Dependence on parameters for the Dirichlet problem with superlinear nonlinearities (Q5929991)

The authors study the nonlinear Hamilton equations NEWLINE\[NEWLINE \frac{d}{dt}L_{x'}(t,x'(t))+V_x(t,x(t))=0,\qquad\text{ a.e. in} [0,T], \tag{1} NEWLINE\]NEWLINE where \(T>0\) is arbitrary, \(L,V:{\mathbb R}\times{\mathbb R}^n\to{\mathbb R}\) are convex, Gateaux differentiable with respect to the second variable and measurable functions in \(t\). They are looking for solutions to (1) being a pair \((x,p)\) of absolutely continuous functions \(x,p:[0,T]\to{\mathbb R}^n,\) subject to the Dirichlet boundary conditions for the second function, that is, \(p(0)=0=p(T),\) such that NEWLINE\[NEWLINE \frac{d}{dt}p(t)+V_x(t,x(t))=0, \qquad p(t)=L_{x'}(t,x'(t)). NEWLINE\]NEWLINE Using the duality theory, the existence of solutions is proved without exploiting the deformation lemmas, the Ekeland variational principle or PS-type conditions. Furthermore, a measure of a duality gap between primal and dual functionals for approximate solutions to (1) in the superlinear case is obtained. Continuous dependence on parameters is also discussed.