Uniqueness of non-trivial spherically symmetric black hole solution in special classes of F(R) gravitational theory

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We show, in detail, that the only non-trivial black hole (BH) solutions for a neutral as well as a charged spherically symmetric space-times, using the class F(R)=R±F1(R), must-have metric potentials in the form [Formula presented] and [Formula presented]. These BHs have a non-trivial form of Ricci scalar, i.e., [Formula presented] and the form of [Formula presented]. We repeat the same procedure for (Anti-)de Sitter, (A)dS, space-time and got the metric potentials of neutral as well as charged in the form [Formula presented] and [Formula presented], respectively. The Ricci scalar of the (A)dS space-times has the form [Formula presented] and the form of [Formula presented]. We calculate the thermodynamical quantities, Hawking temperature, entropy, quasi-local energy, and Gibbs-free energy for all the derived BHs, that behaves asymptotically as flat and (A)dS, and show that they give acceptable physical thermodynamical quantities consistent with the literature. Finally, we prove the validity of the first law of thermodynamics for those BHs.

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