#### Abstract

In this article, we propose some weaker orthogonal type of contraction mappings in the setting of metric spaces endowed with an orthogonal relation, as well as certain sufficient criteria for the existence of fixed points for this class of mappings. To establish all of our results in the manuscript, we just used Wardowski’s strictly increasing property. Using the aforementioned results as an application, we demonstrate that the Volterra type integral equation has a solution and is stable in the Hyers-Ulam-Rassias-Wright sense.

#### 1. Introduction

Gordji et al.  recently introduced the idea of orthogonal sets and extended the Banach contraction principle. They also demonstrated how their findings can be used to ensure the existence and uniqueness of solutions to first-order differential equations. Using Wardowski’s -contraction notion , numerous studies showed the existence of fixed points (see ). The purpose of this study is to improve the concept of Wardowski’s contraction in metric spaces that are not complete in the sense of Gordji et al. . In the context of a metric space equipped with an orthogonal relation, we describe some weaker orthogonal type contraction mappings. For this class of mappings, we also employ strictly increasing properties to construct some necessary requirements for the presence of fixed points. This, we feel, is a significant improvement over many previously published findings. We also enrich this paper with a nontrivial example. We use the aforementioned results to show that the Volterra type integral equation has a solution and is stable in the Hyers-Ulam-Rassias-Wright sense.

Definition 1.1 (see ). Let be a nonempty set and be a binary relation defined on . is called an orthogonal set (O-set), if satisfies the following condition:
there exists such that (for all , ) or (for all , ).

Example 1.2. Let . Define if and only if . It’s clear that applies to all and is an O-set.

More fascinating examples may be found at .

Definition 1.3 (see [1, 19]). A sequence is referred to as a strongly orthogonal sequence (SO-sequence) if (for all ; ) or (for all ; ).
If (for all ) or (for all ), the sequence is referred to as an orthogonal sequence (O-sequence).

Definition 1.4 (see [1, 19]). Let represent an O-set, a metric space, and an orthogonal metric space (OMS). A mapping is said to be strongly orthogonally continuous (SO-continuous) at , if we have , for each SO-sequence in with . In addition, is called SO-continuous on if is SO-continuous for each .
A mapping is called orthogonally continuous (-continuous) at if we have , for each O-sequence in with . A mapping is also called -continuous on if is -continuous for each .

Every continuous mapping is clearly -continuous, while the opposite is not true (see [1,19]).

Definition 1.5 (see [1, 19]). A self mapping on an orthogonal set is called -preserving if , then . Also, is called a weakly -preserving if , then or .

Every -preserving mapping is clearly weakly -preserving, but not the other way around (see ).

Definition 1.6 (see ). If every Cauchy SO-sequence is convergent, then an OMS is called strongly orthogonally complete (SO-complete).

Every complete metric space is clearly SO-complete, while the converse is not true (see ).

#### 2. Fixed Point Results

Here, first we introduce some weaker orthogonal -type contraction mappings in the context of an incomplete metric, as defined in [1,19], and then construct fixed points of mappings meeting such a class of contractions.

For the purpose of simplicity, we’ll suppose that an expression - has the value .

We refer to as the family of all functions that satisfy:

For every , implies .

So there are both and for all because it is known from mathematical analysis that the following is true for all (see )

Remark 2.1. Consider to be a strictly increasing function. Then there are following two possible outcomes:(1)(2), for some (for more details see [18,20,21]).As a result, each strictly increasing function fulfils one of the two conditions either (1) or (2) (see (Aljančić  Proposition 1, Section 8)). Wardowski’s second and third requirements are thus unnecessary.

Example 2.2. Let functions defined by:(1),(2),(3),Then , for all .

To begin, we’ll require the following result, which is an orthogonal metric space extension of .

Lemma 2.3. Letbe a SO-sequence in an OMSsuch thatIf a SO-sequenceis not a Cauchy SO-sequence in, then there isand two sequences of positive integersandsuch thatand the following SO-sequences tend towhen:

Proof. If is not a Cauchy SO-sequence, then there exist and two sequences of positive integers and such thatfor all positive integers . To prove (4), assume thatObviously, and . Then by the well-ordering principle, the minimum element of exists and denoted by , and clearly (4) holds. ThenUsing (2), we conclude thatFurther,as well asTaking the limit and using (2) and (7), we getAlso,Now, from (11) and (12) it follows thatOn the same lines, we can prove that the remaining two sequences in (3) tend to .

In this paper the function will be strict increasing and .

The following assumption is required in our results. Let be a self mapping on a SO-complete OMS .

Property. there is a SO-sequence defined by , for an orthogonal element with or , such that and or for all , then or for all .

Definition 2.4. Let be a self mapping on an OMS if there exist functions and , such that for all with satisfying the following hypotheses:
(c1) ,
(c2) , for all ,
either
(c3) .
or
(c4) implies
or
(c5) , where

(respectively, ).
is called an orthogonal -contraction if it satisfies (c1)-(c3).
is called an orthogonal -Suzuki type contraction if it satisfies (c1), (c2), (c4).
is called a generalized orthogonal -contraction (respectively, weak orthogonal -contraction) if it satisfies (c1), (c2), (c5).

