Abstract
In ordered Banach spaces, characterizations of ordered -weak-ANODD set-valued mappings are introduced and studied, which is applied to giving an approximate solution for a new class of general nonlinear mixed-order quasi-variational inclusions involving ⊕ operator. By using the resolvent operator associated with an -weak-ANODD set-valued mapping and fixed point theory, an existence theorem of solutions and an approximation algorithm for this kind of inclusions are established and discussed in ordered Banach spaces, and the relation between the first-valued point and the solution of the problems is shown. The results obtained seem to be general in nature. MSC:49J40, 47H06.
Highlights
Generalized nonlinear ordered variational inequalities have wide applications in many fields including, for example, mathematics, physics, optimization and control, nonlinear programming, economics, and engineering sciences.The variational inclusion, which was introduced and studied by Hassouni and Moudafi [ ], is a useful and important extension of the variational inequality
Huang and Fang [ ] introduced the concept of generalized m-accretive mapping, studied the properties of the resolvent operator with the generalized m-accretive mapping; and Huang and Fang [ ] studied a class of generalized monotone mappings, maximal η-monotone mappings, and defined an associated resolvent operator in. They developed some iterative algorithms to approximate the solution of a class of general variational inclusions involving maximal η-monotone operators
In [ ], Fang and Huang introduced another class of generalized monotone operators, H-monotone operators, defined an associated resolvent operator, established the Lipschitz continuity of the resolvent operator, and studied a class of variational inclusions in Hilbert spaces using the resolvent operator associated with H-monotone operators
Summary
Generalized nonlinear ordered variational inequalities (ordered equations) have wide applications in many fields including, for example, mathematics, physics, optimization and control, nonlinear programming, economics, and engineering sciences. In the author introduced and studied the approximation algorithm and the approximation solution for a class of generalized nonlinear ordered variational inequalities and ordered equations to find x ∈ X such that A(g(x)) ≥ θ (A(x) and g(x) are single-valued mappings) in ordered Banach spaces [ ]. Proof Let X be a real ordered Banach space, let A be a γ -order non-extended mapping, and let M be an α-weak-non-ordinary difference mapping with respect to A, for each x, y ∈ X, there exist a constant α > and vx ∈ M(A(x)) and vy ∈ M(A(y)) such that. Let A be a γ -order non-extended mapping, and M be an ordered (αA, λ)-weak-ANODD mapping with respect to JMA ,λ
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