Incomplete Metric With The Application For Integral Equations of Volterra Type

Chandok, Sumit; Paunović, Ljiljana; Radenović, Stojan

doi:https://doi.org/10.1155/2022/5959903

Mathematical Problems in Engineering

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Function Space Optimization, Iterative Algorithms, and their Applications

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Research Article | Open Access

Volume 2022 | Article ID 5959903 | https://doi.org/10.1155/2022/5959903

Incomplete Metric With The Application For Integral Equations of Volterra Type

Sumit Chandok,¹Ljiljana Paunović,²and Stojan Radenović³

Academic Editor: Ahmed Zeeshan

Received21 Jan 2022

Revised05 Mar 2022

Accepted02 Apr 2022

Published27 May 2022

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. [1] 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 [2–4], numerous studies showed the existence of fixed points (see [5–18]). 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. [1]. 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 [1]). 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 [1].

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 [1]).

Definition 1.6 (see [19]). 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 [19]).

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 [20])

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 [21] 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 [22].

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. [23]).

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 . From (44), we havefor all . So using Theorem 2.5, we get the result.

Remark 3.3. On the same lines of Theorem 3.2, we can prove other coupled fixed point results.

We get the following result by taking in Theorems 2.5 and 2.6.

Corollary 3.4. Letbe a self mapping on a SO-complete OMS. Assume the following assumptions hold:(i)there exists some, such that for allwith,,orwhere ,(ii)is-preserving,(iii)Property.Then has a unique fixed point.

The following outcome is a direct result of Corollary 3.4.

Corollary 3.5. Let be a self mapping on a SO-complete OMS . Assume the following assumptions hold:(i)is-preserving,(ii)Property,(iii)there exists some,such that for allwith,,any of the following contracting conditions are true:

In each of these circumstances, has a unique fixed point.

Proof. The proof follows directly from Corollary 3.4, as each functions , , , and , where is strictly increasing on .

For in Theorem 2.7, we have the following result.

Corollary 3.6. Let be a self mapping on a SO-complete OMS . Assume the following assumptions hold:(i)there exists some, such that for allwith,,whereand,(ii)is-preserving,(iii)Property.Then has a unique fixed point.

Corollary 3.7. Let be a self mapping on a SO-complete OMS . Assume the following assumptions hold:(i)is-preserving,(ii)Property,(iv)there exists some,such that for allwith,, the following contractive conditions holdwhere .

Then in each of these cases has a unique fixed point.

Proof. The proof immediately follows from Corollary 3.6, as each functions , , and , where is strictly increasing on .

Taking , as a result of Corollary 3.5, we get the following result.

Corollary 3.8. Let be a self mapping on a SO-complete OMS . Assume the following assumptions hold:(i)there exists some, such that for allwith,,where ,(ii)is-preserving,(iii)Property.

Then the mapping has a unique fixed point.

Corollary 3.9. Letbe a self mapping on a complete metric space. Assume that there exists some, such that for all,satisfies,where . Then the mapping has a unique fixed point.

Proof. Define a binary relation on by if and only if
Since satisfies (52), we have , for any fixed and for all . Thus is an O-set and it is easy to see the O-completeness of . Furthermore, is -continuous, -preserving and satisfies (46). Hence using Corollary 3.4, we get the result.

Example 3.10. Let with usual metric . Define the binary relation on by if where . Then is O-complete OMS. Define the mapping byLet . We may assume that , without loss of generality. Then the following cases are satisfied:

Case I. If and , then and .

Case II. If and , then .

Case III. If and then and .

Case IV. If and then , and .
From all these cases, we obtain for all with .
It is easy to see that is -preserving and -continuous. Also 0 is a fixed point of the mapping .

Remark 3.11. On the lines of Corollary 3.9, we can easily say that our results extend the corresponding results of [2–4, 12, 15, 18]. Our results are more general than the results of many researchers (see [2–4, 12, 14, 18] and references cited therein) as we use only strictly increasing condition of Wardowski’s function. Our theorems are, therefore, legitimate generalisations of Wardowski’s fixed point theorem.

4. Applications

The application of the acquired results is demonstrated in this section.

4.1. Solution of Volterra type Integral equation

Here, we show how to apply the existence of a fixed point for -contractions can be applied to the following Volterra type equation:where , , , .

The following assumptions must be made in order to obtain our claims:

(A1) , are SO-continuous functions.

