The Strong Convergence Theorems for the Split Equality Fixed Point Problems in Hilbert Spaces

Size: px
Start display at page:

Download "The Strong Convergence Theorems for the Split Equality Fixed Point Problems in Hilbert Spaces"

Transcription

1 Applied Mathematical Sciences, Vol. 2, 208, no. 6, HIKARI Ltd, The Strong Convergence Theorems for the Split Equality Fixed Point Problems in Hilbert Spaces Jun Niu, Zheng Zhou, Jian-Qiang Zhang, and Li-Juan Qin 2 College of Statistics and Mathematics Yunnan University of Finance and Economics Kunming, Yunnan, 65022, P.R. China Corresponding author 2 Department of Mathematics, Kunming University Kunming, 65024, P.R. China Copyright c 208 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The purpose of this paper is to investigate the split equality fixed point problem for quasi-asymptotically pseudo-contractive mappings in Hilbert spaces. And without assumption of semi-compactness, the strong convergence of the sequence generated by the proposed iterative scheme is obtained. The results presented in this paper improve and extend some recent corresponding results announced. Keywords: Split equality fixed point problems; Hilbert spaces; quasiasymptotically pseudo-contractive mappings; Strong convergence Introduction The split feasibility problem (SF P ) was first introduced in 994 by Censor and Elfving [] in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. It has been shown the it can be used in many areas of applications such as imagine restoration, computer tomograph and radiation therapy treatment planning [3-5].

2 256 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Let C and Q be nonempty closed convex subsets of real Hilbert spaces H and H 2, respectively, and A : H H 2 be a bounded linear operator. The split feasibility problem (SF P ) is formulated as: finding x C, such that Ax Q. (.) It is easy to see that x C is a solution of (SF P ) if and only if it solves the following fixed point equation x = P C (I γa (I P Q )A)x, x C, where P C (resp. P Q ) is the (orthogonal) projection from H (resp. H 2 ) onto C (resp. Q), γ > 0, and A is the adjoint of A. In 203, Moudafi [7,8] proposed a new split feasibility problem which is also called split equality problem (SEP ). Let H, H 2 and H 3 be real Hilbert spaces. A : H H 3 and B : H 2 H 3 be two bounded linear operators, C H and Q H 2 be two nonempty closed convex sets. The split equality problem (SEP ) is formulated as: finding x C, y Q such that Ax = By. (.3) Obviously, if B = I (identity mapping on H 2 ) and H 3 = H 2, then the split equality problem (SEP ) reduces to the split feasibility problem (SF P ). In (.3), when C and Q are the sets of fixed points of two nonlinear operator T and S, and C and Q are nonempty closed convex,respectively, the split equality problem is called split equality fixed point problem (SEF P P ). This is finding x C = F (T ), y Q = F (S) such that Ax = By. (.4) Since each closed and convex subset of a Hilbert space may be considered as a fixed point set of a projection on the subset, hence the split equality fixed point problem is a generalization of the split equality problem. The split equality problem (SEP ) and split equality fixed point problem (SEF P P ) have been studied by many authors [9-7]. To solve the (SEF P P ), Modaufi [9] presented the following simultaneous iterative method and obtained weak convergence theorem: { xn+ = T (x n γa (Ax n By n )), y n+ = S(y n + βb (Ax n By n )), (.5) where T and S are two firmly quasi-nonexpansive operators.

3 Strong convergence theorems 257 In 205, Che and Li [6] proposed the following iterative algorithm for finding a solution of the (SEF P P ) of strictly pseudo-nonexpansive mappings: u n = x n γ n A (Ax n By n ), x n+ = α n x n + ( α n )T u n, v n = y n + γ n B (Ax n By n ), y n+ = α n y n + ( α n )Sv n. They also obtained the weak convergence of the iterative scheme (.6). (.6) In 205, Chang et al. [7] proposed an iterative algorithm to establish the strong convergence and weak convergence results of the (SEF P P ) of L- Lipschitzian and quasi-pseudo-contractive mappings: u n = x n γ n A (Ax n By n ), x n+ = α n x n + ( α n )(( ξ n )I + ξ n T (( η n )I + η n T ))u n, v n = y n + γ n B (Ax n By n ), y n+ = α n y n + ( α n )(( ξ n )I + ξ n S(( η n )I + η n S))v n, (.7) In the above research work, to prove the strong convergence, the semicompactness on mappings is needed. In 205, Zhang et al. [8] introduced an new iterative algorithm to solve split common fixed point problem of asymptotically nonexpansive mappings and proved its strong convergence without assumption of semi-compactness on mappings in Hilbert spaces: z n = x n + λa (T n 2 I)Ax n, y n = α n z n + ( α n )T n z n, C n+ = {v C n : y n v k n z n v, z n v k n x n v }, x n+ = P Cn+ x, n. (.8) In 206, Tang et al. [9] used the following hybrid projection algorithm to solve split equality fixed point problem (SEF P P ) for L-Lipschitzian and quasi-pseudo-contractive mappings in Hilbert spaces and proved its strong convergence theorem without assumption of semi-compactness on mappings: u n = x n γ n A (Ax n By n ), w n = α n x n + ( α n )(( ξ)i + ξt (( η)i + ηt ))u n, v n = y n + γ n B (Ax n By n ), z n = α n y n + ( α n )(( ξ)i + ξs(( η)i + ηs))v n, C n+ Q n+ = {(p, q) C n Q n : w n p 2 + z n q 2 x n p 2 + y n q 2 }, x n+ = P Cn+ x, y n+ = P Qn+ y, (.9)

4 258 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Motivated and inspired by the researches going on this direction, the purpose of this paper is to introduce an new iterative algorithm to solve split equality fixed point problem for quasi-asymptotically pseudo-contractive mappings in Hilbert spaces. And without assumption of semi-compactness on mappings, the strong convergence of the sequence generated by the proposed iterative scheme is obtained. The results presented in this paper improve and extend some recent corresponding results announced. 2 Preliminaries Throughout this paper, we use the notations to denote the strong convergence, and to denote the weak convergence. Let C be a closed convex subset of H. For every point x H, there exists a unique nearest point in C, denoted by P C x satisfying x P C x x y, y C. (2.) The operator P C is called the metric projection mapping of H onto C. The metric projection P C is characterized by the following inequalities: and y P C x, x P C x 0, x H, y C, (2.2) y P C x 2 + x P C x 2 x y 2, y C, x H. (2.3) In a real Hilbert space H, it is also well known that for any x, y H λx+( λ)y 2 = λ x 2 +( λ) y 2 λ( λ) x y 2, λ [0, ] (2.4) and 2 x, y = x 2 + y 2 x y 2. Definition 2.. An operator T : C C is said to be (I) pseudo-contractive if T x T y, x y x y 2, for all x, y C, or equivalently, T x T y 2 x y 2 + (I T )x (I T )y 2, x, y C. (II) quasi-pseudo-contractive if F (T ) and T x p 2 x p 2 + T x x 2, x, y C, p F (T ). (III) asymptotically pseudo-contractive if there exists a sequence {l n } [, + ) with lim l n = such that T n x T n y 2 l n x y 2 + (I T n )x (I T n )y 2, x, y C, for each n,

