Abstract
In this paper, we introduce a system of generalized implicit variational inclusions which consists of three variational inclusions. We design an iterative algorithm with error terms based on relaxed resolvent operator due to Ahmad et al. (Stat Optim Inf Comput 4:183–193, 2016) for approximating the solution of our system. The convergence of the iterative sequences generated by the iterative algorithm is also discussed. An example is given which satisfy all the conditions of our main result.
Highlights
A widely studied problem known as variational inclusion problem have many applications in the fields of optimization and control, economics and transportation equilibrium, engineering sciences, etc
We introduced and study a system of three variational inclusions and we call it system of generalized implicit variational inclusions in real Hilbert spaces
We design an iterative algorithm with error terms based on relaxed resolvent operator for solving system of generalized implicit variational inclusions
Summary
A widely studied problem known as variational inclusion problem have many applications in the fields of optimization and control, economics and transportation equilibrium, engineering sciences, etc. A set-valued mapping M : X → 2X is a said to be (I − H )-monotone if, M is monotone, H is relaxed Lipschitz continuous and [(I − H ) + M](X) = X, where > 0 is a constant. Theorem 2 Let H : X → X be an r-relaxed Lipschitz continuous mapping, I : X → X be an identity mapping and M : X → 2X be a set-valued, (I − H )-monotone mapping. Theorem 3 For each i ∈ {1, 2, 3}, let Xi be a Hilbert space, Ii : Xi → Xi be the identity mappings and Hi, gi : Xi → Xi be the single-valued mappings such that gi is ξi-strongly monotone, gi-Lipschitz continuous and Hi is Hi-Lipschitz continuous, ri-relaxed Lipschitz continuous. ≤ g1 x1n − x1n−1 + H1 g1 x1n − x1n−1 + 1 F11 x1n − x1n−1 + 1 F12 x2n − x2n−1 + 1 F13 x3n − x3n−1
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