About the p-adic Yosida equation inside a disk

Let K be a complete ultrametric algebraically closed field and let ℳ(d(0, Rℒ)) be the field of meromorphic functions inside the disk d(0,R−) = {x ∈ K ∣ ∣x∣ < R}. Let ℳb(d(0, Rℒ)) be the subfield of bounded meromorphic functions inside d(0,R−) and let ℳu(d(0, Rℒ)) = ℳ(d(0, Rℒ)) ∖ ℳb(d(0, Rℒ)) be the subset of unbounded meromorphic functions inside d(0,R−). Initially, we consider the Yosida Equation: , where m ∈ ℕ* and F(X) is a rational function of degree d with coefficients in ℳb(d(0, Rℒ)). We show that, if d ≥ 2m + 1, this equation has no solution in ℳu(d(0, Rℒ)).Next, we examine solutions of the above equation when F(X) is apolynomial with constant coefficients and show that it has no unbounded analytic functions in d(0,R−). Further, we list the only cases when the equation may eventually admit solutions in ℳu(d(0, Rℒ)). Particularly, the elliptic equation may not.