Proximal point algorithms for fixed point problem and convex minimization problem

A Modified Proximal Point Algorithm for Finite Families of Minimization Problems and Fixed Point Problems of Asymptotically Quasi-nonexpansive Multivalued Mappings

“…Lemma 2.2. (see, e.g., [10], [15]) Let H be defined as in Assumption 2.1. Then, for every s, t ∈ H and for every µ ∈ [0, 1], the following inequality holds…”

“…29) Lemma 2.7. (see, e.g., [15], [37] ) Let {s n } be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence s n k of {s n } such that…”

“…20 ) converges weakly and strongly to the common solutions of the minimization problem of (1. 18 ) and fixed point problem of nonspreading-type multivalued mapping T in the framework of a real Hilbert space H. Most recently, EL-yekheir, Mendy and Sow [15] introduced and studied the following modified PPA: Let K be a closed and convex subset of a real Hilbert space H, f : K −→ (−∞, +∞) a proper convex and lower semicontinuous function, g : K −→ H an d-Lipschitizian mapping, B : K −→ H an α-strongly monotone and L-Lipschitizian operator and T : K −→ CK(K) a multivalued quasi-nonexpansive mapping. The PPA-Ishikawa iteration method is defined as follows:…”

“…Inspired and moltivated by the results in [15] and [34], it is natural to ask the following question: Question 1.1. Can we construct a modified proximal point algorithm that generalizes (1.…”

Approximating Common Solution of Minimization Problems Involving Asymptotically Quasi-Nonexpansive Multivalued Mappings

“…The fixed-point algorithm (19) for solving problem (18) was first introduced in 1970 by Martinet [14], and was called a proximinal point algorithm (for short, PPA). In recent times, many researchers have studied and generalized (19), and many interesting results have been obtained for different classes of nonlinear single-valued and multivalued mappings: Rockfeller [15] solved problem ( 18) using ( 19); Marino and Xu [12], and subsequently Phuengrattan and Lerkchaiyaphum [16], obtained weak and strong convergence to the common solution of the minimization problem and fixed-point problem using the modified version of (19) in the setting of real Hilbert spaces.…”