Solvability of a fourth-order boundary value problem with periodic boundary conditions. II (Q1173911)

The paper deals with the boundary value problem \(u^{(4)}+f(t,u,u',u'',u''')=e(t)\), \(u(0)-u(1)=u'(0)-u'(1)=u''(0)- u''(1)=u'''(0)-u'''(1)=0\). It is assumed that \(f\) satisfies Carathéodory conditions, there exist real numbers \(r\), \(R\), \(a\), \(A\) such that \(f(x,u,v,w,y)\geq A\) for a.e. \(t\in[0,1]\), all \((v,w,y)\in\mathbb{R}^ 3\) and all \(u\geq R\) and \(f(x,u,v,w,y)\leq a\) for a.e. \(t\in[0,1]\), all \((v,w,y)\in\mathbb{R}^ 3\) and all \(u\leq r\). Moreover, \(f\) satisfies some additional inequalities. The author gives sufficient conditions for the existence of a solution to this problem for any \(e\in L^ 1(0,1)\) such that \(a<\int^ 1_ 0 e(t)dt<A\). The main tool used in proofs is the Leray-Schauder continuation theorem.