# Ulam Type Stability for a Coupled System of Boundary Value Problems of Nonlinear Fractional Differential Equations

Authors: Aziz Khan; Kamal Shah; Yongjin Li; Tahir Saeed Khan
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We discuss existence, uniqueness, and Hyers-Ulam stability of solutions for coupled nonlinear fractional order differential equations (FODEs) with boundary conditions. Using generalized metric space, we obtain some relaxed conditions for uniqueness of positive solutions for the mentioned problem by using Perov’s fixed point theorem. Moreover, necessary and sufficient conditions are obtained for existence of at least one solution by Leray-Schauder-type fixed point theorem. Further, we also develop some conditions for Hyers-Ulam stability. To demonstrate our main result, we provide a proper example.


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## Body

## 1. Introduction

In last few decades, FODEs become area of interest for the researcher because of high quality accuracy and usability in various fields of science and technology. A lot of physical and natural phenomena can be modeled through FODEs which provide better result than integer order differential equations. Due to this, FODEs are regarded as a special tool for molding. Numerous applications of FODEs can be studied in various disciplines like chemical technology, viscoelasticity, industrial robotics, mathematical economy, turbulent filtration in porous media, fractals theory, ecology, economics, plasma physics, metallurgy, electromagnetic theory, biology, signal and image processing, control theory, electric technology, chemical reaction design, potential theory, radio physics, aerodynamics, pharmacokinetics, and so on; further details are available in literature [1–7]. In last few decades, the existence theory has been given great attention by the researchers. In the concerned theory, they studied existence, uniqueness, and multiplicity of solutions by using different techniques of nonlinear analysis. Therefore, theory on existence and uniqueness of solutions to nonlinear FODEs has been explored very well; see [8–12]. Systems of FODEs have been considered in large numbers of research articles, because most of physical, biological, and chemical phenomena can be modeled in the form of systems of FODEs. For example, Su [13] studied existence of solutions for coupled system of fractional differential equations with two-point boundary value problems given as(1)Dθ1μ1z=F1z,μ2z,Dςμ2z,Dθ2μ1z=F2z,μ1z,Dνμ1z,0<z<1,μ10=μ11=μ20=μ21=0,where Dθ1,Dς,Dθ2, and Dν denote Riemann-Liouville derivatives, 1<θ1,θ2≤2, F1,F2:[0,1]×R×R→R are continuous functions, and μ,ν satisfy θ1-ς and θ2-ν≥1. Wang et al. [14] investigate existence and uniqueness of positive solutions to a coupled system of fractional differential equations with three-point boundary conditions. The corresponding problem is given as follows: (2)Dθ1μ1z=F1z,μ2z,Dθ2μ2z=F2z,μ1z,0<z<1,μ10=0,μ20=0,μ11=aμ1η,μ21=bμ2η,where functions F1,F2:[0,1]×[0,∞)→[0,∞) are continuous, 1<θ1,θ2≤2,0≤a,b≤1,0<η<1, and Dθ1,Dθ2 represent Riemann-Liouville derivatives. Liu et al. [15] studied existence of positive solutions to a coupled system of nonlinear FODEs with integral boundary conditions. The considered problem is given in the following sequel: (3)Dθ1μ1z=F1z,μ1z,μ2z,Dθ2μ2z=F2z,μ1z,μ2z,0<z<1,μ10=0,μ11=∫01ϕzμ1zdz,0<z<1μ20=0,μ21=∫01φzμ2zdz,0<z<1,where Dθ1,Dθ2 are in sense of Riemann-Liouville derivatives, F1,F2:(0,1)×(0,+∞)×(0,+∞)→(0,+∞), and ϕ,φ∈L1(0,1) are nonlocal functions. The existence and uniqueness of solutions of FODEs are an active area of research for the last few decades. For some remarkable work, we refer the reader to [8, 9, 13, 16–20].Another qualitative aspect which is very important from the numerical and optimization point of view is devoted to stability analysis of FODEs. The stability of fractional differential equations has gained great attention from the researchers very recently. Different kinds of stability include exponential, Mittag-Leffler, and Lyapunov stabilities; see [21–23]. One of the most relaxed methods for stability for functional equations was introduced by Ulam [24] and Hyers [25] which is known as Hyers-Ulam stability. The aforesaid stability has been very well investigated for ordinary differential and integral equations as well as functional equations; see [26–29]. But for FODEs, the concerned stability is not properly investigated. Very few papers can be found in literature in which some initial and boundary value problems of FODEs have been considered; see [7, 23, 30–32].Motivated by the aforementioned contributions of researchers, we discuss the existence and uniqueness of solutions for coupled system of nonlinear FODEs with boundary conditions involving fractional integral and derivative. Further, we also investigate the Hyers-Ulam stability for the proposed problem designed by(4)Dθ1μ1z=F1z,μ1z,μ2z0<z<1,Dθ2μ2z=F2z,μ1z,μ2z,0<z<1,μ10=μ1′0=μ1′′0=⋯=μ1n-20=μ11=0,μ20=μ2′0=μ2′′0=⋯=μ2n-20=μ21=0,where the derivative D is in sense of Riemann-Liouville, n-1<θ1,θ2≤n,n≥2, and the functions F1,F2:I×R×R→R are continuous. We use Perov’s fixed point theorem [33] and Leray-Schauder fixed point theorem to develop some results for existence of at least one solution for our proposed coupled nonlinear FODEs with boundary conditions. Further, we establish some conditions for Hyers-Ulam type stability to the considered problem. The whole analysis is then demonstrated by providing a proper example.


