A four-element boundary-value problem with piecewise-continuous coefficients and shift on a compound contour (Q790989)

The boundary value problem \[ a(t)\phi^+[\alpha(t)]+b(t)\overline{\phi^+[\alpha(t)]}=c(t)\phi^- (t)+d(t)\overline{\phi^-\tau)}+g(t), \] \(t\in \Gamma\), in the space \(L_ p(\Gamma,\rho) (1<p<\infty)\) is considered under the following assumptions: \(\Gamma =\cup^{m-1}_{i=0}\Gamma_ i\) is a composite Lyapunov contour, the coefficients a,b,c,d are piecewise continuous functions, \(g\in L_ p(\Gamma,\rho),\) the shift \(\alpha\) maps \(\Gamma '=\cup^{\ell -1}_{i=0}\Gamma_ i\) onto itself preserving the orientation and maps \(\Gamma ''=\cup^{m-1}_{i=\ell}\Gamma_ i\) onto itself changing the orientation. The author gives necessary and sufficient conditions for the Noether property and a formula for the index of this problem.