# A Coupled System of Nonlinear Fractional Differential Equations with Multipoint Fractional Boundary Conditions on an Unbounded Domain

Authors: Guotao Wang; Bashir Ahmad; Lihong Zhang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper investigates the existence of solutions for a coupled system of nonlinear fractional differential equations withm-point fractional boundary conditions on an unbounded domain. Some standard fixed point theorems are applied to obtain the main results. The paper concludes with two illustrative examples.


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## Body

## 1. Introduction

In the last few decades, the subject of fractional calculus has gained considerable popularity and importance as it finds its applications in numerous fields of science and engineering. Some of the areas of recent applications of fractional models include fluid mechanics, solute transport or dynamical processes in porous media, material viscoelastic theory, dynamics of earthquakes, control theory of dynamical systems, and biomathematics. In the afore-mentioned areas, there are phenomena with estrange kinetics involving microscopic complex dynamical behaviour that cannot be characterized by classical derivative models. It has been learnt through experimentation that most of the processes associated with complex systems have nonlocal dynamics possessing long-memory in time, and the integral and derivative operators of fractional order do have some of these characteristics. Thus, due to the modeling capabilities of fractional integrals and derivatives for complex phenomena, the fractional modelling has emerged as a powerful tool and has accounted for the rapid development of the theory of fractional differential equations. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes [1]. The presence of memory term in such models not only takes into account the history of the process involved but also carries its impact to present and future development of the process. For more details and applications, we refer the reader to the books [2–6]. For some recent work on the topic, see [7–27] and references therein.The study of coupled systems involving fractional differential equations is also important as such systems occur in various problems of applied nature. For some recent results on systems of fractional differential equations, see [28–35].Much of the work on fractional differential equations has been considered on finite domain and there are few papers dealing with infinite domain [36–43]. In this paper, we discuss the existence and uniqueness of the solutions of a coupled system of nonlinear fractional differential equations with m-point boundary conditions on an unbounded domain. Precisely, we consider the following problem:(1.1)Dpu(t)+f(t,v(t))=0,2<p<3,Dqv(t)+g(t,u(t))=0,2<q<3,u(0)=u′(0)=0,Dp-1u(+∞)=∑i=1m-2βiu(ξi),v(0)=v′(0)=0,Dq-1v(+∞)=∑i=1m-2γiv(ξi),
where t∈J=[0,+∞), f,g∈C(J×ℝ,ℝ), 0<ξ1<ξ2<⋯<ξm-2<+∞,   Dp and  Dq denote Riemann-Liouville fractional derivatives of order p and q, respectively, and βi>0,andγi>0 are such that 0<∑i=1m-2βiξip-1<Γ(p) and 0<∑i=1m-2γiξiq-1<Γ(q).


## 2. Preliminaries

For the convenience of the readers, in this section we first present some useful definitions and lemmas.Definition 2.1 (see [5]).
The Riemann-Liouville fractional derivative of orderδ for a continuous function f is defined by
(2.1)Dδf(t)=1Γ(n-δ)(ddt)n∫0t(t-s)n-δ-1f(s)ds,n=[δ]+1,
provided that the right-hand side is pointwise defined on(0,∞).Definition 2.2 (see [5]).
The Riemann-Liouville fractional integral of orderδ for a function f is defined as(2.2)Iδf(t)=1Γ(δ)∫0t(t-s)δ-1f(s)ds,δ>0,provided that such integral exists.For the forthcoming analysis, we define the spaces(2.3)X={u∈C[0,+∞):sup⁡t∈J⁡|u(t)|1+tp-1<+∞},Y={v∈C[0,+∞):sup⁡t∈J⁡|v(t)|1+tq-1<+∞}equipped with the norms(2.4)‖u‖X=sup⁡t∈J⁡|u(t)|1+tp-1,(2.5)‖v‖Y=sup⁡t∈J⁡|v(t)|1+tq-1.ObviouslyX and Y are Banach spaces.Lemma 2.3 (see [38]).
Leth∈C([0,+∞)). For  2<α<3, the fractional boundary value problem
(2.6)Dαu(t)+h(t)=0,u(0)=u′(0)=0,Dα-1u(+∞)=∑i=1m-2βiu(ξi)has a unique solution(2.7)u(t)=∫0+∞G(t,s)h(s)ds,where(2.8)G(t,s)=G*(t,s)+G**(t,s),with(2.9)G*(t,s)=1Γ(α){tα-1-(t-s)α-1,0≤s≤t<+∞,tα-1,0≤t≤s<+∞.(2.10)G**(t,s)=∑i=1m-2βitα-1Γ(α)-∑i=1m-2βiξiα-1G*(ξi,s).Lemma 2.4 (see [38]).
For(s,t)∈[0,+∞)×[0,+∞),  G(t,s)/1+tα-1≤L1, where
(2.11)L1=1Γ(α)+∑i=1m-2βiξm-2α-1Γ(α)(Γ(α)-∑i=1m-2βiξiα-1).