5 Strong convergence theorems 259 or equivalently, T n x T n y, x y l n + x y 2, x, y C, for each n. 2 (IV) quasi-asymptotically pseudo-contractive if F (T ) and there exists a sequence {l n } [, + ) with lim l n = such that T n x p 2 l n x p 2 + T n x x 2, x, y C, p F (T ), for each n, Definition 2.2. An operator T : C C is said to be uniformly L- Lipschitzian, if there exists some L > 0 such that T n x T n y L x y, x, y C, for each n. Definition 2.3. An operator T : C C is said to be demi-closed at zero, if for any sequence {x n } with x n x and lim x n T (x n ) = 0, then x = T x. Lemma 2.4. Let H be a real Hilbert space, T : H H be a uniformly L- Lipschitzian and {l n }-quasi-asymptotically pseudo-contractive mapping with L and {l n } [, ) and lim l n =. Let {K n : H H} be a sequence of mappings defined by: If 0 < a < ξ < η < b < conclusions hold: K n := ( ξ)i + ξt n (( η)i + ηt n ). (2.5) M+ + 2 and M = sup l n, then the following +L 4 2 n (M+) 2 () F (T ) = F (T n (( η)i + ηt n )) = F (K n ) for each n ; (2) If T is demi-closed at zero, then K is also demi-closed at zero; (3) For each n and x H, u F (T ) = F (K n ), K n x u k n x u, where k n = + ξ(l n )( + ηl n ), {k n } [, + ) and lim k n =. Proof. () If u F (T ), i.e., u = T u, we have T n (( η)i + ηt n )u = T n (( η)u + ηt n u) = T n u = u. This shows that u F (T n (( η)i + ηt n )). Conversely, if u F (T n (( η)i + ηt n )) for all n, i.e., u = T n (( η)i + ηt n )u. Put (( η)i + ηt n ))u = v, then T n v = u. Now we prove that u = v. In fact, we have u v = u (( η)i +ηt n ))u = η u T n u = η T n v T n u Lη u v.

6 260 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Since 0 < η < M+ + 2 (M+) 2 4 +L 2 < L, we have 0 < Lη <. Then we easily obtain u = v, i.e., u F (T ). This shows that F (T ) = F (T n (( η)i + ηt n )) for all n. It is obvious that u F (K n ) if and only if u F (T n (( η)i + ηt n )). Then the conclusion () is proved. (2) For any sequence {u n } satisfying u n u and u n K u n 0. We show that u F (K ). From conclusion (), it suffices to prove u F (T ). In fact, since T is L-Lipchitz, we have u n T u n u n T (( η)i + ηt )u n + T (( η)i + ηt )u n T u n ξ u n ( ξ)u n ξt (( η)i + ηt )u n + Lη u n T u n = ξ u n K u n + Lη u n T u n. Simplifying it, we have u n T u n ξ( Lη) u n K u n 0. Since T is demi-closed at 0, we have u F (T ) = F (K). The conclusion (2) is proved. (3) For all u F (T ), it follows from Definition 2.(IV) that T n (( η)i + ηt n )x u 2 l n ( η)x + ηt n x u 2 + T n (( η)i + ηt n )x (( η)i + ηt n )x 2 = l n ( η)(x u) + η(t n x u) 2 + T n (( η)i + ηt n )x (( η)i + ηt n )x 2. (2.6) Since T is L-Lipschitz, we have T n (( η)i+ηt n )x T n x L ( η)x+ηt n x x = Lη T n x x. (2.7) From (2.4) and (2.7) we have ( η)(x u) + η(t n x u) 2 = ( η) x u 2 + η T n x u 2 η( η) T n x x 2 ( η) x u 2 + η(l n x u 2 + T n x x 2 ) η( η) T n x x 2 = ( + η(l n )) x u 2 + η 2 T n x x 2. (2.8)

7 Strong convergence theorems 26 Using (2.4), we have T n (( η)i + ηt n )x (( η)i + ηt n )x 2 = ( η)(t n (( η)x + ηt n x) x) + η(t n (( η)x + ηt n x) T n x) 2 = ( η) T n (( η)x + ηt n x) x 2 + η T n (( η)x + ηt n x) T n x 2 η( η) T n x x 2 ( η) T n (( η)x + ηt n x) x 2 η( η η 2 L 2 ) T n x x 2. (2.9) Substituting (2.8) and (2.9) into (2.6), we obtain T n (( η)i + ηt n )x u 2 l n ( + η(l n )) x u 2 + l n η 2 T n x x 2 +( η) T n (( η)x + ηt n x) x 2 η( η η 2 L 2 ) T n x x 2 = l n ( + η(l n )) x u 2 + ( η) T n (( η)x + ηt n x) x 2 η( η η 2 L 2 l n η 2 ) T n x x 2. (2.0) Since η < get M+ + 2 (M+) 2 4 +L 2 we deduce η η 2 L 2 l n η 2 > 0. From (2.0) we T n (( η)i + ηt n )x u 2 l n ( + η(l n )) x u 2 Combine (2.4) and (2.) we have +( η) T n (( η)x + ηt n x) x 2. (2.) K n x u 2 = ( ξ)x + ξt n (( η)x + ηt n x) u 2 = ( ξ) x u 2 + ξ T n (( η)x + ηt n x) u 2 ξ( ξ) T n (( η)x + ηt n x) x 2 ( ξ) x u 2 + ξl n ( + η(l n )) x u 2 +(ξ( η) ξ( ξ)) T n (( η)x + ηt n x) x 2 = ( + ξ( + ηl n )(l n )) x u 2 ξ(η ξ) T n (( η)x + ηt n x) x 2. This together with ξ < η implies that K n x u 2 k n x u 2, x H, u F (K n ), n, where k n = + ξ(l n )( + ηl n ). In view of that {l n } [, + ) and lim l n =, we have {k n } [, + ) and lim k n =. The conclusion (3) is proved.

8 262 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin 3 Main Results Theorem 3.. Let H,H 2 and H 3 be three real Hilbert spaces, A : H H 3 and B : H 2 H 3 be two bounded linear operators with adjoints A and B, respectively. Let T : H H and S : H 2 H 2 be two uniformly L- Lipschitzian and {l n }-quasi-asymptotically pseudo-contractive mappings with F (T ) and F (S). For any given initial points x C = H, y Q = H 2, the sequence {(x n, y n )} is defined as follows: u n = x n γ n A (Ax n By n ), v n = y n + γ n B (Ax n By n ), z n = α n x n + ( α n )K n u n, s n = α n y n + ( α n )G n v n, C n+ Q n+ = {(x, y) C n Q n : z n x 2 + s n y 2 ( + ( α n )(kn 2 ))( x n x 2 + y n y 2 )}, x n+ = P Cn+ x, y n+ = P Qn+ y, (3.) where K n = ( ξ)i +ξt n (( η)i +ηt n ), G n = ( ξ)i +ξs n (( η)i +ηs n ), k n = + ξ(l n )( + ηl n ) and the following conditions are satisfied: () L, and n= (l n ) < ; (2) {α n } (0, ), lim inf α n > 0; (3) γ n (0, max( A 2, B 2 )) with lim inf γ n > 0, n ; (4) 0 < a < ξ < η < b < M+ + 2, M = sup l n, n. +L 4 2 n (M+) 2 If Ω = {(p, q) F (T ) F (S) such that Ap = Bq}, T and S are demiclosed at zero, then the sequences {(x n, y n )} converges strongly to a point (x, y ) Ω. Proof. Since k n = + ξ(l n )( + ηl n ), ξ <, {l n } [, ) and n= (l n ) <, we have n= (k n ) = n= ξ(l n )( + ηl n ) < n= (l n )(l n +) <. Further, due to {α n } (0, ), we also have α n )(k 2 n ) n= (k n )(k n + ) <. We shall divide the proof into five steps. Step. We show that C n Q n is closed and convex for each n. n= ( Putting ρ n = + ( α n )(k 2 n ). Since C = H and Q = H 2, so C Q is closed and convex. Suppose that C n Q n is closed and convex. For any (x, y) C n+ Q n+, we have z n x 2 + s n y 2 ( + ( α n )(k 2 n ))( x n x 2 + y n y 2 ),

