Abstract
Extending the Presic type operators to modular spaces, we introduce generalised Presic type -contractive mappings and strongly -contractive mappings in a modular metric space and establish fixed-point theorems for such contractions in modular spaces. Ulam–Hyers stability of the fixed-point equation involving Presic type operators is also discussed. Our results extend and generalise some known results in the literature. The results are supported by appropriate example and an application to Caratheodory type integral equation.
1. Introduction
Maurice Ren Fréchet [1] introduced the general and axiomatic form of distance as “—space.” Felix Hausdorff [2] reexamined it as a metric space in the setting of points which has been refined, discussed, and generalised in numerous ways. Bakhtin [3], Branciari [4], George et al. [5], and Mitrović and Radenović [6] introduced the notions of a -metric, a rectangular metric, a rectangular -metric, and -metric, respectively. The basic concepts and theory of modular space was formulated in [7]. Later, in [8], the authors established fixed-point theorems in a modular function space. In 2008, Chistyakov [9] introduced the notion of a modular metric and the corresponding modular space. Chistyakov, in [10], established the existence of fixed point for contractive maps and strongly contractive maps in modular metric spaces. Umit et al. [11] introduced Bogin type -contractions and proved fixed-point theorems for such contractions in a -complete modular metric space and provided application to antiperiodic boundary value problems. Furthermore, we see that Chaipunya et al. [12] introduced Geraghty type theorems and Turkoglu and Kilinc [13] introduced Caristi type theorems in a modular metric space and gave its applications in integral equations. For more results of fixed-point theorems in modular metric space and some applications of fixed-point theorems, readers may refer to [14–20]. On the other hand, Presic [21, 22] extended the Banach contraction principle to the product of a finite number of metric spaces and later Ciric and Presic [23] generalised the result of Presic [21] and proved the following.
Theorem 1. Let be a metric space, be a positive integer, and be a mapping satisfying the following condition:where are arbitrary elements in and . Then, there exists some such that . Moreover, if are arbitrary points in and for , then the sequence is convergent and if , then . If, in addition, satisfies , for all . Then, is the unique point satisfying .
Later, the above results were further extended and generalised to a -metric space, cone metric space, and cone -metric space by many authors, and fruitful applications were found for such results (see [24–29]). The aim of this work is to introduce generalised Presic type contractions (which includes Ciric–Presic type contraction and Presic type contractions) in modular metric spaces and establish fixed-point theorems for such contraction mappings in a -complete modular metric space. Our results show symmetric transformation of the well-known Ciric–Presic theorem [23] and Presic theorem [21] from ordinary metric space to a modular metric space. We have introduced Ulam–Hyers stability of fixed-point equations involving Presic type operators in a modular metric space. We have also provided an application of our result to prove the existence of solution of an integral equation of Caratheodory type functions. Our results extend and generalise many known results in the literature.
2. Preliminaries
The following basic concepts of modular spaces are from [9, 10]. Let and be a given function. For any , we write as .
Definition 1. (see [10]). We call a modular metric on if for all and , the following relations hold: (mm1) (mm2) (mm3) In this case, is said to be a metric modular on . Instead of (mm1), if satisfies onlyfor all and , then is said to be pseudomodular on .
Also, is strict modular on if instead of (mm1), it satisfies (2) and for , there exists a number (possibly depending on and ), such thatClearly, if is strict modular, then it is modular, which in turn implies is pseudomodular on .
A metric modular on is convex if it satisfies the (stronger) inequality, in lieu of (mm3):for all .
The main property of a modular on a set is that (see Section 2.3 of [9]), given , the function is nonincreasing on . If , then (mm1) and (mm3) imply
Definition 2. (see [10]). Given a pseudomodular on , the two setsare said to be modular spaces (around ).
Clearly, . Note that if is metric modular on , then modular space can be equipped with a metric generated by given byfor any . Moreover, if is a convex modular on , then by Section 3.5 and Theorem 3.6 of [9], ; i.e., the two modular spaces coincide, and this common set can be endowed with a metric :
Henceforth, and represent modular metric spaces induced by .
