FORWARD-BACKWARD SPLITTING ALGORITHM WITH SELF-ADAPTIVE METHOD FOR FINITE FAMILY OF SPLIT MINIMIZATION AND FIXED POINT PROBLEMS IN HILBERT SPACES

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonex-pansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are de-signed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.