9 Strong convergence theorems 263 which is equivalent to 2ρ n x n ρ n x 2z n + x, x ρ n x n 2 + ρ n y n y 2 z n 2. So, we know that C n+ is closed. Similarly,we can prove that Q n+ is closed. Therefore C n+ Q n+ is closed. Besides, it is easy to prove that C n+ Q n+ also is a convex. Therefore C n+ Q n+ is a closed and convex for any n. Step 2. We prove that Ω C n Q n for any n. For any given (p, q) Ω, then p F (T ), Q F (S) and Ap = Bq. It follows from (3.), we have u n p 2 = x n γ n A (Ax n By n ) p 2 = x n p 2 + γ n A (Ax n By n ) 2 2γ n x n p, A (Ax n By n ) x n p 2 + γ 2 n A 2 Ax n By n 2 2γ n x n p, A (Ax n By n ) = x n p 2 + γ 2 n A 2 Ax n By n 2 2γ n Ax n Ap, Ax n By n.(3.4) Similarly, from (3.), we have v n q 2 y n q 2 +γ 2 n B 2 Ax n By n 2 +2γ n By n Bq, Ax n By n. Adding up (3.4) and (3.5) and noting Ap = Bq, we have that (3.5) u n p 2 + v n q 2 x n p 2 + y n q 2 + γ 2 n( A 2 + B 2 ) Ax n By n 2 2γ n Ax n Ap By n + Bq, Ax n By n = x n p 2 + y n q 2 +γ n (γ n ( A 2 + B 2 ) 2) Ax n By n 2. (3.6) Since γ n (0, max( A 2, B 2 )), γ n A 2 < and γ n B 2 <. This implies that γ n ( A 2 + B 2 ) 2 < 0. Therefore (3.6) can be written as u n p 2 + v n q 2 x n p 2 + y n q 2. (3.7) According to condition (4) and Lemma 2.4, we know F (T ) = F (K n ), F (S) = F (G n ), n ; K n u n p k n u n p, G n v n q k n v n q, n. Then, it follows from (3.), condition (2) and Lemma 2.4, we obtain that z n p 2 = α n x n + ( α n )K n u n p 2 = α n x n p 2 + ( α n ) K n u n p 2 α n ( α n ) K n u n x n 2 α n x n p 2 + ( α n )k 2 n u n p 2 α n ( α n ) K n u n x n 2 α n x n p 2 + ( α n )k 2 n u n p 2. (3.8)

10 264 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Similarly, it follows from (3.), condition (2) and Lemma 2.4 that s n q 2 α n y n q 2 + ( α n )k 2 n v n q 2 α n ( α n ) G n v n y n 2 α n y n q 2 + ( α n )k 2 n v n q 2. (3.9) Adding up (3.8) and (3.9), we have z n p 2 + s n q 2 α n ( x n p 2 + y n q 2 ) Substituting (3.7) into (3.0), we can obtain that +( α n )k 2 n( u n p 2 + v n q 2 ). (3.0) z n p 2 + s n q 2 α n ( x n p 2 + y n q 2 ) +( α n )k 2 n( x n p 2 + y n q 2 ) = (α n + ( α n )k 2 n)( x n p 2 + y n q 2 ) = ( + ( α n )(k 2 n ))( x n p 2 + y n q 2 ). Therefore, we know that (p, q) C n Q n and Ω C n Q n for any n. Step 3. sequences. We show that the sequences {x n } and {y n } are two Cauchy Since Ω C n+ Q n+ C n Q n and (x n+, y n+ ) = (P Cn+ x, P Qn+ y ) C n+ Q n+ C n Q n, then for any n and (p, q) Ω, we have x n+ x p x, n, (3.) y n+ y q y, n. (3.2) Hence, {x n } and {y n } are bounded. For any n, by using (2.3), we have x n+ x n 2 + x x n 2 = x n+ P Cn x 2 + x P Cn x 2 x n+ x 2, and y n+ y n 2 + y y n 2 = y n+ P Qn y 2 + y P Qn y 2 y n+ y 2, which imply that { x n x } and { y n y } are nondecreasing. By virtue of the boundedness of {x n } and {y n }, lim x n x and lim y n y exist. For positive integers m, n with m n, from (x n, y n ) = (P Cn x, P Qn y ) C m Q m and (2.3), we have x m x n 2 + x x n 2 = x m P Cn x 2 + x P Cn x 2 x m x 2, (3.3) y m y n 2 + y y n 2 = y m P Qn y 2 + y P Qn y 2 y m y 2. (3.4)

11 Strong convergence theorems 265 Since lim x n x and lim y n y exist, it follows from (3.3) and (3.4) that x m x n 2 x m x 2 x x n 2 0, y m y n 2 y m y 2 y y n 2 0. Therefore {x n } and {y n } are two Cauchy sequences. Step 4. We show that lim Ax n By n = 0, lim K n u n x n = 0 and lim G nv n y n = 0. Since (x n+, y n+ ) = (P Cn+ x, P Qn+ y ) C n+ Q n+ C n Q n, we have z n x n 2 + s n y n 2 ( z n x n+ + x n+ x n ) 2 +( s n y n+ + y n+ y n ) 2 2 z n x n x n+ x n 2 +2 s n y n y n+ y n 2 (2 + 2ρ n )( x n+ x n 2 + y n+ y n 2 ) 0. So we know that lim z n x n = 0 and lim s n y n = 0. Again since {x n } and {y n } are bounded, we know that {z n } and {s n } are bounded. From (3.6), (3.8) and (3.9), we have z n p 2 + s n q 2 α n x n p 2 + ( α n )kn u 2 n p 2 α n ( α n ) K n u n x n 2 + α n y n q 2 +( α n )kn v 2 n q 2 α n ( α n ) G n v n y n 2 = α n ( x n p 2 + y n q 2 ) +( α n )kn( u 2 n p 2 + v n q 2 ) α n ( α n )( K n u n x n 2 + G n v n y n 2 ) (α n + ( α n )kn)( x 2 n p 2 + y n q 2 ) +( α n )knγ 2 n (γ n ( A 2 + B 2 ) 2) Ax n By n 2 α n ( α n )( K n u n x n 2 + G n v n y n 2 ) = ( + ( α n )(kn 2 ))( x n p 2 + y n q 2 ) +( α n )knγ 2 n (γ n ( A 2 + B 2 ) 2) Ax n By n 2 α n ( α n )( K n u n x n 2 + G n v n y n 2 ). (3.5)

