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Models of static wormholes within the $f(\textit{R}, \textit{T})$ extendedtheory of gravity are investigated and, in particular, the family$f(\textit{R}, \textit{T}) = R + \lambda T$, with $T = \rho + P_{r} + 2P_{l}$being the trace of the energy-momentum tensor. Models corresponding todifferent relations for the pressure components (radial and lateral), andseveral equations of state (EoS), reflecting different matter content, areworked out explicitly. The solutions obtained for the shape functions of thegenerated wormholes obey the necessary metric conditions, as manifested inother studies in the literature. The respective energy conditions reveal thephysical nature of the wormhole models thus constructed. It is found, inparticular, that for each of those considered, the parameter space can bedivided into different regions, in which the exact wormhole solutions fulfillthe null~(NEC) and the weak energy conditions~(WEC), respectively, in terms ofthe lateral pressure. Moreover, the dominant energy condition~(DEC) in terms ofboth pressures is also valid, while $\rho + P_{r} + 2P_{l} = 0$. A similarsolution for the theory $P_{r} = \omega_{1} \rho + \omega_{2} \rho^{2}$ isfound numerically, where $\omega_{1}$ and $\omega_{2}$ are either constant orfunctions of $r$, leading to the result that the NEC in terms of the radialpressure is also valid. For non-constant $\omega_{i}$ models, attention isfocused on the behavior $\omega_{i} \propto r^{m}$. To finish, the question isaddressed, how $f(R) = R+\alpha R^{2}$ will affect the wormhole solutionscorresponding to fluids of the form $P_{r} = \omega_{1} \rho + \omega_{2}\rho^{2}$, in the three cases mentioned above. Issues concerning thenonconservation of the matter energy-momentum tensor, the stability of thesolutions obtained, and the observational possibilities for testing thesemodels are discussed in the last section.

Wormhole models in $f(\textit{R}, \textit{T})$ gravity