Existence and uniqueness of positive mild solutions for nonlocal evolution equations (Q891943)

The authors consider the following semilinear evolution equations on a Banach space \(E\) \[ u'(t)+Au(t)=f(t,u(t)),\quad t\geq 0 \] with nonlocal initial conditions \[ u(0)=\sum_{k=1}^{\infty}c_ku(t_k), \] where \(-A\) generates a \(C_0\)-semigroup \(T (t)(t \geq 0)\) on \(E\), \(0<t_1<t_2<\cdots<t_k<\cdots\) with \(t_k\to \infty\) as \(k\to \infty\) and \(f,c_k\) satisfy some other conditions. The existence and uniqueness of mild solution for the associated linear evolution equation nonlocal problem is established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness of positive mild solutions for the addressed nonlinear evolution equation nonlocal problem are obtained by using the monotone iterative method without the assumption of lower and upper solutions. The theorems proved in this paper are interesting, and improve some related results in ordinary differential equations and partial differential equations. An example is also given to illustrate the abstract results.