Monotone iterative method for differential systems with coupled integral boundary value problems

By establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique, we investigate the differential systems with coupled integral boundary value problems. Sufficient conditions are established for the existence of an extremal system of solutions of the given problem.

MSC:34B15.

We will devote the paper to considering the existence of a solution of coupled integral boundary value conditions for the second-order ordinary differential system (ODS for short)

where A and B are right continuous on [0,1), left continuous at t=1 and nondecreasing on [0,1], A(0)=B(0)=0; {\int }_0^{1}u(s)dA(s) and {\int }_0^{1}u(s)dB(s) denote the Riemann-Stieltjes integrals of u with respect to A and B, respectively.

The theory of differential system with coupled boundary value conditions is an important branch of nonlinear analysis. It is worth mentioning that a differential system with coupled boundary value conditions appears often in investigations connected with mathematical physics, mathematical biology, biochemical system and so on (see [1–3]). One of the basic problems considered in the theory of differential system with coupled boundary value conditions is to establish convenient conditions guaranteeing the existence of solutions of those equations. However, the theory of coupled boundary value problems for a differential system is still in the initial stages.

The monotone iterative technique combined with the method of upper and lower solutions is a powerful tool for proving the existence of solutions of differential equations/system (see [4–9] and the references therein), the advantage and importance of the technique needs no special emphasis [10–12]. The basic idea of this method is that using the upper and lower solutions as an initial iteration, one can construct monotone sequences from the corresponding linear differential equations/system, and these sequences converge monotonically to the maximal and minimal solutions of the nonlinear differential equations/system. When the method is applied to a differential system with coupled boundary value conditions, it usually needs suitable differential inequalities as a comparison principle. The results in this paper are inspired by [13]. Here, we establish differential inequalities as a comparison principle, i.e., Lemma 2.2. Then, we give a different proof for the existence and uniqueness of the solutions for linear coupled boundary value conditions for a differential system, i.e., Lemma 2.5. Finally, by use of the monotone iterative technique and the method of upper and lower solutions, we obtain the existence result of extremal solutions for (1.1).

Let

We make assumptions involving {\kappa }_1, {\kappa }_2 and κ as follows.

(H₁) {\kappa }_1>0, {\kappa }_2>0, \kappa >0.

Definition 2.1 (u_0,v_0)\in E\times E is called a lower system of solutions of differential system (1.1) if

Analogously, ({\alpha }_0,{\beta }_0)\in E\times E is called an upper system of solutions of differential system (1.1) if

In what follows, we assume that

and define that sector

where the vectorial inequalities mean that the same inequalities hold between their corresponding components.

We present new comparison results and lemmas which are crucial for our discussion.

Lemma 2.1 [11, 14]

Suppose that . If there exist M\in C[[a,b],R^{+}] and p\in C^2[a,b] satisfying

then p(t)\le 0, t\in [a,b].

Lemma 2.2 Suppose that (H₁) holds. Let x,y\in E satisfy

where M(t),N(t)\in C[[0,1],R^{+}]. Then x(t)\le 0, y(t)\le 0, t\in [0,1].

Proof Suppose the contrary. By Lemma 2.1, one easily sees that there are only three cases to consider:

Case 1. x(1)\le 0 and y(1)>0. By Lemma 2.1, x(t)\le 0 for all t\in [0,1]. Then y(1)\le {\int }_0^{1}x(t)dB(t)\le 0, which contradicts y(1)>0.

Case 2. y(1)\le 0 and x(1)>0. By Lemma 2.1, y(t)\le 0 for all t\in [0,1]. So, we have x(1)\le {\int }_0^{1}y(t)dB(t)\le 0, which contradicts x(1)>0.

Case 3. x(1)>0 and y(1)>0. There are \xi ,\eta \in [0,1) such that

and

It follows from (2.2) that

Hence, we have

and

Therefore,

and

The last two inequalities give

which implies that 1\le {\kappa }_1{\kappa }_2, a contradiction. Hence, x(t)\le 0, y(t)\le 0, t\in [0,1]. □

Lemma 2.3 Suppose that (H₁) holds. Let x,y\in E satisfy

where M(t),N(t)\in C[[0,1],R^{+}]. Then x(t)=y(t)=0, t\in [0,1].

