Random iterative algorithms and almost sure stability in Banach spaces

Abstract

Many popular iterative algorithms have been used to approximate fixed point of contractive type operators. We define the concept of generalized φ-weakly contractive random operator T on a separable Banach space and establish Bochner integrability of random fixed point and almost sure stability of T with respect to several random Kirk type algorithms. Examples are included to support new results and show their validity. Our work generalizes, improves and provides stochastic version of several earlier results by a number of researchers.

Random fixed point theorems in Banach spaces applied to a random nonlinear integral equation of the Hammerstein type

“…Furthermore, we applied our results in proving the existence of a solution of a nonlinear integral equation of the Hammerstein type. Our results generalize several known results in the literature, including the results of Khan et al [22] and Zhang et al [37]. Moreover, our results unify, extend, and generalize several deterministic fixed point theorems in stochastic version, including the results of Akewe et al [4], Akewe and Okeke [3], Chugh et al [13], and Karahan and Ozdemir [21] among others.…”

“…Proof The proof of Theorem 3.3 follows similar lines as in the proof of Theorem 3.1. [22], Okeke and Abbas [27], and Zhang et al [37]. Moreover, our results extend and generalize several deterministic fixed point theorems in stochastic version, including the results of Chugh et al [13] and Karahan and Ozdemir [21] among others.…”

“…There results unify and generalize the results of Zhang et al [37]. Recently, Khan et al [22] introduced a generalized random operator and proved some convergence and stability results for those kinds of random operators in separable Banach spaces. Their results generalize and improve several known random fixed point results in the literature, including the results of Okeke and Abbas [27].…”

“…In this research, we defined a new random operator called the generalized φ-weakly contraction of the rational type. This new random operator includes those studied by Khan et al [22] and Zhang et al [37] as special cases. We also introduced the random versions of some known faster fixed point iterative schemes (see [13,21]).…”

Stability and Strong Convergence Results for Random Jungck-Kirk-Noor Iterative Scheme

“…The theory of random fixed point theorems was initiated in 1950 by Prague school of probabilistic. After the classical results of Bharucha-Reid [3] in 1976, where he gave sufficient conditions for a stochastic analogue of Schouder's fixed point theorem for random operators, the theory of random fixed points received unprecedented attention by several researchers and many interesting results have appeared in the literature see [7,16,18,28].Špaček [29] and Hanš [8] established stochastic analogue of the Banach fixed point theorem in a separable metric space. Itoh [13] in 1979, generalized and extendedŠpaček and Hanš's theorem to a multivalued contraction random operators.…”