On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators. (English) Zbl 0902.49009

The standard projection technique in Hilbert spaces has been used to construct the solution of the problem: Find \(x\in H\) with \(f(x)\in K\), \(w\in S(x)\), \(z\in T(x)\) and such that the following generalized variational inequality holds: \[ (w- z, v- f(x))\geq 0,\quad \forall v\in K, \] where \(K\subset H\) is a closed convex subset of a Hilbert space \(H\), \(f: H\to H\) is a given strongly monotone and Lipschitz continuous operator, the multivalued mappings \(S, T: H\to 2^H\) are assumed to satisfy the appropriate conditions of relaxed monotonicity and Lipschitz continuity.


49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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[1] Glowinski, R.; Lions, J. L.; Tremolieres, R., Numerical Analysis of Variational Inequalities (1981), North-Holland: North-Holland Amsterdam · Zbl 0508.65029
[2] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001
[3] Saaty, T. L., Modern Nonlinear Equations (1981), Dover: Dover New York · Zbl 0148.28202
[4] Verma, R. U., On external approximation-solvability of nonlinear equations, PanAmer. Math. J., 2, 23-42 (1992) · Zbl 0759.65035
[5] Verma, R. U., General approximation-solvability of nonlinear equations involving A-regular operators, Z. Anal. Anwendungen, 13, 89-96 (1994) · Zbl 0792.65036
[6] Verma, R. U., Nonlinear demiregular approximation solvability of equations involving strongly accretive operators, Proc. Amer. Math. Soc., 123, 217-222 (1995) · Zbl 0815.65077
[7] Verma, R. U., Iterative algorithms for variational inequalities and associated nonlinear equations involving relaxed Lipschitz operators, Appl. Math. Lett., 9, 61-63 (1996) · Zbl 0864.65039
[8] Yao, J.-C., Applications of variational inequalities to nonlinear analysis, Appl. Math. Lett., 4, 89-92 (1991) · Zbl 0734.49003
[9] Zeidler, E., Nonlinear Functional Analysis and Its Applications, II/B (1990), Springer-Verlag: Springer-Verlag New York
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