Solvability for Two-Point Boundary Value Problems

Suppose that hgL1 0,p., ggC R, R., and lim g t.rt.s0. With the < t <ª` Saddle Point Theorem, the solvability is proved for the two-point boundary value problem yuYsuqg u.yh x., u 0.su p.s0, under the condition that p p p F y`.Hsin x dx-Hh x.sin x dx-F q`.Hsin x dx, 0 0 0 where F y`.slim suptªy`F t., F q`.slim inftªq` F t., and ¡2 tF t.s~¢tH0g s. dsyg t., t/0, g 0., ts0.