On subnormal transcendental meromorphic solutions of delay differential equations with higher order derivative

The existence of subnormal transcendental meromorphic solutions for the following two classes of delay differential equations

$ \omega(z+1)-\omega(z-1)+a(z)\left(\frac{\omega^{(k)}(z)}{\omega(z)}\right)^n = R(z,\omega(z)) = \frac{P(z,\omega(z))}{Q(z,\omega(z))} $

and

$ (\omega(z) \omega(z + 1) - 1)(\omega(z) \omega(z - 1) - 1) + a(z) \left( \frac{\omega^{(k)}(z)}{\omega(z)} \right)^n = R(z,\omega(z)) = \frac{P(z,\omega(z))}{Q(z,\omega(z))} $

is studied, where $ k\geq 1 $ and $ n\geq 1 $ are integers, $ a(z) $ is non-zero rational, and $ P(z, \omega) $, $ Q(z, \omega) $ are polynomials in $ \omega $ with coefficients rational functions in $ z $, and the roots of $ Q(z, \omega) $ are non-zero functions of $ z $ and are distinct from the roots of $ P(z, \omega) $. The necessary conditions for the existence of solutions of these equations mentioned above are characterized by using the degrees of $ P(z, \omega) $ and $ Q(z, \omega) $. Moreover, some examples are provided to illustrate these results.