Non-(strong, geometrically) ergodicity criteria for discrete time Markov chains on general state

For discrete-time Markov chains on general state spaces, we establish criteria for non-ergodicity and non-strong ergodicity, and derive sufficient conditions for non-geometric ergodicity via the theory of minimal nonnegative solutions. Our criteria are formulated based on the existence of solutions to inequalities involving the chain's one-step transition kernel. Meanwhile, these practical criteria are applied to a type of examples, which can effectively characterize the non-ergodicity and non-strong ergodicity of a specific class of single birth (death) processes.