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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

In this article, we study the existence of piecewise-continuous mild solutions for the initial value problems for a class of semilinear evolution equations. These equations have non-instantaneous impulses in Banach spaces and the corresponding solution semigroup is noncompact. We assume that the nonlinear term satisfies certain local growth condition and a noncompactness measure condition. Also we assume the non-instantaneous impulsive functions satisfy some Lipschitz conditions. An example is given to illustrate our results.

In this article, the existence and multiplicity results of positive periodic solutions are obtained for the second-order functional differential equation with infinite delay $$ u''(t)+b(t)u'(t)+a(t)u(t)=c(t)f(t,u_t),\quad t\in \mathbb{R} $$ where $a, b, c$ are continuous $\omega$-periodic functions, $u_t\in C_B$ is defined by $u_t(s)=u(t+s)$ for $s\in(-\infty,0]$, $C_{B}$ denotes the Banach space of bounded continuous function $\phi:(-\infty,0]\to\mathbb{R}$ with the norm $\|\phi\|_B=\sup_{s\in(-\infty,0]}|\phi(s)|$, and $f: \mathbb{R}\times C_B\to [0,\infty)$ is a nonnegative continuous functional. The existence conditions concern with the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed point index theory in cones.

The aim of this article is to study the existence and uniqueness of strong solutions for a class of semilinear evolution equations with nonlocal initial conditions. The discussions are based on analytic semigroup theory and fixed point theorems. An example illustrates the main results.