| I. Yermachenko and F. Sadyrbaev. Types of solutions of the second order Neumann problem: multiple solutions |
| The Neumann boundary value problem $x'' = f (t,x,x')
\; (i),$ $\quad x' (0) = 0 = x'(1) \; (ii)$ is studied. Suppose that an equation $(L_2
x)(t):= \frac{d}{dt}(p(t)\,x') + q(t)\,x =F (t,x,x') \quad (iii),$ where $F$ is bounded, is equivalent to
$(i)$ in some $(t,x,x')$-domain $D$ and solutions of the quasi-linear problem $(iii),$ $(ii)$ satisfy the estimate
$(t,x(t),x'(t)) \in D \; \forall t \in [0,1].$ We say then that the original problem $(i),$ $(ii)$ allows for
$L_2$-quasilinearization in $D.$ In this case it is solvable. We
show that if the original problem allows for quasilinearization with respect to essentially different linear parts $(L_2 x)(t),$ then it has multiple solutions. Illustrative examples are analyzed. |