Solutions of Euler‐Lagrange equations in fractional mechanics

In the paper Euler‐Lagrange equations of fractional mechanics are studied. The characteristic feature of such equations is mixing of left‐ and right‐sided fractional derivatives due to the rules of integration by parts in fractional calculus. We propose to solve a class of equations of this type using transformation to equivalent fractional integral equations and then applying Banach theorem on fixed point of a contractive mapping. The method is explained in detail for nonlinear fractional oscillator equation in two versions, then the results for a wide class of fractional differential equations are reported.