Extremal solutions for nonlinear functional \(\varphi\)-Laplacian impulsive equations
This interesting article is devoted to second-order nonlinear \(\varphi\)-Laplacian boundary value problems with impulses of the form \[ (\varphi(u'(t)))' =f(t,u(t),u'(t)); \] \[ g_1(u(0),u(T))=0, \] \[ g_2(u(0), u(T),u'(0),u'(T),u)=0; \] \[ I_k(u(t_k),u(t_k^+))=0, \] \[ M_k(u(t_k), u(t_k^+),u'(t_k),u'(t_k^+),u)=0, \] where \(k=1,\dots,p\), the points \(t_1<\cdots<t_p\) in \((0,T)\) are fixed. The boundary and impulsive conditions are nonlinear and some of them can contain functional dependence on the solution, some monotonicity assumptions are imposed on the corresponding maps. The existence of solutions and of minimal and maximal solutions is studied. Reviewer's remark: Problems with nonlinear functional boundary conditions satisfying monotonicity assumptions were studied in a different situation in [\textit{S. A. Brykalov}, Differ. Equations 29, No. 5, 633--639 (1993); translation from Differ. Uravn. 29, No. 5, 744--750 (1993; Zbl 0835.34087)].
scientific article; zbMATH DE number 5159591
scientific article