Existence result of solutions to differential equations of variable-order with nonlinear boundary value conditions

•We gave some examples demonstrating the transformation between derivatives and integrals of variable-order do not hold.•Differential equations of variable-order are a generalization of differential equations of fractional order.•We used Arzela–Ascoli theorem and some lemmas obtained by us to deal with the existence result of solution.

In this work, a differential equation of variable-order with nonlinear boundary value conditions is discussed. By some analysis techniques and Arzela–Ascoli theorem, existence result of solution is obtained.