12 266 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin It follows from (3.5) that ( α n )k 2 nγ n (2 γ n ( A 2 + B 2 )) Ax n By n 2 +α n ( α n )( K n u n x n 2 + G n v n y n 2 ) ( + ( α n )(k 2 n ))( x n p 2 + y n q 2 ) z n p 2 s n q 2 = ( α n )(k 2 n )( x n p 2 + y n q 2 ) + x n p 2 + y n q 2 z n p 2 s n q 2 = ( α n )(k 2 n )( x n p 2 + y n q 2 ) +( x n p + z n p )( x n p z n p ) +( y n q + s n q )( y n q s n q ) ( α n )(k 2 n )( x n p 2 + y n q 2 ) + ( x n p + z n p )( x n z n ) +( y n q + s n q )( y n s n ). Since lim k n = and {α n } (0, ), we have lim ( α n )(kn 2 ) = 0. By virtue of the boundedness of {z n }, {s n }, {x n } and {y n }, lim z n x n = 0 and lim s n y n = 0, we get ( α n )(kn 2 )( x n p 2 + y n q 2 )+( x n p + z n p )( x n z n )+( y n q + s n q )( y n s n ) 0. In addition, it follows from Conditions (2) and (3) that ( α n )knγ 2 n (2 γ n ( A 2 + B 2 )) > 0 and α n ( α n ) > 0. Thus we may get lim Ax n By n = 0; (3.6) lim K nu n x n = 0, for each n =, 2, 3, ; (3.7) lim G nv n y n = 0, for each n =, 2, 3,. (3.8) Step 5. We show that {(x n, y n )} converges strongly to an element of Ω. By (3.6), we have u n x n + v n y n = γ n A (Ax n By n ) + γ n B (Ax n By n ) So, we know γ n ( A + B ) Ax n By n 0. lim u n x n = 0, lim v n y n = 0. (3.9) From (3.7), (3.8) and (3.9), we have K n u n u n K n u n x n + u n x n 0; (3.20) G n v n v n G n v n y n + v n y n 0. (3.2) Since {x n } and {y n } are two Cauchy sequences, there exists x H and y H 2 such that x n x and y n y. From (3.9) we also have u n x

13 Strong convergence theorems 267 and v n y. So it follows from (3.20), (2.2) and Lemma 2.4 that x F (K ) = F (T ) and y F (G ) = F (S). On the other hand, since A and B are two bounded linear operators, we have that Ax n By n Ax By. By using the weakly lower semi-continuity of squared norm, we have Ax By 2 lim inf Ax n By n 2 = lim Ax n By n 2 = 0, thus Ax = By. Therefore, {(x n, y n )} converges strongly to (x, y ) Ω. completed. The proof is Remark 3.2. Since a quasi-pseudo-contractive mapping is a quasi-aymptotically pseudo-contractive mapping, the Theorem 3. extends the main results in [7] and [9] from quasi-pseudo-contractive mappings to quasi-aymptotically pseudo-contractive mappings. The following corollary may be directly concluded from Theorem 3.. Corollary 3.3. Let H,H 2 and H 3 be three real Hilbert spaces, A : H H 3 and B : H 2 H 3 be two bounded linear operators with their adjoints A and B, respectively. Let T : H H and S : H 2 H 2 be two uniformly L- Lipschitzian and quasi-pseudo-contractive mappings, F (T ), and F (S). For given initial value x C = H, y Q = H 2, and let {(x n, y n )} be defined as follows: u n = x n γ n A (Ax n By n ), v n = y n + γ n B (Ax n By n ), z n = α n x n + ( α n )Ku n, s n = α n y n + ( α n )Gv n, (3.23) C n+ Q n+ = {(x, y) C n Q n : z n x 2 + s n y 2 x n x 2 + y n y 2 }, x n+ = P Cn+ x, y n+ = P Qn+ y, where K = ( ξ)i + ξt (( η)i + ηt ), G = ( ξ)i + ξs(( η)i + ηs), and the following conditions are satisfied: () α n (0, ), lim inf α n > 0, L ; (2) γ n (0, max( A 2, B 2 )), lim inf γ n > 0; (3) 0 < a < ξ < η < b < + +L 2. If Ω = {(p, q) F (T ) F (S) such that Ap = Bq}, T and S are demiclosed at zero, then the sequences {(x n, y n )} converges strongly to a point (x, y ) Ω.

14 268 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin Acknowledgements. This work was supported by the Scientific Research Foundation of Postgraduate of Yunnan University of Finance and Economics (207YUFEYC035) and the Scientific Research Foundation of Yunnan Province Education Department (204C00Y). References [] Y. Censor, T. Elfving, A mulitprojection algorithm using Bregman projections in a product space, Number. Algorithms, 8 (994), [2] C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Problems, 8 (2002), [3] Y. Censor, T. Bortfeld, N. Martin, A. Trofimov, A unified approach for inversion problem in intensity-modulated radiation therapy, Phys. Med. Biol., 5 (2006), [4] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The mulitple-sets split feasibility problem and its applocations for inverse problems, Inverse Problems, 2 (2005), [5] Y. Censor, A. Motova, A. Segal, Perturbed projections and subgradient projections for the mulit-sets split feasibility problem, J. Math. Anal. Appl., 327 (2007), [6] Y. Censor, A. Segal, The split common fixed point problem for directed operators, J. Convex Anal., 6 (2009), [7] A. Moudafi, A relaxed alternating CQ-algorithm for convex feasibility problems, Nonlinear Analysis: Theory, Methods and Applications, 79 (203), [8] A. Moudafi, Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J. Nonlinear Convex Anal., 5 (204), no. 4, [9] A. Moudafi, E. Al-Shemas, Simultaneous iterative methods for split equality problems and applications, Trans. Math. Program. Appl., (203), -.

15 Strong convergence theorems 269 [0] J. Zhao, S. He, Alternating mann iterative algorithms for the split common fixed point problem of quasi-nonexpansive mappings, Fixed Point Theory Appl., 203 (203), [] Z.L. Ma, W. Duan, R.J. Liu, Split Equality Fixed Point Problem for Strictly Pseudocontractive Mappings, International Mathematical Forum, 9 (204), no. 35, [2] X.J. Zi, Z.L. Ma, L. Wang, Y.F. Ma, Split Equality Fixed Point Problem for Asymptotically Nonexpansive Mappings, Applied Mathematical Sciences, 8 (2040, no. 30, [3] C.E. Chidume, P. Ndambomve, A.U. Bello, The split equality fixed point problem for demi-contractive mappings, Journal of Nonlinear Analysis and Optimization, 6 (205), no., [4] J. Zhao, S.N. Wang, Viscosity approximation methods for the split equality common fixed point problem of quasi-nonexpansive operators, Acta Mathematica Scientia, 36B (206), no. 5, [5] C.E. Chidume, P. Ndambomve, A.U. Bello, M.E. Okpala, The Multiplesets Split Equality Fixed Point Problem for Countable Families of Multivalued Demi-contractive Mappings, International Journal of Mathematical Analysis, 9 (205), no. 0, [6] H.T. Che, M.X. Li, A simultaneous iterative method for split equality problems of two finite families of strictly pseudononspreading mappings without prior knowledge of operator norms, Fixed Point Theory and Applications, 205, (205), no.. [7] S.S. Chang, L. Wang, L.J. Qin, Split equality fixed point problem for quasi-pseudo-contractive mappings with applications, Fixed Point Theory Appl., 205 (205), no.. [8] X.F. Zhang, L.Wang, Z.L. Ma, L.J.Qin, The strong convergence theorems for split common fixed point problem ofasymptotically nonexpansive mappings in Hilbert spaces, Journal of Inequalities and Applications, 205 (205),. [9] J.F. Tang, S.S. Chang, C.F. Wen, J. Dong, Hybrid projection algorithm concerning split equality fixed point problem for quasi-pseudo-contractive

