A boundary value problem for implicit vector differential inclusions without assumptions of lower semicontinuity

Abstract

Given a multifunction $F:[a,b]\times\mathbf{R}^{n}\times\mathbf{R}^{n}\to2^{\mathbf{R}}$ and a function $h:X\to\mathbf{R}$ (with $X\subseteq \mathbf{R}^{n}$ ), we consider the following implicit two-point problem: find $u\in W^{2,p}([a,b], \mathbf{R}^{n})$ such that $\scriptsize{ \bigl\{\begin{array}{l} h(u^{\prime\prime}(t))\in F(t,u(t),u^{\prime}(t))\quad \mbox{a.e. in } [a,b], u(a)=u(b)=0_{\mathbf{R}^{n}}. \end{array}\bigr. } $ We prove an existence theorem where, for each $t\in[a,b]$ , the multifunction $F(t,\cdot ,\cdot )$ can fail to be lower semicontinuous even at all points $(x,y)\in\mathbf {R}^{n}\times \mathbf{R}^{n}$ . The function h is assumed to be continuous and locally nonconstant.