A strong convergence theorem for generalized-Φ-strongly monotone maps, with applications

Abstract

Let X be a uniformly convex and uniformly smooth real Banach space with dual space X *. In this paper, a Mann-type iterative algorithm that approximates the zero of a generalized-Φ-strongly monotone map is constructed. A strong convergence theorem for a sequence generated by the algorithm is proved. Furthermore, the theorem is applied to approximate the solution of a convex optimization problem, a Hammerstein integral equation, and a variational inequality problem. This theorem generalizes, improves, and complements some recent results. Finally, examples of generalized-Φ-strongly monotone maps are constructed and numerical experiments which illustrate the convergence of the sequence generated by our algorithm are presented.

Iterative algorithms for solutions of Hammerstein equations in real Banach spaces

“…With this, Chidume and Zegeye [25] were able to prove strong convergence of an iterative algorithm defined in the Cartesian product space E to a solution of the Hammerstein equation (1.1). Extensions of these early results of Chidume and Zegeye [25] were obtained by several authors (see, e.g., Chidume and Zegeye [26,27], Chidume and Djitte [15][16][17], Chidume and Ofoedu [21], Chidume and Shehu [22,23], Chidume et al [12,20] Djitte and Sene [29], Ofoedu and Onyi [39], Ofoedu and Malonza [38], Shehu [43], Minjibir and Mohammed [34], and the references contained therein).…”

A hybrid inertial algorithm for approximating solution of convex feasibility problems with applications

“…Many problems in applications can be transformed into the form of the inclusion (1.2). For example, problems arising from convex minimization, variational inequality, Hammerstein equations, and evolution equations can be transformed into the form of the inclusion (1.2) (see, e.g., Chidume et al [8,14], Rockafellar [37]).…”

“…One of the classical methods for approximating solution(s) of (1.2) in Hilbert spaces is the celebrated proximal point algorithm (PPA) introduced by Martinet [28] and studied extensively by Rockafellar [37] and a host of other authors. Concerning the iterative approximation of solution(s) of (1.2) in more general Banach space, see, e.g., [6,11,14,20,32].…”

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).…”