Second-order functional differential equations with past, present and future dependence
Existence of extremal quasi-solutions to the problem \[ u''(t) = f(t, u(t), u'(t), u|_{I_{0}}, u ,u ) \;\text{for a.a.}\;\; t \in I_0 \] together with the boundary conditions \[ B_-(t, u(t), u|_{I_{0}}, u ,u) = 0 \;\text{for}\;\text{all} \;I_-, \;\; B_+(t, u(t), u|_{I_{0}}, u ,u) = 0 \;\text{for}\;\text{all} \;I_+ \] is established. Here \(I_0 = [a,b],\; I_-= [a-r_-,a],\; I_+ = [b, b+r_+], I = I_- \cup I_0 \cup I_+, r_-, r_+ \geq 0\). \(f\) is assumed to be Carathéodory with respect to its real variables and monotone with respect to its functional arguments. The boundary functionals are also assumed to satisfy monotonicity conditions with respect to their functional arguments. The space \(\chi = \{\gamma: I \rightarrow \mathbb{R}: \gamma_|{}_{I_{0}} \in C(I_{0}), \gamma|I_{\pm} \;\) is bounded\;\}, is endowed with the pointwise ordering and the supremum norm. The quasi-solutions of the above problem are elements of \(\chi\) satisfying \[ v''(t) = f(t, v(t), v'(t), v|_{I_{0}}, u ,w ) \;\text{for \;a.a.}\;\; t \in I_0, \] \[ B_{\pm}(t, v(t), v|_{I_{0}}, v ,w) = 0 \;\text{for \; all} \;I_{\pm} \] and \[ w''(t) = f(t, w(t), w'(t), w|_{I_{0}}, w ,v ) \;\text{for \;a.a.}\;\; t \in I_0,\; B_{\pm}(t, w(t), w|_{I_{0}}, w ,v) = 0 \;\text{for}\;\text{all} \;I_{\pm} \] Under some technical assumptions the existence of extremal quasi-solutions in \([\alpha, \beta]\), where \(\alpha\) and \(\beta\) are lower and upper solutions of the above problem, is proved. The above class of problems includes equations with delay, advance and both delay and advance. An example is discussed as an application of the result is also presented.
scientific article; zbMATH DE number 5887173
scientific article