For the nonlocal Davey-Stewartson I equation, the Darboux transformation is considered and explicit expressions of the solutions are obtained. Like other nonlocal equations, many solutions of this equation may have singularities. However, by suitable choice of parameters in the solutions of the Lax pair, it is proved that the solutions obtained from seed solutions which are zero and an exponential function of t respectively, by a Darboux transformation of degree n are global solutions of the nonlocal Davey-Stewartson I equation. The derived solutions are soliton solutions when the seed solution is zero, in the sense that they are bounded and have n peaks, and "dark cross soliton" solutions when the seed solution is an exponential function of t, in the sense that they are bounded and their norms change fast along some crossing straight lines.

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