## 3. Main Results

This section is devoted to some existence and uniqueness results for problem (1.1).Define the space(3.1)Z={(u,v)∣u∈X,v∈Y}equipped with the norm(3.2)‖(u,v)‖Z=max⁡{‖u‖X,‖v‖Y}.ClearlyZ is a Banach space.Let an operatorQ:Z→Z be defined by(3.3)Q(u,v)=(Q1(v),Q2(u))=(∫0+∞G1(t,s)f(t,v(s))ds,∫0+∞G2(t,s)g(t,u(s))ds),whereG1(t,s)=G11(t,s)+G12(t,s),G2(t,s)=G21(t,s)+G22(t,s), with(3.4)G11(t,s)=1Γ(p){tp-1-(t-s)p-1,0≤s≤t<+∞,tp-1,0≤t≤s<+∞,G12(t,s)=∑i=1m-2βitp-1Γ(p)-∑i=1m-2βiξip-1G11(ξi,s),G21(t,s)=1Γ(q){tq-1-(t-s)q-1,0≤s≤t<+∞,tq-1,0≤t≤s<+∞,G22(t,s)=∑i=1m-2γitq-1Γ(q)-∑i=1m-2γiξiq-1G21(ξi,s).Observe that the problem (1.1) has a solution if and only if the operator Q defined by (3.3) has a fixed point.Lemma 3.1.
For(s,t)∈[0,+∞)×[0,+∞), one has
(3.5)G1(t,s)1+tp-1≤L,G2(t,s)1+tq-1≤L,
where
(3.6)L=max⁡{1Γ(p)+∑i=1m-2βiξm-2p-1Γ(p)(Γ(p)-∑i=1m-2βiξip-1),1Γ(q)+∑i=1m-2γiξm-2q-1Γ(q)(Γ(q)-∑i=1m-2γiξiq-1)}.Theorem 3.2.
Assume that
(H1)
there exist nonnegative functionsa(t),b(t)∈C[0,+∞) such that
(3.7)|f(t,x)|≤a(t)|x|+b(t),t∈[0,+∞),∫0+∞(1+tq-1)a(t)dt<1L,∫0+∞b(t)dt<+∞;(H2)
there exist nonnegative functionsc(t),d(t)∈C[0,+∞) such that
(3.8)|g(t,y)|≤c(t)|y|+d(t),t∈[0,+∞),∫0+∞(1+tp-1)c(t)dt<1L,∫0+∞d(t)dt<+∞.
Then the system (1.1) has a solution.Proof.
Let us take(3.9)R>max⁡{L∫0+∞b(s)ds1-L∫0+∞(1+sq-1)a(s)ds,L∫0+∞d(s)ds1-L∫0+∞(1+sp-1)c(s)ds},
and define
(3.10)BR={(u,v)∈Z∣‖(u,v)‖Z≤R}.
Obviously, BR is a bounded closed and convex set of Z.
As a first step, we show that the operatorQ is BR→BR.
For any(u,v)∈BR, we have(3.11)‖Q1v‖X=supt∈J⁡11+tp-1|∫0+∞G1(t,s)f(s,v(s))ds|≤supt∈J⁡11+tp-1∫0+∞G1(t,s)(a(s)|v(s)|+b(s))ds≤L∫0+∞(1+sq-1)a(s)ds‖v‖Y+L∫0+∞b(s)ds<L∫0+∞b(s)ds1-L∫0+∞(1+tq-1)a(s)ds<R.
Similarly, we can get
(3.12)‖Q2u‖Y=sup⁡t∈J11+tq-1|∫0+∞G2(t,s)g(s,u(s))ds|≤L∫0+∞(1+sp-1)c(s)ds‖u‖X+L∫0+∞d(s)ds<L∫0+∞d(s)ds1-L∫0+∞(1+sp-1)c(s)ds<R.
That is, ∥Q(u,v)∥Z≤R. Thus, QBR⊂BR.
