Abstract
This paper deals with a new system of nonlinear variational inclusion problems involving $(A,\eta)$-maximal relaxed monotone and relative $(A,\eta)$-maximal monotone mappings in 2-uniformly smooth Banach spaces. Using the generalized resolvent operator technique, the approximation solvability of the proposed problem is investigated. An iterative algorithm is constructed to approximate the solution of the problem. Convergence analysis of the proposed algorithm is investigated. Similar results are also proved for other system of variational inclusion problems involving relative $(A,\eta)$-maximal monotone mappings and $(H,\eta)$-maximal monotone mappings.
Highlights
Variational inequalities have been well studied and generalized to different directions due to its large association with partial differential equations and optimization problems
Variational inclusion problem is a natural generalization of variational inequality problem and it is of recent interest
Agarwal and Verma [2] solved a new system of variational inclusion problems involving (A, η)-maximal relaxed monotone mappings and relative (A, η)-maximal monotone mappings in Hilbert space
Summary
Variational inequalities have been well studied and generalized to different directions due to its large association with partial differential equations and optimization problems. Verma [16] solved a new class of set valued variational inclusions involving (A, η)-monotone operators in Hilbert space He studied the notion of (A, η)-maximal relaxed monotonicity in Verma [17]. Sahu et al [15] have studied a class of A-monotone implicit variational inclusion problems in semi-inner product spaces. Giles [8] had shown that if the underlying space X is a uniformly convex smooth Banach space it is possible to define a semi-inner product uniquely. By Hahn Banach theorem, for each x ∈ X, there exists at least one functional fx ∈ X∗ such that ⟨x, fx⟩ = ∥x∥2 Given any such mapping f from X into X∗, it has been verified that [y, x] = ⟨y, fx⟩ defines a semi-inner product.
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