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...

Consider the minimal Sturm-Liouville operator A = Amin generated by the differential expression

Recently a great deal of interest has been focused on the application of HPM for the solution of many different problems. The technique has been applied with great success to obtain the solution of a large variety of nonlinear problems in both ordinary and partial differential equations and integro-differential equations[1-10]. In this work we apply HPM to approximate eigenvalues and eigenfunct...

In this article we have discovered a close relationship between the (algebraic) Bethe Ansatz equations of the spin s XXZ model of a finite size and the q-Sturm-Liouville problem. We have demonstrated that solutions of the Bethe Ansatz equations give rise to the polynomial solutions of a second order q-difference equation in terms of Askey-Wilson operator. The more general form of Bethe Ansatz e...

We prove a Filippov-Gronwall type inequality for solutions of a nonconvex secondorder differential inclusion of Sturm-Liouville type.

By using the Leggett-Williams fixed theorem, we establish the existence of multiple positive solutions for second-order nonhomogeneous Sturm-Liouville boundary value problems with linear functional boundary conditions. One explicit example with singularity is presented to demonstrate the application of our main results.

In this paper we consider eigenvalue problems on time scales involving linear Hamiltonian dynamic systems. We give conditions that ensure that the eigenvalues of the problem are isolated and bounded below. The presented results are applicable also to Sturm-Liouville dynamic equations of higher order, and further special cases of our systems are linear Hamiltonian differential systems as well as...