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This research applies the generalized technique of gravitational decoupling to the Bardeen black hole, producing novel black hole solutions in the context of Rastall theory. We proceed by decomposition of the field equations corresponding to an additional matter source into two sets, for further considerations. The metric functions of the Bardeen black hole are adopted to specify the first set. The second one, which is subject to an extra source, is resolved considering a linear equation of state of matter. Through the integration of the solutions of these sets, we develop two expanded models and conduct an in-depth analysis of their distinct physical characteristics, governed by specific parameters. We investigate thermodynamic quantities like density, anisotropic pressure, energy bounds, asymptotic flatness, and thermodynamical properties like the Hawking temperature, entropy, and specific heat, etc. Both models are asymptotically flat but violate the energy bounds. Furthermore, the density, radial pressure, and Hawking temperature demonstrate consistent and acceptable behavior. Ultimately, the thermodynamic stability is affirmed through the analysis of specific heat and the Hessian matrix.
