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...

In this paper, we derive some identities for the Hausdorff measures of noncompactness of certain matrix operators on the sequence spaces X(r,s) of generalized means. Further, we apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for such operators to be compact. Mathematics subject classification (2010): 46B15, 46B45, 46B50.

We study the existence of mild solutions for semilinear fractional differential equations with nonlocal initial conditions in $L^p([0,1],E)$, where $E$ is a separable Banach space. The main ingredients used in the proof of our results are measure of noncompactness, Darbo and Schauder fixed point theorems. Finally, an application is proved to illustrate the results of this work. 

In this paper, using the concept of measure of noncompactness, which is a very useful and powerful tools in nonlinear functional analysis, metric fixed point theory and integral equations, we introduce a new contraction on a Banach space. For this purpose by using of a measure of noncompactness on a finite product space, we obtain some generalizations of Darbo’s fixed-point theorem. Then, with ...

G. Darbo [Rend. Sem. Math. Univ. Padova, 24 (1955) 84--92] used the measure of noncompactness to investigate operators whose properties can be characterized as being intermediate between those of contraction and compact operators. In this paper, we apply the Darbo's fixed point theorem for solving infinite system of linear equations in some sequence spaces.  

A long-standing question in the theory of measures noncompactness is that for Kuratowski measure α defined on a metric space M, and every bounded subset B⊂M, there countable B0⊂B such α(B0)=α(B)? In this paper, we give an affirmative answer to above. It done by showing each nonempty set B Banach space, so strongly finitely representable B0, free ultrafilter U affinely isometric ultrapower [co(B...

We consider the second-order nonlinear difference equations of the form ∆(rn−1∆xn−1) + pn f (xn−k) = hn. We show that there exists a solution (xn), which possesses the asymptotic behaviour ‖xn − a ∑n−1 j=0 (1/r j) + b‖ = o(1), a,b ∈ R. In this paper, we extend the results of Agarwal (1992), Dawidowski et al. (2001), Drozdowicz and Popenda (1987), M. Migda (2001), and M. Migda and J. Migda (1988...