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=??r?x?+??y,a?x?b and combined with separated boundary condition cos??₁?y'?a?-sin??₁?p?a?y"?a?=0,cos??₂?y'?b?-sin??₂?p?b?y"?b?=0, cos??₃?y?a?-sin??₃?[p?x?y"?x?]'?a?=0,cos??₄?y?b?-sin??₄?[[?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 ?y:=y?4?+qy=??y,y?-1?=y?1?=y"?-1?=y"?1?=0,y?L_|?|²[-1,1] on[-1,1],combined with condition that q and ? are real-valued functions satisfying ??x??0 a.e.x?[-1,1],q,??L¹[-1,1] and ?,?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 ?y:=?p?x?y"?"+q?x?y=???x?y,x?[0,1], B1y:=y?0?cos?₁-py"'?0?sin?₁=0, B2y:=y?1?cos?₂-py"'?1?sin?₂=0, B3y:=y'?0?cos?₃-py"?0?sin?₃=0, B4y:=y'?1?cos?₄-py"?1?sin?₄=0 combined with condition that q and ? are real-valued functions satisfying p?x?>0,w?x??0 a.e.x?[0,1],1/p,q,??L¹[0,1], and ??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.