On measures of noncompactness in the space of functions defined on the half-axis with values in a Banach space

Solvability of an infinite system of integral equations on the real half-axis

“…In this paper we are going to study a more general in nite system of nonlinear integral equations in the same Banach space BC(R+, l∞) but with the use of another measure of noncompactness. Such an approach allows us to obtain an existence result concerning the mentioned in nite system, but under weaker assump-tions than the existence result obtained in [2]. Thus, our result creates a generalization of the existence result contained in paper [2].…”

“…It can be shown that the function µa is a measure of noncompactness in the space BC(R+, E) (cf. [2]). The kernel ker µa of the measure µa consists of all bounded subsets X of the space BC(R+, E) such that functions from X are uniformly continuous and equicontinuous on R+ (equivalently we can say that functions from X are equiuniformly continuous on R+) and tend to zero at in nity with the same rate.…”

“…The present paper is a continuation of papers [1,2], where we constructed measures of noncompactness needed in our study. Particularly, in paper [2] we constructed measures of noncompactness in the Banach space BC(R+, E) containing functions de ned, continuous and bounded on the interval R+ and taking values in a given Banach space E. The construction of those measures depends strongly on a given measure of noncompactness in the space E. Additionally, in the mentioned paper [2] we applied one of the constructed measures of noncompactness to the study of the solvability of an in nite system of integral equations in the space BC (R+, l∞).…”

“…Such an approach allows us to obtain an existence result concerning the mentioned in nite system, but under weaker assump-tions than the existence result obtained in [2]. Thus, our result creates a generalization of the existence result contained in paper [2]. It is also worthwhile mentioning that in paper [1] it was also studied the solvability of an in nite system of integral equations in the Banach space BC(R+, E) but we assumed that E is a Banach space with a regular measure of noncompactness being equivalent to the so-called Hausdor measure of noncompactness.…”

“…However, in the papers published up to now the authors investigated mainly in nite systems of integral equations in the Banach space C([ , T], E) consisting of functions de ned and continuous on the bounded interval [ , T] with values in a Banach space E. The generalization to the space BC(R+, E) is rather a quite new branch of the theory of in nite systems of integral equations (cf. [1,2]).…”