Next, we show thatQ is completely continuous. By continuity of f,g,G1, and G2, it follows that Q is continuous. On the other hand, by a similar process used in [38], we can easily prove that the operators Q1andQ2 are equicontinuous. Therefore it follows that QBR is an equicontinuous set. Also, it is uniformly bounded as QBR⊂BR. Thus, we conclude that Q is a completely continuous operator. Hence, by Schauder fixed point theorem, there exists a solution of (1.1). This completes the proof.Theorem 3.3.
Assume that
(H3)
there exist0<ρ1<1 and nonnegative functions a1(t),b1(t)∈C[0,+∞) such that
(3.13)|f(t,x)|≤a1(t)|x|ρ1+b1(t),t∈[0,+∞),∫0+∞(1+tq-1)a1(t)dt<+∞,∫0+∞b1(t)dt<+∞.(H4)
there exist0<ρ2<1 and nonnegative functions c1(t),d1(t)∈C[0,+∞) such that
(3.14)|g(t,y)|≤c1(t)|y|ρ2+d1(t),t∈[0,+∞),∫0+∞(1+tp-1)c1(t)dt<+∞,∫0+∞d1(t)dt<+∞.
Then the system (1.1) has a solution.Proof.
In this case, we take(3.15)R>max⁡{(∫0+∞)1/(1-ρ2)2L∫0+∞b1(s)ds,(2L∫0+∞(1+sq-1)a1(s)ds)1/(1-ρ1),2L∫0+∞d1(s)ds,(2L∫0+∞(1+sp-1)c1(s)ds)1/(1-ρ2)}.
The rest of the proof is similar to that of Theorem 3.2. So we omit it.Remark 3.4.
By takingρ1,ρ2>1 (instead of 0<ρ1<1, 0<ρ2<1) in (H3) and (H4), one can show that (1.1) has a solution.Theorem 3.5.
Assume that
(H5) the functions f and g satisfy Lipschitz condition; that is, there exist nonnegative functions K1(t) and K2(t) such that(3.16)|f(t,x)-f(t,y)|≤K1(t)|x-y|,t∈[0,+∞),|g(t,x)-g(t,y)|≤K2(t)|x-y|,t∈[0,+∞).
Then the problem (1.1) has a unique solution if
(3.17)μ=L∫0+∞K1(s)(1+sq-1)ds<1,τ=L∫0+∞K2(s)(1+sp-1)ds<1.Proof.
For any(u1,v1),(u2,v2)∈Z, we have
(3.18)‖Q1v2-Q1v1‖X=sup⁡t∈J⁡11+tp-1|∫0+∞G1(t,s)[f(s,v2(s))-f(s,v1(s))]ds|≤sup⁡t∈J⁡∫0+∞G1(t,s)1+tp-1K1(s)|(v2-v1)(s)|ds≤L∫0+∞K1(s)(1+sq-1)ds‖v2-v1‖Y=μ‖v2-v1‖Y.
Similarly, it can be shown that(3.19)‖Q2u2-Q2u1‖Y=sup⁡t∈J⁡11+tq-1|∫0+∞G1(t,s)(g(s,u2(s))-f(s,u2(s)))ds|≤L∫0+∞K2(s)(1+sp-1)ds‖u2-u1‖X=τ‖u2-u1‖X.
Thus, we get(3.20)‖Q(u2,v2)-Q(u1,v1)‖Z≤max⁡{μ,τ}‖(u2,v2)-(u1,v1)‖Z.
Obviously, Q is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. This completes the proof.


