On Some Novel Methods for Solving the Generalized Fermat–Torricelli Problem in Hilbert Spaces

In 1643 P. de Fermat introduced the problem of finding a point in the plane such that the sum of its Euclidean distances to the three given points is minimal. Recently, Reich and Tuyen (J Optim Theory Appl 196(1): 78–97, 2023) extended this problem in Hilbert space setting and named it ‘generalized Fermat–Torricelli problem’. They introduced some iterative methods for approximating the solution of this problem. This paper aims to formulate some novel iterative methods with inertial effect for approximating the solution of this generalized Fermat–Torricelli problem in real Hilbert spaces. First, we prove the weak and strong convergence of the proposed methods with the mild conditions on the control parameters. Then, we establish the weak and strong convergence results for approximating the solution of a variant of the split feasibility problem with multiple output sets. Finally, we provide some numerical examples to justify the validity and efficiency of the proposed iterative methods. The results in this paper improve and generalize the work of Reich and Tuyen (J Optim Theory Appl 196(1): 78–97, 2023).