16 270 Jun Niu, Zheng Zhou, Jian-Qiang Zhang and Li-Juan Qin mappings with application to optimization problem, J. Nonlinear Sci. Appl., 9 (206), Received: February 3, 208; Published: February 5, 208

The Split Common Fixed Point Problem for Asymptotically Quasi-Nonexpansive Mappings in the Intermediate Sense

The Split Common Fixed Point Problem for Asymptotically Quasi-Nonexpansive Mappings in the Intermediate Sense International Mathematical Forum, Vol. 8, 2013, no. 25, 1233-1241 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2013.3599 The Split Common Fixed Point Problem for Asymptotically Quasi-Nonexpansive

More information

On the split equality common fixed point problem for quasi-nonexpansive multi-valued mappings in Banach spaces

On the split equality common fixed point problem for quasi-nonexpansive multi-valued mappings in Banach spaces Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (06), 5536 5543 Research Article On the split equality common fixed point problem for quasi-nonexpansive multi-valued mappings in Banach spaces

More information

The Split Hierarchical Monotone Variational Inclusions Problems and Fixed Point Problems for Nonexpansive Semigroup

The Split Hierarchical Monotone Variational Inclusions Problems and Fixed Point Problems for Nonexpansive Semigroup International Mathematical Forum, Vol. 11, 2016, no. 8, 395-408 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6220 The Split Hierarchical Monotone Variational Inclusions Problems and

More information

The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces Applied Mathematical Sciences, Vol. 11, 2017, no. 12, 549-560 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.718 The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive

More information

Iterative algorithms based on the hybrid steepest descent method for the split feasibility problem

Iterative algorithms based on the hybrid steepest descent method for the split feasibility problem Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (206), 424 4225 Research Article Iterative algorithms based on the hybrid steepest descent method for the split feasibility problem Jong Soo

More information

Strong Convergence Theorem of Split Equality Fixed Point for Nonexpansive Mappings in Banach Spaces

Strong Convergence Theorem of Split Equality Fixed Point for Nonexpansive Mappings in Banach Spaces Applied Mathematical Sciences, Vol. 12, 2018, no. 16, 739-758 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ams.2018.8580 Strong Convergence Theorem of Split Euality Fixed Point for Nonexpansive

More information

Research Article Some Krasnonsel skiĭ-mann Algorithms and the Multiple-Set Split Feasibility Problem

Research Article Some Krasnonsel skiĭ-mann Algorithms and the Multiple-Set Split Feasibility Problem Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 513956, 12 pages doi:10.1155/2010/513956 Research Article Some Krasnonsel skiĭ-mann Algorithms and the Multiple-Set

More information

New Iterative Algorithm for Variational Inequality Problem and Fixed Point Problem in Hilbert Spaces

New Iterative Algorithm for Variational Inequality Problem and Fixed Point Problem in Hilbert Spaces Int. Journal of Math. Analysis, Vol. 8, 2014, no. 20, 995-1003 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4392 New Iterative Algorithm for Variational Inequality Problem and Fixed

More information

Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou

Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou J. Korean Math. Soc. 38 (2001), No. 6, pp. 1245 1260 DEMI-CLOSED PRINCIPLE AND WEAK CONVERGENCE PROBLEMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS Shih-sen Chang, Yeol Je Cho, and Haiyun Zhou Abstract.

More information

Iterative common solutions of fixed point and variational inequality problems

Iterative common solutions of fixed point and variational inequality problems Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 1882 1890 Research Article Iterative common solutions of fixed point and variational inequality problems Yunpeng Zhang a, Qing Yuan b,

More information

A NEW ITERATIVE METHOD FOR THE SPLIT COMMON FIXED POINT PROBLEM IN HILBERT SPACES. Fenghui Wang

A NEW ITERATIVE METHOD FOR THE SPLIT COMMON FIXED POINT PROBLEM IN HILBERT SPACES. Fenghui Wang A NEW ITERATIVE METHOD FOR THE SPLIT COMMON FIXED POINT PROBLEM IN HILBERT SPACES Fenghui Wang Department of Mathematics, Luoyang Normal University, Luoyang 470, P.R. China E-mail: wfenghui@63.com ABSTRACT.

More information

GENERAL NONCONVEX SPLIT VARIATIONAL INEQUALITY PROBLEMS. Jong Kyu Kim, Salahuddin, and Won Hee Lim

GENERAL NONCONVEX SPLIT VARIATIONAL INEQUALITY PROBLEMS. Jong Kyu Kim, Salahuddin, and Won Hee Lim Korean J. Math. 25 (2017), No. 4, pp. 469 481 https://doi.org/10.11568/kjm.2017.25.4.469 GENERAL NONCONVEX SPLIT VARIATIONAL INEQUALITY PROBLEMS Jong Kyu Kim, Salahuddin, and Won Hee Lim Abstract. In this

More information

Strong Convergence Theorems for Nonself I-Asymptotically Quasi-Nonexpansive Mappings 1

Strong Convergence Theorems for Nonself I-Asymptotically Quasi-Nonexpansive Mappings 1 Applied Mathematical Sciences, Vol. 2, 2008, no. 19, 919-928 Strong Convergence Theorems for Nonself I-Asymptotically Quasi-Nonexpansive Mappings 1 Si-Sheng Yao Department of Mathematics, Kunming Teachers

More information

A Note of the Strong Convergence of the Mann Iteration for Demicontractive Mappings

A Note of the Strong Convergence of the Mann Iteration for Demicontractive Mappings Applied Mathematical Sciences, Vol. 10, 2016, no. 6, 255-261 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.511700 A Note of the Strong Convergence of the Mann Iteration for Demicontractive

More information

Strong convergence theorems for total quasi-ϕasymptotically

Strong convergence theorems for total quasi-ϕasymptotically RESEARCH Open Access Strong convergence theorems for total quasi-ϕasymptotically nonexpansive multi-valued mappings in Banach spaces Jinfang Tang 1 and Shih-sen Chang 2* * Correspondence: changss@yahoo.

