Abstract
In this paper, we first provide a refinement result for the abstract continuation theorem for $k$-set contractions. The new version of the theorem is equivalent to the usual one and it better adapts to study the existence of solutions for nonlinear differential equations. Then we discuss a new class of coupled system of implicit fractional boundary value problem. The nonlinear terms of equations involving perturbations, and the boundary conditions are constituted by periodic and anti-periodic boundary conditions. Based on the abstract continuation theorem for $k$-set contractions, an interesting existence result is obtained. Finally, an example is constructed for illustrating the application of our main results.
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