Abstract
We investigate the existence of mild solutions for abstract semilinear measure driven equations with nonlocal conditions. We first establish some results on Kuratowski measure of noncompactness in the space of regulated functions. Then we obtain some existence results for the abstract measure system by using the measure of noncompactness and a corresponding fixed point theorem. The usual Lipschitz-type assumptions are avoided, and the semigroup related to the linear part of the system is not claimed to be compact, which improves and generalizes some known results in the literature.
Highlights
In this paper, we consider the following semilinear measure driven differential system with nonlocal condition: dx(t) = Ax(t) + f t, x(t) dg(t), t ∈ J, ( )x( ) = p(x), where J = [, a] with a >
In Section, we review some concepts and results about the Lebesgue-Stieltjes integral and regulated functions and the Kuratowski measure of noncompactness, which will be used throughout this paper
5 Conclusions In this paper, the issue on abstract semilinear measure driven equations in Banach spaces with nonlocal conditions has been addressed for the first time, which can model a large class of hybrid systems with Zeno behavior
Summary
We obtain some existence results for semilinear measure driven system with nonlocal conditions ( ) by applying the Kuratowski measure of noncompactness and a corresponding fixed point theorem. In Section , we review some concepts and results about the Lebesgue-Stieltjes integral and regulated functions and the Kuratowski measure of noncompactness, which will be used throughout this paper. In Section , some results of the Kuratowski measure of noncompactness and regulated functions are established and applied to investigate the existence for the semilinear measure system ( ).
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
