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Further generalization of generalized quasilinearization method. (English) Zbl 0817.34005

The paper deals with the IVP \(x'= f(t, x)\), \(x(0)= x_ 0\), on \(J= [0, T]\), \(f\in C[J\times {\mathbf R}, {\mathbf R}]\). If \(f(t, x)\) is uniformly convex in \(x\) for all \(t\in [0, T]\), then the method of quasilinearization provides a monotone increasing sequence of approximate solutions that converges quadratically to the unique solution. The main result is the following theorem which shows that it is possible to develop monotone sequences that converge to the solution quadratically when \(f\) admits a decomposition into a sum of two functions \(F\) and \(G\) with \(F+ \psi\) concave and \(G+ \phi\) convex for some concave function \(\psi\) and for some convex function \(\phi\):
Theorem 2.1. Let \(\alpha_ 0, \beta_ 0\in C'[J,{\mathbf R}]\) such that \(\alpha_ 0\leq \beta_ 0\) on \(J\), and \[ \Omega= \{(t, x): \alpha_ 0(t)\leq x\leq \beta_ 0(t),\;t\in J\}. \] Assume that \(\text{A}_ 1)\) \(\alpha_ 0, \beta_ 0\in C'[J,{\mathbf R}]\) such that \(\alpha_ 0'\leq f(t, \alpha_ 0)\), \(\beta_ 0'\geq f(t, \beta_ 0)\) and \(\alpha_ 0\leq \beta_ 0\) on \(J\); \(\text{A}_ 2)\) \(f\in C[\Omega,{\mathbf R}]\), \(f\) admits a decomposition \(f= F+ G\) where \(F_ x\), \(G_ x\), \(F_{xx}\), \(G_{xx}\) exist and are continuous satisfying \(F_{xx}(t, x)+ \psi_{xx}(t, x)\leq 0\) and \(G_{xx}(t, x)+ \phi_{xx}(t, x)\geq 0\) on \(\Omega\), where \(\phi, \psi\in C[\Omega,{\mathbf R}]\), \(\phi_ x(t, x)\), \(\psi_ x (t,x)\), \(\phi_{xx}(t, x)\), \(\psi_{xx}(t, x)\) exist, are continuous and \(\psi_{xx}(t, x)< 0\), \(\phi_{xx}(t, x)> 0\) on \(\Omega\). Then there exist monotone sequences \(\{\alpha_ n(t)\}\), \(\{\beta_ n(t)\}\) which converge uniformly to the unique solution of above IVP and the convergence is quadratic. The proof of this theorem is presented clearly and accurately.

MSC:

34A45 Theoretical approximation of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34A40 Differential inequalities involving functions of a single real variable
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