Bilender P. Allahverdiev[a], Hüseyin Tuna[b] and Hamlet A. Isayev[c]
Existence of solutions for impulsive singular nonlinear q-Sturm-Liouville problems on infinite intervals
Pages: 389-410
Received: 26 June 2023
Accepted in revised form: 14 February 2024
Mathematics Subject Classification: 39A13, 34B16, 34B37, 34B40.
Keywords: \(q\)-difference equations, singular nonlinear boundary-value problems, boundary-value problems with impulses.
Authors address:
[a],[c]: Department of Mathematics, Khazar University, AZ1096 Baku, Azerbaijan
[a],[b]: Research Center of Econophysics, UNEC-Azerbaijan State University of Economics, Baku, Azerbaijan
[b]: Department of Mathematics, Mehmet Akif Ersoy University, 15030 Burdur, Turkey
Abstract:
We are concerned with the impulsive singular nonlinear \(q\)-Sturm-Liouville
equation (with impulsive conditions at \(d\) and boundary conditions at
\(\pm\infty\))
\[
\left[ -\frac{1}{q}\mathcal{D}_{q^{-1}}\mathcal{D}_{q}+v(\eta)\right]
y(\eta)=\Upsilon\left( \eta,y\right) ,~\eta\in J,
\]
where \(J:=J_{1}\cup J_{2},\) \(J_{1}:=(-\infty,d),\) \(J_{2}:=(d,\infty),\)
\(d>0\ \) and \(\ y=y\left( \eta\right) \) is a sought solution. Under various
assumptions on \(\Upsilon \) we prove the existence an uniqueness of solutions
for this problem.