Quasilinearization and approximate quasilinearization for multipoint boundary value problems (Q1081737)

This paper is concerned with multipoint boundary value problems of the form \(x^{(n)}(t)=f(t,x(t),x'(t),...,x^{(q)}(t)),\) \(x(a_ i)=A_{1,i},\) \(x'(a_ i)=A_{2,i},...,x^{(k_ i)}\) \((a_ i)=A_{k_ i+1,i},\) \(1\leq i\leq r\), \(a_ 1<a_ 2<...<a_ r\), \(0\leq k_ i\), \(\sum k_ i+r=n\), \(r\geq 2\), \(0\leq q\leq n-1\). A quasilinear iterative scheme is defined by means of \[ x^{(n)}_{m+1}(t)=f(t,x_ m(t))+\sum^{q}_{i=0}(x^{(i)}_{m+1}(t)-x_ m^{(i)}(t))\frac{\partial}{\partial x_ m^{(i)}(t)}f(t,x_ m(t)) \] \[ x_{m+1}(a_ i)=A_{1,i},\quad x'_{m+1}(a_ i)=A_{2,i},...,\quad x_{m+1}^{(k_ i)}(a_ i)=A_{k_ i+1,i},\quad 1\leq i\leq r,\quad m=0,1,... \] where x(t) stands for \((x(t),x'(t),...,x^{(q)}(t))\), and \(x_ 0(t)\) is a convenient approximate solution of the problem. Conditions are indicated for the convergence of the scheme described above. Estimates for the sequence \(\| x_{m+1}(t)-x_ m(t)\|\) are derived. An approximate quasilinearization procedure is also discussed.