Research Article | Open Access
Solutions for -Point BVP with Sign Changing Nonlinearity
We study the existence of positive solutions for the following nonlinear -point boundary value problem for an increasing homeomorphism and homomorphism with sign changing nonlinearity: , , , , where is an increasing homeomorphism and homomorphism and . The nonlinear term may change sign. As an application, an example to demonstrate our results is given. The conclusions in this paper essentially extend and improve the known results.
In this paper, we study the existence of positive solutions of the following nonlinear -point boundary value problem with sign changing nonlinearity: where is an increasing homeomorphism and homomorphism and ; with and satisfy
(), ;() does not vanish identically on any subinterval of and satisfies.
Definition 1.1. A projection is called an
increasing homeomorphism and homomorphism, if the following conditions are
(i)if , then , for all ;(ii) is a continuous bijection and its inverse mapping is also continuous;(iii), for all .
The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il'in and Moiseev [1, 2]. Motivated by the study of [1, 2], Gupta  studied certain three-point boundary value problems for nonlinear ordinary differential equations. Since then, more general nonlinear multipoint boundary value problems have been studied by several authors. We refer the reader to [4–12] for some references along this line. Multipoint boundary value problems describe many phenomena in the applied mathematical sciences. For example, the vibrations of a guy wire of a uniform cross-section and composed of parts of different densities can be set up as a multipoint boundary value problems (see Moshinsky ); many problems in the theory of elastic stability can be handled by the method of multipoint boundary value problems (see Timoshenko ).
In 2001, Ma  studied -point boundary value problem (BVP):where , and , . Author established the existence of positive solutions under the condition that is either superlinear or sublinear.
In , we considered the existence of positive solutions for the following nonlinear four-point singular boundary value problem with -Laplacian:where . By using the fixed-point theorem of cone, the existence of positive solution and many positive solutions for nonlinear singular boundary value problem -Laplacian is obtained.
Recently, Ma et al.  used the monotone iterative technique in cones to prove the existence of at least one positive solution for -point boundary value problem (BVP):where , , .
In , Wang and Hou investigated the following -point BVP:where , , with and satisfy , .
However, in all the above-mentioned paper, the authors discuss the boundary value problem (BVP) under the key conditions that the nonlinear term is positive continuous function. Motivated by the results mentioned above, in this paper we study the existence of positive solutions of -point boundary value problem (1.1) for an increasing homeomorphism and homomorphism with sign changing nonlinearity. We generalize the results in [4–12].
2. The Preliminary Lemmas
In this section, we present some lemmas which are important to our main results.
Lemma 2.1. Let and hold. Then for , the problemhas a unique solution if and only if can be express as the following equation:where satisfywhereDefine , then there exists a unique satisfying (2.3).
Proof. The method of the proof is similar to [5, Lemma 2.1], we omit the details.
Lemma 2.2. Let and hold. If , the unique solution of the problem (2.1) satisfies
Lemma 2.3. Let and hold. If , the unique solution of the problem (2.1) satisfieswhere
Proof. ClearlyThis implies thatIt is easy to see that , for any with . Hence is a decreasing function on . This means that the graph of is concave down on . So we haveTogether with and on , we getThe proof of Lemma 2.3 is completed.
Lemma 2.4 (see ). Let be a cone in a Banach space . Let be an open
bounded subset of with and . Assume that is a compact
map such that for . Then the following results hold.
(1)If , , then .(2)If there exists such that , for all and all , then .(3)Let be open in such that . If and , then has a fixed point in . The same results hold, if and .
Let , then is Banach space, with respect to the norm . Denotewhere is the same as in Lemma 2.3. It is obvious that is a cone in .
We define and
Lemma 2.5 (see ). defined above
has the following properties:
(a)(b) is open relative to (c) if and only if (d) If , then for . Now, for the convenience, one introduces the following notations:
3. The Main Result
In the rest of the section, we also assume the following conditions.
There exist with such thatThere exist with such thatThere exist with and such thatThere exist with such thatThere exist with such thatThere exist with such that
Our main results are the following theorems.
Theorem 3.1. Assume that hold. Then BVP (1.1) has at least three positive solutions.
Theorem 3.2. Assume that hold. Then BVP (1.1) has at least two positive solutions.
Theorem 3.3. Assume that hold and also assume that or hold. Then BVP (1.1) has at least a positive solution.
Theorem 3.4. Assume that hold and also assume that or hold. Then BVP (1.1) has at least two positive solutions.
Proof of Theorem 3.1. Without loss of
generality, we suppose that hold. Denoteit is easy to check that
Now define an operator by settingwhereBy Lemma 2.3, we have . So by applying Arzela-Ascoli's theorem, we can obtain that is relatively compact. In view of Lebesgue's dominated convergence theorem, it is easy to prove that is continuous. Hence, is completely continuous.
Now, we consider the following modified BVP (1.1):Obviously, BVP (3.10) has a solution if and only if is a fixed point of the operator . From the condition , we have
Next, we will show that .
In fact, by , for , we haveThis implies that for . By Lemma 2.4(1), we have
Furthermore, we will show that .
Let , for , then . We claim that
In fact, if not, there exist and such that
By and Lemma 2.1, we have for ,so thatThen, we have thatThis implies that , this is a contradiction. Hence, by Lemma 2.4(2), it follows that
Finally, similar to the proof of , we can show that .
By Lemma 2.5(a) and and , we have . It follows from Lemma 2.4(3) that has three positive fixed points in , respectively. Therefore, BVP (3.10) has three positive solutions in , respectively.
Then, BVP (3.10) has three positive solutions , which means that are also the positive solutions of BVP (1.1).
Proof of Theorem 3.4. We show that condition implies
condition . Let , then there exists such that since . DenoteThen we haveThis implies that and holds.
Similarly condition implies
By an argument similar to that Theorem 3.1, we can obtain the result of Theorem 3.4. The proof of Theorem 3.4 is completed.
Example 4.1. Consider the
following five-point boundary value problem with -Laplacian:where :
It is easy to check that is continuous. It follows from a direct calculation that
Choose , it is easy to check that and It follows that satisfies the condition of Theorem 3.3, then problems (1.1) have at least two positive solutions.
Remark 4.2. Let , the problem is second-order -point boundary value problem.
Remark 4.3. Let , the problem is boundary value problem with -Laplacian operators.
Hence our results generalize boundary value problem with -Laplacian operators.
This research is supported by the Doctor of Scientific Startup Foundation of Shandong University of Finance of China (08BSJJ32).
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