## 4. Example

Example 4.1.
Consider the following multipoint boundary value problem on an unbounded domain:(4.1)D9/4u(t)+sin⁡tln⁡(1+|v(t)|)(1+t7/4)(2+t)2+(1+cos⁡2t)e-t=0,D11/4v(t)+e-5tsin⁡|u(t)|3(1+t5/4)(1+t)2+4(t+4)2=0,u(0)=u′(0)=0,D5/4u(+∞)=25u(14)+110u(1),v(0)=v′(0)=0,D7/4v(+∞)=310u(14)+15u(1).Heret∈[0,+∞), p=9/4, q=11/4, ξ1=1/4, ξ2=1, β1=2/5, β2=1/10, γ1=3/10, and γ2=1/5. One has(4.2)f(t,v(t))=sin⁡tln⁡(1+|v(t)|)(1+t7/4)(2+t)2+(1+cos⁡2t)e-t,g(t,u(t))=e-5tsin⁡|u(t)|3(1+t5/4)(1+t)2+4(t+4)2.Fora(t)=1/(1+t7/4)(2+t)2, b(t)=2e-t, c(t)=1/3(1+t5/4)(1+t)2, d(t)=4/(t+4)2, by direct calculation we find that(4.3)L=max⁡{1Γ(p)+∑i=1m-2βiξm-2p-1Γ(p)(Γ(p)-∑i=1m-2βiξip-1),1Γ(q)+∑i=1m-2γiξm-2q-1Γ(q)(Γ(q)-∑i=1m-2γiξiq-1)}=max⁡{1Γ(9/4)+(2/5)+(1/10)Γ(9/4)(Γ(9/4)-2/5(1/4)5/4-(1/10)),1Γ(11/4)+(3/10)+(1/5)Γ(11/4)(Γ(11/4)-3/10(1/4)7/4-(1/5))}=1.341213,|f(t,x)|≤a(t)|x|+b(t),|g(t,y)|≤c(t)|y|+d(t),t∈[0,+∞),∫0+∞(1+tq-1)a(t)dt=12<1L=0.745594,∫0+∞b(t)dt=2<+∞,∫0+∞(1+tp-1)c(t)dt=13<1L=0.745594,∫0+∞d(t)dt=1<+∞.Thus all conditions of Theorem3.2 are satisfied. Therefore, by Theorem 3.2, the couple system of nonlinear fractional differential (4.1) has at least one solution.Example 4.2.
Consider the following problem on an unbounded domain:(4.4)Dpu(t)+M1(t)sin⁡v(t)+N1(t)=0,Dqv(t)+M2(t)1+u2(t)+N2(t)=0,u(0)=u′(0)=0,Dp-1u(+∞)=25u(14)+110u(1),v(0)=v′(0)=0,Dq-1v(+∞)=310u(14)+15u(1).
Here t∈[0,+∞), 2<p, q<3, ξ1=1/4, ξ2=1, β1=2/5, β2=1/10, γ1=3/10, and γ2=1/5, M1(t),M2(t),N1(t),N2(t)∈C([0,+∞),ℝ).
With(4.5)f(t,v(t))=M1(t)sin⁡v(t)+N1(t),g(t,u(t))=M2(t)1+u2(t)+N2(t),
we have
(4.6)|f(t,x)-f(t,y)|=|M1(t)||sin⁡x-sin⁡y|≤|M1(t)||x-y|,t∈[0,+∞),|g(t,x)-g(t,y)|=|M2(t)||11+x2-11+y2|≤|M2(t)||x-y|,t∈[0,+∞),where K1(t)=|M1(t)|, K2(t)=|M2(t)|. So, the condition (H5) holds. Let us assume that
(4.7)μ=L∫0+∞|M1(s)|(1+sq-1)ds<1,τ=L∫0+∞|M2(s)|(1+sp-1)ds<1.
For example, condition (4.7) holds if we take
(4.8)p=94,q=114,M1(t)=1(1+t7/4)(2+t)2,M2(t)=13(1+t5/4)(1+t)2.
Thus all the conditions of Theorem 3.5 are satisfied. Therefore, by the conclusion of Theorem 3.5, the coupled system (4.4) has a unique solution.


## 5. Conclusion

We have shown the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations with multipoint fractional boundary conditions on a semi-infinite domain. Our existence results are based on Schauder’s fixed point theorem, while the uniqueness result is obtained by applying Banach’s contraction mapping principle. The existence of solutions for (1.1) has been addressed for different kinds of growth conditions. Our approach is simple and can easily be applied to a variety of problems. This has been demonstrated by solving two examples.



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*Source: 248709-2012-06-06.xml*