The proof of Lemma 2.3 is easy, so we omit it.

Consider the differential system of BVPs

where g,h\in C[0,1].

Lemma 2.4 Assume that (H₁) holds. Then (x,y)\in E\times E is a system of solutions of BVPs (2.3) if and only if (x,y)\in C[0,1]\times C[0,1] is a system of solutions of the integral equation

where

Proof First, suppose that (x,y)\in E\times E is a system of solutions of BVPs (2.3). It is easy to see that (2.3) is equivalent to the system of integral equations

Integrating (2.5) and (2.6) with respect to dB(t) and dA(t) respectively on [0,1] gives

Therefore,

and so

Substituting (2.7) into (2.5) and (2.6), we have

which is equivalent to system (2.4).

Conversely, assume that (x,y)\in C[0,1]\times C[0,1] is a system of solutions of an integral equation. Direct differentiation on (2.4) implies

Making use of the fact k(0,s)=k(1,s)=0 for s\in [0,1], we obtain

Simple computations yield

Hence, (x,y)\in E\times E is a system of solutions of BVPs (2.3). □

Consider the linear differential system of BVPs

where M(t),N(t)\in C[[0,1],R^{+}] and g,h\in C[0,1].

Lemma 2.5 Assume that (H₁) holds. Then there exists a unique system of solutions (x,y) to BVPs (2.8).

Proof It follows from Lemma 2.4 that (2.8) is equivalent to the operator equation

where

and

By using standard arguments, we can easily show that T:C[0,1]\times C[0,1]\to C[0,1]\times C[0,1] is linear completely continuous. By Lemma 2.3, the operator equation (x,y)=T(x,y) has only the zero solution. Then, by the Fredholm theorem, for given (\tilde{g},\tilde{h})\in C[0,1]\times C[0,1], the operator equation (x,y)=T(x,y)+(\tilde{g},\tilde{h}) has only one solution in C[0,1]\times C[0,1]. Hence, (2.8) has exactly one system of solutions (x,y)\in E\times E. □

In this section, on the basis of Lemma 2.2 and Lemma 2.5, using the monotone iterative technique, we shall show an existence theorem of a solution of (1.1).

We list the following assumptions for convenience.

(H₂) f_1(t,x,y) is nondecreasing in y and there exists M\in C[[0,1],R^{+}] such that

where u_0(t)\le x_2\le x_1\le {\alpha }_0(t), v_0(t)\le y\le {\beta }_0(t).

(H₃) f_2(t,x,y) is nondecreasing in x and there exists N\in C[[0,1],R^{+}] such that

where u_0(t)\le x\le {\alpha }_0(t), v_0(t)\le y_2\le y_1\le {\beta }_0(t).

Theorem 3.1 Assume that (u_0,v_0), ({\alpha }_0,{\beta }_0) are lower and upper systems of solutions of problem (1.1) such that (2.1) holds, f_1,f_2\in C[[0,1]\times R^2] and satisfies (H₁)-(H₃). Then there exist monotone iterative sequences \{(u_n,v_n)\}, \{({\alpha }_n,{\beta }_n)\} which converge uniformly on [0,1] to the extremal solutions of problem (1.1) in the sector Ω.

Proof \mathrm{\forall }(\xi ,\eta )\in \mathrm{\Omega }, consider (2.8) with

By Lemma 2.5, (2.8) has a unique system of solutions (x,y)\in E\times E. Denote an operator S:\mathrm{\Omega }\to E\times E by (x,y)=S(\xi ,\eta ), and

Then the operator S has the following properties:

1. (i)

   (u_0,v_0)\le S(u_0,v_0), S({\alpha }_0,{\beta }_0)\le ({\alpha }_0,{\beta }_0).

