On Picard boundary value problems for second-order asymptotically homogeneous equations (Q1279680)

Let \(a,b,c,d\in L^\infty(0,1)\). The author says that \((a,b,c,d)\) (respectively, \((a,b,-\infty,d))\) is nonresonant if for every \(\alpha,\beta\in L^\infty(0,1)\) with \(a\leq\alpha\leq b\) and \(c\leq\beta\leq d\) (respectively, \(\beta\leq d)\), the problem \[ x''+\alpha(t)x'+\beta(t)x=0,\;x(0)=x(1)=0, \] has no nontrivial solution. Consider the problem \[ x''+f(t,x,x')+g(t,x,x')+h(t,x,x')=0,\;x(0)=x(1)=0,\tag{P} \] where \(f,g,h\) are Carathéodory like functions. The main result is: If \((a,b,c,d)\) is nonresonant and \[ a(t)\leq\liminf_{| y|\to\infty}f(t,x,y)/y\leq\limsup_{| y|\to\infty}f(t,x,y)/y\leq b(t), \] \[ c(t)\leq\liminf_{| x|\to\infty}g(t,x,y)/x\leq\limsup_{| x|\to\infty}g(t,x,y)/x\leq d(t), \] uniformly in \(x\in\mathbb{R}\), respectively in \(y\in\mathbb{R}\), for a.e. \(t\in(0,1)\), then (P) has at least one Carathéodory solution. An analogue result holds if \((a,b,-\infty,d)\) is nonresonant. Applications and some other problems are included.