Strong convergence of an inertial algorithm for maximal monotone inclusions with applications

Abstract

An inertial iterative algorithm is proposed for approximating a solution of a maximal monotone inclusion in a uniformly convex and uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a solution of the inclusion. Moreover, the theorem proved is applied to approximate a solution of a convex optimization problem and a solution of a Hammerstein equation. Furthermore, numerical experiments are given to compare, in terms of CPU time and number of iterations, the performance of the sequence generated by our algorithm with the performance of the sequences generated by three recent inertial type algorithms for approximating zeros of maximal monotone operators. In addition, the performance of the sequence generated by our algorithm is compared with the performance of a sequence generated by another recent algorithm for approximating a solution of a Hammerstein equation. Finally, a numerical example is given to illustrate the implementability of our algorithm for approximating a solution of a convex optimization problem.

An Inertial Algorithm for Solving Hammerstein Equations

“…[22][23][24][25] concerning existence and uniqueness results for the Hammerstein Equation ( 20) involving monotone mappings. Recently, Chidume et al [10] established existence result for (20) involving accretive maps and concerning approximation of solutions of the Hammerstein Equation (20), see, e.g., [22,[26][27][28][29][30][31][32] and the references therein. Now, we use Theorem 4 to approximate solutions of Equation (20).…”

Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization

“…It is easily deducible from Lemma 3 that if T is S-pseudocontractive, then a coincidence point of T and S corresponds to a zero of the monotone map S − T. Hence, well known existence results for zeros of monotone operators naturally carry over for coincidence points of operators of this class. The crucial role of zeros of monotone operators in the analysis of solutions to minimization problems (see for example [39][40][41][42]) underscores the interplay between coincidence point problems and optimization problems.…”

Approximation method for monotone inclusion problems in real Banach spaces with applications

“…It is well known that iterative methods involving monotone operators have slow convergence properties. In the literature, the study of convergence properties of iterative algorithms has become an area of contemporary interest (see, e.g., [11][12][13][14][15][16][17]). One method that is now studied enormously is the inertial extrapolation technique which dates back to the early result of Polyak [18] in the context of convex minimization.…”