Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution

where λ ∈ R is a real parameter, Ty ≡ (py′′)′ − qy′, p is a positive twice continuously differentiable on [0, 1], q is a positive continuously differentiable function on [0, 1], r is a nonnegative continuous function on [0, 1] such that r(x)≢ 0 on any subinterval of [0, 1], α, β, c, and δ are any real numbers between 0 and π/2. However, α � c � 0, β � δ � (π/2) cannot be the case; likewise, α, β, c, and δ variables cannot take the value π/2 values at the same time. (e function h is represented as h � af + g, where f, g ∈ C([0, 1] × R5) and satisfy the following conditions: there exists f0 and f0 such that f0 � liminf |u|⟶0 f(x, u, s, λ)

Problems (1)-(2) were considered in [4] (see also [5]) in the case when r(x) is strictly positive on [0, 1], h � f + g, and f satisfies the condition where M is a positive constant. In [4], a global bifurcation result for problems (1)-(2) under these conditions was obtained. e purpose of this paper is to seek an answer to the question that "what happens if the function r(x) is not strictly positive in the range [0, 1]?" Several nonlinear eigenvalue problems for the Sturm-Liouville equation are considered in the literatures [6][7][8][9][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. e 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 h ≡ 0. Moreover, we establish a global bifurcation result for problems (1)-(2) for f ≡ 0. 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.

Preliminary
By B.C., denote the set of functions that satisfy the boundary conditions (2) and consider the following linear eigenvalue problem: obtained from (1)-(2) by setting h ≡ 0. Let (λ, y) be an eigenpair of problem (6). By multiplying both sides of the equation in (1) by y, integrating the resulting equality from 0 to 1, we get where Since (8). it follows from (7) that λ > 0 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.
To preserve the nodal properties along the global continuance of solutions to problems (1)-(2), Aliyev [4] constructed the sets S ] k , k ∈ N, ] ∈ +, − { }, of functions of E 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 k ∈ N and each ] ∈ +, − { }, the sets S + k and S − k are disjoint and open subsets of E.

2
International Journal of Differential Equations Consider the following eigenvalue problem:
Remark 1. By eorem 1, it follows from eorem 1.2 and Remark 4.1 in [4] that the eigenvalues of problem (9) are real and simple and form an unboundedly increasing sequence λ k,ψ ∞ k�1 . Moreover, the eigenfunction y k, ψ (x) corresponding to the eigenvalue λ k,ψ lies in S k .
Let C denote the closure of the set of nontrivial solutions of the nonlinear problems (1)-(2) in R × E.
e proof can be solved similarly to eorem 1.1 in [4] by using eorem 1 and Lemma 1.