On the continuous dependence on parameters of solutions of the fourth order periodic problem (Q2747072)

Fourth-order periodic problems of the form NEWLINE\[NEWLINE{d\over dt} \Biggl({d\over dt} f_{\ddot u}(t,u,\dot u,\ddot u,\omega)- f_{\dot u}(t, u,\dot u,\ddot u,\omega)\Biggr)+ f_{u}(t,u,\dot u,\ddot u,\omega)= 0,\tag{\(*\)}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0)- u(T)= \dot u(0)- \dot u(T)= 0,\quad v(0)- v(T)= \dot v(0)-\dot v(T)= 0,NEWLINE\]NEWLINE and NEWLINE\[NEWLINEu^{(4)}= {d\over dt} F_{\dot u}(t, u,\dot u,\omega)- F_u(t,u,\dot u,\omega),\tag{\(**\)}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0)- u(T)= \dot u(0)- \dot u(T)= \ddot u(0)- \ddot u(T)= u^{(3)}(0)- u^{(3)}(T)= 0,NEWLINE\]NEWLINE are studied: here, \(\omega(t)\in L^\infty([0, T],\mathbb{R}^r)\) is a parameter, and \(v(t)= f_{\ddot u}(t,u(t),\dot u(t), \ddot u(t),\omega(t))\).NEWLINENEWLINENEWLINESufficient conditions guaranteeing the existence of solutions and sufficient conditions guaranteeing a continuous dependence of the solution on the parameter are obtained for both problems. One of the assumptions in the theorems on \((**)\) is: There are \(a(t)\in C(\mathbb{R}^+, \mathbb{R}^+)\) and \(b(t)\in C^1([0, T],\mathbb{R})\) such that \(|F(t, p_0, p_1,\omega)|\leq a(|(p_0, p_1)|) b(t)\), \(|F_{p_i}(t, p_0,p_1,\omega)|\leq a(|(p_0, p_1)|)b(t)\), \(i= 0,1\), for all \((p_0,p_1,\omega)\in \mathbb{R}^n\times \mathbb{R}^n\times M\), \(M\subset \mathbb{R}^r\) and almost every \(t\in [0,1]\). A similar assumption is used in the results on \((*)\). The proofs of the theorems obtained are based on variational methods.