# Generalised Presic Type Operators in Modular Metric Space and an Application to Integral Equations of Caratheodory Type Functions

Authors: Fahad Sameer Alshammari; K. P. Reshma; Rajagopalan R.; Reny George
Journal: Journal of Mathematics
 (2021)

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## Abstract

Extending the Presic type operators to modular spaces, we introduce generalised Presic typew-contractive mappings and strongly w-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.


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## Body

## 1. Introduction

Maurice Ren Fréchet [1] introduced the general and axiomatic form of distance as “L—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 b-metric, a rectangular metric, a rectangular b-metric, and bvs-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 w-contractions and proved fixed-point theorems for such contractions in a w-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.
LetX,d be a metric space, k be a positive integer, and T:Xk⟶X be a mapping satisfying the following condition:(1)dTx1,x2,…,xk,Tx2,x3,…,xk+1≤λ⋅maxdx1,x2,dx2,x3,…dxk,xk+1,where x1,x2,…,xk+1 are arbitrary elements in X and λ∈0,1. Then, there exists some x∈X such that x=Tx,x,…,x. Moreover, if x1,x2,…,xk are arbitrary points in X and for n∈N,xn+k=Txn,xn+1,…,xn+k−1, then the sequence xn is convergent and if limn⟶∞xn=x, then x=Tx,x,…,x. If, in addition, T satisfies DTu,u,…u,Tv,v,…v<du,v, for all u,v∈X. Then, x is the unique point satisfying x=Tx,x,…,x.Later, the above results were further extended and generalised to ab-metric space, cone metric space, and cone b-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 w-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 X≠ϕ and w:0,∞×X×X⟶0,∞ be a given function. For any λ∈R,x,y∈X, we write wλ,x,y as wλx,y.Definition 1.
(see [10]). We call w a modular metric on X if for all λ,μ>0 and x,y,z∈X, the following relations hold:(mm1)wλx,y=0⇔x=y(mm2)wλx,y=wλ,y,x(mm3)wλ+μx,y≤wλx,z+wμz,y
In this case,w is said to be a metric modular on X. Instead of (mm1), if w satisfies only(2)wλx,x=0,for all λ>0 and x∈X, then w is said to be pseudomodular on X.
Also,w is strict modular on X if instead of (mm1), it satisfies (2) and for x,y∈X, there exists a number λ>0 (possibly depending on x and y), such that(3)wλx,y=0⟹x=y.
Clearly, ifw is strict modular, then it is modular, which in turn implies w is pseudomodular on X.
A metric modularw on X is convex if it satisfies the (stronger) inequality, in lieu of (mm3):(4)wλ+μx,y≤λλ+μwλx,z+μλ+μwμz,y,for all λ,μ>0.
The main property of a modularw on a set X is that (see Section 2.3 of [9]), given x,y∈X, the function wλx,y is nonincreasing on 0,∞. If 0<μ<λ, then (mm1) and (mm3) imply(5)wλx,y≤wλ−μx,x+wμx,y=wμx,y.Definition 2.
(see [10]). Given a pseudomodular w on X, the two sets(6)Xw=Xwx0=x∈X:wλx,x0⟶0asλ⟶∞,Xw∗=Xw∗x0=x∈X:there existsλ=λx>0such thatwλx,x0<∞,are said to be modular spaces (around x0).Clearly,Xw⊂Xw∗. Note that if w is metric modular on X, then modular space Xw can be equipped with a metric generated by w given by(7)dwx,y=infλ>0:wλx,y≤λ,for any x,y∈Xw. Moreover, if w is a convex modular on X, then by Section 3.5 and Theorem 3.6 of [9], Xw=Xw∗; i.e., the two modular spaces coincide, and this common set can be endowed with a metric dw∗:(8)dw∗x,y=infλ>0:wλx,y≤1.Henceforth,Xw,w and Xw∗,w represent modular metric spaces induced by w.Definition 3.
(see [10]). Let w be a metric modular on X. We have the following:(i)
A sequencexn in Xw is w-convergent (or modular convergent) to some x∈X, if there exists λ>0, such that limn⟶∞wλxn,x=0(ii)
A sequencexn in X is w-Cauchy if ∃λ>0, possibly dependent on xn, such that wλxm,xn=0 as m,n⟶∞(iii)
X is w-complete if every w-Cauchy sequence is w-convergentFrom [10], we have that Xw and Xw∗ are closed with respect to w convergence, if w is pseudomodular on X. Also, if w is strict, then the modular limit, if exists, is unique. If w is (not necessarily convex) modular on X, then vλx,y=wλx,y/λ is always convex modular on X.


## 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.LetR+ be the set of real numbers, and consider a function Φ:R+k⟶R+ such that(a)
Φ is increasing in all variables(b)
Φu,u,u,…,u≤u, for all u∈RDefinition 4.