## 2. Preliminaries

Here we provide some results and definitions for our proposed coupled nonlinear FODEs with boundary conditions from literature [1–3].Definition 1.
The fractional integral of orderθ1∈R+ of a function F:(0,∞)→R is defined by (5)Iθ1Fz=1Γθ1∫0zz-ħθ1-1Fħdħ,provided that the integral on right is converging.Definition 2.
The Riemann-Liouville fractional order derivative of a functionF:(0,∞)→R is defined by (6)Dθ1Fz=1Γn-θ1dndtn∫0zz-ħn-θ1-1Fħdħ,where n=[θ1]+1 and [θ1] represents the integer part of θ1, provided that the right side is pointwise defined on (0,∞).Lemma 3.
The following result holds for fractional derivative and integral:(7)Iθ1Dθ1Fz=Fz+c1zθ1-1+c2zθ1-2+c3zθ1-3+⋯+cnzθ1-n,for arbitrary ci∈R,i=1,2,…,n.Lemma 4 (see [20]).
Letχ be a Banach space with P⊆χ closed and convex. Let ð be a relatively open subset of P with 0∈ð and T:ð¯→ð be a continuous and compact (completely continuous) mapping. Then either (1)
the mappingT has a fixed point in ð¯ or(2)
there existμ1∈∂ð and λ∈(0,1) with λ=λTμ1.Definition 5.
For a nonempty setZ, a mapping d:Z×Z→Rn is called a generalized metric on Z if the following conditions hold: (M1)
dμ1,μ2=0Rn⇔μ1=μ2,∀μ1,μ2∈X.(M2)
d(μ1,μ2)=d(μ2,μ1),∀μ1,μ2∈X, (symmetric property).(M3)
d(x,y)≤d(x,μ2)+d(μ2,μ1)+d(μ1,y),∀x,y,μ1,μ2∈X (tetrahedral inequality).
Note. The properties such as convergent sequence, cauchy sequence, and open/closed subset are the same for generalized metric spaces as held for the usual metric spaces.Definition 6.
For ann×n matrix A, the spectral radius is defined by ρ(A)=max⁡{ηi,i=1,2,…,n}, where ηi,(i=1,2,…,n) are the eigenvalues of matrix A.Lemma 7 (see [33]).
Let(Z,d) be a complete generalized metric space and let T:Z→Z be an operator such that there exists a matrix A∈M with dTμ1,Tμ2≤Adμ1,μ2,forallμ1,μ2∈Z. If ρ(A)<1, then T has a fixed point μ∗∈Z; further for any μ0 the iterative sequence μn+1=Tμn converges to μ0.Definition 8.
