This work concerns with the solvability of third-order periodic fully problems a weighted parameter, where nonlinearity must verify only local monotone condition and no periodic, coercivity or super sublinearity restrictions are assumed, as usual in literature. The arguments based on new type lower upper solutions method, not necessarily well ordered. A Nagumo growth Leray–Schauder’s topologica...

A higher order Lagrangian or an affine Hamiltonian is totally singular if its vertical Hessian vanishes. A natural duality relation between totally singular Lagrangians and affine Hamiltonians is studied in the paper. We prove that the energy of a totally singular affine Hamiltonian has as a dual a suitable first order totally singular Lagrangian. Relations between the solutions of Euler and Ha...