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A second-order boundary value problem with nonlinear and mixed boundary conditions: existence, uniqueness, and approximation. (English) Zbl 1204.34025

Summary: A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered
\[ Lx=f(t,x,x'),\quad t\in (a,b), \]
\[ g(x(a),x(b),x'(a),x'(b)) = 0,\quad x(b)=x(a) \]
in which \(L\) is a formally self-adjoint second-order differential operator. Under appropriate assumptions on \(L\), \(f\), and \(g\), existence and uniqueness of solutions is established by the method of upper and lower solutions and Leray-Schauder degree theory. The general quasilinearization method is then applied to this problem. Two monotone sequences converging quadratically to the unique solution are constructed.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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