Uniqueness theorem of differential system with coupled integral boundary conditions
Yujun Cui
B1,2, Wenjie Ma
2, Xiangzhi Wang
3and Xinwei Su
41State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology,
Qingdao 266590, P.R. China
2Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, P.R. China
3Jinan Technician College, Jinan 250200, P.R. China
4School of Science, China University of Mining and Technology, Beijing 10083, P.R. China
Received 7 November 2017, appeared 12 February 2018 Communicated by Jeff R. L. Webb
Abstract. The paper is devoted to study the uniqueness of solutions for a differential system with coupled integral boundary conditions under a Lipschitz condition. Our approach is based on the Banach’s contraction principle. The interesting point is that the Lipschitz constant is related to the spectral radius corresponding to the related linear operators.
Keywords: differential system, coupled integral boundary conditions, spectral radius, Banach’s contraction principle.
2010 Mathematics Subject Classification: 34B15.
1 Introduction
In this paper, we consider the uniqueness of solutions for the following differential system with coupled integral boundary conditions
−x00(t) = f(t,x(t),y(t)), t ∈(0, 1),
−y00(t) = g(t,x(t),y(t)), t ∈(0, 1),
x(0) =y(0) =0, x(1) =α[y], y(1) =β[x]
(1.1)
whereα[x],β[x]are bounded linear functionals onC[0, 1]given by α[x] =
Z 1
0 x(t)dA(t), β[x] =
Z 1
0 x(t)dB(t)
involving Riemann–Stieltjes integrals, in particular, A,Bare non-decreasing functions, sodA, dBare positive Stieltjes measures.
BCorresponding author. Email: cyj720201@163.com
Differential system with coupled boundary conditions arise from the study of reaction- diffusion equations and Sturm–Liouville problems, and have extensive applications in various fields of sciences and engineering such as the heat equation and mathematical biology.
The existence of solutions or positive solutions of differential system with coupled bound- ary conditions has been studied by many researchers, see [1–4,6–10,13] for some recent work.
For example, by using the Guo–Krasnosel’skii fixed-point theorem, the existence of positive solution of the following singular system with coupled four-point boundary value conditions are obtained [1]
−x00(t) = f1(t,x(t),y(t)), t ∈(0, 1),
−y00(t) = f2(t,x(t),y(t)), t∈(0, 1),
x(0) =y(0) =0, x(1) =αy(ξ), y(1) = βx(η).
In [8], Infante, Minh ´os and Pietramala, by means of classical fixed point index theory, provided a general theory for existence of positive solutions for coupled systems.
The uniqueness of solutions can be an important problem for boundary value problems of differential equation or differential system. This problem has been investigated by many authors by use of techniques of nonlinear analysis. We refer the reader to [3,4] for some recent uniqueness results for differential system, to [5,12,14] for differential equation. In [3], by means of the Guo–Krasnosel’skii fixed-point theorem and mixed monotone method, Cui, Liu and Zhang investigated the uniqueness of positive solutions of singular system (1.1) in the case that the nonlinearities f andgmay be singular att =0, 1.
However, to our best knowledge, there are fewer results concerned the uniqueness of solutions for differential systems with coupled integral boundary conditions. So, we consider the uniqueness of solutions for differential system (1.1) under a Lipschitz condition on f and g. By using Banach’s contraction principle, a new result on the uniqueness of solutions for differential system (1.1) is obtained. It is worthwhile to mention that the Lipschitz constant is related to the spectral radius corresponding to the related linear operators.
Throughout the paper, we assume that the following conditions hold.
(H1) α[t] =R1
0 tdA(t)>0,β[t] =R1
0 tdB(t)>0, κ=1−α[t]β[t]>0.
(H2) f,g:[0, 1]×R2→Rare continuous.
2 Preliminaries
Let C[0, 1] be the Banach space of continuous functions endowed with the norm kxk = maxt∈[0,1]|x(t)|and let P1be the cone of nonnegative functions inC[0, 1]given by
P1={x∈C[0, 1]:x(t)≥0,∀ t∈ [0, 1]}.
Thus E= C[0, 1]×C[0, 1] is a Banach space with the norm defined byk(x,y)kE = max{kxk, kyk}, andP=P1×P1is a cone inE.
