# Applicability of Mönch’s Fixed Point Theorem on Existence of a Solution to a System of Mixed Sequential Fractional Differential Equation

Authors: Muath Awadalla
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we study the existence and uniqueness of the solution for a coupled system of mixed fractional differential equations. The main results are established with the aid of “Mönch’s fixed point theorem.” In addition, an applied example that supports the theoretical results reached through this study is included.


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## Body

## 1. Introduction

Fractional calculus has an extended history, going all the way back to Leibniz’s 17th-century explanation of the derivative order in 1965. Mathematicians use fractional calculus to study how derivatives and integrals of noninteger order work and how they change over time. Subsequently, the subject attracted the interest of numerous famous mathematicians, including Fourier, Laplace, Abel, Liouville, Riemann, and Letnikov. For current and wide-ranging analyses of fractional derivatives and their applications, we recommend the monographs [1, 2], and the recently mentioned papers [3, 4].Many problems in various scientific branches can be successfully studied using partial differential equations, such as theoretical physics, biology, viscosity, electrochemistry, and other physical processes see [5–9]. For example, but not limited, the authors in [10] employed the fractional derivative of the ψ-Caputo type in modeling the logistic population equation, through which they were able to show that the model with the fractional derivative led to a better approximation of the variables than the classical model. In addition, the authors in [11] employed the fractional derivative of the ψ-Caputo type and used the kernel Rayleigh, to improve the model again in modeling the logistic population equation.The obvious difference between the ordinary differential equation and the fractional differential equation is that the latter is an equation that contains fractional derivatives and also comes in a relationship so that the definition of the fractional derivative is an integral equation on the other side of this equation. Fractional derivatives have drawn the attention of researchers in various fields of research. One of the main goals of solving these equations is to investigate whether these derivatives will help in the future in improving the accuracy of predicting the values of variables in various mathematical models in all sciences, whether in scientific or human aspects.Before starting this research for solutions to these problems, which are recently considered in the applied sciences, verifying the issue of the existence and uniqueness of such equations is an indispensable thing. To study these conditions, most of the researchers use the most important fixed point theorems in the Banach space, such as the Banach contraction principle and Leray- Schauder theorem see [12–18].In 2016, Aljoudi et al. [19] published a study investigating the existence results for the following boundary value problem (sequential Hadamard type).
(1)DH1ν1+ηDH1ν1−1ξω=φ1ω,ξω,ζω,HD1r1ζω,DH1ν2+ηDH1ν2−1ζω=φ2ω,ξω,HD1r2ξω,ζω,ξ1=0,ξe=IHθ1ζε1,ζ1=0,ζe=IHθ2ξε2,where DH1.,ν1,ν2∈1,2,r1,r2∈0,1 is the Hadamard fractional derivative, and IHθ1 is the Hadamard fractional integral with order θ1,θ2>0, φ1,φ2∈C1,e×ℝ3,ℝ,ε1,ε2∈1,e.In 2017, Ahmad and Ntouyas [20] published a study investigating the existence results for the following initial value problem
(2)DCH1ν1DCH1ν2xt−f1t,xt=f2t,xt,t∈1,b,xt=φt,t∈1−τ,1,DCH1ν2xt=μ∈ℝ,where DCH1νi,νi∈0,1,i=1,2 is the Hadamard fractional derivative, fi∈1,b×C−τ,0,ℝ⟶ℝ,φ∈C1−τ,1,ℝ,xt∈C−τ,0,ℝ,wherextγ=xt+γ,γ∈−τ,0.Many researchers went deeper in their research beyond the issue of verifying the issue of the existence of a solution to such equations and studied the issue of the stability of these solutions, it can be seen in [21, 22]. Furthermore, many specialists in this field have taken an interest in hybrid partial differential equations see [23–26].Newly, interest in fractional calculus has increased from a purely mathematical theory and from an applied point of view in various sciences. Focusing on the theory, there are many experts in this field who have studied the existence of solutions for many types’ fractional differential equations (FDEs) using the most famous fixed-point theories such as Banach’s principle and nonlinear Leary-Schauder alternative. While a few of them tried other theories to examine the existence of solutions to these problems, Derbazi and Baitiche [27] publish one of these scientific papers.The aim of this paper is to investigate the existence of solutions for the following nonlinear sequential fractional differential equation subject to the Dirichlet boundary conditions.(3)CDα1CHDβ1ψt=ςt,ψt,φt,CDα2CHDβ2φt=ξt,ψt,φt,ψa=ψT=0,φa=φT=0,where DCαi,DCHβi are the Caputo and Caputo-Hadamard fractional derivatives of order 0<αi,βi≤1,i=1,2.,a≤t≤T.In this work, we will try to follow the researchers and specialists in this field, by working to prove the existence of a solution to the problem presented above. In which the work will be presented in this format: Section2 contains some basic results for fractional calculus. Section 3 shows an important result for the establishment of our main findings, and after that, we present our main findings. In Section 4, an applied example is obtained illustrating what has been obtained in the theoretical aspect of this manuscript. In Section 5, a conclusion and future work section is introduced.


