Strong convergence of a forward–backward splitting method with a new step size for solving monotone inclusions

Abstract

In this paper, our interest is in investigating the monotone inclusion problems in the framework of real Hilbert spaces. To solve this problem, we propose a new modified forward-backward splitting method using the viscosity method (Moudafi in J Math Anal Appl 241(527):46-55, 2000). Under some mild conditions, we establish the strong convergence of the iterative sequence generated by the proposed algorithm. The advantage of our algorithm is that it does not require the co-coercivity of the single-valued operator. Our result improves related results in the literature. Finally, the performances of our proposed method are presented through numerical experiments in signal recovery. Keywords Forward-backward splitting method • Tseng's splitting method • Viscosity method • Zero point Mathematics Subject Classification 47H06 • 47H09 • 47H10 • 47J25 • 65K15 0 ∈ T x, (1) Dedicated to Professor Pham Ky Anh on the Occasion of his 70th Birthday. Communicated by F. T. Carlos Conca.

Iterating Fixed Point via Generalized Mann’s Iteration in Convex b‐Metric Spaces with Application

“…e paper furthered the research already done on the topic of F-contractions and fixed point theory. In future, these results will be formulated in the structure of Hilbert spaces and orthogonal partial b-metric spaces, and its application in convex minimization and fractional differential equations will be investigated in continuation to the work already done in [21][22][23][24][25].…”

A Fast Image Restoration Algorithm Based on a Fixed Point and Optimization Method

“…It is well-known that fixed point theory has relevant applications in many branches of analysis [1][2][3][4][5][6][7][8][9] and it can be applied to solving many areas of science and applied science, engineering, economics and medicine, such as image/signal processing [10][11][12][13][14][15][16][17] and modeling intensity modulated radiation theory treatment planning [18][19][20]. Many real life problems can be equivalently formulated as fixed point problems, meaning that one has to find a fixed point of some operators.…”

A Novel Forward-Backward Algorithm for Solving Convex Minimization Problem in Hilbert Spaces

“…Then, the sequence generated by (6) weakly converges to a solution of VIP. This method is often called Tseng's extragradient method and has received great attention by researchers due to its convergence speed (see, for example, [25][26][27][28]). Following this research direction, the main challenge is to design novel algorithms that can speed up the convergence rate compared to Algorithms 1-3.…”