Lemma 2.1([2]). Let u,v∈C[0, 1], then the system of BVPs
(−x00(t) =u(t), −y00(t) =v(t), t ∈[0, 1], x(0) =y(0) =0, x(1) =α[y], y(1) =β[x]
has integral representation
x(t) =
Z 1
0 G1(t,s)u(s)ds+
Z 1
0 H1(t,s)v(s)ds, y(t) =
Z 1
0 G2(t,s)v(s)ds+
Z 1
0 H2(t,s)u(s)ds, where
G1(t,s) = α[t]t κ
Z 1
0 k(s,τ)dB(τ) +k(t,s), H1(t,s) = t κ
Z 1
0 k(s,τ)dA(τ), G2(t,s) = β[t]t
κ Z 1
0 k(s,τ)dA(τ) +k(t,s), H2(t,s) = t κ
Z 1
0 k(s,τ)dB(τ), k(t,s) =
(t(1−s), 0≤ t≤s ≤1, s(1−t), 0≤ s≤t ≤1.
Employing Lemma 2.1, we can reformulate BVP (1.1) as a fixed point for the following integral equations:
x(t) =
Z 1
0 G1(t,s)f(s,x(s),y(s))ds+
Z 1
0 H1(t,s)g(s,x(s),y(s))ds, y(t) =
Z 1
0 G2(t,s)g(s,x(s),y(s))ds+
Z 1
0 H2(t,s)f(s,x(s),y(s))ds.
Define an operatorSby
S(x,y) = (S1(x,y),S2(x,y)), (x,y)∈E, (2.1) where operatorsS1,S2 :E→C[0, 1]are defined by
S1(x,y)(t) =
Z 1
0 G1(t,s)f(s,x(s),y(s))ds+
Z 1
0 H1(t,s)g(s,x(s),y(s))ds, t∈ [0, 1], S2(x,y)(t) =
Z 1
0
G2(t,s)g(s,x(s),y(s))ds+
Z 1
0
H2(t,s)f(s,x(s),y(s))ds, t∈ [0, 1]. Then the existence of a solution of differential system (1.1) is equivalent to the existence of a fixed point of SonE.
It is well known that the functionk(t,s)has the following properties:
t(1−t)s(1−s)≤k(t,s)≤t(1−t), ∀ t,s∈[0, 1]. From this and(H1), fort,s∈[0, 1], we have
G1(t,s)≤ t+α[t]t κ
Z 1
0 dB(τ), H1(t,s)≤ t κ
Z 1
0 dA(τ), G2(t,s) =t+ β[t]t
κ Z 1
0 dA(τ), H2(t,s) = t κ
Z 1
0 dB(τ), and
G1(t,s)≥ α[t]t κ
Z 1
0 k(s,τ)dB(τ)≥ α[t]s(1−s) κ
Z 1
0 τ(1−τ)dB(τ)·t, G2(t,s)≥ β[t]t
κ Z 1
0
k(s,τ)dA(τ)≥ β[t]s(1−s) κ
Z 1
0
τ(1−τ)dA(τ)·t,
H1(t,s)≥ s(1−s) κ
Z 1
0 τ(1−τ)dA(τ)·t, H2(t,s)≥ H2(t,s) = s(1−s)
κ Z 1
0 τ(1−τ)dB(τ)·t.
Therefore we have
Gi(t,s)≤ρt, Hi(t,s)≤ρt, i=1, 2, (2.2) and
Gi(t,s)≥νts(1−s), Hi(t,s)≥ νts(1−s), i=1, 2, (2.3) where
ρ=max α[t]
κ β[1] +1, β[t]
κ α[1] +1,1 κβ[1],1
κα[1]
, ν=min
α[t]
κ β[t(1−t)], β[t]
κ α[t(1−t)], 1
κβ[t(1−t)], 1
κα[t(1−t)]
.
LetR+ = [0,+∞). For a= (a,b,c,d)∈ R4+ with a2+b2+c2+d2 6= 0, define an operator T: E→Eby
Ta(x,y) = (Ta,1(x,y),Ta,2(x,y)), (2.4) where operatorsTa,1,Ta,2:E→C[0, 1]are defined by
Ta,1(x,y)(t) =
Z 1
0 G1(t,s)(ax(s) +by(s))ds+
Z 1
0 H1(t,s)(cx(s) +dy(s))ds, t∈[0, 1], Ta,2(x,y)(t) =
Z 1
0 G2(t,s)(cx(s) +dy(s))ds+
Z 1
0 H2(t,s)(ax(s) +by(s))ds, t∈[0, 1]. It is not difficult to verify thatTa:E→Eis a completely continuous linear operator.
Definition 2.2([11]). LetEbe a Banach space,P⊂ Ebe a cone inE. Lete ∈P\{θ}, a mapping T: P→Pis called e−positive if for every nonzerox ∈Pa natural number n=n(x)and two positive numbercx,dx can be found such that
cxe≤ Tnx ≤dxe.