## 2. Preliminaries

This part is dedicated to presenting some definitions, postulates, and theorems related to the fixed point concept of solutions of differential equations, which will be used to verify the existence of a solution to the system of equations given by Equation (3).Definition 1 (see [7]).
The Hadamard fractional integral of orderν for a continuous function φ is defined as
(4)IHνφω=1Γν∫aωlnωτq−11τφτdτ,ν>0.Definition 2 (see [7]).
The Hadamard fractional derivative of orderν>0 for a continuous function φ:a,∞⟶ℝ is defined as
(5)DHνφω=δnIHνφω,
n−1<ν<n,n=ν+1, where δ=ωd/dω,ν denotes the integer part of the real number v.Definition 3 (see [5]).
The Caputo-Hadamard fractional derivative of orderν for at least n−times differentiable function φ:a,∞⟶ℝ is defined as
(6)DCHνφω=1Γn−ν∫aωlnωτn−ν−1δngττdτ.Lemma 4 (see [20]).
Letu∈Cδna,T,ℝ,whereCδna,T=u:a,T⟶ℝ:δn−1u∈Ca,T, then IHνDHνuω=uω−∑k=1ncklnω/aν−k, and
(7)IHνDCHνuω=uω−∑k=0n−1cklnωak.Denote the Banach space of all continuous functionsz from a,T into Q by Ca,T,Q,accompanied by the norm:z∞=supa≤t≤Tzt.Definition 5 (see [28]).
The Kuratowski measure of noncompactnessk·.
Defined on bounded setU of Banach space Q is
(8)kU≔infr>0:U=∪i=1mUianddiamUi≤rfor1≤i≤m.Lemma 6 (see [28]).
Given the Banach spaceQ with U, V are two bounded proper subsets of Q, then the following properties hold true
(I)
IfU⊂V,thenkU≤kV;(II)
kU=kU¯=kconv¯U;(III)
Uisrelativlycompact⇄kU=0;(IV)
kδU=δkU,δ∈ℝ;(V)
kU∪V=maxkU,kV;(VI)
kU+V≤kU+kV,U+V=xx=u+v,u∈U,v∈V;(VII)
kU+y=kU,∀y∈Q.Lemma 7 (see [29]).
Given an equicontinuous and bounded setS⊂Ca,T,Q, then the function.
ω↦kSω is continuous on a,T, kCS=maxω∈a,TkSω, and
(9)k∫aTxτdτ≤∫aTkxτdτ,Sτ=xτ:x∈S.Definition 8 (see [3]).
Given the functionψ:a,T×Q⟶Q,ψ satisfy the Carathéodory conditions, if the following conditions applies:
(I)ψω,z is measurable in ω for z∈Q;
(II)ψω,z is continuous in z∈Q for ω∈a,T.Theorem 9 (Mönch’s fixed point theorem [4]).
Given a bounded, closed, and convex subsetΩ⊂Q, such that 0∈Ω, let also Τ be a continuous mapping of Ω into itself.
IfS=conv¯ΤS,orS=ΤS∪0,thenkS=0, satisfied ∀S⊂Ω, then Τ has a fixed point.