Letw be metric modular on X and T:Xwk∗⟶Xw∗ (where Xwk∗denotesXw∗×Xw∗×⋯ktimes) for some positive integer k. Then, we have the following:(i)
T is a generalised Presic type modular contraction (or a generalised Presic w-contraction), if(9)wξλTx1,x2,…,xk,Tx2,x3,…,xk+1≤Φwλx1,x2,wλx2,x3,…,wλxk,xk+1,for some0<ξ<1 and λ0>0 possibly depending on ξ and for all 0<λ≤λ0 and x1,x2,…,xk+1∈Xw∗.(ii)
T is a generalised Presic type strongly modular contraction (or a generalised Presic strong w-contraction), if(10)wξλTx1,x2,…,xk,Tx2,x3,…,xk+1≤ξ⋅Φwλx1,x2,wλx2,x3,…,wλxk,xk+1,for some0<ξ<1 and λ0>0 possibly depending on ξ and for all 0<λ≤λ0 and x1,x2,…,xk+1∈Xw∗.(iii)
T is a Ciric–Presic type modular contraction (or a Ciric–Presic w-contraction), if(11)wξλTx1,x2,…,xk,Tx2,x3,…,xk+1≤maxwλx1,x2,wλx2,x3,…,wλxk,xk+1,for some0<ξ<1 and λ0>0 possibly depending on ξ and for all 0<λ≤λ0 and x1,x2,…,xk+1∈Xw∗.(iv)
T is a Ciric–Presic type strongly modular contraction (or a Ciric–Presic strong w-contraction), if(12)wξλTx1,x2,…,xk,Tx2,x3,…,xk+1≤ξ⋅maxwλx1,x2,wλx2,x3,…,wλxk,xk+1,for some0<ξ<1 and λ0>0 possibly depending on ξ and for all 0<λ≤λ0 and x1,x2,…,xk+1∈Xw∗.(v)
T is a Presic type modular contraction (or a Presic w-contraction), if there exists βi≥0,i=1,2,…,k, ∑i=1kβi<1, such that(13)wξλTx1,x2,…,xk,Tx2,x3,…,xk+1≤β1wλx1,x2+β2wλx2,x3+⋯+βkwλxk,xk+1,for some 0<ξ<1 and λ0>0 possibly depending on ξ and for all 0<λ≤λ0 and x1,x2,…,xk+1∈Xw∗.Theorem 2.
Letw be a strict metric modular on X and Xw∗,w be w-complete. For any positive integer k, let T:Xwk∗⟶Xw∗ be a generalised Presic type strongly w-contractive mapping. If there exists x1,x2,…,xk in X, such that wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, then T has a fixed point; that is, there exists an x∈X such that x=Tx,x,…,x. Moreover, for any x1,x2,…,xk in X, with wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, the sequence xn given by xk+n=Tx1,x2,…,xk converges to a fixed point of T. Furthermore, if T satisfies(14)wξλTu,u,…,u,Tv,v,…,v<wλu,v,for all u,v∈X or if ξ∈0,1/k, then the fixed point x is unique.Proof.
For arbitrary elementsx1,x2,…,xk in X with wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, define the sequence xn in X given by xn+k=Txn,xn+1,…xn+k−1, n∈N. By the w-contractivity of T, there exist two numbers 0<ξ<1 and λ0=λ0ξ such that condition (10) holds. Let αnξpλ=wξpλxn,xn+1, where λ≤λ0. Then, ξpλ<λ<λ0 for any positive integer p. Using the method of induction, we will prove that for any nonnegative integer p,(15)αnξpλ≤Rξλθn,where θ=ξ1/k and Rξλ=maxα1λ/θ,α2λ/θ2,…,αkλ/θk. Clearly, by the definition of Rξλ, (15) is true for n=1,2,…,k and p=0. Let the k+1 inequalities αnξpλ≤Rξλθn,αn+1ξpλ≤Rξλθn+1,…,αn+kξpλ≤Rξλθn+k hold true. Then,(16)αn+kξp+1λ=wξp+1λxn+k,xn+k+1=wξp+1λTxn,xn+1,…,xn+k−1,Txn+1,xn+2,…,xn+k≤ξΦwξpλxn,xn+1,wξpλxn+1,xn+2,…,wξpλxn+k−1,xn+k=ξΦαnξpλ,αn+1ξpλ,…,αn+k−1ξpλ≤ξΦRξλθn,Rξλθn+1,…,Rξλθn+k−1≤ξRξλθn=Rξλθn+k,(17)αn+k+1ξpλ=wξpλxn+k+1,xn+k+2=wξpλTxn+1,xn+2,…,xn+k,Txn+2,xn+3,…,xn+k+1≤ξΦwξp−1λxn+1,xn+2,wξp−1λxn+2,xn+3,…,wξp−1λxn+k,xn+k+1=ξΦαn+1ξp−1λ,αn+2ξp−1λ,…,αn+kξp−1λ=ξΦαn+1ξpλ,αn+2ξpλ,…αn+kξpλ≤ξΦRξλθn+1,Rξλθn+2,…,Rξλθn+k≤ξRξλθn+1=Rξλθn+k+1.
Thus, inductive proof of (15) is complete. Set λ1=1−ξλ0; then, ξiλ1<λ1<λ0. Let integers n and m be such that n>m. We set λ=λn,m=ξmλ1+ξm+1λ1+⋯+ξn−1λ1=ξm1−ξn−m/1−ξλ1. Then, we have(18)wλxm,xn≤wξmλ1xm,xm+1+wξm+1λ1xm+1,xm+2+⋯+wξn−1λ1xn−1,xn≤Rξλ1θm+Rξλ1θm+1+⋯+Rξλ1θn≤Rξλ1θm1+θ+θ2+⋯=Rξλ1θm1−θ.
Taking into account thatλ0=λ1/1−ξ>ξm1−ξn−m/1−ξλ1=λ for all n>m, we get(19)wλ0xm,xn≤wλxm,xn≤Rλ0ξθm1−θ⟶0,asm⟶∞.
Therefore,xn is w-Cauchy in Xw∗. By w-completeness of Xw∗, there exists x in Xw∗ such that limn⟶∞wλ0xn,x=0, and by strictness of w, the limit is unique. Let ϵ>0 be arbitrary. Then, we can find a natural number N such that wλ0xn,xn+1<ϵ and wλ0xn,x<ϵ for all n≥N. Then, for any integer n≥N, we have(20)w1+kξλ0x,Tx,x,…,x≤wλ0x,xn+k+wkξλ0xn+k,Tx,x,…,x=wλ0x,xn+k+wξλ0+ξλ0+⋯ktimesTxn,xn+1,…,xn+k−1,Tx,x,…,x≤wλ0x,xn+k+wξλ0Tx,x,…,x,Tx,x,…,xn+wξλ0Tx,x,⋯x,xn,Tx,x,…,xn,xn+1+⋯+wξλ0Tx,xn,…,xn+k−2,Txn,xn+1,…,xn+k−1≤wλ0x,xn+k+ξΦwλ0x,x,wλ0x,xn+ξΦwλ0x,x,wλ0x,xn,wλ0xn,xn+1+⋯+ξΦwλ0x,xn,wλ0xn,xn+1,…,wλ0xn+k−2,xn+k−1<ϵ.
