Proof Mining for Nonlinear Operator Theory: Four Case Studies on Accretive Operators, the Cauchy Problem and Nonexpansive Semigroups

We present the first applications of proof mining to the theory of partial differential equations as well as to set-valued operators in Banach spaces, in particular to abstract Cauchy problems generated by set-valued nonlinear operators that fulfill certain accretivity conditions. In relation to (various versions of) uniform accretivity we introduce a new notion of modulus of accretivity. A central result is an extraction of effective bounds on the convergence of the solution of the Cauchy problem to the zero of the operator that generates it. We also provide an example of an application for a specific partial differential equation. For such operators as well as for operators fulfilling the so-called $\phi$-expansivity property, again in general real Banach spaces, we give computable rates of convergence of their resolvents to their zeros. We give two applications of proof mining to nonlinear nonexpansive semigroups, analysing two completely different proofs of essentially the same statement and obtaining completely different bounds. More specifically we obtain effective bounds for the computation of the approximate common fixed points of one-parameter nonexpansive semigroups on a subset of a Banach space and (for a convex subset) we give corollaries on their asymptotic regularity with respect to Krasnoselskii's and Kuhfittig's iteration schemata. The bounds obtained in all the above works are all not only effective, but also highly uniform and of low complexity. We finally include a short comment on a different perspective of a (potential) proof-theoretic application to partial differential equations, namely a reverse mathematical study of a proof for the existence of a weak solution of the Navier-Stokes equations motivating future work