right(l),l)\right)-\varphi \left(p\left(H\right(k),k),p\left(H\right(l),l)\right).\hfill \end{array}$$

(57)

This indicates that

$$\underset{t\in [0,1]}{\overset{}{sup}}\mid \left(Hk\right)\left(t\right)\mid +\underset{t\in [0,1]}{\overset{}{sup}}\mid \left(Hl\right)\left(t\right)\mid \le \frac{1}{2}\left(p\left(H\right(k),k)+p\left(H\right(l),l)\right)-\varphi \left(p\left(H\right(k),k),p\left(H\right(l),l)\right),$$

(58)

and so

$$p(H\left(k\right),H\left(l\right))\le \frac{1}{2}\left(p\left(H\right(k),k)+p\left(H\right(l),l)\right)-\varphi \left(p\left(H\right(k),k),p\left(H\right(l),l)\right)$$

(59)

for all $k,l\in \mathsf{\Omega }$. Thus, all hypotheses of Theorem 4 are satisfied; therefore, H has a unique fixed point. This indicates that there exists a unique solution of nonlinear Fredholm integral Equation (49). □

4. Conclusions

Examining the presence and inimitableness of fixed points of self-mappings defined on orthogonal metric spaces has become very popular in the last decade. As a result, many researchers reached more conclusions on this matter. However, the existence of fixed points for self-mapping satisfying Kannan-type contraction conditions in orthogonal metric spaces had never been investigated until now. Therefore, studies in this direction are necessary.

In this paper, the notion of $\varphi$-Kannan orthogonal p-contractive conditions in orthogonal complete metric spaces was presented. On this basis, some fixed-point results were debated. Some explanatory examples were delivered, which indicated the currency of the suppositions and grade of benefit of the suggested conclusions. Lastly, sufficient cases for the presence of a solution to nonlinear Fredholm integral equations were investigated.

In the future, researchers could examine the presence and inimitableness of the fixed points of generalizations of the contraction principle in orthogonal metric spaces equipped with w-distance while taking into account the work in this study by means of auxiliary functions.