Let X be a real uniformly convex Banach space and C a closed convex nonempty subset of X. Let {Ti... more Let X be a real uniformly convex Banach space and C a closed convex nonempty subset of X. Let {Ti}ri1 be a finite family of nonexpansive self-mappings of C. For a given x1 ∈ C, let {xn} and {xin}, i 1, 2,..., r, be sequences defined x0n xn, x1n a1n1 T1x 0 n 1 − a1n1 x
In this paper, we introduce a new iterative scheme for finding a common element of the set of sol... more In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a mixed equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a real Hilbert space. Using the demi-closedness principle for nonexpansive mapping, we prove that the iterative sequence converges strongly to a common element of the above three sets under some control conditions. Our main result extends a recent result of Ceng, Wang and Yao [L.C. Ceng, C.Y. Wang and J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008) 375-390].
We present the strong convergence theorems for the viscosity iterative scheme for finding a commo... more We present the strong convergence theorems for the viscosity iterative scheme for finding a common element of the solution set of the system of general variational inequalities for two arbitrary nonlinear mappings and the fixed point set of a nonexpansive mapping in real 2uniformly smooth and uniformly convex Banach spaces. Furthermore, we apply our main result with the problem of approximating a zero point of accretive operators and a fixed point of strictly pseudocontractive mappings in Banach spaces. The main results presented in this paper improve and extend some results in the literature.
Strong Convergence of the Modified Mann Iterations in a Banach Space
... One way to overcome this difficulty is to use Mann's iteration method that produces a se... more ... One way to overcome this difficulty is to use Mann's iteration method that produces a sequence {x n } via the recursive manner: (1). View the ... For example, Reich [13] proved that if X is a uniformly convex Banach space with a Fréchet differentiable norm and if {α n } is chosen such ...
Strong Convergence of the Modified Mann Iterations in a Banach Space
The Mann iterations for nonexpansive mappings have only weak convergence even in a Hilbert space.... more The Mann iterations for nonexpansive mappings have only weak convergence even in a Hilbert space. We propose two modifications of the Mann iterations in a uniformly smooth Banach space, one for nonexpansive mappings and the other for the resolvent of accretive operators. The two modified Mann iterations are proved to have strong convergence.
The purpose of this paper is to study a new iterative scheme for a finite family of asymptoticall... more The purpose of this paper is to study a new iterative scheme for a finite family of asymptotically nonexpansive mappings in the intermediate sense. Weak and strong convergence theorems for the iterative scheme in a uniformly convex Banach space are established under some conditions which are weaker than demicompactness or completely continuous. Our results improve and generalize the recent known results in the literature.
In this paper, we study a new iterative method for finding a common element of the set of solutio... more In this paper, we study a new iterative method for finding a common element of the set of solutions of a new general system of variational inequalities for two different relaxed cocoercive mappings and the set of fixed points of a nonexpansive mapping in real 2-uniformly smooth and uniformly convex Banach spaces. We prove the strong convergence of the proposed iterative method without the condition of weakly sequentially continuous duality mapping. Our result improves and extends the corresponding results announced by many others. MSC: 46B10; 46B20; 47H10; 49J40 Keywords: a new general system of variational inequalities; relaxed cocoercive mapping; strong convergence Ax-Ay ≤ L xy , ∀x, y ∈ C; (ii) accretive if there exists j(xy) ∈ J(xy) such that Ax-Ay, j(xy) ≥ , ∀x, y ∈ C; (iii) α-inverse strongly accretive if there exist j(xy) ∈ J(xy) and α > such that Ax-Ay, j(xy) ≥ α Ax-Ay , ∀x, y ∈ C;
International Journal of Mathematics and Mathematical Sciences, 2009
LetXbe a real uniformly convex Banach space andCa closed convex nonempty subset ofX. Let{Ti}i=1rb... more LetXbe a real uniformly convex Banach space andCa closed convex nonempty subset ofX. Let{Ti}i=1rbe a finite family of nonexpansive self-mappings ofC. For a givenx1∈C, let{xn}and{xn(i)},i=1,2,…,r, be sequences definedxn(0)=xn,xn(1)=an1(1)T1xn(0)+(1-an1(1))xn(0),xn(2)=an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn,…, xn+1=xn(r)=anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)+⋯+an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))xn,n≥1, whereani(j)∈[0,1]for allj∈{1,2,…,r},n∈ℕandi=1,2,…,j. In this paper, weak and strong convergence theorems of the sequence{xn}to a common fixed point of a finite family of nonexpansive mappingsTi (i=1,2,…,r)are established under some certain control conditions.
We introduce a new system of general variational inequalities in Banach spaces. The equivalence b... more We introduce a new system of general variational inequalities in Banach spaces. The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established. By using this equivalent formulation, we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces.
We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-n... more We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces. The new iterative scheme includes the multistep Noor iterations with errors, modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor, and Khan and Takahashi scheme as special cases. Our results generalize and improve the recent ones announced by Khan et al. (2008), H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006), and many others.