## 3. Existence Results

LetB=ψt,φtψ,φ∈Ca,T,ℝ×Ca,T,ℝ. Obviously, the defined set B is a Banach space with ψ,φB=ψ∞+φ∞.The measurable functionsψ,φ∈Ca,T,ℝ×Ca,T,ℝ are said to be solutions of problem Equation (3) if they satisfy problem (3) associated with the given boundary conditions, our next lemma will introduce the solutions of Equation (3), which indeed needed to investigate the existence results.Lemma 10.
Ifp,q∈Ca,T,ℝ, then the solution of
(10)DCα1DCHβ1ψt=pt,DCα2DCHβ2φt=qt,ψa=ψT=0,φa=φT=0.
With0<αi,βi≤1,i=1,2.a≤t≤T, is given by
(11)ψt=1Γα1Γβ1∫at∫arlntrβ1−1r−xα1−1pxdxdrr−lnt/anT/aβ11Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1pxdxdrr,(12)φt=1Γα2Γβ2∫at∫arlntrβ2−1r−xα2−1qxdxdrr−lnt/anT/aβ21Γα2Γβ2∫aT∫arlnTrβ2−1r−xα2−1qxdxdrr.Proof.
ApplyIRLαi,i=1,2 to Equation (10), respectively, implies
(13)DCHβ1ψt=IRLα1pt+c0,c0∈ℝ,(14)DCHβ2φt=IRLα2qt+d0,d0∈ℝ.
Now, applyIHβi,i=1,2 to Equation (13) and Equation (14), respectively, implies
(15)ψt=IHβ1IRLα1pt+c0lnt/aβ1Γβ1+1+c1,c0,c1∈ℝ,(16)φt=IHβ2IRLα2qt+d0lnt/aβ2Γβ2+1+d1,d0,d1∈ℝ.
Using the conditionsψa=0,φa=0 in Equation (15) and Equation (16), respectively, yields c1,d1 are both zeros. Again the conditions ψT=0,φT=0 in Equation (15) and Equation (16), respectively, give
(17)c0=−Γβ1+1lnT/aβ1IHβ1IRLα1pT,d0=−Γβ2+1lnT/aβ2IHβ2IRLα2qT.
Back substitutingci,di,i=1,2 obtained above in equations Equation (15) and Equation (16), we get
ψt=IHβ1IRLα1pt−lnt/a/lnT/aβ1IHβ1IRLα1pT, and φt=IHβ2IRLα2qt−lnt/a/lnT/aβ2IHβ2IRLα2qT. The proof is completed.To begin formulating theoretical results regarding the problem of having a solution to the system of fractional differential equations given by Equation (3). We will force the following conditions to be hold true.(C1). Assume the functionsς,ξ:a,T×ℝ2⟶ℝ satisfy Carathéodory conditions.(C2).∃lς,lξ∈L∞a,T,ℝ+, and there exist a nondecreasing continuous function ϑς,ϑξ:ℝ+⟶ℝ+, such that ,∀t∈a,T,∀ψ,φ∈B, we have
(18)ςt,ψ,φ∞≤lςtϑςψ∞+φ∞,ξt,ψ,φ∞≤lξtϑξψ∞+φ∞.(C3). LetS⊂B×B, be a bounded set, and ∀t∈a,T, then
(19)κςt,S≤lςtκS,κξt,S≤lξtκS.Also, one can use the fact thatr−aα1≤T−aα1, to deduce that
(20)Ξ1=supa≤t≤T1Γα1Γβ1∫at∫arlntrβ1−1r−xα1−1dxdrr+lnt/alnT/aβ11Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1dxdrr≤2T−aα1lnT/aα1Γα1+1Γβ1+1,Ξ2=supa≤t≤T1Γα2Γβ2∫at∫arlntrβ2−1r−xα2−1dxdrr+lnt/alnT/aβ21Γα2Γβ2∫aT∫arlnTrβ2−1r−xα2−1dxdrr≤2T−aα2lnT/aα2Γα2+1Γβ2+1.Theorem 11.
Assume that the conditions (C1), (C2), and (C3) are satisfied. IfmaxΞ1l¯ς,Ξ2l¯ξ<1, then there exist at least one solution for the boundary value problem Equation (3) on a,T.Proof.
