We study the effect of antiferromagnetic interactions on the single spin-flip Glauber dynamics of two different one-dimensional Ising models with spin $\ifmmode\pm\else\textpm\fi{}1$. The first model is an Ising chain with antiferromagnetic exchange interaction limited to nearest neighbors and subject to an oscillating magnetic field. The system of master equations describing the time evolution of sublattice magnetizations can easily be solved within a linear field approximation and a long time limit. Resonant behavior of the magnetization as a function of temperature (stochastic resonance) is found, at low frequency, only when spins on opposite sublattices are uncompensated owing to different gyromagnetic factors (i.e., in the presence of a ferrimagnetic short range order). The second model is the axial next-nearest-neighbor Ising (ANNNI) chain, where an antiferromagnetic exchange between next-nearest neighbors is assumed to compete with a nearest-neighbor exchange interaction of either sign. The long time response of the model to a weak, oscillating magnetic field is investigated in the framework of a decoupling approximation for three-spin correlation functions, which is required to close the system of master equations. The calculation, within such an approximate theoretical scheme, of the dynamic critical exponent
z, defined as $1∕\ensuremath{\tau}\ensuremath{\approx}{(1∕\ensuremath{\xi})}^{z}$ (where $\ensuremath{\tau}$ is the longest relaxation time and $\ensuremath{\xi}$ is the correlation length of the chain), suggests that the
T=0 single spin-flip Glauber dynamics of the ANNNI chain is in a different universality class than that of the unfrustrated Ising chain.
We consider an axial Ising model with competing interactions up to third neighbors within a low-temperature approximation in terms of interacting and nonreversing domain walls. Assuming small third-neighbor coupling, we find a close relation between the transfer matrix of this classical model and the Hamiltonian of an S=1/2 anisotropic Heisenberg chain in a uniform magnetic field. This connection allows us to obtain the phase boundaries of the ferromagnetic and other low-temperature modulated states. The ground-state energy for arbitrary magnetization values of the Heisenberg chain is shown to provide information about the density of walls or wave number. In contrast with the two-dimensional axial next-nearest-neighbor Ising model, this quantity displays plateaus and reentrant behavior for some relations among the coupling constants. An alternative way to evaluate correlation functions in the antiferromagnetic Heisenberg chain is presented.