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Time-invariant hybrid (HscrTI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on HscrTI algorithms with majority-based (MB) binary message-passing constituents, we use density evolution (DE) and finite-length simulation to analyze the performance and the convergence properties of these algorithms over (memoryless) binary symmetric channels. To apply DE, we generalize degree distributions to have the irregularity of both the code and the decoding algorithm embedded in them. A tight upper bound on the threshold of MB HscrTI algorithms is derived, and it is proven that the asymptotic error probability for these algorithms tends to zero, at least exponentially, with the number of iterations. We devise optimal MB HscrTI algorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB HscrTI algorithms are more robust and can better cope with unknown channel conditions, and thus can be practically more attractive.