Degenerate Three-Point Boundary Conditions

Abstract

For the differential equation $$y^{\prime \prime \prime }(x)=\lambda y(x) $$ on the interval $$[0,1] $$ , we consider the three-point eigenvalue problem $$a_i y^{(i-1)}(0)+y(c)+b_i y^{(i-1)}(1)=0$$ , $$i=1,2,3 $$ , where $$\lambda $$ is the spectral parameter, the point $$c\in (0,1) $$ is fixed, and $$a_i $$ and $$b_i $$ , $$i=1,2,3$$ , are some complex numbers. Necessary and sufficient conditions that the coefficients $$a_i $$ and $$b_i $$ must satisfy for the indicated three-point problem to have degenerate boundary conditions are obtained.