Continuity of solutions to operator equations with respect to a parameter

Let $A(k)u(k)=f(k) (1)$ be an operator equation, $X$ and $Y$ are Banach spaces, $k\inΔ\subset\C$ is a parameter, $A(k):X\to Y$ is a map, possibly nonlinear. Sufficient conditions are given for continuity of $u(k)$ with respect to $k$. Necessary and sufficient conditions are given for the continuity of $u(k)$ with respect to $k$ in the case of linear operators $A(k)$.

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