Asϵ⟶0, we get w1+kξλ0x,Tx,x,…,x=0. By the strictness of w, Tx,x,…,x=x. For uniqueness, suppose there exists y∈Xw∗, where y≠x, such that Ty,…,y=y. Then, using (14), we get(21)wξλ0x,y=wξλ0Tx,…,x,Ty,…,y<wλ0x,y,a contradiction (as ξλ0<λ0). Thus, the uniqueness of x is established.
Now, ifξ∈0,1/k, then(22)wξλ0x,y=wξλ0Tx,…,x,Ty,…,y≤ξλ0kξλ0wξλ0/kTx,x,…,x,Tx,x,…,y+ξλ0kξλ0wξλ0/kTx,x,…,x,y,Tx,x,…,x,y,y+⋯+ξλ0kξλ0wξλ0/kTx,y,…,y,Ty,y,…,y≤ξ1kΦwλ0/kx,x,wλ0/kx,y+ξ1kΦwλ0/kx,x,wλ0/kx,y,wλ0/ky,y+⋯+ξ1kΦwλ0/kx,y,wλ0/ky,y≤ξkkwλ0/kx,y<wλ0/kx,y,a contradiction as ξλ0<λ0/k. Hence, Tx,x,…,x=x is unique in X.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.
Letw be a strict convex modular on X and Xw∗,w be w-complete. For any positive integer k, let T:Xwk∗⟶Xw∗ be a generalised Presic w-contractive mapping. If there exists x1,x2,…,xk in X, such that wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, then T has a fixed point; that is, there exists an x∈X such that x=Tx,x,…,x. Moreover, for any x1,x2,…,xk in X, with wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, the sequence xn given by xk+n=Tx1,x2,…,xk converges to a fixed point of T. Furthermore, if T satisfies(23)wξλTu,u,…,u,Tv,v,…,v<wλu,v,for all u,v∈X or if ξ∈0,1/k, then the fixed point x is unique.Proof.
We takeuλx,y=λ.wλx,y for all x,y∈X and λ>0. Then, for all λ,μ>0 and x,y,z∈X, we have(24)uλ+μx,y=λ+μwλ+μx,y≤λ+μλλ+μwλx,z+μλ+μwμy,z=λ⋅wλx,z+μ⋅wμy,z=uλx,z+uμy,z.
So,u is strict modular on X. Also, by (9), for all 0<λ≤λ0 and x1,x2,…,xk+1∈Xw∗, we have(25)uξλTx1,x2,…,xk,Tx2,x3,…,xk+1=ξλwξλTx1,x2,…,xk,Tx2,x3,…,xk+1≤ξλΦwλx1,x2,wλx2,x3,…,wλxk,xk+1=ξΦλwλx1,x2,λwλx2,x3,…,λwλxk,xk+1=ξΦuλx1,x2,uλx2,x3,…,uλxk,xk+1.
Thus,T is a generalised Presic strong w-contraction mapping for the modular u. Clearly, wξpλx1,x2<∞,wξpλx2,x3<∞,…,wξpλxk−1,xk<∞,wξpλxk,Tx1,x2,…,xk<∞ implies uξpλx1,x2<∞,uξpλx2,x3<∞,…,uξpλxk−1,xk<∞,uξpλxk,Tx1,x2,…,xk<∞. By Theorem 2, there exists x∈X such that u1+kξλ0x,Tx,x,…,x=0. The remaining part of the proof follows the same line as in Theorem 2.TakingΦu1,u2,…,uk=maxu1,u2,…,uk in Theorems 1 and 2, we get the following.Theorem 4.
Letw be strict metric modular on X and Xw∗,w be w-complete. For any positive integer k, let T:Xwk∗⟶Xw∗ be a Ciric–Presic strongly w-contractive mapping. If there exists x1,x2,…,xk in X, such that wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, then T has a fixed point; that is, there exists an x∈X such that x=Tx,x,…,x. Moreover, for any x1,x2,…,xk in X, with wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, the sequence xn given by xk+n=Tx1,x2,…,xk converges to a fixed point of T. Furthermore, if T satisfies(26)wξλTu,u,…,u,Tv,v,…,v<wλu,v,for all u,v∈X or if ξ∈0,1/k, then the fixed point x is unique.Theorem 5.
Letw be strict convex modular on X and Xw∗,w be w-complete. For any positive integer k, let T:Xwk∗⟶Xw∗ be a Ciric–Presic w-contractive mapping. If there exists x1,x2,…,xk in X, such that wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, then T has a fixed point; that is, there exists an x∈X such that x=Tx,x,…,x. Moreover, for any x1,x2,…,xk in X, with wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, the sequence xn given by xk+n=Tx1,x2,…,xk converges to a fixed point of T. Furthermore, if T satisfies(27)wξλTu,u,…,u,Tv,v,…,v<wλu,v,for all u,v∈X or if ξ∈0,1/k, then the fixed point x is unique.TakingΦu1,u2,…,uk=β1u1+β2u2,…,βkuk, with 0≤βi,∑i=1kβi<1 in Theorem 3, we get the following.Theorem 6.
Letw be strict convex modular on X and Xw∗,w be w-complete. For any positive integer k, let T:Xwk∗⟶Xw∗ be a Presic type w-contractive mapping. If there exists x1,x2,…,xk in X, with wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, then T has a fixed point; that is, there exists an x∈X such that x=Tx,x,…,x. Moreover, for any x1,x2,…,xk in X, with wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, the sequence xn given by xk+n=Tx1,x2,…,xk converges to a fixed point of T. Furthermore, if T satisfies(28)wξλTu,u,…,u,Tv,v,…,v<wλu,v,for all u,v∈X or if ξ∈0,1/k, then the fixed point x is unique.Remark 1.
