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Nonlocal boundary value problems with resonant or non-resonant conditions
Boundary Value Problems volume 2013, Article number: 238 (2013)
We study solvability of nonlocal boundary value problems for second-order differential equations with resonance or non-resonance. The method of proof relies on Schauder’s fixed point theorem. Some examples are presented to illustrate the main results.
In this paper, we investigate the existence of solutions for the following boundary value problem:
where , for , , , .
Nonlocal boundary value problems, studied by Il’in and Moiseev , have been addressed by many authors; see, for example, [2–9] and references therein. In the related literature, (1.1) is called resonance when , and non-resonance when . For the boundary value problems at resonance, researchers usually use the continuity method or nonlinear alternative, which involves a complicated a priori estimate for the solution set; see [7, 10–12]. However, it is very difficult to obtain a related estimate for general differential equations. Here we list only a classical result about nonlocal boundary value problems at resonance of the form
where is continuous, is continuous and for , , .
Theorem 1.1 
Suppose that there are two constants such that
(A1) for any , ;
(A2) there exist constants , such that
(A3) for ,
Then (1.2) has at least one solution.
For (1.1), condition (A2) in Theorem 1.1 implies that for . It follows that (1.1) has infinitely many solutions ( is the solution of (1.1)). At this point, Theorem 1.1 has little significance for (1.1). Moreover, there are few papers considering multiple results at resonance. For the case with non-resonance, there is an extensive literature; see [14–17] and the references therein.
The main purpose of this article is to discuss the existence of solutions of equation (1.1) by means of Schauder’s fixed point theorem. We only need to consider the behavior of g and f on some closed sets. Consequently, information on the location of the solution is obtained and multiple results are obtained if g and f satisfy the given conditions on distinct regions. Our approach is valid for the cases at resonance or non-resonance. In addition, some of our conditions are easily certified (see Corollaries 3.1 and 3.2).
The paper is organized as follows. Section 2 introduces an important lemma. Section 3 is devoted to the existence results of (1.1). In Section 4 we extend some results of Section 3 to the general boundary conditions.
In this section, we consider the following boundary value problem for the linear differential equation:
where , , for , and .
Boundary value problem (2.1) has a unique solution .
If on J and , then on J.
If for all , then on J; if for all , then on J.
If () on J, then on J.
Define an operator by , where ; then A is completely continuous.
Proof (1) Any solution of the differential equation can be written as
where , are constants and is a particular solution of . From the boundary conditions, we obtain that
Since the above system has a unique solution , (2.1) has a unique solution .
The conclusion is obvious.
Here we only prove the case of . We consider two cases.
Case 3.1 Assume that for all . We show that , . From (2.1), we obtain that
which implies that is nonincreasing on J. Noting , we obtain that is nonincreasing. Thus .
Since is continuous, by the intermediate value theorem, there exists such that .
If , one can obtain from the monotonicity of that on . Hence,
which implies that . Hence, on J.
If , then , which implies that . Hence, on J.
Case 3.2 There exist such that and . We assume that . Otherwise, for all . Similar to Case 3.1, one can show that for , which is impossible.
If there is such that for , it is easy to check that for all , a contradiction. Since , there exists such that . Noting that and , we obtain that
which is a contradiction.
From Cases 3.1 and 3.2, one can easily obtain that for all .
Since , using the conclusion of (3), we have
Noting that , , we obtain that
If , then and . Thus on J.
Assume that . There exist such that
Noting that and , we have
From (2.3), (2.4) and (2.5), we obtain that on J.
Let in . For any , there exists such that
Noting that , using the conclusion of (4), we obtain that
which implies that A is continuous. Let D be a bounded set in . Then there is such that for all . From the conclusion of (4), we obtain that , which implies that is a uniformly bounded set. Since , h are bounded for and
there exists such that
which implies that is equicontinuous. It follows that is relatively compact in and A is a completely continuous operator. The proof is complete. □
Remark 2.1 Let and
By a direct computation, one can obtain that and
The following well-known Schauder fixed point theorem is crucial in our arguments.
Lemma 2.2 
Let X be a Banach space and be closed and convex. Assume that is a completely continuous map; then T has a fixed point in D.
3 Main results
The following theorem is the main result of the paper.
Theorem 3.1 Assume that there exist constants , such that , , is nondecreasing in and . Further suppose that
Then (1.1) has at least one solution x with .
