Carbone, A.; Conti, G.; Marino, G. A nonlinear boundary-value problem for multivalued differential systems. (English) Zbl 0739.34024 Atti Semin. Mat. Fis. Univ. Modena 38, No. 2, 493-509 (1990). Let \(I=[a,b]\) be a compact interval; \(A(t,x)\) continuous on \(I\times R^ n\); \(F:I\times R^ n\to R^ n\) a compact, convex-valued multivalued map; \(\omega(t,x)\) a continuous function defined on an open connected set \(\Omega\subset R^ 2\); \(L:C(I,R^ n)\to R^ n\) a linear and continuous operator; \(H:C(I,R^ n)\to R^ n\) a continuous operator. Assume that 1) \(\| F(t,x)\|\leq\beta(t)+\alpha(t)\| x\|\) for some \(\alpha,\beta\in L^ 1\); 2) \(\| A(t,x)x\|+\| F(t,x)\|\leq\omega(t,\| x\|)\); 3) \(\| H(u)-L\int^ \tau_ aE_ u(\tau,s)f_ u(s)ds\|\leq\eta\), \(\eta>0\), \(\forall u\in C(I,R^ n)\) and every selection \(f_ u\); 4) for every \(u\in C(I,R^ n)\) fixed, there exists a linear operator \(\Lambda_ u:R^ n\to\hbox{Ker} {\mathcal D}_ u\) such that \(u\to\Lambda_ u\) is continuous with respect to \(u\); \(\exists\lambda>0\) such that \(\|\Lambda_ u\|\leq m\) \(\forall u\in C(I,R^ n)\); \((I-L^ u_ 0\Lambda_ u)(H(u)-L\int^ \tau_ aE_ u(\tau,s)f_ n(s)ds)=0\) for every selection \(f_ u\). Then the problem \(x'\in A(t,x)x+F(t,x) Lx=H(x)\) admits at least one solution. Reviewer: A.I.Kolosov (Khar’kov) MSC: 34B15 Nonlinear boundary value problems for ordinary differential equations 34A60 Ordinary differential inclusions Keywords:nonlinear boundary value problem; multivalued map; linear operator PDFBibTeX XMLCite \textit{A. Carbone} et al., Atti Semin. Mat. Fis. Univ. Modena 38, No. 2, 493--509 (1990; Zbl 0739.34024)