We consider a certain higher order difference equation together with conjugate boundary conditions. A characterization of the values of the parameter is carried out so that the boundary value problem has a positive solution. Next, if the parameter is equal to one, criteria are developed for the existence of two positive solutions of the boundary value problem. In addidition, for particular cases we also offer upper and lower bounds for these positive solutions. Several examples are included to dwell upon the importance of the results obtained.

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