Multi-point boundary value problem of first order impulsive functional differential equations
A monotone iterative technique is developed for first-order functional differential equations with impulses and nonlinear multi-point boundary conditions \[ x'(t)=f(t,x(t),x(\theta(t))),\qquad t \neq t_k, \qquad t \in J= [0,T], \] \[ \Delta x(t_k)= I_k(x(t_k)), \qquad k=1,2,\dots,p, \] \[ g(x(m_1),\dots, x(m_r))=0, \] where \(f \in C(J \times \mathbb{R}^2,\mathbb{R})\), \(g \in C(\mathbb{R}^n,\mathbb{R})\), \(\theta \in C(J,J)\), \(0 \leq \theta \leq T\), \(I_k \in C(\mathbb{R},\mathbb{R})\), \(0<t_1<t_2<\dots<t_p<T\), \(0=m_1<m_2<\dots<m_r=T\), \(m_i \neq t_j\) for \(i=1,\dots,r\), \(j=1,\dots,p\), \(\Delta x(t_k)=x(t_k^+)-x(t_k^-)\), and \(\Delta x'(t_k)=x'(t_k^+)-x'(t_k^-)\) for \(k=1,\dots,p\). For this purpose, the authors provide the concepts of upper and lower solutions, prove comparison results and discuss the existence of a unique solution to related linear boundary value problems for first-order functional differential equations with impulses by means of the method of upper and lower solutions. They establish an integral characterization of solutions and apply Schauder's fixed point theorem to derive the existence of extremal solutions to the problem of interest between the lower and upper solutions. The monotone method is developed for the case \[ g(\alpha(m_1),\dots, \alpha(m_r))\leq 0 \qquad \text{ and } \qquad g(\beta(m_1),\dots, \beta(m_r))\geq 0, \] but also for \[ g(\alpha(m_1),\dots, \alpha(m_r))> 0 \qquad \text{ and } \qquad g(\beta(m_1),\dots, \beta(m_r))< 0, \] where \(\alpha \leq \beta\) are, respectively, lower and upper solutions to the multi-point boundary value problem. An example is provided as an illustration of the main results of the paper.
impulsive functional differential equations
nonlinear multi-point boundary conditions
scientific article; zbMATH DE number 5627270