Abstract
The purpose of this paper is to define 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. (Filomat 31(12):3611–3626, 2017) and Zhang et al. (Appl. Math. Mech. 32(6):805–810, 2011) as special cases. We prove some convergence, existence, and stability results in separable Banach spaces. Moreover, we produce some numerical examples to demonstrate the applicability of our analytical results. Furthermore, we apply our results in proving the existence of a solution of a nonlinear integral equation of the Hammerstein type.
Highlights
The process of solving some real life problems is characterized with uncertainties, ambiguities, and difficulties
Motivated by the results above, we introduce a 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
6 Conclusion In this research, we defined a new random operator called the generalized φ-weakly contraction of the rational type
Summary
The process of solving some real life problems is characterized with uncertainties, ambiguities, and difficulties. The survey article by Bharucha-Reid [12] in 1976, in which he established the sufficient conditions for a stochastic analogue of Schauder’s fixed point theorem for random operators, gave wings to random fixed point theory This area of research over the years has received the attention of several well-known mathematicians leading to the development of several interesting techniques to obtaining the solution of nonlinear random systems (see, e.g., [7,8,9,10, 15,16,17, 19, 20, 22, 23, 27,28,29,30, 33,34,35, 37])
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