Unilateral Global Bifurcation for Fourth-Order Problems and Its Applications

We will establish unilateral global bifurcation result for a class of fourth-order problems. Under some natural hypotheses on perturbation function, we show that (λk, 0) is a bifurcation point of the above problems and there are two distinct unbounded continua, C k and C k , consisting of the bifurcation branch Ck from (μk, 0), where μk is the kth eigenvalue of the linear problem corresponding to the above problems. As the applications of the above result, we study the existence of nodal solutions for the following problems: x + kx + lx = rh(t)f(x), 0 < t < 1, x(0) = x(1) = x(0) = x(1) = 0, where r ∈ R is a parameter and k, l are given constants; h(t) ∈ C([0, 1], [0,∞)) with h(t) ̸ ≡ 0 on any subinterval of [0, 1]; and f : R→ R is continuous with sf(s) > 0 for s ̸ = 0.We give the intervals for the parameter r ̸ = 0 which ensure the existence of nodal solutions for the above fourth-order Dirichlet problems if f0 ∈ [0,∞] or f∞ ∈ [0,∞], where f0 = lim|s|→0f(s)/s and f∞ = lim|s|→+∞f(s)/s.We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.


Introduction
The deformations of an elastic beam in equilibrium state with fixed both endpoints can be described by the fourth-order boundary value problem   +   +  = ℎ ()  () , 0 <  < 1,  (0) =  (1) =   (0) =   (1) = 0, (1) where  ∈ R is a parameter, ,  are given constants, and  : R → R is continuous.When  =  = 0, since problem (1) cannot transform into a system of second-order equation, the treatment method of second-order system does not apply to problem (1).Thus, there exists some difficulty studying problem (1) even in the case of  =  = 0.
In recent years, there has been considerable interest in the above BVP (1) mainly because of their interesting applications.For example, Agarwal and Chow [1] ( =  = 0) first investigated the existence of the solutions of problem (1) by contraction mapping and iterative methods.Subsequently, when  =  = 0, by fixed point theory on cones, Ma et al. [2,3], Yao [4,5], Zhai et al. [6], and Webb et al. [7] studied the existence of positive solutions of problem (1).
On the other hand, by applying the bifurcation techniques of Rabinowitz [8,9], Gupta and Mawhin [10], Lazer and McKenna [11], Liu and O'Regan [12], and Ma et al. [13][14][15] studied the existence of nodal solutions for the fourth-order BVP where both ends were simply supported, and Rynne [16] investigated the nodal properties of the solutions for a general 2th-order problem.
In 2013, when ,  satisfy (1) and (2), Shen and He [32] also studied bifurcation from interval and the existence of positive solutions for problem (1) by applying Rabinowitz's global bifurcation theorem [9].Now, consider the following operator equation: where  is a compact linear operator and  : R ×  →  is compact with  = (‖‖) at  = 0 uniformly on bounded  intervals, where  is a real Banach space with the norm ‖ ⋅ ‖.
Dancer [33] has shown that there are two distinct unbounded continua  +  and  −  , consisting of the bifurcation branch   of S emanating from (, 0), which satisfy either that  +  and  −  are both unbounded or  +  ∩  −  ̸ = {(, 0)}.This result has been extended to one-dimensional -Laplacian problem by Dai and Ma [34].The above results [34] have been improved partially by Dai [35] with nonasymptotic nonlinearity at 0 or ∞.Later, Dancer's result [33] has been also extended to the periodic -Laplacian problems by Dai et al. [36].In 2013, Dai and Han [37] established Dancer-type unilateral global bifurcation results for fourth-order problems of the deformations of an elastic beam in equilibrium state where both ends are simply supported by Dancer [33].
In this paper, based the spectral theory of [30,31], we will establish Dancer-type unilateral global bifurcation results about the continuum of solutions for the following fourthorder eigenvalue problem: where ℎ satisfies (2), and the perturbation function  : (0, 1) × R 2 → R is continuous with (, , 0) ≡ 0 and satisfies the following hypotheses uniformly for  ∈ (0, 1) and  on bounded sets.
denote the set of functions in  which have exactly  − 1 interior nodal (i.e., nondegenerate) zeros in (0, 1) and are positive near  = 0, set  −  = − +  , and   =  +  ∪  −  .They are disjoint and open in .Let Φ ]  = R ×  ]  , ] ∈ {+, −}, and Φ  = R ×   under the product topology.Let S denote the closure in R ×  of the set of nontrivial solutions of (1) and let S ± denote the subset of S with  ∈ Φ ±  and S ± = S + ∪ S − .Under condition (9), we will show that (  , 0) is a bifurcation point of (8) and there are two distinct unbounded continua,  +  and  −  , consisting of the bifurcation branch   from (  , 0), where   is the th eigenvalue of problem (2).Based on the above result, we investigate the existence of nodal solutions for problem (1).
Remark 2. By applying disconjugate operator theory [25,26], the authors [13,14,16] also established the spectrum structure of the corresponding linear eigenvalue problems.On the basis of the above spectrum structure, the authors [13,14,16] studied the existence of nodal solutions of the above problem by applying Rabinowitz's global bifurcation theorem [8].
The rest of this paper is arranged as follows.In Section 2, we will establish unilateral global bifurcation results.In Section 3, we will investigate the existence of nodal solutions for problem (1) under the linear growth condition on .
Then  is a closed operator and  −1 :  →  is completely continuous.
Next, we prove that the first choice of the alternative of Theorem 3 is the only possibility.To do it, we give the following lemma.
Proof.By the proof of Theorem 3.1 in [16, p. 467] (see also Corollary 1.12 and the proof of Theorem 2.3, together with the remark following that proof, in [16]), we easily obtain the result.
Definition 7 (see [38]).Let  be a Banach space and let {  |  = 1, 2, . ..} be a family of subsets of .Then the superior limit D of {  } is defined by Lemma 8 (see [38]).Each connected subset of metric space  is contained in a component, and each connected component of  is closed.Lemma 9 (see [39]).Let  be a Banach space and let {  |  = 1, 2, . ..} be a family of closed connected subsets of .Assume that (i) there exist   ∈   ,  = 1, 2, . .., and  * ∈ , such that Then there exists an unbounded component  in D and  * ∈ .