Consider a Banach spaceW1×W2 such that T1,T2:W1×W2→W1×W2 be two operators. Then the operatorial system provided by(8)μ1z=T1μ1,μ2z,μ2z=T2μ1,μ2zis called Hyers-Ulam stable if we can find Λi(i=1,2,3,4)>0, such that, for each εi(i=1,2)>0 and for each solution (μ1∗,μ2∗)∈E of the inequalities given by(9)μ1∗-T1μ1∗,μ2∗∞≤ε1,μ2∗-T2μ1∗,μ2∗∞≤ε2,there exists a solution (μ1¯,μ2¯)∈E of system (8) which satisfies(10)μ1∗-μ1¯∞≤Λ1ε1+Λ2ε2,μ2∗-μ2¯∞≤Λ3ε1+Λ4ε2.Theorem 9 (see [26]).
Considering a Banach spaceW1×W2 with T1,T2:W1×W2→W1×W2 being two operators such that(11)T1μ1,μ2-T1μ1∗,μ2∗∞≤Λ1μ1-μ1∗∞+Λ2μ2-μ2∗∞,T2μ1,μ2-T2μ1∗,μ2∗∞≤Λ3μ1-μ1∗∞+Λ4μ2-μ2∗∞,∀μ1,μ2,μ1∗,μ2∗∈W1×W2,and if the matrix Λ1Λ2Λ3Λ4 converges to zero, then the operatorial system (8) is Hyers-Ulam stable.Lemma 10.
An equivalent Fredholm integral representation of the system of boundary value problems (4) is given by(12)μ1z=∫01Gθ1z,ħF1ħ,μ1ħ,μ2ħdħ,μ2z=∫01Gθ2z,ħF2ħ,μ1ħ,μ2ħdħ,where Gθ1,Gθ2 are Green’s functions given by(13)Gθ1z,ħ=z1-ħθ1-1Γθ1,0≤z≤ħ≤1,z1-ħθ1-1Γθ1-z-ħθ1-1Γθ1,0≤ħ≤z≤1,Gθ2z,ħ=z1-ħθ2-1Γθ2,0≤z≤ħ≤1,z1-ħθ2-1Γθ2-z-ħθ2-1Γθ2,0≤ħ≤z≤1.Proof.
Applying the operatorIθ1 on the first equation of (4) and using Lemma 3, we have(14)μ1z=-Iθ1F1z,μ1z,μ2z+c1zθ1-1+c2zθ1-2+c3zθ1-3+⋯+cnzθ1-n,ci∈R,fori=1,2,…,n.The boundary conditions μ1(0)=μ1′(0)=μ1′′(0)=⋯=μ1(n-2)(0)=μ1(1)=0 and they yield cn=⋯=c2=0 due to singularity and c1=Iθ1F1(1,μ1(1),μ2(1)). Hence, (14) takes the form(15)μ1z=zθ1-1Γθ1∫011-ħθ1-1F1ħ,μ1ħ,μ2ħdħ-1Γθ1∫0tz-ħθ1-1F1ħ,μ1ħ,μ2ħdħ=∫01Gθ1z,ħF1ħ,μ1ħ,μ2ħdħ.Similarly, by the same process with the second equation of the system, we obtain the second part of (12).Lemma 11 (see [18]).
Green’s functionG=(Gθ1,Gθ2) of system (12) has the following properties: (C1)
G(z,ħ) is continuous function on the unit square for all (z,ħ)∈[0,1]×[0,1].(C2)
G(z,ħ)≥0 for all z,ħ∈[0,1] and G(z,ħ)>0 for all z,ħ∈(0,1).(C3)
max0≤z≤1G(z,ħ)=G(1,ħ),∀ħ∈[0,1].(C4)
There exists a constantγ∈(0,1) such that (16)minz∈θ,1-θGz,ħ≥γħG1,ħforθ∈0,1,ħ∈0,1whereγ=min⁡γθ1=zħθ1-1,γθ2=zħθ2-1.


