This paper is the three dimensional extension of the two dimensional work in Nakao et al. (Reliable Comput 9(5):359-372, 2003) and Watanabe et al. (J Math Fluid Mech 6:1-20, 2004) on a computer assisted proof of the existence of nontrivial steady state solutions for Rayleigh-Bénard convection based on the fixed point theorem using a Newton like operator. The differences are emerging of complicated types of bifurcation, direct attack on the problem without stream functions, and increased complexity of numerical computation. The last one makes it hard to proceed the verification of solutions corresponding to the points on bifurcation diagram for three dimensional case. Actually, this work should be the first result for the three dimensional Navier-Stokes problems which seems to be very difficult to solve by theoretical approaches.
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