Fork=1, Theorems 4 and 5 reduce to Theorem 10 and Theorem 5.4 of [10].Remark 2.
Theorems4 and 5 extend the results of Ciric and Presic [23] to a modular space.Remark 3.
Theorem6 extends the result of Presic [21] to a modular space.Example 1.
LetX=0,1. Define the mapping w:0,∞×X×X⟶0,∞ by wλa1,a2=4/15λ2a1−a2. Note that wλa1,a2<∞ for all a1,a2∈X, then X=Xw∗ and Xw∗,w is a complete modular metric space. Let Φ:R2⟶R be given by Φt1,t2=maxt1,t2. Let T:Xw∗×Xw∗⟶Xw∗ given by Ta1,a2=32/1000a1+a2. Then,(29)wξλTa1,a2,Ta2,a3=wξλ321000a1+a2,321000a2+a3=415ξ2λ2321000a1+a2−32100a2+a3=415ξ2λ2.321000a1+a2−a2+a3=415ξ2λ2.321000a1−a2+a2−a3=815ξ2λ2.321000a1−a2+a2−a32=2ξ2.3210004/15λ2a1−a2+4/15λ2a2−a32≤2ξ2.321000maxwλa1,a2,wλa2,a3.
Ifξ=0.4∈0,1/2, then w0.4λTa1,a2,Ta2,a3≤0.4Φwλa1,a2,wλa2,a3 for all a1,a2,a3∈Xw. Thus, T is a generalised Presic type strongly modular contraction and Ciric–Presic type strongly modular contraction with constant ξ=0.4. Theorems 2 and 4 are applicable, and 0,0∈Xw is the unique fixed point of T.Example 2.
LetX=0,1. Define the mapping w:0,∞×X×X⟶0,∞ by wλa1,a2=a1−a2/λ. Note that wλa1,a2<∞ for all a1,a2∈X; then, X=Xw∗ and Xw∗,w is a complete convex modular metric space. Let Φ:R3⟶R be given by Φt1,t2,t3=t1+t2+t3, if ti≠tj for some i,j and Φt,t,t=t.
LetT:Xw∗×Xw∗×Xw∗⟶Xw∗ be given by Ta1,a2,a3=1/1.01a1+a2+a3. Then,(30)wξλTa1,a2,a3,Ta2,a3,a4=wξλ11.01a1+a2+a3,11.01a2+a3+a4=1ξλ.11.01a1+a2+a3−a2+a3+a4=1ξλ.11.01a1−a2+a2−a3+a3−a4.
Ifξ=1/1.01∈0,1, then w1/1.01λTa1,a2,a3,Ta2,a3,a4≤Φwλa1,a2,wλa2,a3,wλa3,a4 for all a1,a2,a3,a4∈Xw. Thus, T is a generalised Presic type modular contraction, with constant ξ=1/1.01. Thus, T satisfies the condition of Theorem 3 and 0,0,0∈Xw is the unique fixed point of T.


## 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.
Letw be a metric modular on X and T:Xwk∗⟶Xw∗ (where Xwk∗denotesXw∗×Xw∗×⋯ktimes) for some positive integer k. If for all x1,x2,…,xk∈Xwk∗ with wλx1,x2<∞,wλx2,x3<∞,…,wλxk−1,xk<∞,wλxk,Tx1,x2,…,xk<∞, the sequence xk+n given by xk+n=Txn,xn+1,…,xk+n−1 converges and the limit (which depends upon x1,x2,…,xk) is a fixed point of T, then we say that T is a weakly Picard–Presic operator (in short, WPPO). If the fixed point of T is unique, then we say that T is a Picard–Presic operator.Hereafter, we will make use of the following notations:(31)Tnx1,x2,…,xk=xk+n,T∞x1,x2,…,xk=limn⟶∞Tnx1,x2,…,xk=limn⟶∞xn+k.Definition 6.
Letw be metric modular on X, T:Xwk∗⟶Xw∗ be a weakly Picard–Presic operator, and c>0 be a real number. We say that T is a c-weakly Picard–Presic operator (in short, c−WPPO) if for all x=x1,x2,…,xk∈Xwk∗, 1≤i≤k, and for some 0<ξ<1 and λ0>0 possibly depending on ξ and for all 0<λ≤λ0,(32)wλxi,T∞x≤cmaxwλx1,x2,wλx2,x3,…,wλxk−1,xk,wλxk,Tx.LetR+ be the set of nonnegative real numbers, and consider a function Ψ:R+k⟶R+ such that Ψ is continuous at 0 and Ψ0=0.Definition 7.
Letw be a metric modular on X; T:Xwk∗⟶Xw∗ be a weakly Picard–Presic operator. We say that T is a Ψ-weakly Picard–Presic operator (in short, Ψ−WPPO) if for all x=x1,x2,…,xk∈Xwk∗, 1≤i≤k, and for some 0<ξ<1 and λ0>0 possibly depending on ξ and for all 0<λ≤λ0,(33)wλxi,T∞x≤Ψmaxwλx1,x2,wλx2,x3,…,wλxk−1,xk,wλxk,Tx.Proposition 1.
