In this paper, we have investigated Tsallis holographic dark energy (infrared cutoff is the Hubble radius) in homogeneous and anisotropic Bianchi type-III Universe within the framework of Saez–Ballester scalar–tensor theory of gravitation. We have constructed non-interaction and interaction dark energy models by solving the Saez–Ballester field equations. To solve the field equations, we assume a relationship between the metric potentials of the model. We developed the various cosmological parameters (namely deceleration parameter q, equation of state parameter $$\omega _t$$ω t , squared sound speed $$v_s^2$$v s 2 , om-diagnostic parameter Om(z) and scalar field $$\phi $$ϕ ) and well-known cosmological planes (namely $$\omega _t-\omega _t^{'}$$ω t - ω t ′ plane, where $$'$$′ denotes derivative with respect to ln(a) and statefinders ( $$r-s$$r - s ) plane) and analyzed their behavior through graphical representation for our both the models. It is also, quite interesting to mention here that the obtained results are coincide with the modern observational data.

In this paper, we derive f(R) gravity field equations with the help of a spatially homogeneous and anisotropic Bianchi type-III space–time, in the presence of a bulk viscous fluid, containing one-dimensional cosmic strings. Here we obtained the solutions of the field equations, both in the presence and in the absence of cosmic strings, under some specific plausible physical conditions. In particular, cosmological models with bulk viscous strings in f(R) theory of gravity are obtained by using the special law of variation for Hubble’s parameter proposed by Berman (Nuovo Cimento, B74, 182 (1983)). Various physical and kinematical properties of the models are also discussed.