# Fixed Point Results of Dynamic ProcessDˇϒ,μ0 through FIC-Contractions with Applications

Authors: Amjad Ali; Eskandar Ameer; Muhammad Arshad; Hüseyin Işık; Mustafa Mudhesh
Journal: Complexity
 (2022)

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## Abstract

This article constitutes the new fixed point results of dynamic process D(ϒ,μ0) through FIC-integral contractions of the Ciric kind and investigates the said contraction to iterate a fixed point of set-valued mappings in the module of metric space. To do so, we use the dynamic process instead of the conventional Picard sequence. The main results are examined by tangible nontrivial examples which display the motivation for such investigation. The work is completed by giving an application to Liouville‐Caputo fractional differential equations.


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## Body

## 1. Introduction and Preliminaries

In the recent past, the study of metric fixed point theory untied a portal to a new area of pure and applied mathematics, the fixed point theory and its application. This concept was prolonged by either extending metric space into its extensions or by modifying the structure of the contractions (see [1–7]). The most classical structure known as Banach contraction principle (or contraction) theorem was introduced by a Polish mathematician Banach in 1922 [8]. The applications of fixed points of Banach contraction mappings defined for different kinds of spaces is the guarantee of the existence and uniqueness of solutions of differential and integral type equations. The variety of these nonlinear problems imposes the search for more and better tools, which are recently very remarkable in the literature. One of such tools was recently conveyed by Wardowski [9], where the author originated a new class of contractive mapping called F-contraction.Nadler [10] using the idea of Pompeiu–Hausdorff metric discussed the Banach contraction mappings for set-valued functions rather than single-valued functions. Let Δ,δ be a metric space. For μ1,μ2∈Δ and A,B⊆Δ, define the Pompeiu–Hausdorff metric H^ induced by δ on CBΔ as follows:(1)H^A,B=maxsupμ1∈ADˇμ1,B,supμ2∈BDˇμ2,A,for each A,B∈CBΔ, where CBΔ denotes the set of all nonempty closed bounded subsets of Δ and Dˇμ1,B=infμ2∈Bδμ1,μ2. An element μ∈Δ is called a fixed point of a set-valued mapping, i.e., ϒ:Δ⟶CBΔ, then μ∈ϒμ. Also, denote the family of nonempty compact subsets of Δ by KΔ.Some well-known results are related to this section and hereafter.Lemma 1.
LetA and B be nonempty proximal subsets of a metric space Δ,δ. If α∈A, then(2)δα,B≤HA,B.Lemma 2 (see [11]).
LetΔ,δ be a metric space and a sequence μii∈N in Δ,δ such that(3)limi⟶∞δμi,μi+1=0is not a Cauchy sequence. Then, there exists ε>0 and subsequences of positive integers μij and μlj, μij>μlj>j such that(4)δμij,μlj,δμij+1,μij,δμij,μlj−1,δμij+1,μlj−1,δμij+1,μlj+1⟶ε+,asj⟶+∞.Definition 1 (see [12]).
Letϒ:Δ⟶NΔ be a multivalued mapping and μ0∈Δ be arbitrary and fixed. Define(5)DˇΥ,μ0=μjj∈N∪0:μj∈ϒμj−1,for allj∈N.
