# Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution

Authors: Fatma Aydin Akgun
Journal: International Journal of Differential Equations
 (2021)

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## 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.


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## Body

## 1. Introduction

We consider the following nonlinear eigenvalue problem:(1)ℓy≡pxy″″−qxy′′=λrxy+hx,y,y′,y″,y‴,λ,x∈0,1,(2)y′0cosα−py″0sinα=0,y0cosβ+Ty0sinβ=0,y′1cosγ+py″1sinγ=0,y1cosδ−Ty1sinδ=0,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 rx≢0 on any subinterval of 0,1, α,β,γ,andδ are any real numbers between 0 and π/2. However, α=γ=0, β=δ=π/2 cannot be the case; likewise, α,β,γ, and δ variables cannot take the value π/2 values at the same time. The function h is represented as h=af+g, where f,g∈C0,1×ℝ5 and satisfy the following conditions: there exists f0¯ and f0¯ such that(3)f0¯=liminfu⟶0fx,u,s,λu,f0¯=limsupu⟶0fx,u,s,λu,uniformly in x∈0,1, s∈ℝ, s≤m0, λ∈ℝ, where m0 is some small positive constant;(4)gx,u,s,v,w,λ=ou+s+v+wasu+s+v+w⟶0,uniformly in x,λ∈0,1×Λ for any bounded interval Λ⊂ℝ.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 rx is strictly positive on 0,1, h=f+g, and f satisfies the condition(5)fx,u,s,v,w,λu≤M,x∈0,1,u,s,v,w∈ℝ4,u≠0,u+s+v+w≤m0,λ∈ℝ,where M 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 rx is not strictly positive in the range 0,1?”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 Section2, 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.


