Electronic Journal of Differential Equations, 
Vol. 2020 (2020), No. 68, pp. 1-25

Title: Monotone iterative method for retarded evolution equations involving nonlocal
       and impulsive conditions 
       
Authors: Xuping Zhang (Northwest Normal Univ., Lanzhou, China)
         Pengyu Chen  (Northwest Normal Univ., Lanzhou, China)
         Yongxiang Li (Northwest Normal Univ., Lanzhou, China)

Abstract:
 In this article, we apply the perturbation technique and monotone iterative method
 in the presence of the lower and the upper solutions to discuss the existence of the
 minimal and maximal mild solutions to the retarded evolution equations involving
 nonlocal and impulsive conditions in an ordered Banach space X
 $$\displaylines{
 u'(t)+Au(t)= f(t,u(t),u_t),\quad t\in [0,a],\; t\neq t_k,\cr
 u(t_k^+)=u(t_k^-)+I_k(u(t_k)),\quad k=1,2,\dots ,m,\cr
 u(s)=g(u)(s)+\varphi(s),\quad s\in [-r,0],
 }$$
 where $A:D(A)\subset X\to X$ is a closed linear operator and -A
 generates a strongly continuous semigroup T(t) $(t\geq 0)$ on X, a, r>0
 are two constants, $f:[0,a]\times X\times C_0\to X$ is Caratheodory continuous,
 $0<t_1<t_2<\dots<t_m<a$ are pre-fixed numbers, $I_k\in C(X,X)$ for $k=1,2,\dots,m$,
 $\varphi\in C_{0}$ is a priori given history, while the function
 $g:C_{a}\to C_{0}$ implicitly defines a complementary history,
 chosen by the system itself. Under suitable monotonicity conditions and
 noncompactness measure conditions, we obtain the existence of the minimal and
 maximal mild solutions, the existence of at least one mild solutions as well as
 the uniqueness of mild solution between the lower and the upper solutions.
 An example is given to illustrate the feasibility of our theoretical results.
 
Submitted August 11, 2019. Published July 01, 2020.
Math Subject Classifications: 35K90, 47D06, 47H08.
Key Words: Evolution equation; delay; impulsive function; nonlocal condition;
           iterative method; measure of noncompactness.