Sufficient conditions for existence of solutions of nonlinear differential equations with nonlocal boundary conditions
The authors consider the boundary value problems NEWLINE\[NEWLINEy^{(n)}(t)+ \dots +a_1(t)y'(t)+a_0(t)y(t)=f(y(t),y'(t))+\varepsilon G(y,\dots,y^{(n-1)})(t),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sum_{j=1}^nb_{ij}(0)y^{(j-1)}(0)+\sum_{j=1}^nb_{ij}(t_1)y^{(j-1)}(t_1) +\dots+\sum_{j=1}^nb_{ij}(t_N)y^{(j-1)}(t_N)=0\text{ for }i=1,2,\dots,n.NEWLINE\]NEWLINE They provide easily verifiable sufficient conditions for the existence of solutions. These conditions are based on the solution space of the corresponding linear, homogeneous problem and on the size of the nonlinear perturbation. The main results are more general than those in [\textit{J. Rodríguez}, Int. J. Pure Appl. Math. 54, No. 2, 215--226 (2009; Zbl 1184.34029)].
scientific article; zbMATH DE number 6222678