Monotone iterative technique for the initial value problems of impulsive evolution equations in ordered Banach spaces (Q1957584)

The paper deals with the following initial value problem driven by a first-order nonlinear impulsive evolution equation in an ordered Banach space \(E\): \[ u'(t)+Au(t)=f(t,u(t),Gu(t))\, ,\;t\in J\, ,\;t\neq t_k\, ,\, k=1,2,\dots,m \] \[ \Delta u_{|\, t=t_k}=I_k(u(t_k))\, ,\;k=1,2,\dots,m \] \[ u(0)=u_0\in E\;, \] where \(A:D(A)\subset E\rightarrow E\) is a closed linear operator, \(-A\) generates a \(C_0\)-semigroup in \(E\); \(f:J\times E\times E\rightarrow E\) is a nonlinear mapping, \(J=[0,a], \, a>0\); \(Gu(t)=\int_0^t K(t,s)u(s)\,ds\) is a Volterra integral operator with integral kernel \(K\in C(\Delta,I\mathbb{R}^+)\), \(\Delta=\{(t,s):0\leq s\leq t\leq a\}\); \(0<t_1<\dots <t_m< a\); \(\Delta u|_{\, t=t_k}\) represents the jump of \(u\) at \(t=t_k\) and \(I_k:E\rightarrow E\) is an impulse function, \(k=1,2,\dots,m\). In order to discuss the existence (and uniqueness) of mild solutions for the above problem, the author uses a monotone iterative technique in the presence of lower and upper solutions.