Strong convergence and stability of Kirk-multistep-type iterative schemes for contractive-type operators

Hudson Akewe,Adekunle F Olayiwola,Godwin Amechi Okeke

doi:10.1186/1687-1812-2014-45

Hudson Akewe, Adekunle F Olayiwola + Show 1 more

Open Access

https://doi.org/10.1186/1687-1812-2014-45

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Abstract

In this paper, we introduce Kirk-multistep and Kirk-multistep-SP iterative schemes and prove their strong convergences and stabilities for contractive-type operators in normed linear spaces. By taking numerical examples, we compare the convergence speed of our schemes (Kirk-multistep-SP iterative schemes) with the others (Kirk-SP, Kirk-Noor, Kirk-Ishikawa, Kirk-Mann and Kirk iterative schemes) for this class of operators. Our results generalize and extend most convergence and stability results in the literature. MSC:47H09, 47H10.

Highlights

1 Introduction and preliminary definitions The interest in approximating fixed points of various contractive-type operators is increasing. This is because of the close relationship that exists between the problem of solving a nonlinear equation and that of approximating a fixed point of a corresponding contractive-type operator
We show that the Kirk-multistep iterative scheme converges strongly to p of T
Let (E, · ) be a normed linear space, T : E → E be a self-map of E with a fixed point p satisfying the contractive condition

Summary

Main result I

Let E be a normed linear space, T : E → E a self-map of E and x ∈ E. ) has a unique fixed point p; (ii) the Kirk-Noor iterative scheme defined in ) converges strongly to p of T; (iii) the Kirk-Ishikawa iterative scheme defined in Let (E, · ) be a normed linear space, T : E → E be a self-map of E with a fixed point p satisfying the contractive condition. Let (E, · ) be a normed linear space, T : E → E be a self-map of E with a fixed point p satisfying the contractive condition for each x, y ∈ E, ≤ a < and let φ : R+ → R+ be a subadditive monotone increasing function with φ( ) = and φ(Lu) = Lφ(u). For x ∈ E, let {xn}∞ n= be the Kirk-Noor, Kirk-Ishikawa, Kirk-Mann, and Kirk iterative schemes defined by (i) the Kirk-Noor iterative scheme ( . ) is T-stable; (ii) the Kirk-Ishikawa iterative scheme ( . ) is T-stable; (iii) the Kirk-Mann iterative scheme ( . ) is T-stable; (iv) the Kirk iterative scheme ( . ) is T-stable

Example of decreasing function

Observations

Conclusion