Main Results
In this section, we first study the following eigenvalue problem: where  > 0 is a parameter.
(10)  0 = 0 and  ∞ = 0, where Let (), () ∈ (R, R) be such that Let us consider as a bifurcation problem from the trivial solution  ≡ 0 and as a bifurcation problem from infinity.
We add the points {(, ∞) |  ∈ R} to space R × .By [40], we note that problem (34) and problem (35) are the same, and each of them is equivalent to problem (30).By Theorems 3 and 6 and the results of Rabinowitz [41], we have the following Lemma.
Proof of Theorem 13.By Lemma 11 and Remark 12, we only prove D +  and D −  crosses the hyperplane {1} ×  in R × .We only prove the case of D +  since the case of D −  is similar.
In this case, we only need to show that We divide the proof into two steps.
In this case, it follows that           → ∞.
By ( 32) and ( 33), let and then  is nondecreasing and We divide the equation By the continuity and compactness of  −1 , it follows that where  fl lim →∞   , again choosing a subsequence and relabeling if necessary.We claim that  ∈ D +  .
On the other hand, let where Conditions (1) and (2) imply that there exists a positive constant  > 0 such that ℎ() f () >  for any  ∈ (, ) and all  ∈ N. By Lemma 10, we get that   must change its sign in (, ) for  large enough, which is the contradicts.Therefore, for some constant number  > 0 and  ∈ N sufficiently large.
The proof is similar to that for Case 1, so we omit it.
Moreover, by Remark 12 and the problem (1), we can obtain that D ]  ⊂ S ]  .
Proof.Similar to the method of the proof of Theorem 14 and the conclusions of Theorem 18, we can obtain the desired results.
Theorem 18 implies that there exist two sequences of unbounded continua of solutions to problem (71), D Proof.Similar to the method of the proof of Theorem 14 and the conclusions of Theorem 20, we can obtain the desired results.
Remark 29.When  =  = 0, Korman [23] and Rynne [24] investigated the nodal properties of the solutions for problem (1) by applying the bifurcation techniques.Thus, in this sense, our results partially extend and improve the corresponding results of Korman [23] and Rynne [24].

Competing Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
[16, the property .(For the definition of property , see[25, p. 36].)Now, from the proof of [25,Lemma 4](see also the remarks in the final paragraph in[25,  p. 43]; or see the proof of[16, Lemma 3.7]), it follows that, for all  sufficiently large,   must change sign on .
)and then   must change sign on  as  is large enough.