Let (u_1,v_1)=S(u_0,v_0), p(t)=u_0(t)-u_1(t) and q(t)=v_0(t)-v_1(t). By (H₂) and (H₃), we have that

which implies, by virtue of Lemma 2.2, that p(t)\le 0, q(t)\le 0, \mathrm{\forall }t\in [0,1], i.e., (u_0,v_0)\le S(u_0,v_0). A similar argument shows that S({\alpha }_0,{\beta }_0)\le ({\alpha }_0,{\beta }_0). (ii) S is nondecreasing. Let ({\xi }_1,{\eta }_1),({\xi }_2,{\eta }_2)\in \mathrm{\Omega } be such that ({\xi }_1,{\eta }_1)\le ({\xi }_2,{\eta }_2). Suppose that (p,q)=S({\xi }_1,{\eta }_1)-S({\xi }_2,{\eta }_2). By (H₂) and (H₃), we have

which implies by virtue of Lemma 2.2 that p(t)\le 0, q(t)\le 0, \mathrm{\forall }t\in [0,1], i.e., S is nondecreasing. This together with (i) implies that S:\mathrm{\Omega }\to \mathrm{\Omega }.

Now let (u_n,v_n)=S(u_{n-1},v_{n-1}), ({\alpha }_n,{\beta }_n)=S({\alpha }_{n-1},{\beta }_{n-1}), n=1,2,3,\dots . Following (i) and (ii), we have

Using the standard arguments, it is easy to show that \{(u_n,v_n)\} and \{({\alpha }_n,{\beta }_n)\} are uniformly bounded and equicontinuous in Ω. By (3.1) and the Arzela-Ascoli theorem, we have

uniformly on t\in [0,1], and (u_{\ast },v_{\ast }), ({\alpha }^{\ast },{\beta }^{\ast }) satisfy (1.1). Moreover, (u_{\ast },v_{\ast }),({\alpha }^{\ast },{\beta }^{\ast })\in \mathrm{\Omega }. Thus, (u_{\ast },v_{\ast }) and ({\alpha }^{\ast },{\beta }^{\ast }) are solutions of (1.1) in Ω.

Next, we prove that (u_{\ast },v_{\ast }) and ({\alpha }^{\ast },{\beta }^{\ast }) are extremal solutions of (1.1) in Ω. In fact, we assume that (x,y) is any solution of (1.1). That is,

By (H₂) and (H₃), and Lemma 2.2, it is easy by induction to show that

Now, letting n\to \mathrm{\infty } in (3.2), we have (u_{\ast },v_{\ast })\le (x,y)\le ({\alpha }^{\ast },{\beta }^{\ast }). That is, (u_{\ast },v_{\ast }) and ({\alpha }^{\ast },{\beta }^{\ast }) are extremal systems of solutions of (1.1) in Ω. □

Consider the following problems:

where

Obviously,

Take (u_0(t),v_0(t))=(0,0), ({\alpha }_0(t),{\beta }_0(t))=(t,t), then

It shows that (u_0(t),v_0(t)) and ({\alpha }_0(t),{\beta }_0(t)) are lower and upper systems of solutions of (4.1).

On the other hand, by (4.2), we have that

where u_0(t)\le x_1(t)\le x_2(t)\le {\alpha }_0(t), v_0(t)\le y_1(t)\le y_2(t)\le {\beta }_0(t).

Thus, all the conditions of Theorem 3.1 are satisfied. Therefore, by Theorem 3.1, (4.1) has an extremal system of solutions (u_{\ast },v_{\ast }), ({\alpha }^{\ast },{\beta }^{\ast }), which can be obtained by taking limits from some iterative sequences.

The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. The project is supported by the National Natural Science Foundation of China (11371221, 11071141), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, the Postdoctoral Science Foundation of Shandong Province and Foundation of SDUST.

Affiliations

1. School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, P.R. China

   • Yujun Cui

2. Department of Mathematics, Shandong University of Science and Technology, Qingdao, 266590, P.R. China

   • Yujun Cui

3. Department of Statistics and Finance, Shandong University of Science and Technology, Qingdao, 266590, P.R. China

   • Yumei Zou

Corresponding author

Correspondence to Yujun Cui.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read and approved the final manuscript.

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