Periodic Third-Order Problems with a Parameter

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 topological degree theory existence tools. Only solution is studied here it will remain open discussion non-existence multiplicity solutions. Last section contains nonlinear differential model for catatonic phenomena, depending biological and/or chemical parameters.