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Neumann-type boundary value problem associated with Hamiltonian systems. (English) Zbl 07959966

The author investigates some multiplicity results for a Hamiltonian system with Neumann-type boundary conditions. More precisely, the problem under consideration is of the following type: \[ \begin{cases} x'=\nabla_{y}H(t,x,y,u,v)\,,\quad& y'=-\nabla_{x}H(t,x,y,u,v)\,,\\ u'=\nabla_{v}H(t,x,y,u,v)\,,\quad& v'=-\nabla_{u}H(t,x,y,u,v)\,, \end{cases} \] with \[ \begin{cases} y(a)=y(b)=0\,,\\ v(a)=v(b)=0\,. \end{cases} \] Here \[ \begin{aligned} &x=(x_1,...,x_M)\in {\mathbb{R}}^{M},\quad y=(y_1,...,y_M)\in {\mathbb{R}}^{M},\\ &u=(u_1,...,u_L)\in {\mathbb{R}}^{L},\quad\;\; v=(v_1,...,v_L)\in {\mathbb{R}}^{L}, \end{aligned} \] and \(H:[a,b]\times {\mathbb{R}}^{2M+2L}\to {\mathbb{R}}\) is a continuous function, with continuous partial gradients with respect to \(x\), \(y\), \(u\), and \(v\).
It is assumed that, for every \(i=1,\dots,M\), the function \(H=H(t,x,y,u,v)\) is \(\tau_{i}\)-periodic in the variable \(x_{i}\), for some \(\tau_{i}>0\). Moreover, concerning the second system involving \(u'\) and \(v'\), there exist some generalized (constant) lower and upper solutions, with a Nagumo-type condition.
A critical point theorem by Szulkin is then applied in order to show that the the problem has at least \(M+1\) geometrically distinct solutions. The crucial part of this paper is that, in contrast to periodic problem where the Poincaré-Birkhoff Theorem has a significant role, no twist condition is required.
The main ideas in the proof follow the lines of the following papers:
[A. Fonda et al., J. Math. Anal. Appl. 528, No. 2, Article ID 127599, 33 p. (2023; Zbl 1530.37083); A. Fonda and R. Ortega, Rend. Circ. Mat. Palermo (2) 72, No. 8, 3931–3947 (2023; Zbl 1533.37131)]

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
37J51 Action-minimizing orbits and measures for finite-dimensional Hamiltonian and Lagrangian systems; variational principles; degree-theoretic methods
Full Text: DOI

References:

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