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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 389530, 15 pages
Multiple Solutions for a Class of Multipoint Boundary Value Systems Driven by a One-Dimensional -Laplacian Operator
Department of Mathematics, Faculty of Sciences, Razi University, Kermanshash 67149, Iran
Received 20 November 2011; Accepted 11 December 2011
Academic Editor: Kanishka Perera
Copyright © 2012 Shapour Heidarkhani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Employing a recent three critical points theorem due to Bonanno and Marano (2010), the existence of at least three solutions for the following multipoint boundary value system in , , for , is established.
In this work, we consider the following multipoint boundary value system for , where for , , , is a function such that is continuous in for all , is in for every and for all , for and , and denotes the partial derivative of with respect to for .
The study of multiplicity of solutions is an important mathematical subject which is also interesting from the practical point of view because the physical processes described by boundary value problems for differential equations exhibit, generally, more than one solution. In [1–3], Ricceri proposed and developed an innovative minimal method for the study of nonlinear eigenvalue problems. Following that, Bonanno  gave an application of the method to the two-point problem Bonanno also gave more precise versions of the three critical points of Ricceri in [5, 6]. In particular, in , an upper bound of the interval of parameters for which the functional has three critical points is established. Candito  extended the main result of  to the nonautonomous case In , He and Ge extended the main results of [4, 7] to the quasilinear differential equation In , the authors extended the main results of [4, 7, 9] to the quasilinear differential equation with Sturm-Liouville boundary conditions where is a constant, is a positive parameter, . In particular, in , the authors motivated by these works, established some criteria for the existence of three classical solutions of the system (1.1), while in , based on Ricceri’s three critical points theorem , the existence of at least three classical solutions to doubly eigenvalue multipoint boundary value systems was established.
Here and in the sequel, will denote the Cartesian product of space for , that is, equipped with the norm where for .
We say that is a weak solution to (1.1) if and
for every .
A special case of our main result is the following theorem.
Theorem 1.1. Let be two positive continuous functions such that the differential 1-form is integrable, and let be a primitive of such that . Fix , , and assume that then there is such that for each , the problem admits at least two positive classical solutions.
The main aim of the present paper is to obtain further applications of [12, Theorem 2.6] (see Theorem 2.1 in the next section) to the system (1.1), and the obtained results are strictly comparable with those of [9–11], and here we wil give the exact collocation of the interval of positive parameters.
2. Main Results
Our main tool is a three critical points theorem obtained in  (see also [1, 2, 5, 27] for related results), which is a more precise version of Theorem 3.2 of , to transfer the existence of three solutions of the system (1.1) into the existence of critical points of the Euler functional. We recall it here in a convenient form (see ).
Theorem 2.1 (see [12, Theorem 2.6]). Let be a reflexive real Banach space, be a coercive continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on , be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that . Assume that there exist and , with such that,
for each , the functional is coercive.
then, for each , the functional has at least three distinct critical points in .
Put Since for , and the embedding is compact, one has . Moreover, from [10, ], one has
Put for . Let denotes the inverse of for . then, where . It is clear that is increasing on ,
Lemma 2.2 (see [10, Lemma 3.3]). For fixed and , define by then the equation has a unique solution .
Direct computations show the following.
Lemma 2.4. A weak solution to the systems (1.1) coincides with classical solution one. Proof. Suppose that is a weak solution to (1.1), so for every . Note that, in one dimension, any weakly differentiable function is absolutely continuous, so that its classical derivative exists almost everywhere, and that the classical derivative coincides with the weak derivative. Now, using integration by part, from (2.7), we obtain and so for , for almost every . Then, by Lemmas 2.2 and 2.3, we observe for , where is the unique solution of (2.5). Hence, and for , namely is a classical solution to the system (1.1).
For all , we denote by the set Now, we formulate our main result as follows.
Theorem 2.5. Assume that there exist constants for with and , a positive constant and a function such that (A1) , (A2) where (see (2.11)),(A3) uniformly with respect to Then, for each , the system (1.1) admits at least three distinct classical solutions in .Proof. In order to apply Theorem 2.1 to our problem, we introduce the functionals for each , as follows: Since for , is compactly embedded in and it is well known that and are well defined and continuously differentiable functionals whose derivatives at the point are the functionals , , given by for every , respectively, as well as is sequentially weakly upper semicontinuous. Furthermore, Lemma 2.6 of  gives that admits a continuous inverse on , and since is monotone, we obtain that is sequentially weakly lower semiacontinuous (see [29, Proposition 25.20]). Moreover, is a compact operator. From assumption (A1), we get . Since from (2.1) for each , for , we have for each , and so using (2.15), we observe and it follows that Therefore, owing to assumption (A2), we have Furthermore, from (A3), there exist two constants with such that Fix , Then So, for any fixed , from (2.15) and (2.21), we have and thus, which means that the functional is coercive. So, all assumptions of Theorem 2.1 are satisfied. Hence, from Theorem 2.1 with , taking into account that the weak solutions of the system (1.1) are exactly the solutions of the equation and using Lemma 2.4, we have the conclusion.
Now we want to present a verifiable consequence of the main result where the test function is specified.
Define where , then we have the following consequence of Theorem 2.5.
Corollary 2.6. Assume that there exist constants for with and and two positive constants and with such that (B1) for each and ,(B2) , where is given by (2.24) and (see (2.11));(B3) uniformly with respect to .then, for each the systems (1.1) admits at least three distinct classical solutions.
Proof. Set such that for , and . It is easy to see that , and, in particular, one has for , which, employing the condition , gives Since for , the condition (B1) ensures that Therefore, owing to assumption (B2), we have where . Moreover, from assumption (B3) it follows that assumption (A3) is fulfilled. Hence, taking into account that using Theorem 2.5, we have the desired conclusion.
Let us present an application of Corollary 2.6.
Example 2.7. Let be the function defined as for each . In fact, by choosing and , by a simple calculation, we obtain that and , and so with and , we observe that the assumptions (B1) and (B3) in Corollary 2.6 are satisfied. For (B2), So, for every Corollary 2.6 is applicable to the system where .
Here is a remarkable consequence of Corollary 2.6.
Corollary 2.8. Let be a function in such that . Assume that there exist constants for with and and two positive constants and with such that (C1) for each ,(C2)where is given by (2.24),(C3),Then, for each , the systems for , admits at least three distinct classical solutions.
Example 2.9. Let and , . Consider the system Clearly, (H1) and (H2) hold. A simple calculation shows that and . So, by choosing and , we observe that the assumptions (C1) and (C3) in Corollary 2.8 hold. For (C3), since for every , we have Note that . We see that for every Corollary 2.8 is applicable to the system (2.38).
Finally, we prove the theorem in the introduction.
Proof of Theorem 1.1. Since and are positive, then is nonnegative in . Fix for some . Note that , and there is such that and
In fact, one has , where . Hence, there is such that
from Corollary 2.8, with follows the conclusion.
The author would like to thank Professor Juan J. Nieto for valuable suggestions and reading this paper carefully.
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