Non-real Eigenvalues Of Fourth-order Indefinite Linear Boundary Value Problems

In this paper,we consider the non-real eigenvalues and some simple properties of eigencurve for fourth-order indefinite problems.The main works are:In the first chapter,we study some simple properties of eigencurves for the two-parameter fourth-order indefinite differential equation （p（x）y"）"+q（x）y=（λr（x）+μ）y,a≤x≤b and combined with separated boundary condition cos（θ₁）y’（a）-sin（θ₁）p（a）y"（a）=0,cos（θ₂）y’（b）-sin（θ₂）p（b）y"（b）=0, cos（θ₃）y（a）-sin（θ₃）[p（x）y"（x）]’（a）=0,cos（θ₄）y（b）-sin（θ₄）[[（x）y"（x）]’（b）=0 These properties provides fundamental basis for the existence of non-real eigenvalue in the third chapter.In the second chapter,we study the fourth-order indefinite problem τy:=y（4）+qy=λωy,y（-1）=y（1）=y"（-1）=y"（1）=0,y∈L_|ω|²[-1,1] on[-1,1],combined with condition that q and ω are real-valued functions satisfying ω（x）≠0 a.e.x∈[-1,1],q,ω∈L¹[-1,1] and ω,（x） changes sign on [-1,1].We obtain a priori bounds for possible non-real eigenvalues and a sufficient condition for the existence of non-real eigenvalues.In the third chapter,we study the fourth-order linear indefinite problem τy:=（p（x）y"）"+q（x）y=λω（x）y,x∈[0,1], B1y:=y（0）cosθ₁-py"’（0）sinθ₁=0, B2y:=y（1）cosθ₂-py"’（1）sinθ₂=0, B3y:=y’（0）cosθ₃-py"（0）sinθ₃=0, B4y:=y’（1）cosθ₄-py"（1）sinθ₄=0 combined with condition that q and ω are real-valued functions satisfying p（x）>0,w（x）≠0 a.e.x∈[0,1],1/p,q,ω∈L¹[0,1], and ω（x） changes sign on[-1,1].We obtain a priori bounds for possible non-real eigenvalues and a sufficient condition for the existence of non-real eigenvalues.