Abstract
We consider nonlocal differential equations with convolution coefficients of the form−M((a⁎(g∘|u′|))(1))u″(t)=λf(t,u(t)), t∈(0,1), in which the coefficient function M imparts a nonlocal structure to the problem. The function g satisfies p-q growth. A model case occurs when g(t):=tp, where p>1, and a(t)≡1 – i.e., the problem−M(‖u′‖Lpp)u″(t)=λf(t,u(t)), t∈(0,1). By an application of the one-dimensional Sobolev inequality, together with a specially constructed order cone, we are able to demonstrate existence of at least one positive solution to this equation when subjected to Dirichlet boundary conditions. Our methodology utilises a more recently developed topological fixed point theory, which allows for the use of sets that are unbounded in the ambient norm.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: 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.