A second order ODE with a nonlinear final condition (Q2761582)

The object of the paper is the following differential equation of second-order NEWLINE\[NEWLINEu''(t)+ r(t) u'(t)+ g(t,u(t))= f(t),\quad t\in (0,T).\tag{1}NEWLINE\]NEWLINE It is assumed that \(g\) is continuous and satisfies the condition NEWLINE\[NEWLINE[g(t,u)- g(t,v)]/(u- v)\leq c< (\pi/T)^2NEWLINE\]NEWLINE for \(t\in [0,T]\), \(u,v\in\mathbb{R}\), with \(u\neq v\). Moreover, \(r\in H^1(0,T)\) is nondecreasing. Under some additional assumptions, it is proved that the Dirichlet problem \(u(0)= u_0\), \(u(T)= u_T\), for equation (1) is uniquely solvable in the space \(H^2(0,T)\) for any \(f\in L^2(0,T)\). Apart from this, the authors prove also an existence result on equation (1) with the final value conditions \(u(0)= u_0\), \(u(T)= h(u'(T))\), where \(h\) is continuous. A few interesting examples are also provided.