On some new nonlocal solvability theorems for various classes of nonlinear differential equations

Abstract

We study nonlinear boundary value problems of the form

$$ [\Psi u']' + F(x;u',u) = g, u(0) = u(1) = 0 $$

, where Φ is a coercive continuous operator from L _p to L _q , and

$$ F(x;u'',u',u) = g, u(0) = u(1) = 0 $$

; first- and second-order partial differential equations

$$ \Phi (x_1 ,x_2 ;u'_1 ,u'_2 ,u) = 0, \sum\limits_{i = 1}^\infty {[\Psi _i (u'_{x_i } )]'_{x_i } + F(x; \ldots ,u'_{x_i } , \ldots ,u) = g_i } $$

; and general equations F(x; ..., u″ _ii , ...., ...., u′ _i , ...; u) = g(x) of elliptic type.

We consider the corresponding boundary value problems of parabolic and hyperbolic type. The proof is based on various a priori estimates obtained in the paper and a nonlocal implicit function theorem.