Abstract
In this paper, we study the existence of nontrivial solutions for the th Lidstone boundary value problem with a sign-changing nonlinearity. Under some conditions involving the eigenvalues of a linear operator, we use the topological degree theory to obtain our main results.
1. Introduction
In this paper, we investigate the existence of nontrivial solutions for the following th Lidstone boundary value problem with a sign-changing nonlinearity:where the nonlinearity satisfies the following condition:(i)(C0) and there exist three nonnegative functions such that
The Lidstone boundary value problem arises in many different areas of applied mathematics and physics. When , problem (1) describes the deformation of an elastic beam in which both ends are simply supported. Recently, this problem has been extensively studied, and the authors refer the reader to [1–11] and references cited therein. For example, in [1], the authors used a cone-theoretic fixed point theorem to study the existence of nontrivial solutions for the nonlinear Lidstone boundary value problem:where is continuous and is nonnegative. In [2], the authors investigated the existence and uniqueness of positive solutions for the following generalized Lidstone boundary value problem:where and . In view of symmetry, these results demonstrate that problem (4) is essentially identical with Dirichlet boundary condition (1).
Meanwhile, we also note that there are a large number of papers in the literature devoted to sign-changing nonlinearities, and some results can be found in a series of papers [12–32] and the references cited therein. For example, in [12], the authors studied the following higher-order nonlinear fractional boundary value problem involving Riemann–Liouville fractional derivatives:where is a sign-changing nonlinearity. Under some appropriate conditions involving the eigenvalues of the relevant linear operators, they utilized the topological degree to obtain a nontrivial solution for (5). In [13], the authors adopted the similar method in [12] to study the existence of nontrivial solutions for the following system of fractional -difference equations with -integral boundary conditions:where is the -order Riemann–Liouville’s fractional -derivative.
Inspired by the aforementioned works, in this paper, we study the existence of nontrivial solutions for (1) where the nonlinearity is sign-changing. Under some conditions involving the eigenvalues of the revelent linear operators, we use the topological degree to obtain our results.
2. Preliminaries
Let for . Clearly, is a real Banach space and is a solid cone in . In (1), let and from [2, P224], we can obtain that (1) is equivalent to the following integral equation:where
Next, we provide a lemma, which expresses some vital properties of the functions .
Lemma 1. (i) are nonnegative continuous functions on , and (ii) has the inequalities (iii), (iv), and
Proof. We only prove (iii) and (iv). For (iii), holds obviously. From the definition of , we haveFor (iv), when , we haveNoting that , we haveWhen , we haveUsing the symmetry of , we easily haveThis completes the proof.
Let with . Then, we have the following equations:where
Lemma 2. Let , for . Then, if , we have , where
This is a direct result from Lemma 1 (ii), so we omit its proof.
Remark 1. , for .
Lemma 3 (see [33], Theorem 1. [3]). Let be a bounded open set in a Banach space and be a continuous compact operator. If there exists such thatthen the topological degree .
Lemma 4 (see [33], Lemma 4. [1]). Let be a bounded open set in a Banach space with and be a continuous compact operator. Ifthen the topological degree .
3. Main Results
Define the operator by
Moreover, the continuity of implies that is completely continuous and the existence of solutions for (1) is equivalent to that of fixed points of .
Now, we list some assumptions for the functions and : (C1)There exist with such that (C2), uniformly for . (C3)There exist with such thatwhere
Theorem 1. Suppose that (C0)–(C3) hold. Then, (1) has at least one nontrivial solution.
Proof. We divide the following two steps:(i)Step 1. By (C3), there exist and such that for all , with . Substituting this inequality into (20), we have Now for this , we claim Suppose the contrary. Then, there exist and such that Therefore, we obtain Multiply by on both sides and integrate over and use (15) to obtain This indicates that , and thus, , . This contradicts to . Consequently, (26) holds and Lemma 4 yields that Step 2. By virtue of (C2), there exist and such thatfor all and . For any given with , and using (C1), there exists such thatConsequently, when , we haveNoting that when , , are bounded, we can letThen, we can obtainfor all . Note that can be chosen arbitrarily small, and letwhereand . Now, we claimwhere . Suppose that (37) is not satisfied. Then, there exist and such thatLetThen, from Lemma 2, we have . Now, we estimate the norm of . Noting that , we obtainWe calculate . By (38) we haveUsing (C0), Lemma 2 and Remark 1, we findTherefore, (34) enables us to calculateConsequently, we haveNow, we estimate . From (42), we have , and using , we haveAs a result, by means of (43) and (44), we obtainCombining this with (38), we haveDefine . Then, and . Hence, from (14), we havewhich contradicts the definition of . Therefore, (37) holds, and from Lemma 3, we obtainNow, (29) and (49) together imply thatTherefore, the operator has at least one fixed point in . Equivalently, (1) has at least one nontrivial solution. This completes the proof.
4. Conclusion
In this paper, we use the topological degree to study the nontrivial solutions for the th Lidstone boundary value problem (1). To the best of our knowledge, there are few works that deal with the problem where the nonlinear terms may be unbounded and sign-changing. Moreover, it is remarked that the main result is discussed under some conditions concerning the first eigenvalues corresponding to the relevant linear operators. These mean that our main result is an improvement in some related works.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
The study was carried out in collaboration among all authors. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by University Natural Science Foundation of Anhui Provincial Education Department (Grant nos. KJ2019A0666 and KJ2018A0452), the Foundation of Suzhou University (Grant no. 2016XJGG13), Technology Research Foundation of Chongqing Educational Committee (Grant nos. KJQN201800533 and KJQN202000528), and Qilu University of Technology (Shandong Academy of Sciences) Young Doctor Cooperative Funding (Grant no. 2017BSHZ012).