LetT:Xwk∗⟶Xw∗ be a Ciric–Presic strong w-contraction. Then, T is a c−WPPO.Indeed, by Theorem4, T is a WPPO. Now, for any x=x1,x2,…xk∈Xw∗ and 0<ξ<1, choose integers m,n, and r large enough so that ξm+1+ξm+2+⋯ξm+n+1≤1 and ξr+ξr+1+⋯ξr+k−i≤ξm+1. Then, we have(34)wλxi,Tnx≤wξm+1+ξm+2+⋯ξm+n+1λxi,Tnx≤wξm+1λxi,Tx+wξm+2λTx,T2x+⋯wξm+n+1λTn−1x,Tnx≤wξr+ξr+1+⋯ξr+k−iλxi,Tx+wξ2λTx,T2x+⋯wξn+1λTn−1x,Tnx≤wξr+ξr+1+⋯ξr+k−iλxi,xk+1+wξ2λxk+1,xk+2+⋯wξn+1λxn−1,xn≤wξrλxi,xi+1+wξr+1λxi+1,xi+2+⋯wξr+k−i−1λxk−1,xk+wξr+k−iλxk,xk+1+wξ2λxk+1,xk+2+⋯wξn+1λxn−1,xn.Using (15), we get(35)wλxi,Tnx≤Rξλξi/k+ξi+1/k+⋯ξk−1/k+ξk/k+⋯ξn−1/k.Asn⟶∞, we get(36)wλxi,T∞x≤Rξλ1+ξi/k+ξi+1/k+⋯ξk−1/k+ξk/k+⋯≤11−ξ1/kmaxα1λθ,α2λθ2,…,αkλθk≤11−ξ1/kmaxwλx1,x2ξ1/k,wλx2,x3ξ2/k,…,wλxk−1,xkξk−1/k,wλxk,Txξk/k≤1ξ1−ξ1/kmaxwλx1,x2,wλx2,x3,…,wλxk−1,xk,wλxk,Tx.Hence,T is a c−WPPO with c=1/ξ1−ξ1/k.For somex∗∈FixXw∗, the attraction basin of x∗ with respect to T is given by(37)ABTx∗=x=x1,x2,…,xk∈Xwk∗:Tnxis defined for alln∈NandTnx⟶x∗,and the attraction basin of T is given by(38)ABT=∪x∗∈FixTABTx∗.For someT:Xwk∗⟶Xw∗, we consider the fixed-point equation(39)x=Tx,x,…,x,x∈Xw∗,and the inequality(40)wλy,Ty,y,…,y≤ϵ,for allλ>0.Definition 8.
Equation (39) is Ulam–Hyers stable if there exists c>0 such that for each ϵ>0 and each solution y∗ of (40) with y∗,y∗,…,y∗∈ABT, there exists a solution x∗ of fixed-point equation (39) such that wλy∗,x∗≤cϵ for all λ>0.Definition 9.
Equation (39) is generalised Ulam–Hyers stable if there exists a function ϕ:R+⟶R+ increasing, continuous at 0 and ϕ0=0, such that for each ϵ>0 and each solution y∗ of (40) with y∗,y∗,…,y∗∈ABT, there exists a solution x∗ of fixed-point equation (39) such that wλy∗,x∗≤ϕϵ for all λ>0.Theorem 7.
Letw be strict metric modular on X and Xw∗,w be w-complete. For any positive integer k, let T:Xwk∗⟶Xw∗ be a c−WPPO. Then, fixed-point equation (39) is Ulam–Hyers stable.Proof.
Letϵ>0, λ>0, and y∗ be a solution of (39) with y=y∗,y∗,…,y∗∈ABT; that is, wλy∗,Ty≤ϵ. Then, there exists x∗∈FixT; that is, x∗ is a solution of (39) such that Tny⟶x∗. Since T:Xwk∗⟶Xw∗ is a c−WPPO, we have(41)wλy∗,T∞y≤c⋅maxwλy∗,y∗,wλy∗,y∗,…,wλy∗,y∗,wλy∗,Ty≤cϵ.
Thus, fixed-point equation (39) is Ulam–Hyers stable.Remark 4.
Fork=1, 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 ϕ:R+⟶R+ be a continuous, convex, increasing, and unbounded function such that ϕu=0 iff u=0. Moreover, ϕ admits the inverse function ϕ−1:R+⟶R+, which is continuous and strictly increasing, and ϕ−1u=0 iff u=0. Let M be a set of real-valued function on a,b⊂R with a<b; that is,(42)M=u:a,b⟶A,A⊂Ris complete.Definew:0,∞×M×M⟶0,∞ for all γ>0 and u,w∈M by(43)wγu,w=supπ∑i=1mϕuτi+wτi−1−uτi−1+wτiγτi−τi−1τi−τi−1,where the supremum is taken over all partitions π=τii=0n of the interval a,b; that is, a=τ0<τ1<τ2⋯<τn=b. Then, it is known that wγu,w is convex pseudomodular on M(see [9, 10]).For someu0∈R, consider the constant function u0∈M given by u0τ=u0 for all τ∈a,b. Define the convex pseudomodular metric space Mw∗ as(44)Mw∗=Mw∗u0=u∈M:∃γ=γu>0,such thatwγu,u0<∞.The spaceMw∗ is denoted by GVϕa,b and is called the space of mappings of bounded generalised ϕ− variations (see [34]). Then, u∈Mw∗=GVϕa,b if and only if u:a,b⟶R, and there exists a constant γ=γu>0 such that(45)wγu,u0=supπ∑i=1nϕuτi−uτi−1γτi−τi−1τi−τi−1<∞.Clearly, wγ,u,u0 is independent of. u0Consider the setMı given by(46)Mı=u;u:a,b⟶A,ua=u0⊂Mw∗.Lemma 1.
The functionw given in (43) defines a strict convex metric modular on Mı.Proof.
It is enough to show thatwγu,w=0⟹uτ=wτ for all u,w∈Mı and τ∈a,b. Clearly, for any ψ,τ∈a,b, with ψ≠τ,(47)ϕuτ+wψ−uψ+wτγτ−ψτ−ψ≤wγu,w,which implies(48)uτ−wτ+wψ−uψ≤γτ−ψϕ−1wγu,wτ−ψ.
Letwγu,w=0. Then, since ϕ−10=0, we get(49)uτ−wτ+wψ−uψ≤γτ−ψϕ−10=0.Equivalently,(50)uτ−wτ=uψ−wψ.
Solving the system by takingψ=a in (50) and using ua=wa=u0, we obtain uτ=wτ for any τ∈a,b.We define(51)Mw∗ı=Mw∗∩Mı=GVϕa,b∩Mı=u∈GVϕa,b,ua=u0=u∈Mı:∃γ=γu>0,such thatwγu,u0<∞.Then,Mw∗ı is a metric modular space.Lemma 2.
