Iterative Solutions Of Variational Inclusions And Its Application

In this paper,we studied the iterative solution of variational inclusions with different operators and its application and the c convergence of mulit-step iterations for a finite fam-ily of successively asymptotically stronglyΦ-hemicontractive type operations.In the case of weaker conditions, our results extendor improve the existing relevant results.Specifically stated as follows.In the chapter l,we introduces the research background and the relevant results.In the chapter 2,we give the notion of proximal mapping associated with the (A,η)-monotone operator and study a new system of generalized nonlinear implicit quasi-variational inclusions with (A,η)-monotone operator in Banach spaces without usual uniform smoothness or reflexiveness. By using the proximal mapping technique,we con-structed some iterative algorithms to approximate the solutions of a new system of gen-eralized nonlinear implicit quasi-variational inclusions with (A,η)-monotone operators in Banach spaces without usual uniform smoothness or reflexiveness.Then proved the exis-tence of solutions and the convergence of the sequences generated by the algorithms.The results in this chapter extend and improve some well-known results in the literature.More details please read the chapter 2.In the chapter 3,we introduce an iterative scheme for finding a common element of the set of a generalized equilibrium problems,the set of common fixed points of a fam-ily of infinitely non expansive mappings and the set of the variational inequality for a relaxed (α,β)-coercive mapping in Hilbert space.We prove strong convergence of the iter-ative sequence generated by the proposed iterative algorithm to the unique solution of a variational inequality,which is the optimality condition for the minimization problem.The results in this paper extend and improve some well-known results in the literature.More details please read the chapter 3.In the chapter 4,Mainly Studied the composite iterative algorithm for finite family of strict pseudo-contractions and infinite family of nonexpansive mappings in Hilbert spaces. by modifying the normal Mann's iteration process,We construct a new iterative algo-rithm for finite family of strict pseudo-contractions and infinite family of non expansive mappings in Hilbert spaces. Under suitable conditions, a strong convergence theorem for approximating a common fixed point of the above two sets are obtained and this fixed point is also a solution of the variational inequality. Our results extend and improve the recent ones announced by many others.More details please read the chapter 4.In the chapter 5, the problems which modified multi-step Noor iterations with er-rors converges strongly to a common fixed point are investigated for a finite family of uniformly generalized Lipschitz continuous and successively asymptotically stronglyΦ-hemicontractive type operators in uniformly smooth Banach spaces. A s application, it is obtained that the result of Huang Zhenyu at 2007 concerning the equivalence of the convergence criteria between modified Mann iterations with errors and modified multi-step Noor iterations with errors for successivelyΦ-strongly pseudo-contractive operators with bounded range in same spaces. Furthermore, the methods of proofs are quite differ-ent from Huang Zhenyu's. Meanwhile, these results improve and generalize many recent corresponding results obtained by Rhoades B. E., Soltuz S. M.,Huang Z.Y., Bu F.W., Noor M. A.,Huang Z.Y., Bu F.W.,Su K.,Yao Y. H., Chen R. D., Zhou H. Y.,Liu Z.Q., Kim J. K., Kim K.H.,Liu L.S.,Ni R. X.,Xu Y.G. and the other authors.More details please read the chapter 5.