Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

Let $h\in L^{1}[0,1]\cap C(0,1)$ be nonnegative and $f(t,u,v)+h(t)\geq 0$. We study the existence and multiplicity of positive solutions for the nonlinear fourth-order two-point boundary value problem $$ u^{(4)}(t)=f(t,u(t),u^{\prime }(t)),\quad\ 0< t< 1,\quad\ u(0)=u^{\prime }(0)=u^{\prime }(1)=u^{\prime \prime \prime }(1)=0, $$ where the nonlinear term $f(t,u,v)$ may be singular at $t=0$ and $t=1$. By constructing a suitable cone and integrating certain height functions of $f(t,u,v)$ on some bounded sets, several new results are obtained. In mechanics, the problem models the deflection of an elastic beam fixed at the left end and clamped at the right end by sliding clamps.