The metric modular spaceMw∗ı is w-complete.Proof.
Letun⊂Mw∗ı be w-Cauchy. Then,(52)wγun,um⟶0,asn,m⟶∞,for some γ=γun>0. Therefore, for n,m∈Nandt∈a,b, we have(53)unτ−umτ=unτ−umτ−un0+um0≤|unτ+um0−umτ+un0≤γt−0ϕ−1wγun,umt−0⟶0,asn,m⟶∞,which implies that Limn,m⟶∞unτ−umτ=0. Since A is complete, the sequence un converges to some u:a,b⟶A with ua=u0. That is, for all t∈a,b, Limn⟶∞unτ−umτ=0 holds for u∈Mı. It remains to show that Limn⟶∞wγunτ,uτ=0. From lower semicontinuity of w(see [9], p.27), we have(54)wγun,u≤limn⟶∞infwγun,um,for every n∈N. Since un is w-Cauchy, for all ϵ>0, there exists Nϵ∈N such that ∀n,m≥Nϵ,wγun,um<ϵ.
Hence, for alln≥Nϵ,(55)limm⟶∞supwγun,um≤supm≥Nϵwγun,um<ϵ.
Thus, for everyϵ>0, there exists Nϵ∈N such that(56)wγun,u≤Limm⟶∞infwγ,un,um≤Limm⟶∞supwγ,un,um<ϵ.
Then,un is w-convergent to u. As Mw∗ı is closed under the modular convergence, we have u∈Mw∗ı, and thus, Mw∗ı is w-complete.Furthermore, ifϕ satisfies the Orlicz condition at infinity, that is, ϕw/w⟶∞ as w⟶∞, then w1u,0 is called the ϕ-variation of the function u:a,b⟶R; the function u with w1u,0<∞ is said to be of bounded ϕ-variation on a,b and we have(57)wγu,w=wγu−w,0=w1u−wγ,0,γ>0,u,w∈M.For the functionsu:a,b⟶A in the space Mw∗=GVϕa,b, it is known that (see [10, 35])(58)u∈GVϕa,b⇔wγu,0<∞,for someγ=γu>0,(59)⇔u∈ACa,b,u′∈L1a,bwithwγu,0=∫abϕu′τγdτ<∞,where ACa,b is the space of all absolutely continuous real-valued functions on a,b and L1a,b is the space of all Lebesgue integrable functions on a,b.Now, we apply the result given in Theorem2 to the following integral equation:(60)uψ=∫aμψKψ,v,uvdv+∫aσψJψ,v,uvdv+wψ,for ψ∈a,b, where u,w∈Mw∗ı, K,J:a,b×a,b×A⟶R are Caratheodory type functions, μ,σ:a,b⟶a,b, and μa=σa=a. Define the function T:Mw∗ı×Mw∗ı⟶Mw∗ı by(61)Tu1,u2ψ=∫aμψKψ,v,u1vdv+∫aσψJψ,v,u2vdv+wψ,where u1,u2,w∈Mw∗ı. Then, problem (60) is equal to the fixed-point problem:(62)Tu,u=u.Definition 10.
Equation (60) is Ulam–Hyers stable, if there exists c>0 such that for each ϵ>0 and u∗∈Mw∗ı for which(63)supπ∑i=1mϕu∗τi+Tu∗,u∗τi−1−u∗τi−1+Tu∗,u∗τiγτi−τi−1τi−τi−1≤ϵ,there exists a solution u of (60) such that supπ∑i=1mϕu∗τi+uτi−1−u∗τi−1+uτi/γτi−τi−1τi−τi−1≤cϵ, for all τ∈a,b, where the supremum is taken over all partitions π=τii=0n of the interval a,b; that is, a=τ0<τ1<τ2⋯<τn=b and ϕ:R+⟶R+ is a continuous, convex, increasing, and unbounded function such that ϕu=0 iff u=0 admits the inverse function ϕ−1:R+⟶R+, which is continuous and strictly increasing, and ϕ−1u=0 iff u=0.Ifϕ 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 c>0 such that for each ϵ>0 and u∗∈Mw∗ı for which ∫abϕu∗′τ−Kτ,v,u∗μτ−Kτ,v,u∗στ−u′τ/γdτ≤ϵ there exists a solution u of (60) such that ∫abϕu∗′τ−u′τ/γdτ≤cϵ for all τ∈a,b.We will analyse problem (60) under the following conditions:(m1) For everyuj∈R, where j∈N, the functions Kψ,v,uj and Jψ,v,uj are Lebesgue measurable on a,b and there exist points u0,v0∈R such that(64)∫abϕKψ,v,u0γKdv<∞,∫abϕJψ,v,v0γJdv<∞,for someγK=γKψ,v,u0>0 and γJ=γJψ,v,v0,>0, where ϕ is a function satisfying the Orlicz condition at infinity and γ>0.(m2) There exists a constantL>0 such that(65)Kψ,v,u1−Kψ,v,u2+Jψ,v,u2−Jψ,v,u3≤L2b−amaxu1−u2,u2−u3,for almost all ψ,v∈a,b and u1,u2∈A.Theorem 8.
Under the assumptions (m1) and (m2), the operator T is a function from Mw∗ı×Mw∗ı to Mw∗ı and the following inequality holds:(66)wLb−aγTu1,u2,Tu2,u3≤Lb−amaxwγu1,u2,wγu2,u3,for all γ>0 and u1,u2,u3∈Mw∗ı.Proof.
Applying Jensen’s integral inequality with the convex functionϕ,(67)ϕ1b−a∫abuτdτ≤1b−a∫abϕuτdτ,u∈L1a,b,and by the property of the convex function ϕ,(68)ϕauτ≤aϕuτ,0<a<1,uτ∈R,where the integral on the RHS takes values in 0,∞.Step 1.