More information

A generalized forward-backward method for solving split equality quasi inclusion problems in Banach spaces

A generalized forward-backward method for solving split equality quasi inclusion problems in Banach spaces Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 4890 4900 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa A generalized forward-backward

More information

Strong Convergence of the Mann Iteration for Demicontractive Mappings

Strong Convergence of the Mann Iteration for Demicontractive Mappings Applied Mathematical Sciences, Vol. 9, 015, no. 4, 061-068 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.015.5166 Strong Convergence of the Mann Iteration for Demicontractive Mappings Ştefan

More information

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2009, Article ID 728510, 14 pages doi:10.1155/2009/728510 Research Article Common Fixed Points of Multistep Noor Iterations with Errors

More information

Research Article Generalized Mann Iterations for Approximating Fixed Points of a Family of Hemicontractions

Research Article Generalized Mann Iterations for Approximating Fixed Points of a Family of Hemicontractions Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 008, Article ID 84607, 9 pages doi:10.1155/008/84607 Research Article Generalized Mann Iterations for Approximating Fixed Points

More information

Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces

Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 016, 4478 4488 Research Article Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert

More information

Research Article An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem

Research Article An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem Abstract and Applied Analysis, Article ID 60813, 5 pages http://dx.doi.org/10.1155/014/60813 Research Article An Iterative Algorithm for the Split Equality and Multiple-Sets Split Equality Problem Luoyi

More information

Research Article Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive Mappings

Research Article Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive Mappings Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 428241, 11 pages doi:10.1155/2008/428241 Research Article Convergence Theorems for Common Fixed Points of Nonself

More information

INERTIAL ACCELERATED ALGORITHMS FOR SOLVING SPLIT FEASIBILITY PROBLEMS. Yazheng Dang. Jie Sun. Honglei Xu

INERTIAL ACCELERATED ALGORITHMS FOR SOLVING SPLIT FEASIBILITY PROBLEMS. Yazheng Dang. Jie Sun. Honglei Xu Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 200X doi:10.3934/xx.xx.xx.xx pp. X XX INERTIAL ACCELERATED ALGORITHMS FOR SOLVING SPLIT FEASIBILITY PROBLEMS Yazheng Dang School of Management

More information

A general iterative algorithm for equilibrium problems and strict pseudo-contractions in Hilbert spaces

A general iterative algorithm for equilibrium problems and strict pseudo-contractions in Hilbert spaces A general iterative algorithm for equilibrium problems and strict pseudo-contractions in Hilbert spaces MING TIAN College of Science Civil Aviation University of China Tianjin 300300, China P. R. CHINA

More information

Monotone variational inequalities, generalized equilibrium problems and fixed point methods

Monotone variational inequalities, generalized equilibrium problems and fixed point methods Wang Fixed Point Theory and Applications 2014, 2014:236 R E S E A R C H Open Access Monotone variational inequalities, generalized equilibrium problems and fixed point methods Shenghua Wang * * Correspondence:

More information

Research Article Some Results on Strictly Pseudocontractive Nonself-Mappings and Equilibrium Problems in Hilbert Spaces

Research Article Some Results on Strictly Pseudocontractive Nonself-Mappings and Equilibrium Problems in Hilbert Spaces Abstract and Applied Analysis Volume 2012, Article ID 543040, 14 pages doi:10.1155/2012/543040 Research Article Some Results on Strictly Pseudocontractive Nonself-Mappings and Equilibrium Problems in Hilbert

More information

CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES

CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES International Journal of Analysis and Applications ISSN 2291-8639 Volume 8, Number 1 2015), 69-78 http://www.etamaths.com CONVERGENCE OF HYBRID FIXED POINT FOR A PAIR OF NONLINEAR MAPPINGS IN BANACH SPACES

More information

PROXIMAL POINT ALGORITHMS INVOLVING FIXED POINT OF NONSPREADING-TYPE MULTIVALUED MAPPINGS IN HILBERT SPACES

PROXIMAL POINT ALGORITHMS INVOLVING FIXED POINT OF NONSPREADING-TYPE MULTIVALUED MAPPINGS IN HILBERT SPACES PROXIMAL POINT ALGORITHMS INVOLVING FIXED POINT OF NONSPREADING-TYPE MULTIVALUED MAPPINGS IN HILBERT SPACES Shih-sen Chang 1, Ding Ping Wu 2, Lin Wang 3,, Gang Wang 3 1 Center for General Educatin, China

More information

Research Article Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

Research Article Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space Journal of Applied Mathematics Volume 2012, Article ID 435676, 15 pages doi:10.1155/2012/435676 Research Article Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space Bin-Chao Deng,

More information

On nonexpansive and accretive operators in Banach spaces

On nonexpansive and accretive operators in Banach spaces Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 3437 3446 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa On nonexpansive and accretive

More information

The Journal of Nonlinear Science and Applications

The Journal of Nonlinear Science and Applications J. Nonlinear Sci. Appl. 2 (2009), no. 2, 78 91 The Journal of Nonlinear Science and Applications http://www.tjnsa.com STRONG CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS AND FIXED POINT PROBLEMS OF STRICT

More information

STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS

STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS ARCHIVUM MATHEMATICUM (BRNO) Tomus 45 (2009), 147 158 STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS Xiaolong Qin 1, Shin Min Kang 1, Yongfu Su 2,

More information

Two-Step Iteration Scheme for Nonexpansive Mappings in Banach Space

Two-Step Iteration Scheme for Nonexpansive Mappings in Banach Space Mathematica Moravica Vol. 19-1 (2015), 95 105 Two-Step Iteration Scheme for Nonexpansive Mappings in Banach Space M.R. Yadav Abstract. In this paper, we introduce a new two-step iteration process to approximate

More information

STRONG CONVERGENCE OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS IN BANACH SPACES

STRONG CONVERGENCE OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS IN BANACH SPACES Kragujevac Journal of Mathematics Volume 36 Number 2 (2012), Pages 237 249. STRONG CONVERGENCE OF AN IMPLICIT ITERATION PROCESS FOR ASYMPTOTICALLY NONEXPANSIVE IN THE INTERMEDIATE SENSE MAPPINGS IN BANACH

More information

CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES. Gurucharan Singh Saluja

CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES. Gurucharan Singh Saluja Opuscula Mathematica Vol 30 No 4 2010 http://dxdoiorg/107494/opmath2010304485 CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES Gurucharan Singh Saluja Abstract

More information

On the Convergence of Ishikawa Iterates to a Common Fixed Point for a Pair of Nonexpansive Mappings in Banach Spaces

On the Convergence of Ishikawa Iterates to a Common Fixed Point for a Pair of Nonexpansive Mappings in Banach Spaces Mathematica Moravica Vol. 14-1 (2010), 113 119 On the Convergence of Ishikawa Iterates to a Common Fixed Point for a Pair of Nonexpansive Mappings in Banach Spaces Amit Singh and R.C. Dimri Abstract. In

More information

Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings

Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings Mathematica Moravica Vol. 20:1 (2016), 125 144 Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings G.S. Saluja Abstract. The aim of

More information

THROUGHOUT this paper, we let C be a nonempty

THROUGHOUT this paper, we let C be a nonempty Strong Convergence Theorems of Multivalued Nonexpansive Mappings and Maximal Monotone Operators in Banach Spaces Kriengsak Wattanawitoon, Uamporn Witthayarat and Poom Kumam Abstract In this paper, we prove

More information

Synchronal Algorithm For a Countable Family of Strict Pseudocontractions in q-uniformly Smooth Banach Spaces

Synchronal Algorithm For a Countable Family of Strict Pseudocontractions in q-uniformly Smooth Banach Spaces Int. Journal of Math. Analysis, Vol. 8, 2014, no. 15, 727-745 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.212287 Synchronal Algorithm For a Countable Family of Strict Pseudocontractions

