Euler–Lagrange-Type Equations for Functionals Involving Fractional Operators and Antiderivatives

The goal of this paper is to present the necessary and sufficient conditions that every extremizer a given class functionals, defined on set C1[a,b], must satisfy. Lagrange function depends generalized fractional derivative, integral, an antiderivative involving previous operators. We begin by obtaining Euler–Lagrange equation, which condition optimize functional. By imposing convexity over function, we prove it also for optimization. After this, consider variational problems with additional constraints admissible functions, such as isoperimetric holonomic problems. end considering generalization fundamental problem, where order not restricted real values between 0 1, but may take any positive value. some examples illustrate our results.