Recall that a real number λ is an eigenvalue of the operator T if there exists a non-zero elementx ∈Esuch thatTx =λx.
Lemma 2.3 ([11, Theorem 2.5, Lemma 2.1, Theorem 2.10]). Suppose that T : E → E is a e−positive, completely continuous linear operator. If there exist ψ ∈ E\(−P)and a constant c > 0 such that cTψ≥ψ, then the spectral radius r(T)6=0, and r(T)is the unique positive eigenvalue with its eigenfunction in P.
Lemma 2.4. Suppose that (H1)holds. Then for the operator Ta defined by (2.4), there is a unique positive eigenvalue r(Ta)with its eigenfunction in P.
Proof. First, we show that Ta is e−positive with e(t) = (t,t), that is, for any (x,y) ∈ P\{θ}, there existcx,y,dx,y >0 such that
cx,y·e ≤Ta(x,y)≤dx,y·e. (2.5) Let dx,y = ρ(a+c)R1
0 x(s)ds+ρ(b+d)R1
0 y(s)ds. By (2.2), we can derive Ta(x,y)(t) ≤ dx,y· (t,t) = dx,y·e(t). Let cx,y = ν(a+c)R1
0 s(1−s)x(s)ds+ν(b+d)R1
0 s(1−s)y(s)ds. By (2.3), Ta(x,y)(t)≥ cx,y·e(t)holds, in particular, we have Tae(t)≥ ce(t)·e(t). So (2.5) is proved and Lemma2.4 holds follows from Lemma2.3. This completes the proof.
Remark 2.5. Let(ϕ,ψ)be the positive eigenfunction ofTa corresponding tor(Ta), thus Ta(ϕ,ψ) =r(Ta)(ϕ,ψ). (2.6) Then by the proof of Lemma2.4and Definition2.2, there existcϕ,ψ>0 such that
cϕ,ψ·(t,t) =cϕ,ψ·e(t)≤Ta(ϕ,ψ)(t) =r(Ta)·(ϕ(t),ψ(t)), i.e.,
t≤ r(Ta) cϕ,ψ
ϕ(t), t ≤ r(Ta) cϕ,ψ
ψ(t), t∈ [0, 1]. (2.7)
3 Main results
Theorem 3.1. Suppose that there existsa= (a,b,c,d)∈R4+ with a2+b2+c2+d26=0such that
|f(t,u1,v1)− f(t,u2,v2)| ≤a|u1−u2|+b|v1−v2|, ∀t∈ [0, 1], u1,u2,v1,v2 ∈R, (3.1) and
|g(t,u1,v1)−g(t,u2,v2)| ≤c|u1−u2|+d|v1−v2|, ∀ t∈[0, 1], u1,u2,v1,v2∈R. (3.2) If r(Ta)<1, then differential system(1.1)has a unique solution in E.
Proof. It is clear that the fixed points of operatorS coincide with the solutions to differential system (1.1).
For(x,y)∈ E, by (2.2), (2.7), (3.1) and(3.2) we have
|S1(x,y)(t)|
≤
Z 1
0 G1(t,s)f(s,x(s),y(s))ds−
Z 1
0 G1(t,s)f(s, 0, 0)ds
+
Z 1
0 G1(t,s)f(s, 0, 0)ds +
Z 1
0 H1(t,s)g(s,x(s),y(s))ds−
Z 1
0 H1(t,s)g(s, 0, 0)ds
+
Z 1
0 H1(t,s)g(s, 0, 0)ds
≤
Z 1
0 G1(t,s)|f(s,x(s),y(s))− f(s, 0, 0)|ds+
Z 1
0 G1(t,s)|f(s, 0, 0)|ds +
Z 1
0 H1(t,s)|g(s,x(s),y(s))−g(s, 0, 0)|ds+
Z 1
0 H1(t,s)|g(s, 0, 0)|ds
≤ρt
(a+c)
Z 1
0
|x(s)|ds+ (b+d)
Z 1
0
|y(s)|ds +
Z 1
0
|f(s, 0, 0)|ds+
Z 1
0
|g(s, 0, 0)|ds
≤ r(Ta)ρ cϕ,ψ
(a+c)
Z 1
0
|x(s)|ds+ (b+d)
Z 1
0
|y(s)|ds
+
Z 1
0
|f(s, 0, 0)|ds+
Z 1
0
|g(s, 0, 0)|ds
·ϕ(t), t∈ [0, 1]. In the same way, we can prove that
|S2(x,y)(t)| ≤ r(Ta)ρ cϕ,ψ
(a+c)
Z 1
0
|x(s)|ds+ (b+d)
Z 1
0
|y(s)|ds
+
Z 1
0
|f(s, 0, 0)|ds+
Z 1
0
|g(s, 0, 0)|ds
·ψ(t), t ∈[0, 1].
