On Some Functional Equations with Borel Summable Solutions

Abstract

A functional equation P m i¼1 a i uðj i ðzÞÞ ¼ f ðzÞ is considered, where fj i ðzÞg m i¼1 are holomorphic functions in a neighborhood of z ¼ 0 with j i ð0Þ ¼ 0 and f ðzÞ is holomorphic in a sector with vertex z ¼ 0. It is shown under some conditions of fj i ðzÞg m i¼1 , f ðzÞ and fa i g m i¼1 that the equation has a formal power series solution that is Borel summable.

A functional equation with Borel summable solutions and irregular singular solutions

“…We refer the proof of Lemma 4.1 to [5]. In the following of this section we assume Condition B.…”

“…Let us proceed to obtain an equation that u(ξ) satisfies. The following procedure is similar to that in [5].…”

“…In case {a i (z)} 1≤i≤m are constants, the terms Q ℓ v (ℓ = 0, 1, • • • ) do not appear in [5].…”

“…We study the existence of a solution of formal power series u(z) = ∑ ∞ n=0 c 0 n z n . In case {a i (z)} m i=1 are constants, it is studied inŌuchi [5]. Next we study the homogeneous case f (z) ≡ 0.…”

“…Next we study the homogeneous case f (z) ≡ 0. We obtain the existence of a formal solution in the form of v(z) = e ψ(1/z) z α w(z), where ψ(t) is a polynomial of t and w(z) = ∑ ∞ n=0 c 1 n z n with w(0) = 1, which is not studied in [5]. We also show under certain conditions of {a i (z)} m i=1 , {φ i (z)} m i=1 and f (z) that there exists a Borel summable function u(z) (w(z)) in some sector such that u(z) ∼ u(z) (resp.…”