Abstract
In this paper, we construct an inertial algorithm that approximates a common fixed point of a countable family of multi-valued total asymptotically strict quasi-$\phi$-pseudocontractive maps in real Banach spaces and prove strong convergence of the sequence generated by this algorithm. We provide a numerical example to illustrate the implementability of the proposed algorithm and also show that our algorithm converges faster than some algorithms recently proposed by other authors for solving this class of problem. Furthermore, we present some applications of our theorems. Finally, our theorems are significant improvement on several important recent results.
Highlights
Let E be a smooth real Banach space
We construct an inertial algorithm that approximates a common fixed point of a countable family of multi-valued total asymptotically strict quasi-φ -pseudocontractive maps in real Banach spaces and prove strong convergence of the sequence generated by this algorithm
Let E be a real reflexive, strictly convex and smooth Banach space, A : E → 2E∗ be a maximal monotone operator with A−10 = 0/, for any x ∈ E. y ∈ A−10 and r > 0, we have φ (y, QAr x) + φ (QAr x, x) ≤ φ (y, x), where QAr : E → E is defined by QAr x := (J + rA)−1Jx
Summary
Let E be a smooth real Banach space. Define the Lyapunov functional φ : E × E → R by φ (u, y) = u 2 − 2 u, Jy + y 2 ∀, u, y ∈ E. In 2015, Wang and Yang [2] enlarged the class of operators for which the results in the paper of Qin et al [1] are applicable, by proving for the class of total asymptotically strict quasi-φ - pseudocontraction that the sequence generated by the following algorithm:. Zhang et al [3] proposed the following hybrid projection algorithm for approximating common fixed points of a finite family of closed total asymptotically strict quasi-φ pseudocontractions: u0 ∈ E chosen arbitrary, C0i = C, i = 1, 2, ..., N, yin = J−1. Cn+1 = {v ∈ Cn : φ (v, vn) ≤ φ (v, wn)}, un+1 = ΠCn+1u0, n ≥ 1, Our contribution in this paper, is to construct in certain Banach spaces an inertial iterative algorithm that approximates common fixed points of an infinite family of multi-valued (ki, γni , δni, ψi)-total asymptotically strict quasi-φ -pseudocontractive maps. Corollaries of our theorems are significant improvement on several important recent results announced by other authors, in particular, the results of Chidume et al [10], Zhang et al [3], Wang and Yang [2], Zhang [4], and Qin et al [1] (see concluding remark below)
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