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This paper proposes, in the framework of Hadamard manifolds, two iterative schemes for approximating a solution of a variational inequality problem involving a nonexpansive mapping with a fixed point set of another nonexpansive mapping as constraint. The first scheme is a modified Halpern iteration and the second is a viscosity-type iteration with a weakly contraction mapping. We also discuss some special cases of the mentioned problem. Numerical examples are provided to illustrate the algorithm's numerical behavior. [ABSTRACT FROM AUTHOR]