Definition 3. (see [10]). Let be a metric modular on X. We have the following:(i)A sequence in is -convergent (or modular convergent) to some , if there exists , such that (ii)A sequence in is -Cauchy if , possibly dependent on , such that as (iii) is -complete if every -Cauchy sequence is -convergent
From [10], we have that and are closed with respect to convergence, if is pseudomodular on . Also, if is strict, then the modular limit, if exists, is unique. If is (not necessarily convex) modular on , then is always convex modular on .
3. Main Results
In this section, we introduce Presic type operators in modular metric space and prove existence of unique fixed points for such operators.
Let be the set of real numbers, and consider a function such that(a) is increasing in all variables(b), for all
Definition 4. Let be metric modular on and (where ) for some positive integer . Then, we have the following:(i) is a generalised Presic type modular contraction (or a generalised Presic w-contraction), if for some and possibly depending on and for all and .(ii) is a generalised Presic type strongly modular contraction (or a generalised Presic strong w-contraction), if for some and possibly depending on and for all and .(iii) is a Ciric–Presic type modular contraction (or a Ciric–Presic w-contraction), if for some and possibly depending on and for all and .(iv) is a Ciric–Presic type strongly modular contraction (or a Ciric–Presic strong w-contraction), if for some and possibly depending on and for all and .(v) is a Presic type modular contraction (or a Presic w-contraction), if there exists , , such thatfor some and possibly depending on and for all and .
Theorem 2. Let be a strict metric modular on and be -complete. For any positive integer , let be a generalised Presic type strongly w-contractive mapping. If there exists in , such that , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.
Proof. For arbitrary elements in with , define the sequence in given by , . By the -contractivity of , there exist two numbers and such that condition (10) holds. Let , where . Then, for any positive integer . Using the method of induction, we will prove that for any nonnegative integer ,where and . Clearly, by the definition of , (15) is true for and . Let the inequalities hold true. Then,Thus, inductive proof of (15) is complete. Set ; then, . Let integers and be such that . We set . Then, we haveTaking into account that for all , we getTherefore, is -Cauchy in . By -completeness of , there exists in such that , and by strictness of , the limit is unique. Let be arbitrary. Then, we can find a natural number such that and for all . Then, for any integer , we haveAs , we get . By the strictness of , . For uniqueness, suppose there exists , where , such that . Then, using (14), we geta contradiction (as ). Thus, the uniqueness of is established.
Now, if , thena contradiction as . Hence, is unique in .
In the next result, using a convex modular metric, we prove the existence of a fixed point for a generalised Presic type modular contractive mapping.
Theorem 3. Let be a strict convex modular on and be -complete. For any positive integer , let be a generalised Presic w-contractive mapping. If there exists in , such that , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.
Proof. We take for all and . Then, for all and , we haveSo, is strict modular on . Also, by (9), for all and , we haveThus, is a generalised Presic strong w-contraction mapping for the modular . Clearly, implies . By Theorem 2, there exists such that . The remaining part of the proof follows the same line as in Theorem 2.
Taking in Theorems 1 and 2, we get the following.
Theorem 4. Let be strict metric modular on and be -complete. For any positive integer , let be a Ciric–Presic strongly w-contractive mapping. If there exists in , such that , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.
Theorem 5. Let be strict convex modular on and be -complete. For any positive integer , let be a Ciric–Presic w-contractive mapping. If there exists in , such that , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.
Taking , with in Theorem 3, we get the following.
Theorem 6. Let be strict convex modular on and be -complete. For any positive integer , let be a Presic type w-contractive mapping. If there exists in , with , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.
Remark 1. For , Theorems 4 and 5 reduce to Theorem 10 and Theorem 5.4 of [10].
Remark 2. Theorems 4 and 5 extend the results of Ciric and Presic [23] to a modular space.
Remark 3. Theorem 6 extends the result of Presic [21] to a modular space.
Example 1. Let . Define the mapping by . Note that for all , then and is a complete modular metric space. Let be given by . Let given by . Then,If , then for all . Thus, is a generalised Presic type strongly modular contraction and Ciric–Presic type strongly modular contraction with constant . Theorems 2 and 4 are applicable, and is the unique fixed point of .
Example 2. Let . Define the mapping by . Note that for all ; then, and is a complete convex modular metric space. Let be given by , if for some and .
Let be given by . Then,If , then for all . Thus, is a generalised Presic type modular contraction, with constant . Thus, satisfies the condition of Theorem 3 and is the unique fixed point of .
