Generalized monotone iterative method for nonlinear boundary value problems with causal operators
© Wang and Tian; licensee Springer. 2014
Received: 17 March 2014
Accepted: 28 July 2014
Published: 25 September 2014
This paper discusses nonlinear boundary value problems for causal differential equations where the right-hand side is the sum of two monotone functions. We develop the monotone iterative technique and establish the existence results of the extremal solutions. The results obtained include several special cases and extend previous results; two examples satisfying the assumptions are also presented.
MSC: 34B15, 39B12.
As we all know, the monotone iterative technique is an effective and a flexible method, and it provides a useful mechanism to prove existence results for nonlinear differential equations, for detail see for monographs , papers –, and the references therein. The basic idea of this method is that using the upper and lower solutions as an initial iteration, one can construct monotone sequences from a corresponding linear equations, and these sequences converge monotonically to the minimal and maximal solutions of the nonlinear equations.
In 2004, West  developed this method, considered the generalized monotone iterative method for initial value problems, obtained the existence of extremal solutions for differential equations where the forcing function is the sum of two monotone functions, one of which is monotone non-decreasing and the other is non-increasing.
Recently, this method has been extended to causal differential equations. Its theory has the powerful quality of unifying ordinary differential equations, integro differential equations, differential equations with finite or infinite delay, Volterra integral equations and neutral equations. We refer to the monograph by Lakshmikantham  and papers , .
In 2009, Lakshmikantham discussed in  the following problem with causal operators: , , where , are causal operators. However, we notice that the results are only valid for initial value problems. Motivated by the above excellent work, we extend the notion of casual operators to nonlinear boundary value problems and develop the monotone iterative technique.
where are causal operators.
Note that the nonlinear boundary value problem (1.1) reduce to periodic boundary value problems for , initial value problems for which has been studied in  and other general conditions such as . Thus problem (1.1) can be regarded as a generalization of the boundary value problems mentioned above.
The rest of this paper is organized as follows. In Section 2, we develop the monotone technique for (1.1); four theorems and several special cases are given. In Section 3, we give two examples to illustrate the results obtained. Finally, a brief summary is given in Section 4.
2 Main results
Let , . In order to prove general results, we need the following definitions.
- (1)natural lower and upper solutions if
- (2)coupled lower and upper solutions of type I, if
- (3)coupled lower and upper solutions of type II, if
- (4)coupled lower and upper solutions of type III, if
If we set , , is non-decreasing and is non-increasing, then the natural lower and upper solutions and the coupled lower and upper solutions of type III satisfy type II. Thus, we only need to consider the case of the coupled lower and upper solutions of type I and II for (1.1).
A function is said to be a natural solution if it satisfies (1.1).
- (2)are said to be coupled solutions of type I, if
- (3)U, V are said to be coupled solutions of type II, if
- (4)U, V are said to be coupled solutions of type III, if
Coupled solutions , are said to be couple minimal and maximal solutions of (1.1), if for any coupled solutions U, V, we have .
We suppose that the following hypotheses hold:
H1: are the coupled lower and upper solutions of type I for (1.1) withon J;
H2: the operators Q, S in (1.1) are such that, is non-decreasing in u andis non-increasing in u;
Then there exist two monotone sequences , such that , uniformly and monotonically on J and that ρ, r are coupled minimal and maximal solutions of type I for (1.1). Furthermore, if u is any natural solution of (1.1) such that on J, then on J.
It follows that on J, which implies on J. Similarly, we may obtain on J.
This implies that on J, and .
Obviously, the constructed sequences , are equicontinuous and uniform bounded. Thus, by the Ascoli-Arzela theorem, we have , on J. Since the sequences , are monotone, the entire sequences converge uniformly and monotonically to ρ, r on J, respectively.
That implies , which proves on J. Using similar arguments we can conclude on J. Since , by the principle of induction, holds for all n. Taking the limit as , we have on J proving ρ, r are coupled minimal and maximal solutions of type I for (1.1). Since any natural solution u of (1.1) can be considered as coupled solutions of type I, we also have on J. This completes the proof. □
Let the hypotheses of Theorem 2.1hold. Then, for any natural solution u of (1.1) with, there exist alternating sequences, uniformly on J with. Here ρ, r are the coupled minimal and maximal solutions of type I for (1.1). Also, on J.