Claim:T is well defined on Mw∗ı×Mw∗ı.
Letu1,u2∈Mw∗ı, i.e., u1,u2∈GVϕa,b and u1a=u2a=u0. Since u1,u2∈ACa,b, in the light of (m1) and (m2), the functions K and J are measurable on a,b. Let us prove that K,J∈L1a,b. By Lebesgue’s theorem, uv=u0+∫avu′ψdψ for all v∈a,b and so, by (m2), we have(69)Kψ,v,u1v+Jψ,v,u2v≤Kψ,v,u1v−Kψ,v,u0+Kψ,v,u0+Jψ,v,u2v−Jψ,v,v0+Jψ,v,v0≤L2b−amaxu1v−u0,u2v−v0+Kψ,v,u0+Jψ,v,v0≤L2b−amax∫abu1′vdv+u0−u0,∫abu2′vdv+u0−v0+Kψ,v,u0+Jψ,v,v0.
Therefore,(70)Kψ,v,u1v+Jψ,v,u2v≤L2b−amax∫abu1′vdv+u0−u0,∫abu2′vdv+u0−v0+Kψ,v,u0+Jψ,v,v0,for almost all v∈a,b. Since u1,u2∈Mw∗ı, we have u1,u2∈GVϕa,b and so there exist constants γ1=γ1u1>0 and γ2=γ2u2>0 such that(71)C1≡wγ1u1,u0=∫abϕu1′vγ1dv<∞,C1′≡wγ2u2,u0=∫abϕu2′vγ2dv<∞.
Also by (m1), there exist constantsγK=γKψ,v,v0>0 and γJ=γJψ,v,v0>0 such that(72)C2=∫abϕKψ,v,u0γKdv<∞,C3=∫abϕJψ,v,v0γJdv<∞.
Settingγ0=L2b−aγ1+1+γK+γJ and γ0′=L2b−aγ2+1+γK+γJ, we get(73)L2b−aγ1γ0+1γ0+γKγ0+γJγ0=1,L2b−aγ2γ0′+1γ0′+γKγ0′+γJγ0′=1.
Ifmax∫abu1′vdv+u0−u0,∫abu2′vdv+u0−v0=∫abu1′vdv+u0−u0 by the convexity of ϕ, we find(74)ϕKψ,v,u1v+Jψ,v,u2vγ0≤ϕ1γ0L2b−a∫abu1′vdv+L2b−au0−u0+Kψ,v,u0+Jψ,v,v0=ϕL2b−aγ1γ0∫abu1′vγ1dψ+1γ0L2b−au0−u0+γKγ0Kψ,v,u0γK+γJγ0Jψ,v,v0γJ≤L2b−aγ1γ0ϕ∫abu1′vγ1dψ+1γ0ϕL2b−au0−u0+γKγ0ϕKψ,v,u0γK+γJγ0ϕJψ,v,v0γJ=L2b−aγ1γ0C1+1γ0ϕL2b−au0−u0+γKγ0ϕKψ,v,u0γK+γJγ0ϕJψ,v,v0γJ,and so(75)∫abϕKψ,v,u1v+Jψ,v,u2vγ0dv≤L2b−a2γ1γ0C1+b−aγ0ϕL2b−au0−u0+γKγ0C2+γJγ0C3≡C0<∞.
Now, it follows from (67) that(76)ϕ1b−a∫abKψ,v,u1v+Jψ,v,u2vγ0dv≤1b−a∫abϕKψ,v,u1v+Jψ,v,u2vγ0dv≤C0b−a,which implies(77)∫abKψ,v,u1v+Jψ,v,u2vdτ≤γ0b−aϕ−1C0b−a<∞.
Similarly, if(78)max∫abu1′ψdψ+u0−u0,∫abu2′ψdψ+u0−v0=∫abu2′ψdψ+u0−v0,then(79)∫abϕKψ,v,u1v+Jψ,v,u2vγ0′dv≤L2b−a2γ2γ0′C1′+1γ0′ϕL2b−au0−v0+γKγ0′C2+γJγ0′C3≡C0′b−a,implying(80)∫abKψ,v,u1v+Jψ,v,u2vdv≤γ0′b−aϕ−1C0′b−a<∞.
Hence, in both the cases,(81)∫abKψ,v,u1vdv+∫abJψ,v,u21vdv<∞.
Therefore,K,J∈L1a,b. Thus, the operator T is well defined on Mw∗ı×Mw∗ı and by (61), Tu1,u2∈ACa,b for all u1,u2∈Mw∗ı which implies that Tu1,u2′∈L1a,b and(82)Tu1,u2′ψ=Kψ,v,u1v+Jψ,v,u2v,for almost all v∈a,b.Step 2.
It is clear from (61) and μa=σa=a that given u1,u2,w∈Mw∗ı,(83)Tu1,u2a=∫aμaKψ,v,u1vdv+∫aσaJψ,v,u2vdv+wa=u0,and so Tu1,u2∈Mı=u:a,b⟶R:ua=u0. Now, we will show that Tu1,u2∈Mw∗. By virtue of (59) and (73)–(82), we have(84)wγ0Tu1,u2,u0=∫abϕTu1,u2′vγ0dv≤C0,or(85)wγ0′Tu1,u2,u0=∫abϕTu1,u2′vγ0dv≤C0′,and so T maps Mw∗ı×Mw∗ı⟶Mw∗ı.Step 3.
To obtain (66), let γ>0 and u1,u2,u3∈Mw∗ı. By (58), (59), and (82), we find(86)wLb−aγTu1,u2,Tu2,u3=wLb−aγTu1,u2−Tu2,u3,0=∫abϕTu1,u2−Tu2,u3′vLb−aγdv=∫abϕKψ,v,u1v+Jψ,v,u2v−Kψ,v,u2v+Jψ,v,u3vLb−aγdv=∫abϕKψ,v,u1v−Jψ,v,u2v+Kψ,v,u2v−Jψ,v,u3vLb−aγdv.