## 2. Preliminary

ByB.C., denote the set of functions that satisfy the boundary conditions (2) and consider the following linear eigenvalue problem:(6)ℓy=λrxy,x∈0,1,y∈B.C.,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(7)∫01pxy″x2+qxy′x2dx+Ny=λ∫01rxyx2dx,where(8)Ny=y′02cotα+y02cotβ+y′12cotγ+y12cotδ.Sincer∈C0,1, rx≥0 for x∈0,1 and rx≢0 on any subinterval of 0,1, and α,β,γ,δ∈0,π/2, except for the cases α=γ=0, β=δ=π/2, and α=β=γ=δ=π/2 by (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.Theorem 1.
The eigenvalues of the linear spectral problem (6) are positive and simple and form an unboundedly increasing sequence λkk=1∞. The eigenfunction ykx, k∈ℕ, corresponding to the eigenvalue λk, has exactly k−1 simple zeros in 0,1. Moreover, if (i) yk0yk′0>0, (ii) yk0=0, or (iii) yk′0=0 and yk0yk″0>0, then yk′x, k∈ℕ, has either k−1 or k simple zeros in the interval 0,1, if (iv) yk0yk′0<0, (v) yk′0=0, and yk0yk″0<0, or (vi) yk′0=yk″0=0, then y1′x has no zeros and yk′x for k≥2 has either k−1 or k−2 simple zeros in 0,1.LetE be a Banach space, C30,1∪B.C. with the usual norm u3=u∞+u′∞+u″∞+u‴∞, where u∞=maxx∈0,1ux.To preserve the nodal properties along the global continuance of solutions to problems (1)-(2), Aliyev [4] constructed the sets Skν, k∈ℕ, ν∈+,−, 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 ν∈+,−, the sets Sk+ and Sk− are disjoint and open subsets of E.Consider the following eigenvalue problem:(9)ℓyx+rxψxyx=λrxyx,x∈0,1,y∈B.C.,where ψ∈L10,1.Remark 1.
By Theorem1, 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 λk,ψk=1∞. Moreover, the eigenfunction yk,ψx corresponding to the eigenvalue λk,ψ lies in Sk.Lemma 1.
(see [4], Lemma 1.1). If λ,y∈ℝ×E is a solution of (1)-(2) such that λ,y∈ℝ×∂Skν, then y≡0.LetC denote the closure of the set of nontrivial solutions of the nonlinear problems (1)-(2) in ℝ×E.Iff≡0, then the following global bifurcation results for (1)-(2) are obtained.Theorem 2.
For eachk∈ℕ, there exist continua Ck+ and Ck− of C containing λk,0 that are unbounded in ℝ×E and contained in ℝ×Sk+∩λk,0 and ℝ×Sk−∪λk,0, 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:(10)ℓy=λrxy+rxfx,y3εy,y′,y″,y‴,λ+gx,y,y′,y″,y‴,λ,x∈0,1,y∈B.C.,where ε>0.Ifu+s is sufficiently small, then by virtue of condition (3), we have(11)fx,uεu,s,v,w,λu+s+v+w=fx,uεu,s,v,w,λuεuuεuu+s+v+w≤maxf0¯−1,f0¯+1u+s+v+wε,which is uniformly in x∈0,1 and λ∈ℝ. Hence,(12)rxfx,uεu,s,v,w,λ=ou+s+v+wasu+s+v+w⟶0,is also uniformly in x,λ∈0,1×Λ for any bounded interval Λ⊂ℝ. Then, it follows from Theorem 2 that for each k∈ℕ, there exists unbounded continuances Ck,ε+ and Ck,ε− of solutions of problem (10) such that(13)λk,0∈Ck,ε+⊂ℝ×Sk+∪λk,0λk,0∈Ck,ε−⊂ℝ×Sk−∪λk,0.Let(14)M=maxf0¯,f0¯,Jk=λk−M,λk+M,Jkτ=λk−M−τ,λk+M+τ,Ik=λk−f0¯,λk−f0¯,Ikτ=λk−f0¯−τ,λk−f0¯+τ,and τ0>0 be some fixed sufficiently small number.Lemma 2.
For everyk∈ℕ, each ν∈+,−, and for each sufficiently small κ>0, there exists a solution λ˜,y˜ of problems (1)-(2) such that(15)λ˜∈Jkτ0,y˜∈Skν,y˜3=κ.Proof.
By the above arguments, for anyε>0, problem (10) has a solution λε,yε such that(16)yε∈Skν,yε3=κ.
Let(17)ψεx=−fx,yεxεyεx,yε′x,yε″x,yε′″x,λκ,ε+yεx,ifyεx≠0,0,ifyεx≠0.
Then,λε,yε is a solution of the following nonlinear problem:(18)ℓyx+rxψεxyx=λrxyx+gx,yx,y′x,y″x,y‴x,λ,where x∈0,1,y∈B.C.. By condition (4), the nonlinear problem (18) is linearizable, and the linearization of this problem for y=0 is given by(19)ℓyx+rxψεxyx=λrxyx,x∈0,1,y∈B.C..
In (3), there exists κ0∈0,m0 such that for any y,s∈ℝ2, y+s<κ0 and λ∈ℝ, the following inequality holds:(20)f0¯−τ02<fx,y,s,λy<f0¯+τ02.
Letκ<κ0, then(21)ψεx≤M0yxε≤M0,x∈0,1,where M0=M+τ0/2, follows from (18).