Proof From Lemma 2.1, if x is a solution of (1.1), x satisfies
where is composition of A and H defined as , and the operator H is defined in as
Obviously, a fixed point of is a solution of (1.1). Set . Since the function is nondecreasing in , we obtain that for ,
Using (3.1), we have
for any . Since , and A is one nondecreasing operator, we obtain that for ,
Also, the fact that A is completely continuous and H is continuous gives that is a continuous, compact map. By Lemma 2.2, has at least one fixed point in Ω. The proof is complete. □
Remark 3.1 In Theorem 3.1, the condition that is nondecreasing in can been replaced by the weaker condition
Corollary 3.1 Assume that and the following condition holds:
(H1) There exist constants such that , , and
Then (1.1) has at least one solution x with .
Proof Since , there exists such that the function is nondecreasing in . When , . We directly apply Theorem 3.1 and this ends the proof. □
Corollary 3.2 Assume that and the following condition holds:
(H2) There exists constant such that , , and
Then (1.1) has at least one solution x with .
Proof Since , there exists such that the function is nondecreasing in . Condition (3.1) is satisfied if (3.4) holds. The proof is complete. □
Example 3.1 Consider the differential equation
Let , , , and n be a positive integer. For any and ,
Hence, by Corollary 3.1, (3.5) has a solution . Since n is an arbitrary positive integer, (3.5) has infinitely many solutions.
Example 3.2 Consider the differential equation
Using Corollary 3.2, we obtain that (3.6) has at least a nonnegative solution . Moreover, it is not difficult to show that for any .
In this section, we extend some results in the previous section to the following equation:
where and are linear operators defined as
where and for , , , .
Consider the boundary value problem for the linear differential equation:
where is sufficiently large, . We introduce the following assumptions.
(P1) The condition implies that boundary value problem (4.2) has a unique solution .
(P2) The condition implies that on J.
(P3) The condition () implies that on J.
(P4) The condition () implies that on J.
We say that the boundary condition satisfies P123 if (4.2) satisfies conditions (P1), (P2), (P3), and P134 if (4.2) satisfies conditions (P1), (P3), (P4).
Assume that the boundary condition satisfies P123 and (H1) holds. Then (4.1) has at least one solution x with .
Assume that the boundary condition satisfies P134 and (H2) holds. Then (4.1) has at least one solution x with .
The proof of Theorem 4.1 is similar to that of Theorem 3.1 and we omit it.
Remark 4.1 The solution obtained in Corollary 3.2 or (2) of Theorem 4.1 may be trivial. Further suppose that
Then the solution obtained is nonnegative and nontrivial.
Remark 4.2 The boundary condition satisfies P134 if it satisfies P123.
Remark 4.3 Consider the two-point boundary conditions:
One can easily check that boundary conditions (4.3), (4.4) satisfy P123, and conditions (4.5), (4.6), (4.7) satisfy P134.
Next, we consider the boundary conditions
where α, β, λ, μ, (), () are constants and , , .
Theorem 4.2 Set , .
If , , then the boundary condition satisfies P123.
If , or , , then the boundary condition satisfies P134.
Proof Without loss of generality, we assume that . For any sufficiently large and , the linear differential equation
has a unique solution .
Suppose that on J; if is not true, by the maximum principle, we get that or . If , then . From the boundary conditions, we have
By the maximum principle, we obtain that , which is a contradiction. If , then . From the boundary conditions, we have
By the maximum principle, we obtain that , a contradiction.
If and , , then .
Now suppose that on J and .
If there is such that , noting that , we have
If , from the boundary conditions, we obtain that , , which implies that . The case has been discussed. If , from the boundary conditions, we obtain that , , which implies that . The case has also been discussed. The proof is complete. □
Example 4.1 Consider the differential equation
where , are constants.
Let and . Set , . If n is a sufficiently large, positive integer, then for any , ,
By Theorems 4.1 and 4.2, (4.10) has a solution . Hence, (4.10) has infinitely many solutions.
Example 4.2 Consider the differential equation
where , , are constants.
In fact, and . The boundary conditions in (4.11) satisfy P134 for and P123 for .
Equation (4.11) has a solution and , for all . Set , then for all . By Theorems 4.1 and 4.2, (4.11) has a solution . Now we show that for . Assume that there exists with . Since , is minimum value and , . On the other hand, , a contradiction. If , then . This is impossible.
Equation (4.11) has infinitely many solutions for . Set , , where is an integer. Since for all , (4.11) has a solution . Hence, (4.11) has infinitely many solutions.
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The work is supported by the NNSF of China (11171085) and Hunan Provincial Natural Science Foundation of China.
The authors declare that they have no competing interests.
Authors typed, read and approved the final draft.