# The Uniqueness Theorem of the Solution for a Class of Differential Systems with Coupled Integral Boundary Conditions.

1. Introduction

In this paper, we study the following differential system with coupled integral boundary conditions:

[mathematical expression not reproducible], (1)

where [alpha][u], [beta][u] are bounded linear functionals on C[0, 1] given by

[alpha][u] = [[integral].sup.1.sub.0]u(t)dA(t), [beta][u] = [[integral].sup.1.sub.0]u(t)dB(t), (2)

involving Riemann-Stieltjes integrals defined via positive Stieltjes measures of A, B.

Differential systems with coupled boundary conditions have some applications in various fields of sciences and engineering, for example, the heat equation , reaction-diffusion phenomena , and interaction problems . The existence of solutions for differential system with coupled boundary conditions has received a growing attention in the literature; for details, see [4-21]. For example, Asif and Khan in  obtained the existence of positive solution for singular sublinear system with coupled four-point boundary value conditions by using the Guo-Krasnosel'skii fixed point theorem. In , Cui and Sun discuss the existence of positive solutions of singular superlinear coupled integral boundary value problems by constructing a special cone and using fixed point index theory. In , Cui and Zou proved the existence of extremal solutions of coupled integral boundary value problems by monotone iterative method. In , Infante, Minhos, and Pietramala presented a general theory for existence of positive solutions for coupled systems by use of fixed point index theory.

The question of existence and uniqueness of solution of differential equations and differential systems is an age-old problem and it has a great importance, as much in theory as in applications. This problem has been investigated by use of a variety of nonlinear analyses such as fixed point theorem for mixed monotone operator [7, 15, 22-25], maximal principle , Banach's contraction mapping principle [26-29], and the linear operator theory [27, 30, 31].

For example, the authors  introduced a Banach space using the positive eigenfunction of linear operator related to differential system (1). They established the uniqueness results for differential system (1) under a Lipschitz condition. It should be noted that the Lipschitz constant is related to the spectral radius corresponding to the related linear operators. The obtained results are optimal from the viewpoint of theory. However, it is very difficult to determine the spectral radius for differential system (1) with general functions A(t), B(t).

Motivated by the above works, we investigate the uniqueness of solutions for differential system (1) by using a system of inequalities and the linear operator theory. The main features of this paper are as follows: (1) The main results are mostly implemented to the uniqueness result for coupled boundary value problems. (2) An easy criterion to determine the uniqueness result is obtained by using a system of inequalities. (3) An example shows that the main result provides the same results with weaker conditions.

Throughout the paper, we assume that the following condition hold:

([H.sub.1]) [alpha][t] = [[integral].sup.1.sub.0]tdA(t) > 0, [beta][t] = [[integral].sup.1.sub.0]tdB(t) > 0, [kappa] = 1-[alpha][t][beta][t] > 0.

([H.sub.2]) f,g : [0,1]x [R.sup.2] [right arrow] R are continuous.

2. Preliminaries

Let C[0,1] be the Banach space with the maximal norm given by [parallel]x[parallel] = [max.sub.t[member of][0,1]][absolute value of x(t)]. Let E = C[0,1] xC[0,1], [[parallel](x, y)[parallel].sub.E] = max{[parallel]x[parallel], [parallel]y[parallel]}. Then (E, [[parallel](*,*)[parallel].sub.E]) is a Banach space.

Lemma 1 (see ). Let u, v [member of] C[0,1], then the system of BVPs

-u"(t) = x(t), -v"(t) = y(t), t [member of] [0, 1] , u(0) = v(0) = 0, u(1) = [alpha][v], v(1) = [beta][u] (3)

has integral representation

u(t) = [[integral].sup.1.sub.0][G.sub.1](t,s)x(s)ds + [[integral].sup.1.sub.0][H.sub.1](t,s)y(s)ds, v(t) = [[integral].sup.1.sub.0][G.sub.2](t,s)y(s)ds + [[integral].sup.1.sub.0][H.sub.2](t,s)x(s)ds, (4)

where

[mathematical expression not reproducible] (5)

Lemma 2 (see ). The functions k(t,s), [G.sub.i](t,s), [H.sub.i](t,s) (i = 1,2) satisfy the following properties:

[G.sub.i](t, s) [less than or equal to] [rho]t, [H.sub.i](t,s) [less than or equal to] [rho]t, (6)

[for all]t,s [member of] [0,1], i = 1,2, 0 < k(t,s) [less than or equal to] s(1-s), [for all]t,s [member of] (0,1), (7)

where

[rho] = max{[[alpha][t]/[kappa]][beta] + 1, [[beta][t]/[kappa]][alpha] + 1, [1/[kappa]][beta], [1/[kappa]][alpha]}. (8)

With the help of Lemma 1, BVP (1) can be viewed as a fixed point in E for the completely continuous operator

S(u, v) = ([S.sub.1](u, v),[S.sub.2] (u, v)), (u, v) [member of] E, (9)

where [S.sub.1],[S.sub.2] : E [right arrow] C[0,1] are defined by

[mathematical expression not reproducible] (10)

In order to prove our main result, the following criterion for solving system of inequalities is needed.

