The existence and approximation of extremal solutions to several first-order discontinuous dynamic equations with nonlinear boundary value conditions. (English) Zbl 1148.34012

Otero-Espinar, Victoria; Vivero, Dolores R.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 7, 2027-2037 (2008).

From the introduction: This paper is devoted to proving the existence and approximation of extremal solutions in the sector generated by a lower and an upper solution for five first-order discontinuous dynamic equations with nonlinear functional boundary value conditions which include as particular choices the usual boundary conditions such as the initial and periodic ones.
For every \(i\in\{1,\dots,5\}\), we consider the first-order dynamic equation
\[ \begin{cases} L_iu(t) = N_iu(t);\quad \Delta\text{-a.a. } t\in D^0=[t_0, T)_{\mathbb T},\\B(u(t_0),u)=0,\end{cases}\tag{\(P_i\)} \]
where \(L_i, N_i : AC(D)\to L^1_\Delta(D^0)\), \(AC(D)\) denotes the class of all absolutely continuous functions on \(D = [t_0,T]_{\mathbb T}\).

MSC:

34B15	Nonlinear boundary value problems for ordinary differential equations
39A10	Additive difference equations
34A45	Theoretical approximation of solutions to ordinary differential equations
34A36	Discontinuous ordinary differential equations

Keywords:

time scales; first-order dynamic equation; weak solution; lower and upper solution; nonlinear boundary value conditions

References:

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