Inverse problems of recovering the coefficients of Sturm–Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: 1) from the sequences of eigenvalues and norming constants; 2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

For general Sturm-Liouville operators with separated boundary conditions, we prove the following: If E1,2 ∈ R and if u1,2 solve the differential equation Huj = Ejuj , j = 1, 2 and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection P(E1,E2)(H) of H equals the number of zeros of the Wronskian of u1 and u2.

Abstract This paper consider the existence and multiplicity of solutions for the first order Hamiltonian systems satisfying Sturm-Liouville boundary conditions without convexity assumption. The gradient of Hamiltonian function is generalized asymptotically linear. We find critical points of the corresponding functional by verifying the assumptions of Theorems about critical points given by Bart...

In this paper, the fractional Sturm-Liouville problems, in which the second order derivative is replaced by a fractional derivative, are derived by the Homotopy perturbation method. The fractional derivatives are described in the Caputo sense. The present results can be implemented on the numerical solutions of the fractional diffusion-wave equations. Numerical results show that HPM is effectiv...