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Decomposition conditions for two-point boundary value problems. (English) Zbl 0971.47029
This article deals with Dirichlet, Neumann, periodic and antiperiodic problems for the equation \(x''= f(t,x,x')\). Applying the abstract continuation type theorem of Petryshin on \(A\)-proper mappings the author proves the approximation solvability of the problems listed above; in particular, the classical Galerkin method is justified. It is assumed that the function \(f\) has a form \(f(t,x,p)= g(t,x,p)+ h(t,x,p)\), where \(g(t,x,p)\) satisfies conditions which garantee the nonnegativity of integrals \(\int^1_0 xg(t,x,p) dt\) for corresponding subspace of \(C^2([0,1])\), and \(h(t,x,p)\) satisfies inequalities of type \(|h(t,x,p)|\leq a|x|+ b|p|\) with suitable \(a> 0\), \(b>0\).

MSC:
47E05 General theory of ordinary differential operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
65J15 Numerical solutions to equations with nonlinear operators
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