Solvability of Extended General Strongly Mixed Variational Inequalities

Thakur, Balwant Singh

Research Article

Solvability of Extended General Strongly Mixed Variational Inequalities

Balwant Singh Thakur¹

Pt. Ravishankar Shukla University

Abstract

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.

How to Cite

B. S. Thakur, Solvability of Extended General Strongly Mixed Variational Inequalities, Int. J. Anal. Appl. 4 (2014), 78–86.

BibTeX

@article{Thakur2014,
  author  = {Thakur, Balwant Singh},
  title   = {Solvability of Extended General Strongly Mixed Variational Inequalities},
  journal = {Int. J. Anal. Appl.},
  volume  = {4},
  year    = {2014},
  pages   = {78--86},
  doi     = {}
}

References

[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) http://eudml.org/doc/123617.
[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) http://eudml.org/doc/122114.
[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).