Research On The 2×2Sturm-Liouville Problem With Eigenparameter Boundary Condition

The present problem on a 2 x 2 Sturm-Liouville problem with eigenparameter boundary condition is researched.thereu(x),w(x),v(x)∈C2[0,π].By analysising and calculating,an entire function w(λ) of this problem is obtained,whose zero set coincedents the set of eigenvalue of the corresponding eigenvalue problem.On this premise, furthermore the rank of the eigenvalue equals the order of the zero is proved, which is important for the trace formula and the expansion theorem.Then using the residue method,a trace formula for above problem is obtained.Finally the existence of resolvent is proved through Green function.Madding use the theory of completely continuous,the expansion theorem is obtained.