4. Ulam–Hyers Stability
In this section, we discuss the Ulam–Hyers stability of fixed-point equations involving Presic type operators in a modular metric space. We begin with the following concepts and definitions.
Definition 5. Let be a metric modular on and (where ) for some positive integer . If for all with , the sequence given by converges and the limit (which depends upon ) is a fixed point of , then we say that is a weakly Picard–Presic operator (in short, ). If the fixed point of is unique, then we say that is a Picard–Presic operator.
Hereafter, we will make use of the following notations:
Definition 6. Let be metric modular on , be a weakly Picard–Presic operator, and be a real number. We say that is a -weakly Picard–Presic operator (in short, ) if for all , , and for some and possibly depending on and for all ,
Let be the set of nonnegative real numbers, and consider a function such that is continuous at 0 and .
Definition 7. Let be a metric modular on ; be a weakly Picard–Presic operator. We say that is a -weakly Picard–Presic operator (in short, ) if for all , , and for some and possibly depending on and for all ,
Proposition 1. Let be a Ciric–Presic strong w-contraction. Then, is a .
Indeed, by Theorem 4, is a . Now, for any and , choose integers , and large enough so that and . Then, we have
Using (15), we get
As , we get
Hence, is a with .
For some , the attraction basin of with respect to is given byand the attraction basin of is given by
For some , we consider the fixed-point equationand the inequality
Definition 8. Equation (39) is Ulam–Hyers stable if there exists such that for each and each solution of (40) with , there exists a solution of fixed-point equation (39) such that for all .
Definition 9. Equation (39) is generalised Ulam–Hyers stable if there exists a function increasing, continuous at 0 and , such that for each and each solution of (40) with , there exists a solution of fixed-point equation (39) such that for all .
Theorem 7. Let be strict metric modular on and be -complete. For any positive integer , let be a . Then, fixed-point equation (39) is Ulam–Hyers stable.
Proof. Let , , and be a solution of (39) with ; that is, . Then, there exists ; that is, is a solution of (39) such that . Since is a , we haveThus, fixed-point equation (39) is Ulam–Hyers stable.
Remark 4. For , Definitions 6–9 reduce to Definitions 1, 2, 7, and 8, respectively, of [30].
5. Application to Integral Equation
Recently, some interesting applications of fixed-point theorems in proving the existence of solutions of various generalised integral equations and differential equations have been found (see [31–33]. Moreover, in [33], the Ulam–Hyers stability of nonlinear implicit fractional differential equations with Riemann–Liouville fractional derivative was discussed. In this section, we present an application of our results to prove the existence of solution of a generalised integral equation of Caratheodory type and give the conditions under which such integral equations are Ulam–Hyers stable. Let be a continuous, convex, increasing, and unbounded function such that iff . Moreover, admits the inverse function , which is continuous and strictly increasing, and iff . Let be a set of real-valued function on with ; that is,
Define for all and bywhere the supremum is taken over all partitions of the interval ; that is, . Then, it is known that is convex pseudomodular on (see [9, 10]).
For some , consider the constant function given by for all . Define the convex pseudomodular metric space as
The space is denoted by and is called the space of mappings of bounded generalised variations (see [34]). Then, if and only if , and there exists a constant such thatClearly, is independent of.
Consider the set given by
Lemma 1. The function given in (43) defines a strict convex metric modular on .
Proof. It is enough to show that for all and . Clearly, for any , with ,which impliesLet . Then, since , we getEquivalently,Solving the system by taking in (50) and using , we obtain for any .
We define
Then, is a metric modular space.
Lemma 2. The metric modular space is w-complete.
Proof. Let be -Cauchy. Then,for some . Therefore, for , we havewhich implies that . Since is complete, the sequence converges to some with . That is, for all , holds for . It remains to show that . From lower semicontinuity of (see [9], p.27), we havefor every . Since is -Cauchy, for all , there exists such that .
Hence, for all ,Thus, for every , there exists such thatThen, is -convergent to . As is closed under the modular convergence, we have , and thus, is -complete.
Furthermore, if satisfies the Orlicz condition at infinity, that is, as , then is called the -variation of the function ; the function with is said to be of bounded -variation on and we have
For the functions in the space , it is known that (see [10, 35])where is the space of all absolutely continuous real-valued functions on and is the space of all Lebesgue integrable functions on .