Beginning with introducing the following continuous operatorϒ:B⟶B, asϒ=ϒ1ψ,φt,ϒ2ψ,φt, where
(21)ϒ1ψ,φt=1Γα1Γβ1∫at∫arlntrβ1−1r−xα1−1ςx,ψx,φxdxdrr−lnt/anT/aβ11Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1ςx,ψx,φxdxdrr,ϒ2ψ,φt=1Γα2Γβ2∫at∫arlntrβ2−1r−xα2−1ξx,ψx,φxdxdrr−lnt/anT/aβ21Γα2Γβ2∫aT∫arlnTrβ2−1r−xα2−1ξx,ψx,φxdxdrr.
According to the conditions (C1) and (C2), the operatorϒ is well defined. Then, the following operator equation can be an equivalent equation to the fractional equations given by Equation (11) and Equation (12)
(22)ψ,φ=ϒψ,φ.
Subsequently, proving the existence of the solution to Equation (22) is equivalent to proving the existence of a solution to Equation (3).
LetΘε=ψ,φ∈B:ψ,φ≤ε,ε>0 be a closed bounded convex ball in B with ε≥l¯ςΞ1ϑςε+l¯ξΞ2ϑξε,where l¯ς=supa≤t≤Tlςt,.
For the possibility of applying Mönch’s fixed point theorem, we will proceed in the proof in the form of four steps, and thus, we achieve the desired goal by proving the existence of a solution to the equation given in Equation (3).
Firstly, we show thatϒΘε⊂Θε, for this, we lett∈a,T, and for any ψ,φ∈Θε, we have
(23)ϒ1ψ,φ∞≤1Γα1Γβ1∫at∫arlntrβ1−1r−xα1−1ςx,ψx,φx∞dxdrr+lnt/alnT/aβ11Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1ςx,ψx,φx∞dxdrr.
Based on (C2),∀t∈a,T, observe that
(24)ςt,ψt,φt∞≤lςtϑςψt∞+φt∞≤l¯ςϑςε,then
(25)ϒ1ψ,φ∞≤1Γα1Γβ1∫at∫arlntrβ1−1r−xα1−1lςtϑςψt∞+φt∞dxdrr+lnt/alnT/aβ11Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1lςtϑςψt∞+φt∞dxdrr,≤l¯ςϑςψt∞+φt∞supa≤t≤T1Γα1Γβ1∫at∫arlntrβ1−1r−xα1−1dxdrr+lnt/alnT/aβ11Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1dxdrr≤l¯ςΞ1ϑςε.
Similarly,(26)ϒ2ψ,φ∞≤1Γα2Γβ2∫at∫arlntrβ2−1r−xα2−1lξtϑξψt∞+φt∞dxdrr+lnt/anT/aβ21Γα2Γβ2∫aT∫arlnTrβ2−1r−xα2−1lξtϑξψt∞+φt∞dxdrr≤l¯ξΞ2ϑξε.
Equation (25) and Equation (26) imply that
(27)ϒψ,φB=ϒ1ψ,φ∞+ϒ2ψ,φ∞≤l¯ςΞ1ϑςε+≤l¯ξΞ2ϑξε≤ε.
This proves thatϒΘε⊂Θε.
Secondly, we need to show the continuity forϒ to see this, we take the sequence un=ψn,φn∈Θε, such that un⟶u=ψ,φasn⟶∞..
Owing to the Carathéodory continuity ofς, it is obvious that
(28)ς·,ψn·,φn·⟶ς·,ψ·,φ·asn⟶∞.
Keeping in mind was given in (C2), one can deduce that(29)lntrβ1−1r−xα1−1ςr,ψnr,φnr−ςr,ψr,φr∞≤l¯ςϑςεlntrβ1−1r−xα1−1.
Together with the Lebesgue dominated convergence theorem and the fact that the function
r↦l¯ςϑςεlnt/rβ1−1r−xα1−1 is the Lebsegue integrable on a,T, we have
(30)1Γα1Γβ1∫at∫arlntrβ1−1r−xα1−1ςr,ψnr,φnr−ςr,ψr,φr∞dxdrr+lnt/alnT/aβ11Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1ςr,ψnr,φnr−ςr,ψr,φr∞dxdrr⟶0asn⟶∞.