## 3. Existence and Hyers-Ulam Stability

DefineU={μ1(z)∣μ1(z)∈C[0,1]} endowed with the Chebyshev norm μ1∞=maxt∈[0,1]μ1z. Further, define the norms (μ1,μ2)W1×W2=μ1∞+μ2∞ and μ1,μ2W1×W2=max⁡{μ1∞,μ2∞}. Then, the product spaces (W1×W2,·W1×W2),(W1×W2,·W1×W2) are Banach spaces. Define the cones P,K⊂W1×W2 by P={(μ1,μ2)∈W1×W2:μ1(z),μ2(z)≥0,∀z∈[0,1]} and K={(μ1,μ2)∈P:minz∈J[μ1(z)+μ2(z)]≥γ(μ1,μ2)W1×W2}, where J=[θ,1-θ],θ∈(0,1).Lemma 12.
Assume thatF1,F2:[0,1]×R×R→R are continuous. Then (μ1,μ2)∈W1×W2 is a solution of (12), if and only if (μ1,μ2)∈W1×W2 is a solution of system of Fredholm integral equations (4).Proof.
The proof of Lemma12 is similar to proof of Lemma 3.1 in [18].DefineT:W1×W2→W1×W2 by(17)Tμ1,μ2z=∫01Gθ1z,ħF1ħ,μ1ħ,μ2ħdħ,∫01Gθ2z,ħF2ħ,μ1ħ,μ2ħdħ=T1μ1,μ2z,T2μ1,μ2z.By Lemma 12 the problem of existence of solutions of the integral equations (12) coincides with the problem of existence of fixed points of T.Lemma 13.
Assume thatF1,F2:[0,1]×[0,∞)×[0,∞)→[0,∞) are continuous. Then T(P)⊂P and T(K)⊂K, where T is defined by (17).Proof.
The relationT(P)⊂P easily follows from the properties (C1) and (C2) of Lemma 11 and all we need to show is that T(K)⊂K holds. For (μ1,μ2)∈K, we have T(μ1,μ2)∈P and in view of property (C4) of Lemma 11, for all z∈J, we obtain(18)T1μ1z,μ2z=∫01Gθ1z,ħF1ħ,κ1ħ,μ2ħdħ≥γθ1∫01Gθ11,ħF1ħ,μ1ħ,μ2ħdħ.Hence, it follows that (19)minz∈J⁡T1μ1z,μ2z≥γθ1T1μ1,μ2∞,∀z∈J.Similarly, we obtain (20)minz∈J⁡T2μ1z,μ2z≥γθ2T2μ1,μ2∞,∀z∈J.It follows that (21)minz∈J⁡T1μ1z,μ2z+T2μ1z,μ2z≥γT1μ1,μ2,T2μ1,μ2W1×W2,∀z∈J,which implies that T(μ1,μ2)∈K.Lemma 14.
Assume thatF1,F2:[0,1]×R×R→R are continuous; then T:P→P is completely continuous.Proof.
We omit the proof, because it is similar to the proof of Lemma3.2 in [18].Lemma 15.
Assume thatF1 and F2 are continuous on [0,1]×R×R→R and there exist ϕi,ψi(i=1,2):(0,1)→[0,∞) such that the following hold: (C1∗)
F1z,μ1,μ2-F1z,μ1¯,μ2¯≤ϕ1zμ1-μ1¯+ψ1zμ2-μ2¯,z∈(0,1), for μ1,μ2,μ1¯,μ2¯≥0;(C2∗)
