We obtain a uniform ergodic theorem for the sequence $\frac1{s(n)} \sum_{k=0}^n(\varDelta s)(n-k)\,T^k$, where $\varDelta$ is the inverse of the endomorphism on the vector space of scalar sequences which maps each sequence into the sequence of its partial sums, $T$ is a bounded linear operator on a Banach space and $s$ is a divergent nondecreasing sequence of strictly positive real numbers, such that $\lim_{n\rightarrow+\infty} s(n+1)/s(n)=1$ and $\varDelta^qs\in\ell_1$ for some positive integer $q$. Indeed, we prove that if $T^{n}/s(n$) converges to zero in the uniform operator topology, then the sequence of averages above converges in the same topology if and only if $1$ is either in the resolvent set of $T$, or a simple pole of the resolvent function of $T$.

In this paper, we prove the existence of $(\omega,c)$-periodic solutions for a nonhomogeneous linear impulsive system by constructing Green functions and adjoint systems, respectively. In addition, we study the existence and uniqueness of $(\omega,c)$-periodic solutions for a semilinear impulsive system via fixed point approach. Two examples are provided to illustrate our results.

By utilizing some scalar inequalities obtained via Hermite's interpolating polynomial, we will obtain lower and upper bounds for the difference in Jensen's inequality and in the Edmundson-Lah-Ribaric inequality in time scale calculus that hold for the class of $n$-convex functions. Main results are then applied to generalized means, with a particular emphasis to power means, and in that way some new reverse relations for generalized and power means that correspond to $n$-convex functions are obtained.

We prove the existence and uniqueness of the solutions for the following impulsive semilinear evolution equations with delays and nonlocal conditions: $$\left\{ \begin{array}{lr} \acute{z} =-Az +F(t,z_{t}), & z\in Z, \quad t \in (0, \tau], t \neq t_k, \\ z(s)+(g(z_{\tau_1},z_{\tau_2},\dots, z_{\tau_q}))(s) = \phi(s), & s \in [-r,0],\\ z(t_{k}^{+}) = z(t_{k}^{-})+J_{k}(z(t_{k})), & k=1,2,3, \dots, p. \end{array} \right.$$ where $0 \lt t_1 \lt t_2 \lt t_3 \lt ··· \lt t_p \lt \tau, 0 \lt \tau_{1} \lt \tau_{2} \lt ··· \lt \tau_{q} \lt r \lt \tau, Z$ is a Banach space $Z$, $z_t$ defined as a function from $[−r, 0]$ to $Z^ \alpha$ by $z_{t}(s) = z(t + s),−r \le s \le 0, g : C([−r, 0];Z^{\alpha}_{q}) \rightarrow C([−r, 0];Z^{\alpha})$ and $J_{k} : Z^{\alpha} \rightarrow Z^{\alpha}, F : [0,\tau] ×C(−r,0;Z^{\alpha}) \rightarrow Z$. In the above problem, $A : D(A)\subset Z \rightarrow Z$ is a sectorial operator in $Z$ with $−A$ being the generator of a strongly continuous compact semigroup $\{T(t)\}_{t \ge 0}$, and $Z^{\alpha}= D(A^{\alpha})$. The novelty of this work lies in the fact that the evolution equation studied here can contain non-linear terms that involve spatial derivatives and the system is subjected to the influence of impulses, delays and nonlocal conditions, which generalizes many works on the existence of solutions for semilinear evolution equations in Banch spaces. Our framework includes several important partial differential equations such as the Burgers equation and the Benjamin-Bona-Mohany equation with impulses, delays and nonlocal conditions.