Yields toϒ1ψn,φnt−ϒ1ψ,φt∞⟶0asn⟶∞.∀t∈a,T, we get
(31)ϒ1ψn,φn−ϒ1ψ,φ∞⟶0asn⟶∞,
that is the operatorϒ1 is continuous.
In a like manner, we have(32)ϒ2ψn,φn−ϒ2ψ,φ∞⟶0asn⟶∞,
Combining (31) and (32), we obtain
(33)ϒψn,φn−ϒψ,φ∞⟶0asn⟶∞.
From equation (33), we conclude that the operator ϒ is continuous.
Third, to verify the equicontinuity for the operatorϒ, let t1,t2∈a,T,t1<t2, and for any ψ,φ∈Θε, then
(34)ϒ1ψ,φt2−ϒ1ψ,φt1∞≤1Γα1Γβ1∫at1∫arlnt2rβ1−1−lnt1rβ1−1r−xα1−1ςx,ψx,φx∞dxdrr+1Γα1Γβ1∫t1t1∫arlnt2rβ1−1r−xα1−1ςx,ψx,φx∞dxdrr+lnt2/aβ1−lnt2/aβ1lnT/aβ1Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1ςx,ψx,φx∞dxdrr,≤l¯ςtϑςε×1Γα1Γβ1∫at1∫arlnt2rβ1−1−lnt1rβ1−1r−xα1−1dxdrr+1Γα1Γβ1∫t1t1∫arlnt2rβ1−1r−xα1−1dxdrr+lnt2/aβ1−lnt1/aβ1lnT/aβ1Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1dxdrr⟶0ast1⟶t2.
Similarly, we get(35)ϒ2ψ,φt2−ϒ2ψ,φt1∞≤l¯ξtϑξε×1Γα2Γβ2∫at1∫arlnt2rβ2−1−lnt1rβ2−1r−xα2−1dxdrr+1Γα2Γβ2∫t1t1∫arlnt2rβ2−1r−xα2−1dxdrr+lnt2/aβ2−lnt2/aβ2lnT/aβ2Γα2Γβ2∫aT∫arlnTrβ2−1r−xα2−1dxdrr⟶0ast1⟶t2.
Note that the R.H.S’s of the above inequalities of Equation (34) and Equation (35) are free of ψ,φ∈Θε, which implies that ϒ is equicontinuous and bounded.
Fourth and finally, we need to satisfy Mönch’s hypothesis, so we letU=U1∩U2,.
whereU1,U2⊆Θε. Moreover, U1,U2 are assumed to be bounded and equicontinuous, such that
(36)U1⊂conv¯ϒ1U1∪0,andU2⊂conv¯ϒ2U2∪0.
Thus, the functionsI1t=κU1t,I2t=κU2t are continuous on a,T.
Based on lemma Equation(10), lemma Equation (11), and (C3), we get
(37)I1t=κU1t≤κconv¯ϒ1U1t∪0≤κϒ1U1t≤κ1Γα1Γβ1∫at1∫arlnt2rβ1−1−lnt1rβ1−1r−xα1−1ςx,ψx,φx∞dxdrr+1Γα1Γβ1∫t1t1∫arlnt2rβ1−1r−xα1−1ςx,ψx,φx∞dxdrr+lnt2/aβ1−lnt2/aβ1lnT/a1Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1ςx,ψx,φx∞dxdrr:ψ,φ∈U1≤1Γα1Γβ1∫at1∫arlnt2rβ1−1−lnt1rβ1−1r−xα1−1κςx,U1xdxdrr+1Γα1Γβ1∫t1t1∫arlnt2rβ1−1r−xα1−1κςx,U1xdxdrr+lnt2/aβ1−lnt2/aβ1lnT/a1Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1κςx,U1xdxdrr≤1Γα1Γβ1∫at1∫arlnt2rβ1−1−lnt1rβ1−1r−xα1−1lςxκU1xdxdrr+1Γα1Γβ1∫t1t1∫arlnt2rβ1−1r−xα1−1lςxκU1xdxdrr+lnt2/aβ1−lnt2/aβ1lnT/a1Γα1Γβ1∫aT∫arlnTrβ1−1r−xα1−1lςxκU1xdxdrr≤Ξ1l¯ςI1∞.
That is
I1≤Ξ1l¯ςI1, but it is assumed that maxΞ1l¯ς,Ξ2l¯ξ<1, which implies that I1∞=0, i.e.,
(38)I1t=0,∀t∈a,T.
In a like manner, we haveI2t=0,∀t∈a,T. So κUt≤κU1t=0 and.
κUt≤κU2t=0, which implies that Ut is relatively compact in B×B. Now, Arzela-Ascoli is applicable, which means that U is relatively compact in Θε., and therefore, using theorem 9, we deduce that the operator ϒ has a fixed point ψ,φ (solution of the problem Equation (3)) on Θε. And that ends the proof.