(A2) there is a strictly increasing SO-sequence satisfying , , , such that for all and such that with defined by or and , we have

Let be a complete normed linear space, which contains all continuous functions that have Bielecki’s norm: .

We’re now in a position to state our initial conclusion on existence.

Theorem 4.1. If (A1) and (A2) hold, the nonlinear integral problem (4.1) has a unique solution in.

Proof. Define the operator asA solution of the (54) will be a fixed point of the operator .
Define the orthogonality relation on by or for all , , in order to satisfy all of the requirements of Theorem 2.5.
Consider of the formHere is -preserving. For each with and , we haveIt follows that , so .
Next we claim that is orthogonal -contraction. Take a function , . Fix and take any with such that . Take note that for each , we getTherefore, we getNext, we see that , and since , for all . Hence, we haveUsing the properties of sequence , we getBy considering the supremum with respect to in the aforementioned inequality, we get orthogonal -contraction. For , the calculations are the same. The proof comes to a finish with the Theorem 2.5.

4.2. Hyers-Ulam-Rassias-Wright Stability

In fixed point theory, generalization of Ulam stability [16,24] has piqued the interest of various scholars (see [25–27]). In this section, we will look at the Hyers-Ulam-Rassias-Wright stability of the integral (54).

The following series representation defines the Wright function (see [28]):for . It is an entire function of order .

If (54) meets the following criteria, it is called Hyers-Ulam-Rassias-Wright stable:

for each and for every solution , there is a constant satisfying

there exists some satisfying and

such that

Theorem 4.2. The fixed point problem, where, is Hyers-Ulam-Rassias-Wright stable, under the hypothesis of Theorem 4.1.

Proof. On the account of Theorem 4.1, we guarantee a unique such that , that is, forms a solution of . Let and be an -solution, that is,Using Theorem 4.1, we haveTherefore, , where . As a result, the (54) is Hyers-Ulam-Rassias-Wright stable.

4.3. Differential equations

We’ll now show that the differential equation below has a solution:where is evaluated at each , is SO-continuous, has a positive solution in , where is a subset of the Banach space of continuous functions , , with the supremum norm .

Define the orthogonality relation on by if and only if or for all .

The (69) can be simplified in the following formFurther, we obtain

To satisfy the hypotheses of Theorem 2.5, we demonstrate that the operator is an orthogonal -contraction on for and , .

For every with or and , we haveHere, we found , which, when combined with the fact that a function , is decreasing provides the followingFurther, using the increasing function , , we getNow, if , we get operator is an orthogonal -contraction for and , .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Acknowledgments

For the research grant, the first author is grateful to NBHM, DAE (grant 02011/11/2020/ NBHM (RP)/R&D-II/7830).

