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We develop a theoretical framework for computer-assisted proofs of theexistence of invariant objects in semilinear PDEs. The invariant objectsconsidered in this paper are equilibrium points, traveling waves, periodicorbits and invariant manifolds attached to fixed points or periodic orbits. Thecore of the study is writing down the invariance condition as a zero of anoperator. These operators are in general not continuous, so one needs to smooththem by means of preconditioners before classical fixed point theorems can beapplied. We develop in detail all the aspects of how to work with theseobjects: how to precondition the equations, how to work with the nonlinearterms, which function spaces can be useful, and how to work with them in acomputationally rigorous way. In two companion papers, we present two differentimplementations of the tools developed in this paper to study periodic orbits.

A Framework for the Numerical Computation and a Posteriori Verification of Invariant Objects of Evolution Equations