Lemma 3. Let a,b,c,d [member of] [0, +[infinity]) with a < 1, d < 1. Then the inequality system

a + b[mu] [less than or equal to] [lambda], c + d[mu] [less than or equal to] [lambda][mu] (11)

has a solution ([lambda],[mu]) with [lambda] [member of] (0,1), [mu] > 0 if and only if a,b,c,d satisfy

(1-d)(1-a) > bc. (12)

Proof.

Necessity. The proof is obviously true for the case: be = 0. So we consider the remaining case be [not equal to] 0. From the first inequality in (11), we get

[mu] [less than or equal to] ([lambda]-a)/b. (13)

Substituting it into the second inequality in (11), we have

c [less than or equal to] ([lambda]-d)[mu] [less than or equal to] ([lambda]-d)[lambda]-a/b. (14)

Thus,

(1-d)(1-a) > ([lambda]-d)([lambda]-a) [greater than or equal to] bc. (15)

Sufficiency. For the case be = 0, we can take [lambda] = max{(d + 1)/2,(d + 1)/2}. So we consider the last case bc [not equal to] 0. Let

[phi](x) = (x-d)(x-a) - bc, x [member of] R. (16)

From the derivative of [phi](x), we conclude that [phi](x) is increasing on [(a + d)/2,1]. This together with the locally sign-preserving property of [phi](x) implies that there exists [lambda] [member of] [(a + d)/2,1) such that

([lambda]-d)([lambda]-a) [greater than or equal to] bc (17)

The above inequality can be rewritten as

c/[lambda]-d [less than or equal to] [k.summation over (j=T)]-a/b. (18)

Hence (11) holds for [mu] [member of] [c/([lambda] - d), ([lambda] - a)/b].

3. Main Result

For notational convenience, let

[mathematical expression not reproducible] (19)

where

[phi](t) = t(1-t)(1+t)/6. (20)

Take [phi](t) = t. By (7), we get

[mathematical expression not reproducible] (21)

[mathematical expression not reproducible] (22)

[mathematical expression not reproducible] (23)

[mathematical expression not reproducible] (24)

By use of (21), (22), (23), and (24), we present the main result of this paper.

Theorem 4. Suppose that there exist four nonnegative constants [a.sub.1], [b.sub.1], [c.sub.1], [d.sub.1] such that the following conditions hold:

[mathematical expression not reproducible], (25)

[a.sub.11][a.sub.1]+[a.sub.12][c.sub.1] < 1, [a.sub.21][d.sub.1]+[a.sub.22][b.sub.1] < 1. (26)

Then differential system (1) has a unique solution in E.

Proof. We divide the proof into several main steps to show that the operator S has a unique point in E under the conditions of Theorem 4.

Step 1. It follows from (25), (26), and Lemma 3 that there exist [lambda] [member of] (0,1), [mu] > 0 such that

([a.sub.11][a.sub.1]+[a.sub.12][c.sub.1])+([a.sub.11][b.sub.1]+[a.sub.12][d.sub.1]) [mu] [less than or equal to] [lambda], ([a.sub.21][c.sub.1]+[a.sub.22][a.sub.1])+([a.sub.21][d.sub.1]+[a.sub.22][b.sub.1])[mu] [less than or equal to] [lambda][mu]. (27)

Let us introduce a linear operator T on E as

T(u,v) = ([T.sub.1](u,v),[T.sub.2](u,v)), (28)

where [T.sub.1], [T.sub.2] : E [right arrow] C[0,1] is given by

[mathematical expression not reproducible] (29)

Take [psi](i) = [mu]t. Now, (21)-(24) and (27) show that

[mathematical expression not reproducible] (30)

[mathematical expression not reproducible] (31)

T([phi],[psi])(t) [less than or equal to] [lambda]([phi](t),[psi](t)). (32)

Then for p [member of] N, by induction, we obtain

[T.sup.P]([phi],[psi])(t) [less than or equal to] [[lambda].sup.p]([phi](t),[psi](t)). (33)

Step 2. For all (u,v) [member of] E with u(t) [greater than or equal to] 0 and v(t) [greater than or equal to] 0, there exists M = M(u,v) [member of] (0, +[infinity]) such that

T(u,v)(t) [less than or equal to] M x ([phi](t)),[psi](t)), t [member of] [0,1]. (34)

Indeed, by Lemma 2, we have

[mathematical expression not reproducible] (35)

So, we can take M = [rho]max{1,1/[mu]}(([a.sub.1] + [c.sub.1])[parallel]u[parallel] + ([b.sub.1] + [d.sub.1])[parallel]v[parallel]) such that (34) holds.