More information

Convergence Theorems of Approximate Proximal Point Algorithm for Zeroes of Maximal Monotone Operators in Hilbert Spaces 1

Convergence Theorems of Approximate Proximal Point Algorithm for Zeroes of Maximal Monotone Operators in Hilbert Spaces 1 Int. Journal of Math. Analysis, Vol. 1, 2007, no. 4, 175-186 Convergence Theorems of Approximate Proximal Point Algorithm for Zeroes of Maximal Monotone Operators in Hilbert Spaces 1 Haiyun Zhou Institute

More information

Existence and convergence theorems for the split quasi variational inequality problems on proximally smooth sets

Existence and convergence theorems for the split quasi variational inequality problems on proximally smooth sets Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (206), 2364 2375 Research Article Existence and convergence theorems for the split quasi variational inequality problems on proximally smooth

More information

Research Article Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem

Research Article Self-Adaptive and Relaxed Self-Adaptive Projection Methods for Solving the Multiple-Set Split Feasibility Problem Abstract and Applied Analysis Volume 01, Article ID 958040, 11 pages doi:10.1155/01/958040 Research Article Self-Adaptive and Relaxed Self-Adaptive Proection Methods for Solving the Multiple-Set Split

More information

Strong Convergence Theorem by a Hybrid Extragradient-like Approximation Method for Variational Inequalities and Fixed Point Problems

Strong Convergence Theorem by a Hybrid Extragradient-like Approximation Method for Variational Inequalities and Fixed Point Problems Strong Convergence Theorem by a Hybrid Extragradient-like Approximation Method for Variational Inequalities and Fixed Point Problems Lu-Chuan Ceng 1, Nicolas Hadjisavvas 2 and Ngai-Ching Wong 3 Abstract.

More information

Weak and strong convergence of a scheme with errors for three nonexpansive mappings

Weak and strong convergence of a scheme with errors for three nonexpansive mappings Rostock. Math. Kolloq. 63, 25 35 (2008) Subject Classification (AMS) 47H09, 47H10 Daruni Boonchari, Satit Saejung Weak and strong convergence of a scheme with errors for three nonexpansive mappings ABSTRACT.

More information

Existence of Solutions to Split Variational Inequality Problems and Split Minimization Problems in Banach Spaces

Existence of Solutions to Split Variational Inequality Problems and Split Minimization Problems in Banach Spaces Existence of Solutions to Split Variational Inequality Problems and Split Minimization Problems in Banach Spaces Jinlu Li Department of Mathematical Sciences Shawnee State University Portsmouth, Ohio 45662

More information

Research Article Iterative Approximation of a Common Zero of a Countably Infinite Family of m-accretive Operators in Banach Spaces

Research Article Iterative Approximation of a Common Zero of a Countably Infinite Family of m-accretive Operators in Banach Spaces Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 325792, 13 pages doi:10.1155/2008/325792 Research Article Iterative Approximation of a Common Zero of a Countably

More information

Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces

Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces Viscosity Iterative Approximating the Common Fixed Points of Non-expansive Semigroups in Banach Spaces YUAN-HENG WANG Zhejiang Normal University Department of Mathematics Yingbing Road 688, 321004 Jinhua

More information

Research Article Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and Applications

Research Article Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and Applications Abstract and Applied Analysis Volume 2012, Article ID 479438, 13 pages doi:10.1155/2012/479438 Research Article Hybrid Algorithm of Fixed Point for Weak Relatively Nonexpansive Multivalued Mappings and

More information

Research Article Algorithms for a System of General Variational Inequalities in Banach Spaces

Research Article Algorithms for a System of General Variational Inequalities in Banach Spaces Journal of Applied Mathematics Volume 2012, Article ID 580158, 18 pages doi:10.1155/2012/580158 Research Article Algorithms for a System of General Variational Inequalities in Banach Spaces Jin-Hua Zhu,

More information

Strong Convergence of an Algorithm about Strongly Quasi- Nonexpansive Mappings for the Split Common Fixed-Point Problem in Hilbert Space

Strong Convergence of an Algorithm about Strongly Quasi- Nonexpansive Mappings for the Split Common Fixed-Point Problem in Hilbert Space Strong Convergence of an Algorithm about Strongly Quasi- Nonexpansive Mappings for the Split Common Fixed-Point Problem in Hilbert Space Lawan Bulama Mohammed 1*, Abba Auwalu 2 and Salihu Afis 3 1. Department

More information

A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces

A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces Bin Dehaish et al. Journal of Inequalities and Applications (2015) 2015:51 DOI 10.1186/s13660-014-0541-z R E S E A R C H Open Access A regularization projection algorithm for various problems with nonlinear

More information

Convergence theorems for mixed type asymptotically nonexpansive mappings in the intermediate sense

Convergence theorems for mixed type asymptotically nonexpansive mappings in the intermediate sense Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 2016), 5119 5135 Research Article Convergence theorems for mixed type asymptotically nonexpansive mappings in the intermediate sense Gurucharan

More information

A New Modified Gradient-Projection Algorithm for Solution of Constrained Convex Minimization Problem in Hilbert Spaces

A New Modified Gradient-Projection Algorithm for Solution of Constrained Convex Minimization Problem in Hilbert Spaces A New Modified Gradient-Projection Algorithm for Solution of Constrained Convex Minimization Problem in Hilbert Spaces Cyril Dennis Enyi and Mukiawa Edwin Soh Abstract In this paper, we present a new iterative

More information

Iterative Convex Optimization Algorithms; Part One: Using the Baillon Haddad Theorem

Iterative Convex Optimization Algorithms; Part One: Using the Baillon Haddad Theorem Iterative Convex Optimization Algorithms; Part One: Using the Baillon Haddad Theorem Charles Byrne (Charles Byrne@uml.edu) http://faculty.uml.edu/cbyrne/cbyrne.html Department of Mathematical Sciences

More information

Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces

Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces Mathematica Moravica Vol. 19-1 2015, 33 48 Convergence to Common Fixed Point for Two Asymptotically Quasi-nonexpansive Mappings in the Intermediate Sense in Banach Spaces Gurucharan Singh Saluja Abstract.