We are now ready to provide our first outcome.

Theorem 2.5. Let be a -preserving, an orthogonal -contraction and satisfy Property on a SO-complete OMS (not necessarily a complete metric space). Then has a unique fixed point.

Proof. Let be such that or . Take , . In this manner, we define a SO-sequence in by for all .
If for some , then . So, we take , and , for all . As is an orthogonal -contraction, for every , we obtainTherefore, , for every .
From (c2), and exist, with for all . As a result, for all we haveAs , sequence is strictly decreasing and converging to some , for all .
Taking in (15), we getwhich is a contradiction and hence .
Now we must demonstrate that is a Cauchy SO-sequence. Assume, on the other hand, that is not a Cauchy SO-sequence. Putting , in (c3), we haveIt implies thatSince the SO-sequence is not a Cauchy SO-sequence, by Lemma 2.3, we have and tend to , as .
Now, using (18) and (c2), there exist and such that we getwhenever . That is,for . Taking , in the last relation, we getwhich contradicts to our assumption. This establishes that the SO-sequence is a Cauchy SO-sequence. Since is SO-complete, there exists such that .
By using our other assumption, or . Using (c3), we getUsing (c2) and , we obtain . Taking , we get , that is is a fixed point of .
Now we’ll show that is the only unique fixed point of . Suppose that is another fixed point of with . By our choice of , or . Since is -preserving, we have and or and . Therefore,which is a contradiction as . Hence and has a unique fixed point.
Now, we give the following result on -Suzuki type contraction which is related to the generalization and the improvement of Theorem 2.5.

Theorem 2.6. Letbe a-preserving, an orthogonal-Suzuki type contraction and satisfy Propertyon a SO-complete OMS. Thenhas a unique fixed point.

Proof. On the similar lines of Theorem 2.5, we may assume that , for all .
Since is an orthogonal -Suzuki type contraction, for every , we have . So from (c4), we getTherefore, we have , for all . Now, using the similar comments in Theorem 2.5, we get .
Now we must demonstrate that is a Cauchy SO-sequence. Assume, on the other hand, that Â is not a Cauchy SO-sequence.
So by Lemma 2.3, we have and tend to , as .
Therefore, it follows that there is some such that , for all with . Then, by substituting , in (c4) for , we haveIt implies thatNow, using (c2) and (26), there exist and such that we getwhenever . That is,for . Taking in the obtained last relation, we getwhich contradicts to our assumption. This establishes that the SO-sequence is a Cauchy SO-sequence. Since is an SO-complete, there exists such that .
Now, we claim that for all ,Now, again we supposee that there is some such thatTherefore, . It impliesIt follows from (31) and (32) thatSince , . Using (c2), we get . Hence by using property of , we getIt follows from (31), (33) and (34),which is a contradiction. Hence (30) holds.
By our assumption, or . So from (30) and (c4), for every , either , or holds. Also, we can rewrite it asFurther, using (c2) and , we getTaking , we get in both the cases.
It is easy to see the uniqueness of a fixed point of .

Theorem 2.7. Letbe a-preserving, generalized orthogonal-contraction and satisfy Propertyon a SO-complete OMS. Thenhas a unique fixed point.

Proof. On the similar lines of Theorem 2.5, we may assume that , for all .
Since is a generalized orthogonal -contraction, for every , we getwhereIt is clear that , otherwise we get a contradiction.
Therefore, , for all . Now, using the similar comments in Theorem 2.5, we get .
Now we must demonstrate that is a Cauchy SO-sequence. Assume, on the other hand, that Â is not a Cauchy SO-sequence. So by Lemma 2.3, we have and tend to , as .
Putting , in (c5), we havewhereUsing Lemma 2.3, we have .
Now, taking , using (40) and (c2), there exist and such that we getwhich contradicts to our assumption. This establishes that the SO-sequence is a Cauchy SO-sequence. Since is-complete, there exists such that .
By using our assumption, or . Using (c5), we getwhere .
Now taking , using (c2) and , we get , that is is a fixed point of . It is easy to see that is a unique fixed point of .

Remark 2.8. Every weak orthogonal -contraction is a generalized orthogonal -contraction. So Theorem 2.7 is also true if we take weak orthogonal -contraction.

#### 3. Consequences of Fixed Point Results

In this section, we discuss some of the ramifications of the preceding section’s findings.

First, we’ll illustrate how our findings allow us to formulate coupled fixed point theorems in O-complete orthogonal metric spaces using our results. The following definition emerges first.

Let be a given mapping. We say that is a coupled fixed point of if and .

Our result is based on the following simple lemma which tells a coupled fixed point is a fixed point (see Samet et al. ).

Lemma 3.1. Letbe a given mapping. Define the mappingby, for all. Then,is a coupled fixed point ofif and only ifis a fixed point of.

Theorem 3.2. Let be a self mapping on a SO-complete OMS . Assume the following assumptions are true:(i)there exists, such that for allwith,, for all,,where ,(ii)is-preserving,(iii)If there exist SO-sequencesdefined by,for orthogonal elementswithor, such that,andor,or, for all, thenor,or, for all.Then has a coupled fixed point.

Proof. Here take is SO -complete OMS. Define the mapping by , for all