Abstract
AbstractIn this paper, we introduce a split hierarchical monotone variational inclusion problem (SHMVIP) which includes split variational inequality problems, split monotone variational inclusion problems, split hierarchical variational inequality problems, etc., as special cases. An iterative algorithm is proposed to compute the approximate solutions of an SHMVIP. The weak convergence of the sequence generated by the proposed algorithm is studied. We present an example to illustrate our algorithm and convergence result.
Highlights
Let H and H be real Hilbert spaces, C ⊆ H and Q ⊆ H be nonempty, closed, and convex sets, A : H → H be a bounded linear operator, and f : H → H and g : H → H be two given operators
If f and g are convex and differentiable, the split variational inequality problem (SVIP) is equivalent to the following split minimization problem: min f (x), subject to x ∈ C, ( . )
If the sets C and Q are the set of fixed points of the operators T : H → H and S : H → H, respectively, the SVIP is called a split hierarchical variational inequality problem (SHVIP)
Summary
Let H and H be real Hilbert spaces, C ⊆ H and Q ⊆ H be nonempty, closed, and convex sets, A : H → H be a bounded linear operator, and f : H → H and g : H → H be two given operators. If the sets C and Q are the set of fixed points of the operators T : H → H and S : H → H , respectively, the SVIP is called a split hierarchical variational inequality problem (SHVIP) It is introduced and studied by Ansari et al [ ]. Let φ : H → H be a given single-valued α-inverse strongly monotone operator and λ ∈. It is well known that when the set-valued mapping B : H ⇒ H is maximal monotone, for each x ∈ H and λ > , there is a unique z ∈ H such that x ∈ (I + λB)z [ , ] In this case, the operator JλB := (I + λB)– is called resolvent of B with parameter λ. This implies that x ∈ (I +λB)(JλB(x)) and y ∈ (I +λB)(JλB(y))
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