On the significance of integral properties of orbits in some superlinear fixed-period problems. (English) Zbl 1025.34018

Gheorghiu, C. I.; Mureşan, A.

Semin. Fixed Point Theory Cluj-Napoca 3, 223-230 (2002).

The nonlinear two-point boundary value problem \[ u_{xx}+ u^0,\;u(0)=u(L) =0,\;p<1, \;L>0 \] is considered. For every odd \(p\geq 3\) the corresponding positive solution has a constant moment, i.e. it does not depend on \(L\). These integral properties mean the existence of a Hamiltonian for the system of initial value differential equations in which the above system has been transformed. Two numerical methods (a 2-stage Runge-Kutta and a Stormer-Verlet), which preserve the Hamiltonian, are used to integrate the system the positive solutions of which agree fairly well with the author’s previous results.

Reviewer: M.Gousidou-Koutita (Thessaloniki)

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
37M15	Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
65P10	Numerical methods for Hamiltonian systems including symplectic integrators
35J60	Nonlinear elliptic equations
65L10	Numerical solution of boundary value problems involving ordinary differential equations

Keywords:

Hamitonian systems; symplectic integrators; constant moment; positive solutions