By virtue of the maximum and minimum property of eigenvalues (see [2], Ch. 6), the k-th eigenvalue λk,ψε of (19) is determined from the relation(22)λk,ψε=maxy∈Vk−1miny∈B.C.∫01pxy″2x+qxy′2xdx+Ny+∫01rxψεxy2xdx∫01rxy2xdx:∫01rxφxyxdx=0,φx∈Vk−1,where Vk−1 is an arbitrary set of linearly independent functions φix, i=1,2,…,k−1. Hence,(23)λk=maxy∈Vk−1miny∈B.C.∫01pxy″2x+qxy′2xdx+Ny∫01rxy2xdx:∫01rxφxyxdx=0,φx∈Vk−1,by (21) and (23), it follows from (22) that(24)λk−M0≤λk,ψε≤λk+M0,i.e.,(25)λk,ψε∈Jkτ02.
Sinceyε3=κ and yε∈Skν by Theorem 2 (also from [12], Ch. 4), we can choose κ0 as(26)λε−λk,ψε<τ02forκ<κ0.
From (25) and (26), we obtain(27)λε∈Jkτ0.
By conditions (3) and (4), it follows from (1) that λε,yε:ε∈0,1 is bounded in ℝ×C40,1. Thus, we can find a sequence εnn=1∞⊂0,1, εn⟶0 as n⟶∞, such that λεn,yεn converges to a solution λ˜,y˜ of the nonlinear eigenvalue problems (1)-(2). Then, by using (16) and (27), we get λ˜∈Jkτ0, y˜3=κ, and y˜∈Skν¯=Skν∪∂Skν.
Consequently, it follows from Lemma1 that y˜∈Skν. The proof of this lemma is complete.
If there exists a sequenceλk,nν,yk,nνn=1∞⊂ℝ×Skν of solutions to this problem which converges to λ,0 in ℝ×E as n⟶∞, then λ,0∈ℝ×E is called a bifurcation point of problems (1)-(2) with respect to the set ℝ×Skν, k∈ℕ, ν∈+,−.Corollary 1.
For everyk∈ℕ and each ν∈+,−, the set of bifurcation points of problems (1)-(2) with respect to the set ℝ×Skν is nonempty.Proof.
Letκnn=1∞⊂0,κ0 be a sequence converging to zero as n⟶∞, then by Lemma 2, for every k∈ℕ, each ν∈+,−, and any n∈ℕ, there exists a solution λk,nν,yk,nν of (1)-(2) such that(28)λk,nν∈Jkτ0,yk,nν∈Skν,yk,nν3=κn.
Then, from the sequenceλk,nν,yk,nνn=1∞, one can select a subsequence λk,nlν,yk,nlνl=1∞ which converges to λ,0, where λ∈Jkτ0. This completes the proof.Lemma 3.
Letλ,0 be a bifurcation point of (1)-(2) with respect to the set ℝ×Skν, k∈ℕ, ν∈+,−, then λ∈Ik.Proof.
Suppose thatλ,0 is a bifurcation point of (1)-(2) with respect to the set ℝ×Skν, then there exists a sequence λk,nν,yk,nνn=1∞ which converges to λ,0 in ℝ×E.
Let we assumeλ∉Ik. If we denote(29)ϵ0=distλ,Ik2,then there exists n0∈ℕ such that for all n≥n0, the following inequality holds:(30)λk,nν−λ<ϵ0.
Therefore,(31)distλk,nν,Ik>ϵ0forn≥n0.
We leavek∈ℕ and ν∈+,− arbitrary and fixed. It is evident that for each n∈ℕ, the pair λk,nν,yk,nν is a solution to the problem;(32)ℓyx+rxψnxyx=λrxyx+gx,yx,y′x,y″x,y‴x,λ,x∈0,1,y∈B.C.,where(33)ψnx=−fx,yk,nνx,yk,nν′x,yk,nν″x,yk,nν′″x,λk,nνyk,nνx,ifyk,nνx≠0,0,ifyk,nνx≠0.
By condition (3), there exists κ˜0∈0,m0 such that for any y,s∈ℝ2, y+s<κ˜0, and λ∈ℝ, the inequality(34)f0¯−ϵ02<fx,y,s,λy<f0¯+ϵ02,holds. It is acceptable to take n1≥n0, thus(35)yk,nν3<κ˜0,for all n≥n1. Then, from (34) and (33), we get(36)−f0¯−ϵ02<ψnx<−f0¯+ϵ02,x∈0,1,n≥n1. By using (23) and replacing ψε by ψn, we get(37)λk−f0¯−ϵ02<λk,ψn<λk−f0¯+ϵ02forn≥n1,(38)distλk,ψn,Ik<ϵ02forn≥n0,where λk,ψn is the kth eigenvalue of the linear spectral problem(39)ℓyx+rxψnxyx=λrxyx,x∈0,1,y∈B.C.
Note that Theorem2 is applicable to the nonlinear problem (32) as a result of condition (4). Since yk,nν∈Skν and yk,nν3⟶0 as n⟶∞, it follows from Theorem 2 that we can choose n2≥n1 so that(40)λk,nν−λk,ψn<ϵ2forn≥n2,i.e.,(41)distλk,nν,λk,ψn<ϵ02forn≥n2,
Then, by (38) and (41), we get(42)distλk,n,Ik≤distλk,n,λk,ψn+distλk,ψn,Ik<ϵ02+ϵ02=ϵ0forn≥n2,which contradicts relation (31). This completes the proof.
For eachk∈ℕ and each ν, we denote by D˜kν the union of all components Dk,λν of C bifurcating from points λ,0∈Ik×0 with respect to the set ℝ×Skν. Note that the set D˜kν is nonempty as a result of Corollary 1 and Lemma 3. It is obvious that the set Dkν=D˜kν∪Jk×0 is connected in ℝ×E.Theorem 3.
For eachk∈ℕ and each ν, the set Dkν is unbounded in ℝ×E and lies in ℝ×Skν∪Ik×0.The proof can be shown similar to Theorem 1.3 in [4] by using Lemma 2, Corollary 1, and Lemma 3.



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