Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution

Abstract

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

1. Introduction

We consider the following nonlinear eigenvalue problem:where is a real parameter, , is a positive twice continuously differentiable on , is a positive continuously differentiable function on , is a nonnegative continuous function on such that on any subinterval of , are any real numbers between 0 and . However, , cannot be the case; likewise, and δ variables cannot take the value values at the same time. The function is represented as , where and satisfy the following conditions: there exists and such thatuniformly in , , , , where is some small positive constant;uniformly in for any bounded interval .

Problems (1)-(2) occur in the study of various processes in mechanics and physics (see [1] and [2]). Particularly, it arises when solving an important problem of aeroelasticity, the loss of stability of a flexible elongated plate under the action of a supersonic gas flow, compressed or stretched by external stresses (see [1] and [3]).

Problems (1)-(2) were considered in [4] (see also [5]) in the case when is strictly positive on , , and satisfies the conditionwhere is a positive constant. In [4], a global bifurcation result for problems (1)-(2) under these conditions was obtained. The purpose of this paper is to seek an answer to the question that “what happens if the function is not strictly positive in the range ?”

Several nonlinear eigenvalue problems for the Sturm–Liouville equation are considered in the literatures [6–10] and the bibliography therein. In these papers, the global bifurcation results were obtained, namely, the unbounded continuance of solutions bifurcating from intervals of the line of trivial solutions and possessing the usual nodal properties was established.

The structure of this paper is as follows. In Section 2, we study the structure of root subspaces and the oscillatory properties of the eigenfunctions of the linear problem obtained from (1)-(2) by setting . Moreover, we establish a global bifurcation result for problems (1)-(2) for . In Section 3, using an approximate problem, we prove the existence of solutions of problems (1)-(2) with small norms and usual nodal properties. Next, we find the bifurcation intervals of the line of trivial solutions concerning the sets with fixed oscillation count. Finally, we establish the existence of global continue of solutions bifurcating from these intervals.

2. Preliminary

By , denote the set of functions that satisfy the boundary conditions (2) and consider the following linear eigenvalue problem:obtained from (1)-(2) by setting . Let be an eigenpair of problem (6). By multiplying both sides of the equation in (1) by , integrating the resulting equality from 0 to 1, we getwhere

Since , for and on any subinterval of , and , except for the cases , , and by (8). it follows from (7) that and all eigenvalues of the problem (6) are positive. Hence, by following the arguments of Banks and Kurowski [11], we can justify the following result.

Theorem 1. The eigenvalues of the linear spectral problem (6) are positive and simple and form an unboundedly increasing sequence . The eigenfunction , , corresponding to the eigenvalue , has exactly simple zeros in . Moreover, if (i) , (ii) , or (iii) and , then , , has either or simple zeros in the interval , if (iv) , (v) , and , or (vi) , then has no zeros and for has either or simple zeros in .

Let be a Banach space, with the usual norm , where .

To preserve the nodal properties along the global continuance of solutions to problems (1)-(2), Aliyev [4] constructed the sets , , , of functions of which possess the nodal properties of eigenfunctions of the spectral problem (6) and their derivatives by using the Prüfer type transformation. It is obvious that for each and each , the sets and are disjoint and open subsets of .

Consider the following eigenvalue problem:where .

Remark 1. By Theorem 1, it follows from Theorem 1.2 and Remark 4.1 in [4] that the eigenvalues of problem (9) are real and simple and form an unboundedly increasing sequence . Moreover, the eigenfunction corresponding to the eigenvalue lies in .

Lemma 1. (see [4], Lemma 1.1). If is a solution of (1)-(2) such that , then .

Let denote the closure of the set of nontrivial solutions of the nonlinear problems (1)-(2) in .

If , then the following global bifurcation results for (1)-(2) are obtained.

Theorem 2. For each , there exist continua and of containing that are unbounded in and contained in and , respectively.

The proof can be solved similarly to Theorem 1.1 in [4] by using Theorem 1 and Lemma 1.

3. Global Bifurcation of Solutions of Problems (1)-(2)

Consider the following approximate problem:where .

If is sufficiently small, then by virtue of condition (3), we havewhich is uniformly in and . Hence,is also uniformly in for any bounded interval . Then, it follows from Theorem 2 that for each , there exists unbounded continuances and of solutions of problem (10) such that

Letand be some fixed sufficiently small number.

Lemma 2. For every , each , and for each sufficiently small , there exists a solution of problems (1)-(2) such that

Proof. By the above arguments, for any , problem (10) has a solution such thatLetThen, is a solution of the following nonlinear problem:where . By condition (4), the nonlinear problem (18) is linearizable, and the linearization of this problem for is given byIn (3), there exists such that for any , and , the following inequality holds:Let , thenwhere , follows from (18).
By virtue of the maximum and minimum property of eigenvalues (see [2], Ch. 6), the -th eigenvalue of (19) is determined from the relationwhere is an arbitrary set of linearly independent functions , . Hence,by (21) and (23), it follows from (22) thati.e.,Since and by Theorem 2 (also from [12], Ch. 4), we can choose asFrom (25) and (26), we obtainBy conditions (3) and (4), it follows from (1) that is bounded in . Thus, we can find a sequence , as , such that converges to a solution of the nonlinear eigenvalue problems (1)-(2). Then, by using (16) and (27), we get , , and .
Consequently, it follows from Lemma 1 that . The proof of this lemma is complete.
If there exists a sequence of solutions to this problem which converges to in as , then is called a bifurcation point of problems (1)-(2) with respect to the set , , .

Corollary 1. For every and each , the set of bifurcation points of problems (1)-(2) with respect to the set is nonempty.

Proof. Let be a sequence converging to zero as , then by Lemma 2, for every , each , and any , there exists a solution of (1)-(2) such thatThen, from the sequence , one can select a subsequence which converges to , where . This completes the proof.

Lemma 3. Let be a bifurcation point of (1)-(2) with respect to the set , , , then .

Proof. Suppose that is a bifurcation point of (1)-(2) with respect to the set , then there exists a sequence which converges to in .
Let we assume . If we denotethen there exists such that for all , the following inequality holds:Therefore,We leave and arbitrary and fixed. It is evident that for each , the pair is a solution to the problem;whereBy condition (3), there exists such that for any , , and , the inequalityholds. It is acceptable to take , thusfor all . Then, from (34) and (33), we get By using (23) and replacing by , we getwhere is the th eigenvalue of the linear spectral problemNote that Theorem 2 is applicable to the nonlinear problem (32) as a result of condition (4). Since and as , it follows from Theorem 2 that we can choose so thati.e.,Then, by (38) and (41), we getwhich contradicts relation (31). This completes the proof.
For each and each , we denote by the union of all components of bifurcating from points with respect to the set . Note that the set is nonempty as a result of Corollary 1 and Lemma 3. It is obvious that the set is connected in .

Theorem 3. For each and each , the set is unbounded in and lies in .

The proof can be shown similar to Theorem 1.3 in [4] by using Lemma 2, Corollary 1, and Lemma 3.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.