On non-monotone approximation schemes for solutions of the second-order differential equations. (English) Zbl 1299.34045

Dobkevich, Maria

Differ. Integral Equ. 26, No. 9-10, 1169-1178 (2013).

Using the method of upper and lower solutions, the author studies the solvability of the mixed-type boundary value problem \[ x'' = f(t,x,x'), \;\;x'(a) = A, \; x(b) = B, \] which arises, e.g., when we are looking for the radially symmetric solutions of the problem \[ \Delta u + \varphi (u) = 0 \;\;\text{in} \;\Omega, \quad u = 0 \;\;\text{on} \;\partial \Omega . \] If the regular upper and lower functions exist, then the studied ODE problem possesses maximal and minimal solutions that can be approximated by monotone iterations. Moreover, there exits a solution of zero type, i.e., the corresponding equation of variations has a solution with no zero point in \((a,b)\). If the problem has only non-zero-type solutions (i.e., solutions of the corresponding equation of variations are of oscillatory types), non-monotone approximations are possible and the limiting solutions preserves the same (similar) type of their approximations.

Reviewer: Gabriela Holubová (Plzeň)

MSC:

34A45	Theoretical approximation of solutions to ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

Laplacian; mixed-type problem; radially symmetric solutions; upper and lower solutions; maximal and minimal solutions; non-monotone approximations