In this paper, we investigate existence of mild solutions to a non-instantaneous integrodifferential equation via resolvent operators in the sense Grimmer Fréchet spaces. Utilizing technique measures noncompactness conjunction with Darbo's fixed point theorem, present sufficient criteria ensuring controllability given problem. An illustrative example is also discussed.

Using the concept of measure of noncompactness, we present some results concerning the existence, uniform local attractivity and global attractivity of solutions for a functional integral equation. Our results improve and extend some previous known results and based on weaker conditions. Some examples which show that our results are applicable when the previous results are inapplicable are also...

The Kuratowski measure of noncompactness α on an infinite dimensional Banach space (X, ‖ · ‖) assigns to each bounded set S in X a nonnegative real α(S) by the formula α(S) = inf{δ > 0 | S = n ⋃ i=1 Si for some Si with diam(Si) ≤ δ, for 1 ≤ i ≤ n <∞}. In general a map β which assigns to each bounded set S in X a nonnegative real and which shares most of the properties of α is called a homogeneo...

&lt;p style='text-indent:20px;'&gt;We discuss the existence and uniqueness of mild solutions for a class quasi-linear fractional integro-differential equations with impulsive conditions via Hausdorff measures noncompactness fixed point theory in Banach space. Mild solution controllability is discussed two particular cases.&lt;/p&gt;

We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete R-trees hyperconvex hulls are uniquely determined. Next we show that hyperconvexity of subsets of normed spaces implies their convexity if and only if the space under consideration is strictly convex. Moreover, we prove a Krein-Milman type theorem for Rtrees. Finally, we discuss a ge...