Each element ofDˇϒ,μ0 is called a dynamic process of ϒ starting point μ0. The dynamic process μjj∈N∪0 onward be written as μj.Example 1 (see [12]).
LetΔ=C0,1 be a Banach space with a norm μ=supr∈0,1μr, μ∈Δ. Let ϒ:Δ⟶2Δ be such that, for every μ∈Δ, ϒμ is a collection of the functions(6)r↦k∫0rμtdt,k∈0,1,that is,(7)ϒμr=k∫0rμtdt:k∈0,1,μ∈Δ,and let μ0r=r, r∈0,1, then the sequence 1/j!j+1!rj+1 is a dynamic process of ϒ with starting point μ0.A mappingϒ:Δ⟶R is said to be Dˇϒ,μ0-dynamic lower semicontinuous at μ∈Δ, if for every dynamic process μj∈Dˇϒ,μ0 and for every subsequence μji of μj convergent to μ, we get ϒμ≤liminfi⟶∞ϒμji. If ϒ is Dˇϒ,μ0-dynamic lower semicontinuous at each μ∈Δ, then ϒ is said to be Dˇϒ,μ0-dynamic lower semicontinuous. If for every sequence μj⊂Δ and μ∈Δ such that μj⟶μ, we have ϒμ≤liminfj⟶∞ϒμj, then ϒ is known as lower semicontinuous.As of now, Branciari [5] generalized the second well-known contraction of Banach contraction mappings is determined, i.e., let Δ,δ be a metric space and a mapping ϒ:Δ⟶Δ such that(8)∫0δϒμ1,Υμ2φsds≤β∫0δμ1,μ2φsdsfor all μ1,μ2∈Δ, where β∈0,1, φ∈Φ, and Φ is the class of all functions φ:0,+∞⟶0,+∞ which is Lebesgue integrable, summable on each compact subset of 0,+∞ and ∫0εφsds>0 for all ε>0. Then, Υ has a fixed point.The following lemmas are helpful for our main results. We shall also suppose thatφ∈Φ.Lemma 3 (see [6]).
Letμii∈N be a nonnegative sequence in such a way that limi⟶+∞μi=μ. Then,(9)limi⟶+∞∫0μiφsds=∫0μφsds.Lemma 4 (see [6]).
Letμii∈N be a nonnegative sequence. Then,(10)limi⟶+∞∫0μiφsds=0⇔limi⟶+∞μi=0.In 2012, Wardowski [9] initiated the term of F-contraction and implemented on fixed point theorem related with F-contraction. So, with the intent that, he generalizes contraction theorem which is a purely altered from many past results in the literature frame.Definition 2 (see [9]).
Letϒ:Δ⟶Δ is called an ℱ-contraction on a metric space Δ,δ, if there exist ℱ∈∇Ϝ and τ∈R+ in such a way that, δϒμ1,ϒμ2>0 implies(11)τ+ℱδϒμ1,ϒμ2≤ℱδμ1,μ2.
For eachμ1,μ2∈Δ, where ∇Ϝ is the class of all functions ℱ:R+⟶R such thatℱiμ1<μ2 implies ℱμ1<ℱμ2 for all μ1,μ2∈R+.ℱii For each sequence μj of positive real numbers,(12)limj⟶∞μj=0ifflimj⟶∞ℱμj=−∞.ℱiii There is k∈0,1 in such a way that limc⟶0+ckℱc=0.From now, we present some well-defined examples ofℱ-contraction that are listed as follows:ℱa:ℱμ=lnμℱb:ℱμ=lnμ+μℱc:ℱμ=−1/μℱd:ℱμ=lnμ2+μOwing toℱi and (11), clearly, we conclude that every F-contraction ϒ is a contractive mapping. Consequently, every F-contraction is a continuous mapping (see more [13]).The main purpose of this manuscript is to introduce the new concept of dynamic iterative processDˇϒ,μ0 based on FIC-integral contractions and prove some related multivalued fixed point results in the class of metric space. To approximate our main results by tangible examples are also determined. At the end, the work is completed by giving an application to Liouville–Caputo fractional differential equations.


