Structure Of Eigenvalues Of Multi-point Boundary Value Problems Of Second-Order Differential Equations

For equations y′′+q(x)y=λy,0≤x≤1, the eigenvalues under Dirichlet or Neu-mann boundary condition make up a real sequence, lower-bounded, tending+∞. RuyunMa analysed the rational point multi-point boundary value problem(MPBVP) in2006. In2007, B.P. Rynne studied MPBVP with bounded positive coefcients. He obtained a pos-itive real sequence tending infinity as the eigenvalue set. Later in2010, Meirong Zhangworked on general MPBVP, concluded the eigenvalues consist of a complex sequence,with lower-bounded real parts tending+∞. There also exists a real eigenvalue sequence.Furthermore, when the coefcients are restricted, the complex eigenvalues count up to atmost a finite number; only real eigenvalues exist with coefcients small enough.All multi-point conditions in the above problems only concern the solution itself.They can be taken as a generalization for Dirichlet condition. Here we study eigenvalueproblems with general potential, and boundary condition that involves the derivative ofthe solution. This type of condition can be viewed as a generalization for Neumann con-dition. We Start from the simple case where the potential is0. By properties of almost pe-riodic function and hyperbolic function, we derive the real eigenvalues are lower-boundedwith no upper limit. Then by comparison of general eigenfunction to the eigenfunctionwith0potential, we prove lower-boundedness for real parts of the eigenvalues for gen-eral potential case. And again by properties of almost periodic functions we get a realeigenvalue sequence. When the coefcients are small, we compare the eigenfunctions ofgeneral case and Neumann problem. By estimates, Rouche theorem, and intermediatevalue theorem, we conclude all large eigenvalues are real, which indicates the number ofcomplex eigenvalues is finite. Further, when coefcients are small enough, by the implic-it function theorem, we get the result that not only are the large eigenvalues real, but alleigenvalues are real, which is in line with the classic Neumann problem.Boundary conditions discussed so far all come in one same shape. One of the twoendpoints is fixed, the other has a multi-point condition. We presented a condition withboth endpoints involving multi-points. And by properties of almost periodic functions, wegave a real eigenvalue sequence tending+∞for both cases with0potential and generalpotential.