Now, we apply the result given in Theorem 2 to the following integral equation:for , where , are Caratheodory type functions, , and . Define the function bywhere . Then, problem (60) is equal to the fixed-point problem:
Definition 10. Equation (60) is Ulam–Hyers stable, if there exists such that for each and for whichthere exists a solution of (60) such that for all , where the supremum is taken over all partitions of the interval ; that is, and is a continuous, convex, increasing, and unbounded function such that iff admits the inverse function , which is continuous and strictly increasing, and iff .
If also satisfies Orlicz condition at infinity, then the above definition is equivalent to the following.
Definition 11. Equation (60) is Ulam–Hyers stable if there exists such that for each and for which there exists a solution of (60) such that for all .
We will analyse problem (60) under the following conditions: (m1) For every , where , the functions and are Lebesgue measurable on and there exist points such that for some and , where is a function satisfying the Orlicz condition at infinity and . (m2) There exists a constant such thatfor almost all and .
Theorem 8. Under the assumptions () and (), the operator is a function from to and the following inequality holds:for all and .
Proof. Applying Jensen’s integral inequality with the convex function ,and by the property of the convex function ,where the integral on the RHS takes values in .
Step 1. Claim: is well defined on .
Let , i.e., and . Since , in the light of (m1) and (m2), the functions and are measurable on . Let us prove that . By Lebesgue’s theorem, for all and so, by (m2), we haveTherefore,for almost all . Since , we have and so there exist constants and such thatAlso by (m1), there exist constants and such thatSetting and , we getIf by the convexity of , we findand soNow, it follows from (67) thatwhich impliesSimilarly, ifthenimplyingHence, in both the cases,Therefore, . Thus, the operator is well defined on and by (61), for all which implies that andfor almost all .
Step 2. It is clear from (61) and that given ,and so . Now, we will show that . By virtue of (59) and (73)–(82), we haveorand so maps .
Step 3. To obtain (66), let and . By (58), (59), and (82), we findBy (m2) and Lebesgue’s theorem, we have for almost all (since ),Therefore,Applying (67) and (68), we getTherefore,
Theorem 9. Under the assumptions , , and , integral equation (60) admits a solution .
Proof. (i) By Lemmas 1 and 2, is strict modular onand the modular spaceis -complete. By Theorem 8, the operator is Ciric–Presic type strongly -contractive. We are left to show that for some , , such that . Clearly, for constant function , we haveSince is a constant function, . Therefore,where . By Theorem 4, the integral operator admits a fixed point and thus problem (60) has a solution.
Theorem 10. Under the assumptions , , and , integral equation (60) is Ulam–Hyers stable.
Proof. Let and for which . Then, we haveBy Theorem 8, the operator is Ciric–Presic type strongly -contractive and so by Proposition 1, is . By Theorem 7, the fixed-point equation (62) is Ulam–Hyers stable; that is, there exists such that and . Thus, we have
We now furnish a numerical example to validate the hypothesis of Theorem 9.
Example 3. Consider the integral equationfor some , , and . We will apply Theorem 9, to prove the existence of a solution of (97).
Proof. Define the operator asNow, set , and . We observe the following:(i)Let for all . For and , we see that the functions and satisfy (m1).(ii)By using the Mathematica software, we found that the following inequalities are true for all (see Figures 1 and 2):Thus, we getand thus, and satisfies (m2), with .
Hence, all the conditions of Theorem 9 are satisfied. It is evident that integral equation (97) has a unique solution defined by .
6. Conclusion
The fixed-point technique is used to solve many mathematical problems as it gets involved with differential and integral equations, integro-differential equations, game theory, economics, and more disciplines. In this work, unique fixed points of generalised Presic type operators defined on modular metric space are obtained and the results are applied in proving the existence of solutions of integral equations involving Caratheodory type functions. Ulam–Hyers stability of fixed-point equations involving Presic type operators in a modular metric space is introduced. Our results pave way for further research in modular metric spaces. Following the techniques of our work, Presic–Bogin-type contractions, Presic–Chatterjee type contractions, and Presic–Hardy–Rogers type contractions can be extended to the setting of a modular metric space.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the Deanship of Scientific Research, Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia, under Grant no. 2020/01/17320.