Step 3. For any given ([u.sub.0], [v.sub.0]) [member of] E, n = 1,2, ..., let ([u.sub.n], [v.sub.n]) = S([u.sub.n-1], [v.sub.n-1]). By Step 2, there exists M > 0 such that

T([absolute value of ([u.sub.1](t)-[u.sub.0](t))], [absolute value of ([v.sub.1](t)- [v.sub.0](t))]) [less than or equal to] M x ([phi](t),[psi](t)), i [member of] [0,1]. (36)

Notice for p [member of] N that

[mathematical expression not reproducible] (37)

Thus, by (33) and (36), we obtain that

[mathematical expression not reproducible] (38)

Thus for n, m [member of] N, we conclude that

[mathematical expression not reproducible] (39)

The above two inequalities ensure that {([u.sub.n], [v.sub.n])} is a Cauchy sequence in E. Since E is complete, there exists ([u.sup.*], [v.sup.*]) [member of] E such that [lim.sub.n[right arrow][infinity]]([u.sub.n], [v.sub.n]) = ([u.sup.*], [v.sup.*]). Therefore, ([u.sup.*], [v.sup.*]) is a fixed point of S that follows from the continuity of operator S.

Step 4. We show that S has a unique fixed point. Suppose there exist two elements ([u.sup.*], [v.sup.*]),([u.sub.*], [v.sub.*]) with S([u.sup.*], [v.sup.*]) = ([u.sup.*], [v.sup.*]) and S([u.sub.*], [v.sub.*]) = ([u.sub.*], [v.sub.*]). By Step 2, there exists M > 0 such that

T([absolute value of [u.sup.*](t)-[u.sub.*](t)], [absolute value of [v.sup.*](t)-[v.sub.*](t)]) [less than or equal to] M([phi](t),[psi](t)), t [member of] [0,1]. (40)

Applying the method used in Step 3 again, for p [member of] N, we get

[absolute value of ([u.sup.*](t)-[u.sub.*](t))] [less than or equal to] M[[[lambda].sup.p]/1-[lambda]][phi](t), [absolute value of ([v.sup.*](t)-[v.sub.*](t))] [less than or equal to] M[[[lambda].sup.p]/1-[lambda][[psi](t). (41)

Hence we get the desired results.

In the following, we give an example to illustrate our theory.

Example 5. Consider the differential system

[mathematical expression not reproducible] (42)

where [h.sub.1], [h.sub.2] [member of] C[0,1]. We have

[mathematical expression not reproducible] (43)

Let

[mathematical expression not reproducible], (44)

then

[mathematical expression not reproducible], (45)

where t [member of] [0,1],[u.sub.1],[u.sub.2],[v.sub.1],[v.sub.2] [member of] R. Hence, there exists a solution ([lambda],[mu]) = (11/12,3/2) of the following inequality system:

([a.sub.11][a.sub.1]+ [a.sub.12][c.sub.1]) + ([a.sub.11][b.sub.1] + [a.sub.12][d.sub.1])[mu] = 5/12 + 1/3[mu] [less than or equal to] [lambda], ([a.sub.21][c.sub.1]+[a.sub.22][a.sub.1])+([a.sub.21][d.sub.1]+[a.sub.22][b.sub.1])[mu] = 2/3 + 5/12[mu] [less than or equal to] [lambda][mu]. (46)

Therefore, according to Theorem 4, the problem (42) has a unique solution.

When the nonlinearity of differential equation and differential system satisfies Lipschitz condition, the usual method to obtain the uniqueness is the well-known Banach's contraction principle. For this purpose, we should add some restriction on the Lipschitz constants to guarantee the norm of a linear operator related to differential equation and differential system less than 1. Next, we discuss the estimate of the norm of a linear operator related to differential system (42).

Take [sigma](t) = 1, [??](t) = 3/2. After standard computation, we get

[mathematical expression not reproducible], (47)

Then

[mathematical expression not reproducible] (48)

So it follows from the definition of the norm for linear operator that

[parallel]T[parallel] [greater than or equal to] [[parallel]T([sigma], [??][parallel].sub.E]/[[parallel]([sigma],[??][parallel].sub.E] = 31/18 > 1. (49)

Thus Example 5 shows that Theorem 4 provides the same results with weaker conditions.

https://doi.org/10.1155/2018/9601868

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (11371221, 11571207), Shandong Natural Science Foundation (ZR2018MA011), and the Tai'shan Scholar Engineering Construction Fund of Shandong Province of China.

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Shuman Meng (1) and Yujun Cui (iD) (1,2)

(1) Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China

(2) State 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, China

Correspondence should be addressed to Yujun Cui; cyj720201@163.com

Received 16 April 2018; Accepted 3 July 2018; Published 11 July 2018

Academic Editor: Jorge E. Macias-Diaz
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