Therefore,Smaps all ofEinto the following vector subspace E1 =
(x,y)∈ E: |x(t)|
ϕ(t) ,
|y(t)|
ψ(t) are bounded fort ∈[0, 1]
. Evidently,E1 is a subspace of EandE1 is an Banach space with the norm
k(x,y)k1=max (
sup
t∈[0,1]
|x(t)|
ϕ(t) , tsup∈[0,1]
|y(t)|
ψ(t) )
. So it suffices to consider the fixed point ofSin E1. Note that
Ta(ϕ,ψ) =r(Ta)(ϕ,ψ) means
r(Ta)ϕ(t) =
Z 1
0 G1(t,s)(aϕ(s) +bψ(s))ds+
Z 1
0 H1(t,s)(cϕ(s) +dψ(s))ds and
r(Ta)ψ(t) =
Z 1
0 G2(t,s)(cϕ(s) +dψ(s))ds+
Z 1
0 H2(t,s)(aϕ(s) +bψ(s))ds.
Let(x1,y1),(x2,y2)∈ E1. Then
|S1(x1,y1)(t)−S1(x2,y2)(t)|
≤
Z 1
0 G1(t,s)f(s,x1(s),y1(s))ds−
Z 1
0 G1(t,s)f(s,x2(s),y2(s))ds +
Z 1
0 H1(t,s)g(s,x1(s),y1(s))ds−
Z 1
0 H1(t,s)g(s,x2(s),y2(s))ds
≤ a Z 1
0 G1(t,s)|x1(s)−x2(s)|ds+b Z 1
0 G1(t,s)|y1(s)−y2(s)|ds +c
Z 1
0 H1(t,s)|x1(s)−x2(s)|ds+d Z 1
0 H1(t,s)|y1(s)−y2(s)|ds
≤ a Z 1
0 G1(t,s)k(x1,y1)−(x2,y2)k1ϕ(s)ds+b Z 1
0 G1(t,s)k(x1,y1)−(x2,y2)k1ψ(s)ds +c
Z 1
0 H1(t,s)k(x1,y1)−(x2,y2)k1ϕ(s)ds+d Z 1
0 H1(t,s)k(x1,y1)−(x2,y2)k1ψ(s)ds
=k(x1,y1)−(x2,y2)k1·Ta,1(ϕ,ψ)(t) =r(Ta)k(x1,y1)−(x2,y2)k1·ϕ(t). In the same way, we can prove that
|S2(x1,y1)(t)−S2(x2,y2)(t)| ≤r(Ta)k(x1,y1)−(x2,y2)k1·ψ(t), t∈[0, 1]. The above two inequalities imply that
kS(x1,y1)−S(x2,y2)k1 ≤r(Ta)k(x1,y1)−(x2,y2)k1, ∀(x1,y1),(x2,y2)∈ E1. Notice thatr(Ta)< 1, the operator Sis a contraction. Hence, it follows from the well known Banach’s contraction principle thatShas a unique fixed point (x,y)∈ E1, which is obviously a unique solution of differential system (1.1). It ends the proof.
From the above argument, we know that the basic space used in the proof of Theorem3.1 is E1, not in E. If we consider differential system (1.1) in E by use of Banach’s contraction principle, the result of Theorem3.1remains true except that the conditionr(Ta)<1 is replaced bykTak<1, where
kTak= sup
(x,y)∈E
kTa(x,y)kE k(x,y)kE . It follows from the well-known Gelfand’s Formula that
r(Ta) = lim
n→∞
qn
kTank ≤ kTak
which concludes that it may be favorable to consider the uniqueness of differential system (1.1) in E1.
In the following, we give two examples to illustrate our main result. Obviously, it is rather difficult to determine the value of r(Ta)in general. In the two examples, we determine the spectral radiusr(Ta)for certain four-point coupled boundary conditions which can be seen as a special cases of coupled integral boundary conditions.