## 2. Main Result

First, we give our main definition.Definition 3.
LetΔ,δ be a metric space, μ0∈Δ, ℱ∈∇Ϝ and φ∈Φ. A set-valued map ϒ:Δ⟶CBΔ is said to be FIC-integral contraction with respect to a dynamic process μi∈Dˇϒ,μ0, if there exists τ:R+⟶R+ such that(13)H^ϒμi,ϒμi+1>0⇒τUμi−1,μi+ℱ∫0H^ϒμi,ϒμi+1φsds≤ℱ∫0Uμi−1,μiφsds,for all i∈N, where(14)Uμi−1,μi=maxδμi−1,μi,Dˇμi−1,Υμi−1,Dˇμi,Υμi,Dˇμi−1,Υμi+Dˇμi,Υμi−12.Remark 1.
For the act of continuing our results, we consider only the dynamic processesμi∈DˇΥ,μ0 satisfying the following structure:(15)δμi,μi+1>0⇒δμi−1,μi>0for eachi∈N.
If the investigated process does not satisfy (15), then there is i0∈N such that(16)δμi0,μi0+1>0and(17)δμi0−1,μi0=0.
Then, we getμi0−1=μi0∈ϒμi0−1 which implies the existence of fixed point due to this consideration of dynamic process that satisfying (15) does not depreciate a generality of our approach.Example 2.
Letℱ:R+⟶R be defined by ℱμ=lnμ. Each set-valued FIC-integral contraction ϒ on a metric space Δ,δ with respect to dynamic process Dˇϒ,μ0 assures that(18)τUμi−1,μi+ℱ∫0H^ϒμi,ϒμi+1φsds≤ℱ∫0Uμi−1,μiφsds.
Upon setting, we have(19)∫0H^ϒμi,ϒμi+1φsds≤e−τUμi−1,μi∫0Uμi−1,μiφsds,for all i∈N, μi∈Dˇϒ,μ0, and ϒμi−1≠ϒμi. In view of the above observations, clearly, for μi0−1,μi0∈Dˇϒ,μ0 such that ϒμi0−1=ϒμi0, the following inequality also holds through Dˇϒ,μ0(20)∫0H^ϒμi0−1,ϒμi0φsds≤e−τUμi0−1,μi0∫0Uμi0−1,μi0φsds,that is, ϒ is a contraction.Theorem 1.
LetΔ,δ be a complete metric space, μ0∈Δ and ϒ:Δ⟶KΔ be a set-valued FIC-integral contraction with respect to the dynamic process μi∈DˇΥ,μ0. Assume thatProof.
In view ofμi∈Dˇϒ,μ0, if there exists i0∈N such that μi0=μi0+1, then the existence of a fixed point is obvious. Therefore, if we let μi∉ϒμi, then Dˇμi,ϒμi>0 for every i∈N. Using (15) and by Lemma 1, one writes(21)ℱ∫0Dˇμi,ϒμiφsds≤ℱ∫0H^ϒμi,ϒμi+1φsds,(22)≤ℱ∫0Uμi−1,μiφsds−τUμi−1,μi=ℱ∫0maxδμi−1,μi,Dˇμi−1,ϒμi−1,Dˇμi,ϒμi,Dˇμi−1,ϒμi+Dˇμi,ϒμi−12.
Moreover, sinceϒμi is compact, we obtain μi+1∈ϒμi such that δμi,μi+1=Dˇμi,ϒμi. Using (21), we have(23)ℱ∫0δμi,μi+1φsds≤ℱ∫0H^ϒμi−1,ϒμiφsds≤ℱ∫0δμi−1,μiφsds−τδμi−1,μi<ℱ∫0δμi−1,μiφsds.
In view of the above observations,δμi,μi+1 is decreasing and hence convergent. We now show that limi⟶∞δμi,μi+1=0. In the light of (D1), there exist σ>0 and i0∈N such that τδμi−1,μi>σ for all i>i0. So, we have(24)ℱ∫0δμi,μi+1φsds≤ℱ∫0δμi−1,μiφsds−τδμi−1,μi≤ℱ∫0δμi−2,μi−1φsds−τδμi−2,μi−1−τδμi−1,μi⋮≤ℱ∫0δμ0,μ1φsds−τδμ0,μ1−⋯−τδμi−1,μi=ℱ∫0δμ0,μ1φsds−τδμ0,μ1+⋅⋅⋅+τδμi0−1,μi0−τδμi0,μi0+1+⋅⋅⋅+τδμi−1,μi≤ℱ∫0δμ0,μ1φsds−i−i0σ.
