Solvability of Extended General Strongly Mixed Variational Inequalities

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Balwant Singh Thakur


In this paper, a new class of extended general strongly mixed variational inequalities is introduced and studied in Hilbert spaces. An existence theorem of solution is established and using resolvent operator technique, a new iterative algorithm for solving the extended general strongly mixed variational inequality is suggested. A convergence result for the iterative sequence generated by the new algorithm is also established.

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  1. Baiocchi,C., Capelo,A.: Variational and Quasi Variational Inequalies. J. Wile and Sons, New York (1984).
  2. Bnouhachem,A., Noor,M.A., Al-Shemas,E.H.: On self-adaptive method for general mixed variational inequalities, Math. Prob. Engineer. (2008), doi: 10.1155/2008/280956.
  3. Brezis,H.: Op ´erateurs maximaux monotone et semi-groupes de contractions dans les espaces de Hilbert. In: North-Holland Mathematics Studies. 5, Notas de matematics, vol. 50, NorthHolland, Amsterdam (1973).
  4. Browder,F.E.: Fixed point theorems for nonlinear semicontractive mappings in Banach spaces. Arch. Rat. Mech. Anal. 21, 259-269 (1966).
  5. Combettes,P.L., Hirstoaga,S.A.: Visco-penalization of the sum of two monotone operators. Nonlinear Anal. 69, 579-591 (2008).
  6. Dhage,B.C.: Remarks on two fixed-point theorems involving the sum and the product of two operators. Comput. Math. Appl. 46, 1779-1785 (2003).
  7. Fucik,S.: Fixed point theorems for a sum of nonlinear mapping. Comment. Math. Univ. Carolinae 9, 133-143 (1968).
  8. Fucik,S.: Solving of nonlinear operator equations in Banach space. Comment. Math. Univ. Carolinae 10, 177-186 (1969).
  9. Glowinski,R., Lions,J.L., Tremolieres,R.: Numerical Analysis of Variational Inequalities. North-Holland, Amesterdam, Holland (1981).
  10. Hassouni,A., Moudafi,A.: Perturbed algorithm for variational inclusions. J. Math. Anal. Appl. 185, 706-712 (1994).
  11. Kirk,W.A.: On nonlinear mappings of strongly semicontractive type. J. Math. Anal. Appl. 27, 409-412 (1969).
  12. Krasnoselskii,M.A.: Two remarks of the method of successive approximations. Uspeki Mat. Nauk 10, 123-127 (1955).
  13. Minty,H.J.: On the monotonicity of the gradient of a convex function. Pacific J. Math. 14, 243-247 (1964).
  14. Noor,M.A.: Strongly nonlinear variational inequalities. C.R. Math. Rep. Acad. Sci. Canad. 4, 213-218 (1982).
  15. Noor,M.A.: On a class of variational inequalities. J. Math. Anal. Appl. 128, 135-155 (1987).
  16. Noor,M.A.: A class new iterative methods for general mixed variational inequalities. Math. Comput. Modell. 31, 11-19 (2000).
  17. Noor,M.A.: Modified resolvent splitting algorithms for general mixed variational inequalities. J. Comput. Appl. Math. 135, 111-124 (2001).
  18. Noor,M.A.: Operator-splitting methods for general mixed variational inequalities. J. Ineq. Pure Appl. Math. 3(5), Art.67, 9p. (2002)
  19. Noor,M.A.: Psueudomontone general mixed variational inequalities. Appl. Math. Comput. 141, 529-540 (2003).
  20. Noor,M.A., Ullah,S., Noor,K.I., Al-Said,E.: Iterative methods for solving extended general mixed variational inequalities. Comput. Math. Appl. 62, 804-813 (2011).
  21. O'Regan, D.: Fixed point theory for the sum of two operators. Appl. Math. Lett. 9, 1-8 (1996).
  22. Petryshyn,W.V.: Remarks on fixed point theorems and their extensions. Trans. Amer. Math. Soc. 126, 43-53 (1967).
  23. Siddiqi,A.H., Ansari,Q.H.: General strongly nonlinear variational inequalities. J. Math. Anal. Appl. 166, 386-392 (1992).
  24. Stampacchia,G.: Formes bilineares sur les ensemble convexes. C. R. Acad. Sci. Paris 285, 4413-4416 (1964).
  25. Verma,R.U.: Generalized auxiliary problem principle and solvability of a class of nonlinear variational inequalities involoving cocoercive and co-Lipschitzian mappings. J. Ineq. Pure Appl. Math. 2(3), Art.27, 9p. (2001)
  26. Webb,J.R.L.: Fixed point theorems for nonlinear semicontractive operators in Banach spaces. J. London Math. Soc. 1, 683-688 (1969).
  27. Weng,X.L.: Fixed point iteration for local stricly pseudo-contractive mappings. Proc. Amer. Math. Soc. 113, 727-731 (1991).
  28. Zeidler,E.: Nonlinear functional analysis and its applications, II/B : Nonlinear monotone operators. Springer, New York (1990).