Abstract: Letting fi(x), i = 1,..., p be function with value in a Clifford algebra An(R). we call F(x) = (f1, f2,.... fp) function vector, where f1, f2,..., fp are vector components for F(x). In this paper, by thinking of the vector-valued analysis, we discuss a four elements nonlinear boundary value problem for several unknown functions vector with shifts and conjugate and a corresponding linear boundary problem.Applying the method of integral equations and Shauder fixed-point theorem and contract mapping theorem, we prove the existence of solution for the nonlinear problem and the existence and uniqueness for the linear problem.

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