Abstract
The study was conducted with the aim to find out random variational-like inclusion involving relaxed monotone, random operators in Hilbert space and construct a new iterative algorithm. The study also proves the existence of random solution of random variational-like inclusion problem and convergence of random iterative sequences generated by random iterative algorithm. The results in this paper unify, improve and extend some well-known results from the literature.
Highlights
Introduction and preliminariesVariational inclusions are one of the significant generalizations of variational inequalities, which have wide applications in mechanics, optimization and control, economics and transportation, equilibrium and engineering sciences
Random operator theory is a fascinating subject needed to study for various classes of random equations
The initial point in the study of fixed point theory belongs to Prague School of Probabilistic in 1950s
Summary
Variational-like inclusions are the generalization of variational inequality problems. (ii) J is said to be -relaxed Lipschitz continuous, if there exists a measurable function :Ω → (0, ∞) such that ⟨J(t, x(t)) − J(t, y(t)), (t, x(t), y(t))⟩ ≤ − (t)‖x(t) − y(t)‖2, ∀ x(t), y(t) ∈ H;. (iii) is said to be Lipschitz continuous if there exists a measurable function :Ω → (0, ∞) such that ‖ (t, x(t), y(t))‖ ≤ (t)‖x(t) − y(t)‖, ∀ x(t), y(t) ∈ H;. (vi) g is said to Lipschitz continuous if there exists a measurable function g:Ω → (0, ∞) such that ‖g(t, x(t)) − g(t, y(t))‖ ≤ g(t)‖x(t) − y(t)‖, ∀ x(t), y(t) ∈ H;. Definition 1.8 Let :Ω × H → R ∪ {+∞} is a proper -subdifferential (may not be convex) functional, :Ω × H × H → H, J:Ω × H → H be the random mappings and I:Ω × H → H be an identity mapping.
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