## 4. Example

In this section, we provide an applied example that supports the theoretical results reached through this study.Defineψ0=ψ=ψ1,ψ2,⋯,ψn,⋯:limn⟶∞ψn=0, it is obvious that z0 is a Banach space withψ∞=supn≥1ψn. For this, we consider the following boundary value problem:
(39)DC0.5DCH0.75ψt=ςt,ψt,φt,t∈1,3,DC0.6DCH0.9φt=ξt,ψt,φt,t∈1,3,ψ1=ψ3=0,0,0,⋯,0,⋯,φ1=φ3=0,0,0,⋯,0,⋯.Here,α1=0.5, β1=0.75, α2=0.6, β2=0.9, a=1, and T=3.Now, let us take for example(40)ςt,ψt,φt=1lnt+1014n+ln1+ψn+φn,n≥1,ξt,ψt,φt=t101n4+tan−11+ψn+φn,n≥1.∀t∈1,3,withψnn≥1,φnn≥1∈ψ0, assumption (C1) of theorem 11 is satisfied. Furthermore,
(41)ςt,ψ,φ∞≤1lnt+1014n+ln1+ψn+φn∞≤1lnt+10ψ+1=lςtϑςψ.Similarly,
(42)ξt,ψ,φ∞≤t101n4+tan−11+ψn+φn∞≤t10ψ+1=lξtϑξψ.That is (C2) of theorem11 is satisfied as well.Next, if we consider the bounded subsetS⊂ψ0×ψ0, we obtain
(43)κςt,S≤lςtκS,κξt,S≤lξtκS,where in our case, we have lςt=1/lnt+9,lξt=t/10; the latter two inequalities show that the condition (C2) of the theorem 11 is satisfied.Finally, we calculate(44)l¯ς=110,Ξ1≤2T−aα1lnT/aα1Γα1+1Γβ1+1=3.6411,l¯ξ=310,Ξ2≤2T−aα2lnT/aα2Γα2+1Γβ2+1=0.9376.Then,maxΞ1l¯ς,Ξ2l¯ξ=max0.3611,0.28128=0.3611<1. So all conditions of theorem 11 satisfied, that is the problem Equation (39) has at least one solutionψ,φ∈C1,3,ψ0×C1,3,ψ0.


## 5. Conclusion

In the current paper, we studied the existence and uniqueness of solution for a coupled system of a mixed fractional differential equations. The main results are established by the aid “Mönch’s fixed point theorem.” In addition, an applied example that supports the theoretical results reached through this study is included. For future work, more investigations can be performed for such a system by applying another type of fractional derivatives to verify the existence and uniqueness issue, stability via Ulam-Hyeres technique is also possible to be verified.



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