F2z,μ1,μ2-F2z,μ1¯,μ2¯≤ϕ2zμ1-μ1¯+ψ2zμ2-μ2¯, for z∈(0,1),μ1,μ2,μ1¯,μ2¯≥0.(C3∗)
ρ(A)<1, where the matrix A∈M2,2(R+) is defined by (22)A=∫01Gθ11,ħϕ1ħdħ∫01Gθ11,ħψ1ħdħ∫01Gθ21,ħϕ2ħdħ∫01Gθ21,ħψ2ħdħ.Then system (12) has a unique positive solution (μ1,μ2)∈P.Proof.
In view of Definition5, we define the generalized metric d by (23)dμ1,μ2,μ1¯,μ2¯=μ1-μ1¯∞μ2-μ2¯∞,∀μ1,μ2,μ1¯,μ2¯∈W1×W2.Obviously (W1×W2,d) is a generalized complete metric space. For any (μ1,μ2),(μ1¯,μ2¯)∈W1×W2 using properties (C3) and (C3∗), we obtain (24)T1μ1,μ2z-T1μ1¯,μ2¯z≤maxz∈0,1⁡∫01Gθ1z,ħF1ħ,μ1ħ,μ2ħ-F1ħ,μ1¯ħ,μ2¯ħdħ≤∫01Gθ11,ħϕ1ħμ1-μ1¯∞+ψ1ħμ2-μ2¯∞dħ,which implies that(25)T1μ1,μ2-T1μ1¯,μ2¯≤∫01G11,ħϕ1ħdħμ1-μ1¯∞+∫01G11,ħψ1ħdħμ2-μ2¯∞.Similarly, we obtain(26)T2μ1,μ2-T2μ1¯,μ2¯≤∫01G21,ħϕ2ħdħμ1-μ1¯∞+∫01G21,ħψ2ħdħμ2-μ2¯∞.Hence, it follows that(27)Tμ1,μ2-Tμ1¯,μ2¯≤Adμ1,μ2,μ1¯,μ2¯,∀μ1,μ2,μ1¯,μ2¯∈W1×W2,where(28)A=∫01Gθ11,ħϕ1ħdħ∫01Gθ11,ħψ1ħdħ∫01Gθ21,ħϕ2ħdħ∫01Gθ21,ħψ2ħdħ.By (C3∗), ρ(A)<1. Hence by Lemma 7, system (12) has a unique positive solution.Lemma 16.
LetF1 and F2 be continuous on [0,1]×R×R→R and there exist ai,bi,ci(i=1,2):(0,1)→[0,∞) satisfying(C4∗)
F1z,μ1z,μ2z≤a1(z)+b1zμ1z+c1zμ2z,z∈(0,1),μ1,μ2≥0;(C5∗)
F2z,μ1z,μ2z≤a2(z)+b2(z)(μ1z+c2μ2z,z∈(0,1),μ1,μ2≥0;(C6∗)
A1=∫01Gθ1(1,ħ)a1(ħ)dħ<∞,B1=∫01Gθ1(1,ħ)[b1(ħ)+c(ħ)]dħ<1;(C7∗)
A2=∫01Gθ2(1,ħ)a2(ħ)dħ<∞,B2=∫01Gθ2(1,ħ)[b2(ħ)+c2(ħ)]dħ<1.Then system (12) has at least one positive solution (μ1,μ2) in (29)Q=μ1,μ2∈P:μ1,μ2<min⁡2A11-2B1,2A21-2B2.Proof.
Chooser=min⁡2A1/(1-2B1),2A2/(1-2B2) and define ð={(μ1,μ2)∈P:(μ1,μ2)<r}. By Lemma 14, the operator T:ð¯→P is completely continuous. Choose λ∈(0,1) and (μ1,μ2)∈∂ð such that (μ1,μ2)=λT(μ1,μ2). Then, by properties (C1), (C3), and (C4∗), we obtain for all z∈[0,1](30)μ1z∞≤λmaxz∈0,1∫01Gθ1z,ħF1ħ,μ1ħ,μ2ħdħ≤λ∫01Gθ11,ħa1ħ+∫01Gθ11,ħb1ħμ1ħ+c1ħμ2ħdħ≤A1+rB1≤r2.Similarly, we obtain μ2∞≤r/2; hence (μ1,μ2)<r, which shows that (μ1,μ2)∉∂ð. Thus, by Schauder fixed point theorem, T has at least one fixed point in ð¯.