By (m2) and Lebesgue’s theorem, we have for almost allv∈a,b (since u1a=u2a=u3a=u0),(87)Kψ,v,u1v+Jψ,v,u2v−Kψ,v,u2v+Jψ,v,u3v≤L2b−amaxu1−u2,u2−u3.
Therefore,(88)Kψ,v,u1v+Jψ,v,u2v−Kψ,v,u2v+Jψ,v,u3v≤L2b−amax∫abu1−u2′ψ,∫abu2−u3′ψ.
Applying (67) and (68), we get(89)ϕKψ,v,u1v−Jψ,v,u2v+Kψ,v,u2v−Jψ,v,u3vLb−aγ≤ϕL2b−amax∫abu1−u2′ψLb−aγdψ,∫abu2−u3′ψLb−aγdv=ϕLb−amax∫abu1−u2′ψLb−aγdψ,∫abu2−u3′ψb−aγdv≤Lb−aϕmax1b−a∫abu1−u2′ψγdψ,1b−a∫abu2−u3′ψγdv≤Lmax∫abϕu1−u2′ψγdψ,∫abϕu2−u3′ψγdψ=Lmaxwγu1,u2,wγu2,u3.
Therefore,(90)wLb−aγTu1,u2,Tu2,u3=∫abϕKψ,v,u1v−Jψ,v,u2v+Kψ,v,u2v−Jψ,v,u3vLb−aγdv≤Lb−amaxwγu1,u2,wγu2,u3.Theorem 9.
Under the assumptionsm1, m2, and 0<Lb−a<1, integral equation (60) admits a solution u∈GVϕa,b.Proof.
(i) By Lemmas1 and 2, w is strict modular on(91)Mı=u;u:a,b⟶A,ua=u0,and the modular space(92)Mw∗ı=GVϕa,b∩Mı=u∈GVϕa,b,ua=u0,is w-complete. By Theorem 8, the operator T:Mw∗ı×Mw∗ı⟶Mw∗ı is Ciric–Presic type strongly w-contractive. We are left to show that for some γ>0, ∃u∈Mw∗ı, such that wγu,Tu<∞. Clearly, for constant function u0∈Mw∗ı, we have(93)wγ0Tu0,u0,u0=wγTu0,u0−u0,0=∫abϕTu0,u0−u0′γ0dτ=∫abϕTu0,u0′γ0dτ=∫abϕKψ,v,u0+Jψ,v,u0γ0dτ≤L2b−a2γ1γ0∫abϕu0′ψγ1dψ+b−aγ0ϕL2b−au0−u0+γKγ0C2+γJγ0C3.
Sinceu0∈Mw∗ı is a constant function, u0′=0. Therefore,(94)wγ0Tu0,u0,u0≤b−aγ0ϕL2b−au0−u0+γKγ0C2+γJγ0C3<∞,where L2b−aγ1+1+γK+γJ=γ0. By Theorem 4, the integral operator T admits a fixed point and thus problem (60) has a solution.Theorem 10.
Under the assumptionsm1, m2, and 0<Lb−a<1, integral equation (60) is Ulam–Hyers stable.Proof.
Letϵ>0 and u∗∈Mw∗ı for which ∫abϕu∗′τ−Kτ,v,u∗μτ−Kτ,v,u∗στ−w′τ/γdτ≤ϵ. Then, we have(95)∫abϕu∗τ−Tu∗μτ,u∗στ′γdτ≤ϵ⟹wλu∗−Tu∗,u∗,0≤ϵ⟹wλu∗,Tu∗,u∗≤ϵ.
By Theorem8, the operator T:Mw∗ı×Mw∗ı⟶Mw∗ı is Ciric–Presic type strongly w-contractive and so by Proposition 1, T is c−WPPO. By Theorem 7, the fixed-point equation (62) is Ulam–Hyers stable; that is, there exists u∈Mw∗ı such that u=Tu,u and ωλu∗,u≤cϵ. Thus, we have(96)ωλu∗,u≤cϵ⟹ωλu∗−u,0≤cϵ⟹∫abϕu∗′τ−u′τγdτ≤cϵ.We now furnish a numerical example to validate the hypothesis of Theorem9.Example 3.
Consider the integral equation(97)xs=−s8+3s24+14arctans2−12arctans22−s28ln1+s44+∫0s2/2ln1+xv4dv+∫0s/2xv41+xvdv,s∈0,1,for some x∈Mw∗ı, A=0,100, and u0=0. We will apply Theorem 9, to prove the existence of a solution of (97).Proof.
Define the operatorT:Mw∗ı⟶Mw∗ı as(98)Txs=−s8+3s24+14arctans2−12arctans22−s28ln1+s44+∫0s2/2ln1+xv2dv+∫0s/2xv21+xvdv,s∈0,1.
Now, setKs,v,xv=ln1+xv/4,Js,v,xv=xv/41+xv,μs=s2/2,σs=s/2, and ws=−s/8+3s2/4+1/4arctans/2−1/2arctans2/2−s2/8ln1+s4/4. We observe the following:(i)
Letϕt=t for all t∈R+. For u0=v0=9 and γK=1,γJ=1, we see that the functions K and J satisfy (m1).(ii)
By using the Mathematica software, we found that the following inequalities are true for allx,y,z∈0,100 (see Figures 1 and 2):(99)ln1+x4−ln1+y4≤x−y4,(100)y41+y−z41+z≤y−z4.
Thus, we get(101)Ks,v,xv−Ks,v,yv+Js,v,yv−Js,v,zv≤ln1+x−ln1+y4+y41+y−z41+z≤12maxx−y,y−z,and thus, K and J satisfies (m2), with L2=1/2.
Hence, all the conditions of Theorem9 are satisfied. It is evident that integral equation (97) has a unique solution x∈Mw∗ı defined by xs=s2.Figure 1
Inequality in (99).Figure 2
Inequality in (100).


## 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.



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