Let X be a real uniformly convex Banach space and C a closed convex nonempty subset of X. Let {Ti... more Let X be a real uniformly convex Banach space and C a closed convex nonempty subset of X. Let {Ti}ri1 be a finite family of nonexpansive self-mappings of C. For a given x1 ∈ C, let {xn} and {xin}, i 1, 2,..., r, be sequences defined x0n xn, x1n a1n1 T1x 0 n 1 − a1n1 x
In this paper, we introduce a new iterative scheme for finding a common element of the set of sol... more In this paper, we introduce a new iterative scheme for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a mixed equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a real Hilbert space. Using the demi-closedness principle for nonexpansive mapping, we prove that the iterative sequence converges strongly to a common element of the above three sets under some control conditions. Our main result extends a recent result of Ceng, Wang and Yao [L.C. Ceng, C.Y. Wang and J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008) 375-390].
We present the strong convergence theorems for the viscosity iterative scheme for finding a commo... more We present the strong convergence theorems for the viscosity iterative scheme for finding a common element of the solution set of the system of general variational inequalities for two arbitrary nonlinear mappings and the fixed point set of a nonexpansive mapping in real 2uniformly smooth and uniformly convex Banach spaces. Furthermore, we apply our main result with the problem of approximating a zero point of accretive operators and a fixed point of strictly pseudocontractive mappings in Banach spaces. The main results presented in this paper improve and extend some results in the literature.
Strong Convergence of the Modified Mann Iterations in a Banach Space
... One way to overcome this difficulty is to use Mann's iteration method that produces a se... more ... One way to overcome this difficulty is to use Mann's iteration method that produces a sequence {x n } via the recursive manner: (1). View the ... For example, Reich [13] proved that if X is a uniformly convex Banach space with a Fréchet differentiable norm and if {α n } is chosen such ...
Strong Convergence of the Modified Mann Iterations in a Banach Space
The Mann iterations for nonexpansive mappings have only weak convergence even in a Hilbert space.... more The Mann iterations for nonexpansive mappings have only weak convergence even in a Hilbert space. We propose two modifications of the Mann iterations in a uniformly smooth Banach space, one for nonexpansive mappings and the other for the resolvent of accretive operators. The two modified Mann iterations are proved to have strong convergence.
The purpose of this paper is to study a new iterative scheme for a finite family of asymptoticall... more The purpose of this paper is to study a new iterative scheme for a finite family of asymptotically nonexpansive mappings in the intermediate sense. Weak and strong convergence theorems for the iterative scheme in a uniformly convex Banach space are established under some conditions which are weaker than demicompactness or completely continuous. Our results improve and generalize the recent known results in the literature.
In this paper, we study a new iterative method for finding a common element of the set of solutio... more In this paper, we study a new iterative method for finding a common element of the set of solutions of a new general system of variational inequalities for two different relaxed cocoercive mappings and the set of fixed points of a nonexpansive mapping in real 2-uniformly smooth and uniformly convex Banach spaces. We prove the strong convergence of the proposed iterative method without the condition of weakly sequentially continuous duality mapping. Our result improves and extends the corresponding results announced by many others. MSC: 46B10; 46B20; 47H10; 49J40 Keywords: a new general system of variational inequalities; relaxed cocoercive mapping; strong convergence Ax-Ay ≤ L xy , ∀x, y ∈ C; (ii) accretive if there exists j(xy) ∈ J(xy) such that Ax-Ay, j(xy) ≥ , ∀x, y ∈ C; (iii) α-inverse strongly accretive if there exist j(xy) ∈ J(xy) and α > such that Ax-Ay, j(xy) ≥ α Ax-Ay , ∀x, y ∈ C;
International Journal of Mathematics and Mathematical Sciences, 2009
LetXbe a real uniformly convex Banach space andCa closed convex nonempty subset ofX. Let{Ti}i=1rb... more LetXbe a real uniformly convex Banach space andCa closed convex nonempty subset ofX. Let{Ti}i=1rbe a finite family of nonexpansive self-mappings ofC. For a givenx1∈C, let{xn}and{xn(i)},i=1,2,…,r, be sequences definedxn(0)=xn,xn(1)=an1(1)T1xn(0)+(1-an1(1))xn(0),xn(2)=an2(2)T2xn(1)+an1(2)T1xn+(1-an2(2)-an1(2))xn,…, xn+1=xn(r)=anr(r)Trxn(r-1)+an(r-1)(r)Tr-1xn(r-2)+⋯+an1(r)T1xn+(1-an(r)(r)-an(r-1)(r)-⋯-an1(r))xn,n≥1, whereani(j)∈[0,1]for allj∈{1,2,…,r},n∈ℕandi=1,2,…,j. In this paper, weak and strong convergence theorems of the sequence{xn}to a common fixed point of a finite family of nonexpansive mappingsTi (i=1,2,…,r)are established under some certain control conditions.
We introduce a new system of general variational inequalities in Banach spaces. The equivalence b... more We introduce a new system of general variational inequalities in Banach spaces. The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established. By using this equivalent formulation, we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces.
We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-n... more We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces. The new iterative scheme includes the multistep Noor iterations with errors, modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor, and Khan and Takahashi scheme as special cases. Our results generalize and improve the recent ones announced by Khan et al. (2008), H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006), and many others.
Uploads
Papers by Suwicha Imnang