Implicit nonlinear discontinuous functional boundary value <f>ϕ</f>-Laplacian problems: extremality results

This paper is devoted to the study of the following implicit nonlinear discontinuous functional boundary value problem(IP) <fen><cp type="lcub" STYLE="S"><ar><r><c CSPAN="1" RSPAN="1" CA="L" RA="T">L0 u(t)=f(t,u,L0 u) for a.e. t∈I=[a,b],</c></r><r><c CSPAN="1" RSPAN="1" CA="L" RA="T">L1(u)=B1(u,L1(u)),</c></r><r><c CSPAN="1" RSPAN="1" CA="L" RA="T">0=L2(u(a),u(b)),</c></r></ar></fen>whereL0 u(t)=−<NU><rm>d</rm></NU>/<rm>d</rm>tϕ(t,u(t),u′(t))−g(t,u,u(t),u′(t)),andL1(u)=L1(u(a),u(b),u′(a),u′(b),u).Supposing that there is a lower solution <f>α</f> and an upper solution <f>β</f>, such that <f>α&les;β</f>, the existence of extremal solutions lying between both functions is proved. [Copyright &y& Elsevier]