Let us setλi=∫0δμi,μi+1φsds>0 for i=0,1,2,… and from (24), we see that limi⟶∞ℱλi=−∞. By means of (ℱii), we have(25)limi⟶∞λi=0.
Also, in the light ofℱiii, there is α∈0,1 such that(26)limi⟶∞λiαℱλi=0.
Furthermore, from (24), we can write for all i>i0(27)λiαℱλi−λiαℱλ0≤λiαℱλ0−i−i0σ−λiαℱλ0=−λiαi−i0σ≤0.
Taking limit asi⟶∞ in (27) and using (26), we have(28)limi⟶∞iλiα=0.
Let us perceive that, from (28), there is i1∈ℕ such that iλiα≤1 for all i≥i1. We have(29)λi≤1i1/α.
Now, in order to show thatμi is a Cauchy sequence, we consider j1, j2∈ℕ such that j1>j2≥i1. From (29) and by virtue of metric condition, we have(30)∫0δμj1,μj2φsds≤∫0δμj1,μj1+1φsds+∫0δμj1+1,μj1+2φsds+⋅⋅⋅+∫0δμj2−1,μj2φsds=λj1+λj1+1+⋅⋅⋅+λj2−1=∑l=j1j2−1λl≤∑l=j1∞λl≤∑l=j1∞1l1/α.
In the light of (30) and view of convergence of series ∑l=j1∞1/l1/α, we see that ∫0δμj1,μj2φsds⟶0. Hence, μi is Cauchy sequence in Δ,δ. Furthermore, for the completeness of Δ, there is μ∗∈Δ such that limi⟶∞μi=μ∗. Since ϒ is compact, then we have ϒμi⟶ϒμ∗. By Lemma 1, one writes(31)Dˇμi,ϒμ∗≤H^ϒμi−1,ϒμ∗.
So,Dˇμ∗,Υμ∗=0 and μ∗∈ϒμ∗. Suppose, on the contrary, μ∗∉ϒμ∗. Then, there exist i0∈ℕ and subsequence μik of μi such that Dˇμik+1,ϒμ∗>0 for each ik≥i0 (otherwise, there is i1∈ℕ such that μi∈ϒμ∗ for every i≥i1, which yields that μ∗∈ϒμ∗). By contractive condition, one writes(32)ℱ∫0Dˇμik+1,ϒμ∗φsds≤ℱ∫0H^ϒμik,ϒμ∗φsds≤ℱ∫0Uμik,μ∗φsds−τUμik,μ∗.
Upon lettingk⟶∞ in (32),(33)ℱ∫0Dˇμ∗,ϒμ∗φsds≤ℱ∫0Dˇμ∗,ϒμ∗φsds−τDˇμ∗,μ∗<ℱ∫0Dˇμ∗,ϒμ∗φsds.which is a contradiction. On the other hand, we see that the mapping Δ∋μi↦δμi,ϒμi is Dˇϒ,μ0-dynamic lower semicontinuous, we have(34)∫0Dˇμ∗,ϒμ∗φsds≤liminfn⟶∞∫0Dˇμik,ϒμikφsds≤liminfn⟶∞∫0Dˇμi,ϒμiφsds=0and by virtue of closedness of ϒμ∗ implies that μ∗∈ϒμ∗ which means that μ∗ is a fixed point of ϒ.Remark 2.
If in Theorem1, instead of the contractive condition (13), we assume the following condition(35)H^ϒμi,ϒμi+1>0⇒τUjμi−1,μi+ℱ∫0H^ϒμi,ϒμi+1φsds≤ℱ∫0Ujμi−1,μiφsds,where j∈1,2,3 and(36)U1μi−1,μi=δμi−1,μi,U2μi−1,μi=maxδμi−1,μi,Dˇμi−1,ϒμi−1,Dˇμi,ϒμi,U3μi−1,μi=maxδμi−1,μi,Dˇμi−1,ϒμi−1+Dˇμi,ϒμi2,Dˇμi−1,ϒμi+Dˇμi,ϒμi−12,for all i∈N, μi∈Dˇϒ,μ0, then there exists a fixed point of the mapping Υ with the assumptions (D1) and (D2) on Theorem 1.Corollary 1.