References

M. E. Gordji, M. Rameani, M. Rameani, M. De La Sen, and Y. J. Cho, “On orthogonal sets and Banach fixed point theorem,” Fixed Point Theory, vol. 18, no. 2, pp. 569–578, 2017.
View at: Publisher Site | Google Scholar
D. Wardowski, “Fixed points of a new type of contractive mappings in complete metric spaces,” Fixed Point Theory and Applications, vol. 2012, no. 1, 2012.
View at: Publisher Site | Google Scholar
D. Wardowski and N. V. Dung, “Fixed points of F-weak contractions on complete metric spaces,” Demonstratio Mathematica, vol. 47, no. 1, pp. 146–155, 2014.
View at: Publisher Site | Google Scholar
D. Wardowski, “Solving existence problems via $F$-contractions,” Proceedings of the American Mathematical Society, vol. 146, no. 4, pp. 1585–1598, 2017.
View at: Publisher Site | Google Scholar
I. Altun, G. Minak, and H. Dag, “Multivalued F-contractions on complete metric space,” J. Nonlinear Convex Anal., vol. 16, no. 4, pp. 659–666, 2015.
View at: Google Scholar
S. Chandok, R. K. Sharma, and S. Radenović, “Multivalued problems via orthogonal contraction mappings with application to fractional differential equation,” Journal of Fixed Point Theory and Applications, vol. 23, no. 2, p. 14, 2021.
View at: Publisher Site | Google Scholar
S. Chandok and S. Radenović, “Existence of solution for orthogonal $${\mathcal {F}}$$-contraction mappings via Picard-Jungck sequences,” The Journal of Analysis, 2021.
View at: Publisher Site | Google Scholar
M. Cosentino and P. Vetroa, “Fixed point results for F-contractive mappings of Hardy-Rogers-type,” Filomat, vol. 28, no. 4, pp. 715–722, 2014.
View at: Publisher Site | Google Scholar
G. Durmaz, G. Minak, and I. Altun, “Fixed points of ordered F-contractions,” Hacettepe J. Math. Stat., vol. 45, no. 1, pp. 15–21, 2016.
View at: Publisher Site | Google Scholar
N. Fabiano, V. Parvaneh, D. Mirković, L. Paunović, and S. Radenović, “On W-contractions of Jungck-Ćirić-Wardowski-type in metric spaces,” Cogent Mathematics & Statistics, vol. 7, no. 1, Article ID 1792699, 2020.
View at: Publisher Site | Google Scholar
G. Minak, A. Helvaci, and I. Altun, “Ćirić type generalized F-contractions on complete metric spaces and fixed point results,” Filomat, vol. 28, no. 6, pp. 1143–1151, 2014.
View at: Google Scholar
H. Piri and P. Kumam, “Some fixed point theorems concerning F-contraction in complete metric spaces,” Fixed Point Theory and Applications, vol. 2014, no. 1, 2014.
View at: Publisher Site | Google Scholar
O. Popescu and G. Stan, “Two fixed point theorems concerning F-contraction in complete metric spaces,” Symmetry, vol. 12, no. 1, p. 58, 2020.
View at: Publisher Site | Google Scholar
K. Sawangsup, W. Sintunavarat, and Y. J. Cho, “Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces,” Journal of Fixed Point Theory and Applications, vol. 22, no. 1, p. 10, 2020.
View at: Publisher Site | Google Scholar
M. Turinici, “Wardowski Implicit Contractions in Metric Spaces,” 2013, https://arxiv.org/abs/1211.3164.
View at: Google Scholar
S. S. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964.
J. Vujaković, S. Mitrović, M. Pavlović, and S. Radenović, “On recent results concerning F-contraction in generalized metric spaces,” Mathematics, vol. 8, no. 5, p. 767, 2020.
View at: Publisher Site | Google Scholar
J. Vujaković and S. Radenović, “On some F-contraction of Piri-Kumam-Dung-type mappings in metric spaces,” Vojnotehnicki glasnik, vol. 68, no. 4, pp. 697–714, 2020.
View at: Publisher Site | Google Scholar
M. Ramezani and H. Baghani, “Contractive gauge functions in strongly orthogonal metric spaces,” International Journal of Nonlinear Analysis and Applications, vol. 8, pp. 23–28, 2017.
View at: Publisher Site | Google Scholar
W. Rudin, Principles of Mathematical Analysis, McGraw-Hill Book Company, New York-San Francisco-Toronto-London, 1964.
S. Aljančić, Uvod U Realnu I Funkcionalnu Analizu, Naucna knjiga, Beograd, 1969.
S. Radenovic and S. Chandok, “Simulation type functions and coincidence points,” Filomat, vol. 32, no. 1, pp. 141–147, 2018.
View at: Publisher Site | Google Scholar
B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for -contractive type mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 4, pp. 2154–2165, 2012.
View at: Publisher Site | Google Scholar
D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences, vol. 27, no. 4, pp. 222–224, 1941.
View at: Publisher Site | Google Scholar
Z. Eidinejad and R. Saadati, “Hyers-Ulam-Rassias-Wright stability for fractional oscillation equation,” Discrete Dynamics in Nature and Society, vol. 2022, Article ID 9412009, 2022.
View at: Publisher Site | Google Scholar
E.-S. El-Hady and S. Ogrekci, “On Hyers-Ulam-Rassias stability of fractional differential equations with Caputo derivative,” Journal of Mathematical and Computational Science, vol. 2021, Article ID 9817668, pp. 325–332, 2021.
View at: Google Scholar
R. K. Sharma and S. Chandok, “Multivalued problems, orthogonal mappings, and fractional integro-differential equation,” Journal of Mathematics, vol. 2020, Article ID 6615478, pp. 1–8, 2020.
View at: Publisher Site | Google Scholar
E. M. Wright, “The generalized Bessel function of order greater than one,” The Quarterly Journal of Mathematics, vol. 11, no. 1, pp. 36–48, 1940.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Sumit Chandok et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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