Example 3.2. Consider the system
−x00(t) =asinx(t) +h1(t), t∈(0, 1),
−y00(t) =ap
y2(t) +1+h2(t), t∈(0, 1), x(0) =y(0) =0, x(1) =y(13), y(1) =3x(14),
(3.3)
where a ∈ R, h1,h2 ∈ C[0, 1]. In this case the integral boundary conditions are given by the functionalsα[y] =y(13)andβ[x] =3x(14).
Let
f(t,x,y) =asinx+h1(t), g(t,x,y) =a q
y2+1+h2(t), then
|f(t,u1,v1)− f(t,u2,v2)| ≤ |a||u1−u2|, ∀ t∈ [0, 1], u1,u2,v1,v2 ∈R and
|g(t,u1,v1)−g(t,u2,v2)| ≤ |a||v1−v2|, ∀ t∈[0, 1], u1,u2,v1,v2∈R.
Thus we haveb=c=0,κ=1−α[t]β[t] = 34.
Take a = (|a|, 0, 0,|a|). Let (ϕ,ψ) be the positive eigenfunction of Ta corresponding to r(Ta), thus
Ta,1(ϕ,ψ) =r(Ta)ϕ, Ta,2(ϕ,ψ) =r(Ta)ψ. (3.4) Letλ= |a|
r(Ta). It follows from (3.4) that
(−ϕ00(t) =λϕ, −ψ00(t) =λψ(t), t∈(0, 1), ϕ(0) =ψ(0) =0, ϕ(1) =ψ(13), ψ(1) =3ϕ(14). By ordinary method, we conclude that(ϕ(t),ψ(t)) = (c1,c2)sin√
λtfor somec1,c2 ∈R. This together with the four-point coupled boundary conditions yields
c1sin√
λ=c2sin
√ λ
3 , c2sin√
λ=3c1sin
√ λ 4 .
So,λis the unique positive solution of the equation sin2√
λ=3 sin
√ λ 3 sin
√ λ
4 , λ∈(0,π2).
We can obtain λ≈ 1.95852 ≈ 3.83584 by MATLAB. Therefore, if |a| < 3.83584, the problems (3.3) has a unique solution.
Example 3.3. Consider the differential system
−x00(t) =acosx(t)−aln(1+y2(t)) +h1(t), t∈(0, 1),
−y00(t) =aarctanx(t)−ay(t) +h2(t), t∈ (0, 1), x(0) =y(0) =0, x(1) =y(12), y(1) =2x(14),
(3.5)
wherea∈R,h1,h2 ∈C[0, 1]. Let
f(t,x,y) =acosx−aln(1+y2) +h1(t), g(t,x,y) =aarctanx−ay+h2(t), then
|f(t,u1,v1)− f(t,u2,v2)| ≤ |a||u1−u2|+|a||v1−v2|, and
|g(t,u1,v1)−g(t,u2,v2)| ≤ |a||u1−u2|+|a||v1−v2|, wheret∈ [0, 1],u1,u2,v1,v2 ∈R.
Takea = (|a|,|a|,|a|,|a|). Let(ϕ,ψ)be the positive eigenfunction of Ta corresponding to r(Ta), thus
Ta,1(ϕ,ψ) =r(Ta)ϕ, Ta,2(ϕ,ψ) =r(Ta)ψ. (3.6) Letλ= |a|
r(Ta). It follows from (3.6) that
(−ϕ00(t) =λϕ(t) +λψ(t), −ψ00(t) =λϕ(t) +λψ(t), t∈ (0, 1), ϕ(0) =ψ(0) =0, ϕ(1) =ψ(12), ψ(1) =2ϕ(14).
By ordinary method, we deduce that ϕ(t) = c21 sin√
2λt+ c22t,ψ(t) = c21sin√
2λt− c22t for somec1,c2 ∈ R. Clearly,c1 6= 0 holds from the non-negativity of functionsϕandψ. Without loss of generality, we assume thatc1=2. Considering the boundary conditions, we have
sin√
2λ+c2 2 =sin
√2λ 2 − c2
4, and
sin√
2λ− c2
2 =2 sin
√2λ 4 + c2
8
! . Therefore,λis the smallest positive solution of the equation
2 sin√
2λ−sin
√2λ
2 −2 sin
√2λ
4 =0, λ∈
0,π2
2
.
With the help of MATLAB, we haveλ≈ 2.0236421 which implies that the problems (3.5) has a unique solution if|a| ≤2.02364.
Acknowledgements
The authors sincerely thank the reviewer for useful comments that have led to the present improved version of the original paper. The Project Supported by the National Natural Sci- ence Foundation of China (11371221, 11371364, 11571207), the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.
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