Research Article | Open Access
Multiple Positive Solutions for Singular Semipositone Periodic Boundary Value Problems with Derivative Dependence
By constructing a special cone in and the fixed point theorem, this paper investigates second-order singular semipositone periodic boundary value problems with dependence on the first-order derivative and obtains the existence of multiple positive solutions. Further, an example is given to demonstrate the applications of our main results.
In this paper, we are concerned with the existence of multiple positive solutions for the second-order singular semipositone periodic boundary value problems (PBVP, for short): where , the nonlinear term may be singular at , , and , also may be negative for some value of , , and .
In recent years, second-order singular periodic boundary value problems have been studied extensively because they can be used to model many systems in celestial mechanics such as the N-body problem (see [1–11] and references therein). By applying the Krasnosel'skii's fixed point theorem, Jiang  proves the existence of one positive solution for the second-order PBVP where is a constant and . Zhang and Wang  used the same fixed point theorem to prove the existence of multiple positive solutions for PBVP (1.2) when is nonnegative and singular at , not singular at , . Lin et al.  only obtained the existence of one positive solution to PBVP (1.1) when is semipositone and singular only at .All the above works were done under the assumption that the first-order derivative is not involved explicitly in the nonlinear term .
Motivated by the works of [5–7], the present paper investigates the existence of multiple positive solutions to PBVP (1.1). PBVP (1.1) has two special features. The first one is that the nonlinearity may depend on the first-order derivative of the unknown function , and the second one is that the nonlinearity is semipositone and singular at , , and . We first construct a special cone different from that in [5–7] and then deduce the existence of multiple positive solutions by employing the fixed point theorem on a cone. Our results improve and generalize some related results obtained in [5–7].
The contents of this paper are distributed as follows. In Section 2, we introduce some lemmas and construct a special cone, which will be used in Section 3. We state and prove the existence of at least two positive solutions to PBVP (1.1) in Section 3. Finally, an example is worked out to demonstrate our main results.
2. Some Preliminaries and Lemmas
Define the set functions where is the conjugate exponent of , where is the Gamma function.
Given , let be the Green function for the equation
Now, the following Lemma follows immediately from the paper .
Lemma 2.1. has the following properties:(G1) is continuous in and for all ;(G2) for all and ;(G3) denote and , then ;(G4) there exist functions such that
where are constants, are independent solutions of the linear differential equation , and ;(G5) is bounded on .
Denote , then .
Remark 2.2. Using paper , we can get when and , obtaining
Let with norm , where . Then is a Banach space. Let , from Lemma 2.1, we know that are both constants and , .
It is easy to conclude that is a cone of and is an open set of .
Lemma 2.3 (see ). Let be a Banach space and a cone in . Suppose and are bounded open sets of such that and suppose that is a completely continuous operator such that(1) and for ; for or(2) and for ; for . Then has a fixed point in .
For convenience, let us list some conditions for later use.(H0) is continuous and there exists a constant such that where , and ;(H1) there exist and with such that (H2) there exists such that where ;(H3) there exists such that
3. Main Results
Theorem 3.1. Assume that conditions (H0)–(H3) are satisfied, then PBVP (1.1) has at least two positive solutions such that , where .
Proof. We consider the following PBVP:
It is easy to see that if and is a positive solution of PBVP (3.1) with for , then is a positive solution of PBVP (1.1) and .
As a result, we will only concentrate our study on PBVP (3.1).
Define an operator by where is the Green function to problem (2.3). (1) We first show that is completely continuous for any .
For any , from (H1), we have . So, by Lemma 2.1 and (3.2),
From (3.5), we have . Therefore, , .
Assume that with . Thus, from (H1), we have where .
Lemma 2.1 and Lebesgue-dominated convergence theorem guarantee that So, is continuous.
For any bounded , From Lemma 2.1 and (H1), for any , we have which means the functions belonging to and the functions belonging to are uniformly bounded on . Notice that which implies that the functions belonging to are equicontinuous on . From Lemma 2.1, we have where are constants, are independent solutions of the linear differential equation , and .
It is easy to see that is continuous in and for and . So, for any , we have
Therefore, Thus, the functions belonging to are equicontinuous on . By Arzela-Ascoli theorem, is relatively compact in .
Hence, is completely continuous for any . (2) We now show that
For any , we have From (H1) and (3.2),
Suppose that there exist and such that , that is, for , This is in contradiction with and (3.13) holds. (3) Next, we show that
Suppose this is false, then there exist and with , that is, for , we have
From (H2), we have Therefore, by (3.18), (3.19), and (H2), it follows that Thus, . This is in contradiction with and (3.17) holds. (4) Choose . From (H3), there exists such that
Now, we show that where .
For any , we have This and (3.21) together with (3.2) imply
Suppose that there exist and such that ,then, for , we have This is in contradiction with and (3.22) holds.
Now, (3.13), (3.17), (3.22), and Lemma 2.3 guarantee that has two fixed points , with . Clear, PBVP (3.1) has at least two positive solutions .
Corollary 3.3. Assume that (H0)–(H2) hold, then PBVP (1.1) has at least one positive solution such that , where .
Corollary 3.4. Assume that (H0) and (H3) hold, and
(H4) there exist such that where . Then PBVP (1.1) has at least one positive solution such that , where .
Example 3.5. Consider the following second-order singular semipositone PBVP:
PBVP (3.27) has at least two positive solutions and for .
To see this, we will apply Theorem 3.1 with , , , , .
From Remark 2.2, it is easy to see that , , , , and .
By simple computation, we easily get and . So (H0) holds.
Taking , , then , and for any , , , Thus, (H1) holds.
Taking , then for , we have So, . That is, (H2) holds.
Let , then it is easy to check that (H3) holds.
Research supported by the Project of Shandong Province Higher Educational Science and Technology Program (J09LA08) and Reward Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2011SF022), China.
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