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Let $f:[a,b]\times {ℝ}^{n+1}\to ℝ$ be a Carath’eodory’s function. Let $\left\{t_h\right\}$, with $t_h\in [a,b]$, and $\left\{x_h\right\}$ be two real sequences. In this paper, the family of boundary value problems $$\left\{\begin{array}{c}{x}^{\left(k\right)}=f\left(t,x,x^{\text{'}},...,x^{\left(n\right)}\right)x^{\left(i\right)}\left(t_i\right)=x_i,i=0,1,...,k-1\hfill \end{array}\right.(k=n+1,n+2,n+3,...)$$ is considered. It is proved that these boundary value problems admit at least a solution for each $k\ge \nu$, where $\nu \ge n+1$ is a suitable integer. Some particular cases, obtained by specializing the sequence $\left\{t_h\right\}$, are pointed out. Similar results are also proved for the Picard problem.