LetΔ,δ be a complete metric space, μ0∈Δ, ℱ∈∇Ϝ, φ∈Φ, and ϒ:Δ⟶KΔ. Assume that there exists τ:R+⟶R+ such that(37)H^ϒμi,ϒμi+1>0⇒τUμi−1,μi−1∫0H^ϒμi,ϒμi+1φsds≤−1∫0Uμi−1,μiφsds,for alli∈N, μi∈Dˇϒ,μ0, where(38)Uμi−1,μi=maxδμi−1,μi,Dˇμi−1,ϒμi−1,Dˇμi,ϒμi,Dˇμi−1,ϒμi+Dˇμi,ϒμi−12.Then, there exists a fixed point of the mappingϒ with the assumptions (D1) and (D2) on Theorem 1.Proof.
If we chooseℱμ=−1/μ, the proof follows from Theorem 1.Corollary 2.
LetΔ,δ be a complete metric space, μ0∈Δ, ℱ∈∇Ϝ, φ∈Φ, and ϒ:Δ⟶KΔ. Assume that there exists τ:R+⟶R+ such that(39)H^ϒμi,ϒμi+1>0⇒τUμi−1,μi+11−exp∫0H^ϒμi,ϒμi+1φsds≤11−exp∫0Uμi−1,μiφsds,for all i∈N, μi∈Dˇϒ,μ0, where(40)Uμi−1,μi=maxδμi−1,μi,Dˇμi−1,ϒμi−1,Dˇμi,ϒμi,Dˇμi−1,ϒμi+Dˇμi,ϒμi−12.Then, there exists a fixed point of the mappingΥ with the assumptions (D1) and (D2) on Theorem 1.Proof.
If we chooseℱμ=1/1−expμ, the proof follows from Theorem 1.
The direct consequence of Theorem1 for single-valued maps is the following.Corollary 3.
LetΔ,δ be a complete metric space, μ0∈Δ, ℱ∈∇Ϝ, φ∈Φ, and ϒ:Δ⟶Δ. Assume that there exists τ:R+⟶R+ such that δϒiμ0,ϒi+1μ0>0 implies(41)τδϒi−1μ0,ϒiμ0+ℱ∫0δϒiμ0,ϒi+1μ0φsds≤ℱ∫0δϒi−1μ0,ϒiμ0φsds,for all i∈N and liminfk⟶l+τk>0 for each l≥0. Suppose also that a mapping Δ∋μ↦δμ,ϒμ is Dˇϒ,μ0-dynamic lower semicontinuous. Then, ϒ has a fixed point.Corollary 4.
LetΔ,δ be a complete metric space, ℱ∈∇Ϝ, φ∈Φ, and ϒ:Δ⟶Δ. Assume that there exists τ:R+⟶R+ such that δϒμ,ϒ2μ>0 implies(42)τδμ,ϒμ+ℱ∫0δϒμ,ϒ2μφsds≤ℱ∫0δμ,ϒμφsds,for all μ∈Δ and liminfk⟶l+τk>0 for each l≥0. Suppose also that a mapping Δ∋μ↦δμ,ϒμ is lower semicontinuous. Then, Υ has a fixed point.Example 3.
LetΔ=0,+∞ with the usual metric, Δ constitutes a complete metric space. Consider a mapping ϒ:Δ⟶KΔ by ϒμ=0,μ/2, μ>0 and τ:R+⟶R+ by(43)τμ=−lnμ,μ∈0,12ln2,μ∈12,∞.
Define dynamic iterative processDˇϒ,μ0: a sequence μi is given by μi=μ0gi−1 for all i∈N with initial point μ0=2 and g=1/2 such that
Continuing the above iterative process, we see that(44)Dˇϒ,μ0=1,12,14,18,…is a dynamic iterative process of ϒ starting from the point μ0=2. Setting φs=1 for all s∈R and ℱs=lns. For μi∈Dˇϒ,μ0 and H^ϒμi,ϒμi+1>0, we have(45)eτμi−1−μi+ℱ∫0μi−1−μi2φsds≤eℱ∫0μi−1−μiφsdseτμi−1−μi+ln∫0μi−1−μi2φsds≤eln∫0μi−1−μiφsdseτμi−1−μieln∫0μi−1−μi2φsds≤eln∫0μi−1−μiφsdseτμi−1−μi∫0μi−1−μi2φsds≤∫0μi−1−μiφsdsμi−1−μi2≤e−τμi−1−μiμi−1−μi,and so(46)τUμi−1,μi+ℱ∫0H^Υμi,Υμi+1φsds≤ℱ∫0Uμi−1,μiφsds.
Hence, all the required hypotheses of Theorem1 are satisfied and hence 0 is a fixed point of ϒ.


