Abstract
In this paper, we consider the following high-order p-Laplacian neutral differential equation with singularity: $$ \bigl(\varphi_{p}\bigl(x(t)-cx(t-\tau)\bigr)^{(n)} \bigr)^{(m)}+f\bigl(x(t)\bigr)x'(t)+g\bigl(t,x(t-\sigma) \bigr)=e(t). $$ By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established.
Highlights
Carathéodory function, i.e., it is measurable in the first variable and continuous in the second variable, and for every < r < s there exists hr,s ∈ L [, T] such that |f (t, x(t))| ≤ hr,s for all x ∈ [r, s] and a.e. t ∈ [, T]. g(t, x) being singular at means that g(t, x) becomes unbounded when x → +. τ and σ are constants and ≤ τ, σ < T; e : R → R is a continuous periodic function with e(t + T) ≡ e(t) and e(t) dt is a positive constant, c is a constant and |c| = ; n, m are positive integers
Differential equations with singularities have been considered from the very beginning of the discipline
The main reason is that singular forces are ubiquitous in applications, gravitational and electromagnetic forces being the most obvious examples
Summary
|f (t)| dt, Let X and Y be real Banach spaces and L : D(L) ⊂ X → Y be a Fredholm operator with index zero, here D(L) denotes the domain of L. This means that Im L is closed in Y and dim Ker L = dim(Y / Im L) < +∞. (H ) There exist constants < D < D such that if x is a positive continuous T -periodic function satisfying g t, x(t) dt = , .
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.