## 4. Hyers-Ulam Stability of (12)

In this section, we obtain some appropriate conditions under which the toppled system under our consideration is Hyers-Ulam stable.Let the following assumption hold:(C8∗)
There exist constantsLi>0,Ki>0,i=1,2, such that (31)F1z,μ1,μ2-F1z,μ¯1,μ¯2≤L1μ1-μ1¯+L2μ2-μ2¯,F2z,μ1,μ2-F2z,μ¯1,μ¯2≤K1μ1-μ1¯+K2μ2-μ2¯.Theorem 17.
Assume that hypothesis(C8∗) holds and T1,T2:W1×W2→W1×W2 such that(32)T1μ1,μ2-T1μ1∗,μ2∗∞≤Λ1μ1-μ1∗∞+Λ2μ2-μ2∗∞,T2μ1,μ2-T2μ1∗,μ2∗∞≤Λ3μ1-μ1∗∞+Λ4μ2-μ2∗∞,for all solutions (μ1,μ2),(μ1∗,μ2∗)∈W1×W2, with (33)Q=Λ1Λ2Λ3Λ4,converges to zero. Then the solution of coupled system (4) is Hyers-Ulam stable.Proof.
Consider(34)T1μ1,μ2z-T1μ1¯,μ2¯z≤maxz∈0,1⁡∫01Gθ1z,ħF1ħ,μ1ħ,μ2ħ-F1ħ,μ1¯ħ,μ2¯ħdħ≤∫01Gθ11,ħL1μ1-μ1¯∞+L2μ2-μ2¯∞dħ,which implies that |T1(μ1,μ2)-T1(μ1¯,μ2¯)|≤Λ1μ1-μ1¯∞+Λ2μ2-μ2¯∞.
According to the previous sentence,Λ1=L1/Γθ1+1,Λ2=L2/Γθ1+1. Similarly, we can also get(35)T2μ1,μ2-T2μ1∗,κ2∗∞≤Λ3μ1-μ1¯∞+Λ4μ2-μ2¯∞,where Λ3=K1/Γθ2+1,Λ4=K2/Γθ2+1. Therefore from (34) and (35), we have the following system of inequalities:(36)T1μ1,μ2-T1μ1∗,μ2∗∞≤Λ1μ1-μ1∗∞+Λ2μ2-μ2∗∞,T2μ1,μ2-T2μ1∗,μ2∗∞≤Λ3μ1-μ1∗∞+Λ4μ2-μ2∗∞,where M=Λ1Λ2Λ3Λ4 which converges to zero. Thus, in view of Theorem 9, the solution of coupled system (4) is Hyers-Ulam stable.


## 5. Examples

Example 1.
Consider the following coupled nonlinear FODEs of boundary conditions:(37)D5/2μ1z+Γ52zμ1z16+z2μ2z32=0,z∈0,1,D5/2μ2z+Γ529z2cosμ1z16π+9zcosμ2z32π=0,z∈0,1,μ10=μ1′0=μ11=0,μ20=μ2′0=μ21=0.Here n=3 and ϕ1(z)=Γ(5/2)(z/16),ψ1(z)=Γ(5/2)(z2/32),ϕ2(z)=Γ(5/2)(9z2/16π),ψ2(z)=Γ(5/2)(9z/32π). Moreover (38)A=∫01Gθ11,ħϕ1ħdħ∫01Gθ11,ħψ1ħdħ∫01Gθ21,ħϕ2ħdħ∫01Gθ21,ħψ2ħdħ=0.04600.00070.00680.0058.Here, ρ(A)=4.61×10-2<1; hence by Lemma 15 the BVP (37) has a unique solution. For F1 and F2, we have a1(z)=0,b1(z)=Γ5/2(z/16),c1(z)=Γ(5/2)(z2/32),a2(z)=0,b2(z)=Γ(5/2)(9z2/16π),c2(z)=Γ(5/2)(9z/32π) and, by simple calculation, we obtain(39)A1=∫01Gθ1z,ħa1ħdħ<∞,A2=∫01Gθ2z,ħa2ħdħ<∞,B1=∫01Gθ1z,ħb1ħ+c1ħdħ<1,B2=∫01Gθ2z,ħb2ħ+c2ħdħ<1.Hence, by using Lemma 16, BVP (37) has at least one positive solution. Further, it is easy to compute the matrix Q which converges to zero and so the solution is Hyers-Ulam stable by using Theorem 17.


## 6. Conclusion

In this paper, we investigate existence and uniqueness of solutions for the nonlinear FODEs with boundary conditions and also investigate Hyers-Ulam stability for the mentioned problem. We use Perov’s fixed point theorem [33] and Leray-Schauder fixed point theorem to develop some results for existence of at least one solution for our proposed coupled nonlinear FODEs with boundary conditions. Further, we establish some conditions for Hyers-Ulam type stability to the considered problem. The whole paper is very easy because of relaxed methods and conditions.



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