New insights into the Riesz space fractional variational problems and Euler-Lagrange equations

In this paper, we investigate the solvability of a constrained variational problem with a Lagrangian dependent on the Riesz-Caputo derivative. Our approach leverages the direct method in the calculus of variations and the theory of fractional calculus. The main objective of this study is to establish a compactness property of the Riesz fractional integral operator, which enables us to discover extremum points of the constrained fractional variational problem without imposing the convexity condition on the fractional operator variable of the associated Lagrangians. Following this, we derive the Euler-Lagrange equations in their weak form, highlighting their significance in determining minimizers of the variational problem. Finally, we explore a compelling application of fractional variational calculus, specifically examining the intriguing relationship between the fractional Sturm-Liouville eigenvalue problem and constrained fractional variational problems. Our findings provide a new perspective on the solvability of constrained fractional variational problems and offer insights into the application of the direct method in such problems.