First we show that and .
Thus we prove , which gives on J. Similarly, we can show .
Hence on J, i.e., on J. A similar argument yields . In order to avoid repetition, we can prove each of the following: , , , and .
Then on J, thus . Similarly, we can obtain on J.
This implies that on J, i.e., on J. Using similar arguments we can show , , . Combining all these arguments, we now have the desired relations (2.7).
This implies that and . Similarly, we obtain , , and .
By employing a reasoning similar to that of Theorem 2.1, we get the sequences , which converge uniformly and monotonically to ρ, r on J, respectively. Thus, ρ, r are coupled solutions of type I for (1.1).
Finally, to prove that ρ, r are coupled minimal and maximal solutions of (1.1), let be any coupled solutions of type I for (1.1). Similar to the proof of the above, if for some positive integer k, we can easily see that on J, by the induction, one has holds on J for all n. Taking the limit as , we have on J proving ρ, r are coupled minimal and maximal solutions of type I for (1.1). Since we have already shown that holds on J for all n. Now taking the limit as , we get on J. This completes the proof. □
we see that it can be seen as (1.1) withreplaced byandreplaced by. Thus we get the same conclusions as for Theorem 2.1and Theorem 2.2.
- (i)there exist the monotone sequences , converging to ρ and r, where are coupled minimal and maximal solutions of (1.1), respectively, satisfying
- (ii)there exist alternating sequences converging to ρ, converging to r, where are coupled minimal and maximal solutions of (1.1), respectively, satisfying
which is the same as (1.1) withreplaced by Nu andreplaced by. Hence the conclusions of Theorem 2.1and Theorem 2.2remain valid.
which is the same as (1.1) withreplaced byandreplaced byand the conclusions of Theorem 2.1and Theorem 2.2hold.
which is the same as (1.1) withreplaced byandreplaced byand the conclusions of Theorem 2.1and Theorem 2.2hold.
Ifis not non-decreasing andis not non-increasing, then for some, such thatis non-decreasing andis non-increasing, we can get the conclusions of Theorem 2.1and Theorem 2.1withreplaced byandreplaced by.
We can always construct coupled upper and lower solutions of type II as in this paper. To avoid repetition, we will merely state the next two theorems without proof since it follows along the same lines as Theorems 2.1 and 2.2.
forand if u is any natural solution of (1.1) such thaton J, thenon J.
In this section, we give two simple but illustrative examples, thereby validating the proposed theorems.
It proves that , are coupled lower and upper solutions of problem (3.1). Then assumptions H1, H2, and H3 hold with . By Theorem 2.1, we obtain the existence of monotone sequences that approximate the extremal solutions of (3.1) in the sector . By Theorem 2.2, we obtain the existence of alternating sequences that also converge to the extremal solutions.
Functions , are coupled lower and upper solutions of type II for problem (3.2). It is easy to see that H1, H2, and H3 hold with . Consider the function , satisfy , , respectively. Therefore, applying Theorem 2.3, by using (2.1) and (2.2), we obtain the existence of monotone sequences that approximate the extremal solutions of (3.2) in the sector . From Theorem 2.4, we obtain the existence of alternating sequences that also converge to the extremal solutions.
In this paper, we have developed monotone iterative method for casual differential equations with nonlinear boundary conditions. The method is based on the new concepts of lower and upper solutions (resp. coupled lower and upper solutions). We have constructed monotone sequences and alternating sequences from a corresponding linear equations. It was proven that these sequences converge uniformly to the coupled minimal and maximal solutions of the problems.
The condition on the function was necessary to carry out the proofs. A future research direction would be to consider this method for nonlinear conditions of or . Also, since the theory of causal differential equations has recently gained more attention, much work can be done on its theoretical research.
The authors would like to thank the reviewers and the editors for their valuable suggestions and comments. This work was supported by the NNSF of China (No. 61073121), the Fundamental Research Funds for the Central Universities (No. 13ZD19), and the Higher School Science Research of Hebei Province of China (No. Z2013038).
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