Abstract. For second order nonlinear differential equations with nonintegrable singularities with respect to the time variable, unimprovable sufficient conditions for solvability and unique solvability of multi-point boundary value problems are established. îâäæñéâ. éâëîâ îæàæï ŽîŽûîòæãæ áæòâîâêùæŽèñîæ àŽêðëèâIJâIJæïŽåãæï ŽîŽæêðâàîâIJŽáæ ïæêàñèŽîëIJâIJæå áîëæåæ ùãèŽáæï éæéŽîå áŽáàâêæèæŽ éîŽãŽèû...

For the functional differential equation u(n)(t) = f(u)(t) we have established the sufficient conditions for solvability and unique solvability of the boundary value problems u(i)(0) = ci (i = 0, . . . , m− 1), ∫ +∞ 0 |u(m)(t)|2dt < +∞ and u(i)(0) = ci (i = 0, . . . , m− 1),

We prove local unique solvability of the wave equation for a large class of weakly singular, locally bounded space-time metrics in a suitable space of generalised functions.

Nonimprovable in a certain sense, sufficient conditions for the solvability and unique solvability of the problem u′(t) = F (u)(t), u(a)− λu(b) = h(u) are established, where F : C([a, b];R) → L([a, b];R) is a continuous operator satisfying the Carathéodory condition, h : C([a, b];R) → R is a continuous functional, and λ ∈ R+. 2000 Mathematics Subject Classification: 34K10.

In their classical work, Ladyzhenskaya and Ural′tseva gave a definition of weak solution for parabolic equations in cylindrical domains. Their definition was broad enough to guarantee the solvability of all such problems but narrow enough to guarantee the uniqueness of these solutions. We give here some alternative definitions which are appropriate to non-cylindrical domains, and we prove the u...