Solution of a nonlinear two-point boundary value problem with Neumann- type boundary data
The authors present a very interesting method to solve the boundary value problem \[ (1)\quad u''(t)=f(t,{\mathcal A}u(t)),\quad u'(a)=A,\quad u'(b)=B, \] where \(A,B\in {\mathbb{R}}\), f: [a,b]\(\times {\mathbb{R}}\to {\mathbb{R}}\) is square integrable in the first argument and Lipschitz continuous in the second argument, and \({\mathcal A}: L_ 2(a,b)\to L_ 2(a,b)\) is Lipschitz continuous. Since for problems of this kind existence or uniqueness of solutions does not hold in general, the authors introduce the auxiliar problem \[ (2)\quad u''(t)+\delta (u)=f(t,{\mathcal A}u(t),\quad u'(a)=A,\quad u'(b)=B,\quad Ju=\lambda, \] where \(\lambda\) is a real parameter, J is the mean value operator \(Ju=(b-a)^{- 1}\int^{b}_{a}u(t)dt\) and \(\delta (u)=Jf(\cdot,{\mathcal A}u(\cdot))-((B- A)/(b-a)).\) A function u in the Sobolev space \(H^ 2(a,b)\) is a solution of (1) if and only if it is a solution of (2) for which \(\delta (u)=0\). Theorem 2.1 states that (2) has a unique solution if \(LA_ 0<3\sqrt{10}/(b-a)^ 2\) where L and \(A_ 0\) are Lipschitz constants of f and \({\mathcal A}\), respectively. Under these assumptions, if \(u_{\lambda}\) is the unique solution of (2) corresponding to the parameter \(\lambda\), problem (1) is reduced to finding the zeros of the function \(\delta_ 0(\lambda)=\delta (u_{\lambda})\). Three examples demonstrate the usefulness of the theory.
scientific article; zbMATH DE number 4109140