## 3. An Application

In this frame of study, we deal with some new aspects of Liouville–Caputo fractional differential equations in module of complete metric space. Several earlier developments on fixed point theory and its applications involving fractional calculus can be found in [14].Define the Liouville–Caputo fractional differential equations based on orderκDˇc,κ by(47)Dˇc,καg=1Γi−κ∫0gg−ti−κ−1αitdt,where i−1<κ<i, i=κ+1, α∈Ci0,+∞, and the collection κ represents positive real number and Γ represents the Gamma function. Let Δ:=CI,R be the space of all continuous real-valued functions on I. And, complete metric space δς:Δ×Δ⟶0,+∞ be given by(48)δςε1,ε2=supa∈Iε1a−ε2a.Now, consider the following fractional differential equations and its integral boundary valued problem:(49)Dˇc,κβg=Lg,βg,where g∈0,1, κ∈1,2 and(50)β0=0,β1=∫0ϑβgdg,ϑ∈0,1,where I=0,1, β∈CI,R and L:I×R⟶R be a continuous function. Let P:Δ⟶Δ be defined by(51)Pvr=1Γκ∫0gg−tκ−1Lt,vtdt−2g2−ϑ2Γκ∫011−tκ−1Lt,vtdt+2g2−ϑ2Γκ∫0ϑ∫0g1g1−t1κ−1Lt1,vt1dt1dt,for v∈Δ and g∈0,1. Now, we start the main result of this section.Theorem 2.
LetL:I×R⟶R be a continuous function, nondecreasing on second variable and there is a nonconstant function τ such that εi∈DˇςΥ,ε0 and g∈0,1 implies(52)Pεi−1r−Pεir≤ΩUεi−1,εir1+τmaxg∈IUεi−1,εir2,where Ω=2κ−1Γκ+1/25κ+2 and(53)Uεi−1,εir=maxεi−1r−εir,εi−1r−ϒεi−1r,εir−ϒεir,εi−1r−ϒεir+εir−ϒεi−1r2.Then, equations (49) and (50) has at least one solution on Δ.Proof.
For everyg∈I and owing to operator P:Δ⟶Δ, one writes
Upon setting, we see that
In the light of above observation, we have
which implies that(54)Pεi−1r−Pεir≤Uεi−1,εir1+τmaxg∈IUεi−1,εir2.
By above virtue, we have(55)δςPεi−1r−Pεir=supa∈IPεi−1r−Pεir≤Uεi−1,εir1+τmaxg∈IUεi−1,εir2.
Furthermore, by contractive condition (13) upon setting of φs=1 for all s∈R and ℱs=−1/s, we have(56)H^ϒεi,ϒεi+1>0⇒τUεi−1,εi+ℱ∫0H^ϒεi,ϒεi+1φsds≤ℱ∫0Uεi−1,εiφsds,for all i∈N, εi∈Dˇςϒ,ε0 and for each given ϵ>0 such that ∫0ϵφsds>0. Thus, all the required hypotheses of Theorem 1 are satisfied, and hence equations (49) and (50) has at least one solution on Δ.Example 4.
Based upon the Liouville–Caputo fractional differential equations based on orderκDˇc,κ. Let us consider the following integral boundary-value problem:(57)Dˇc,32βg=1g+32βg1+βgand(58)β0=0,β1=∫03/4βgdg,ϑ∈0,1,where κ=3/2, ϑ=34, and Lt,vt=1/g+32αg/1+αg. So, the above setting is an example of equations (49) and (50). Hence, here is clearly the pair of equations (57) and (58) has at least one solution.


## 4. Conclusions

In this paper, we have investigated the preexisting results of fixed point for set-valued mappings rather than the conventional mappings. Based upon a Wardowski integral and with a nonnegative Lebesque integrable mapping, we have transformed the conventional theorems of fixed point into the module ofFIC. Instead of the traditional Picard sequence, the dynamic process DˇΥ,μ0 is adopted to iterate the fixed point. Afterwards, the results have been explained by rendering concrete examples, and some foremost corollaries have been deduced from the prime results. Also, we provide illustrative applications to Liouville–Caputo fractional differential equations.



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