Abstract: Magnetic monopoles stand for the static solution arising from a (1+ 3)–
dimensional theory describing the interaction between a real scalar triplet and non–Abelian
gauge field. In this paper, we obtain a two–point boundary value problem of a first–order
ordinary differential equations from the self–dual monopole model. Then we establish
the existence and uniqueness theorem for the problem by using a dynamical shooting
method, we also obtain sharp asymptotic estimates for the solutions at infinity.

Key words: Self-dual monopole, Dynamical shooting method, Existence and uniqueness