6533b872fe1ef96bd12d2dea

RESEARCH PRODUCT

Oscillatory Solutions of Boundary Value Problems

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We consider boundary value problems of the form $$\displaystyle\begin{array}{rcl} & x'' = f(t,x,x'), & {}\\ & x(a) = A,\quad x(b) = B,& {}\\ \end{array}$$ assuming that f is continuous together with f x and fx′. We study also equations in a quasi-linear form $$\displaystyle{x'' + p(t)x' + q(t)x = F(t,x,x').}$$ Introducing types of solutions of boundary value problems as an oscillatory type of the respective equation of variations, we show that for a solution of definite type, the problem can be reformulated in a quasi-linear form. Resonant problems are considered separately. Any resonant problem that has no solutions of indefinite type is in fact nonresonant. The ways of how to detect solutions of definite types are discussed.