On holomorphic solutions of some inhomogeneous linear differential equations in a banach space over a non-archimedean field

Let A be a closed linear operator on a Banach space $ mathfrak{B} $ mathfrak{B} over the field Ω of complex p-adic numbers having an inverse operator defined on the whole $ mathfrak{B} $ mathfrak{B} , and f be a locally holomorphic at 0 $ mathfrak{B} $ mathfrak{B} -valued vector function. The problem of existence and uniqueness of a locally holomorphic at 0 solution of the differential equation y ^(m) − Ay = f is considered in this paper. In particular, it is shown that this problem is solvable under the condition $ mathop {lim }limits_{n to infty } sqrt[n]{{left| {A^{ - n} } right|}} $ mathop {lim }limits_{n to infty } sqrt[n]{{left| {A^{ - n} } right|}} = 0. It is proved also that if the vector-function f is entire, then there exists a unique entire solution of this equation. Moreover, the necessary and sufficient conditions for the Cauchy problem for such an equation to be correctly posed in the class of locally holomorphic functions are presented.