More information

Research Article Strong Convergence of a Projected Gradient Method

Research Article Strong Convergence of a Projected Gradient Method Applied Mathematics Volume 2012, Article ID 410137, 10 pages doi:10.1155/2012/410137 Research Article Strong Convergence of a Projected Gradient Method Shunhou Fan and Yonghong Yao Department of Mathematics,

More information

Remark on a Couple Coincidence Point in Cone Normed Spaces

Remark on a Couple Coincidence Point in Cone Normed Spaces International Journal of Mathematical Analysis Vol. 8, 2014, no. 50, 2461-2468 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.49293 Remark on a Couple Coincidence Point in Cone Normed

More information

Convergence theorems for a finite family. of nonspreading and nonexpansive. multivalued mappings and equilibrium. problems with application

Convergence theorems for a finite family. of nonspreading and nonexpansive. multivalued mappings and equilibrium. problems with application Theoretical Mathematics & Applications, vol.3, no.3, 2013, 49-61 ISSN: 1792-9687 (print), 1792-9709 (online) Scienpress Ltd, 2013 Convergence theorems for a finite family of nonspreading and nonexpansive

More information

Research Article Convergence Theorems for Infinite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces

Research Article Convergence Theorems for Infinite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly Convex Banach Spaces Abstract and Applied Analysis Volume 2012, Article ID 435790, 6 pages doi:10.1155/2012/435790 Research Article Convergence Theorems for Infinite Family of Multivalued Quasi-Nonexpansive Mappings in Uniformly

More information

Split equality problem with equilibrium problem, variational inequality problem, and fixed point problem of nonexpansive semigroups

Split equality problem with equilibrium problem, variational inequality problem, and fixed point problem of nonexpansive semigroups Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 3217 3230 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Split equality problem with

More information

Zeqing Liu, Jeong Sheok Ume and Shin Min Kang

Zeqing Liu, Jeong Sheok Ume and Shin Min Kang Bull. Korean Math. Soc. 41 (2004), No. 2, pp. 241 256 GENERAL VARIATIONAL INCLUSIONS AND GENERAL RESOLVENT EQUATIONS Zeqing Liu, Jeong Sheok Ume and Shin Min Kang Abstract. In this paper, we introduce

More information

Academic Editor: Hari M. Srivastava Received: 29 September 2016; Accepted: 6 February 2017; Published: 11 February 2017

Academic Editor: Hari M. Srivastava Received: 29 September 2016; Accepted: 6 February 2017; Published: 11 February 2017 mathematics Article The Split Common Fixed Point Problem for a Family of Multivalued Quasinonexpansive Mappings and Totally Asymptotically Strictly Pseudocontractive Mappings in Banach Spaces Ali Abkar,

More information

A General Iterative Method for Constrained Convex Minimization Problems in Hilbert Spaces

A General Iterative Method for Constrained Convex Minimization Problems in Hilbert Spaces A General Iterative Method for Constrained Convex Minimization Problems in Hilbert Spaces MING TIAN Civil Aviation University of China College of Science Tianjin 300300 CHINA tianming963@6.com MINMIN LI

More information

Research Article Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings

Research Article Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings Discrete Dynamics in Nature and Society Volume 2011, Article ID 487864, 16 pages doi:10.1155/2011/487864 Research Article Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive

More information

Strong Convergence of Two Iterative Algorithms for a Countable Family of Nonexpansive Mappings in Hilbert Spaces

Strong Convergence of Two Iterative Algorithms for a Countable Family of Nonexpansive Mappings in Hilbert Spaces International Mathematical Forum, 5, 2010, no. 44, 2165-2172 Strong Convergence of Two Iterative Algorithms for a Countable Family of Nonexpansive Mappings in Hilbert Spaces Jintana Joomwong Division of

More information

Split hierarchical variational inequality problems and fixed point problems for nonexpansive mappings

Split hierarchical variational inequality problems and fixed point problems for nonexpansive mappings Ansari et al. Journal of Inequalities and Applications (015) 015:74 DOI 10.1186/s13660-015-0793- R E S E A R C H Open Access Split hierarchical variational inequality problems and fixed point problems

More information

Strong convergence of multi-step iterates with errors for generalized asymptotically quasi-nonexpansive mappings

Strong convergence of multi-step iterates with errors for generalized asymptotically quasi-nonexpansive mappings Palestine Journal of Mathematics Vol. 1 01, 50 64 Palestine Polytechnic University-PPU 01 Strong convergence of multi-step iterates with errors for generalized asymptotically quasi-nonexpansive mappings

More information

Common Fixed Point Theorems for Ćirić-Berinde Type Hybrid Contractions

Common Fixed Point Theorems for Ćirić-Berinde Type Hybrid Contractions International Journal of Mathematical Analysis Vol. 9, 2015, no. 31, 1545-1561 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.54125 Common Fixed Point Theorems for Ćirić-Berinde Type

More information

A Relaxed Explicit Extragradient-Like Method for Solving Generalized Mixed Equilibria, Variational Inequalities and Constrained Convex Minimization

A Relaxed Explicit Extragradient-Like Method for Solving Generalized Mixed Equilibria, Variational Inequalities and Constrained Convex Minimization , March 16-18, 2016, Hong Kong A Relaxed Explicit Extragradient-Like Method for Solving Generalized Mixed Equilibria, Variational Inequalities and Constrained Convex Minimization Yung-Yih Lur, Lu-Chuan

More information

WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES

WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES Fixed Point Theory, 12(2011), No. 2, 309-320 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES S. DHOMPONGSA,

More information

ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES

ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 3, 2018 ISSN 1223-7027 ON A HYBRID PROXIMAL POINT ALGORITHM IN BANACH SPACES Vahid Dadashi 1 In this paper, we introduce a hybrid projection algorithm for a countable

More information

Weak and Strong Convergence Theorems for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Nonself-Mappings

Weak and Strong Convergence Theorems for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Nonself-Mappings Int. J. Nonlinear Anal. Appl. 3 (2012) No. 1, 9-16 ISSN: 2008-6822 (electronic) http://www.ijnaa.semnan.ac.ir Weak and Strong Convergence Theorems for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive

More information

Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces

Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces An. Şt. Univ. Ovidius Constanţa Vol. 19(1), 211, 331 346 Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces Yonghong Yao, Yeong-Cheng Liou Abstract

More information

Fixed Points of Multivalued Quasi-nonexpansive Mappings Using a Faster Iterative Process

Fixed Points of Multivalued Quasi-nonexpansive Mappings Using a Faster Iterative Process Fixed Points of Multivalued Quasi-nonexpansive Mappings Using a Faster Iterative Process Safeer Hussain KHAN Department of Mathematics, Statistics and Physics, Qatar University, Doha 73, Qatar E-mail :

More information

Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces

Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (6), 37 378 Research Article Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups

More information

Viscosity approximation method for m-accretive mapping and variational inequality in Banach space

Viscosity approximation method for m-accretive mapping and variational inequality in Banach space An. Şt. Univ. Ovidius Constanţa Vol. 17(1), 2009, 91 104 Viscosity approximation method for m-accretive mapping and variational inequality in Banach space Zhenhua He 1, Deifei Zhang 1, Feng Gu 2 Abstract

More information

Fixed point theory for nonlinear mappings in Banach spaces and applications

Fixed point theory for nonlinear mappings in Banach spaces and applications Kangtunyakarn Fixed Point Theory and Applications 014, 014:108 http://www.fixedpointtheoryandapplications.com/content/014/1/108 R E S E A R C H Open Access Fixed point theory for nonlinear mappings in

More information

Fixed Points for Multivalued Mappings in b-metric Spaces

Fixed Points for Multivalued Mappings in b-metric Spaces Applied Mathematical Sciences, Vol. 10, 2016, no. 59, 2927-2944 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.68225 Fixed Points for Multivalued Mappings in b-metric Spaces Seong-Hoon

More information

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

More information

On an iterative algorithm for variational inequalities in. Banach space

On an iterative algorithm for variational inequalities in. Banach space MATHEMATICAL COMMUNICATIONS 95 Math. Commun. 16(2011), 95 104. On an iterative algorithm for variational inequalities in Banach spaces Yonghong Yao 1, Muhammad Aslam Noor 2,, Khalida Inayat Noor 3 and

More information

Research Article Modified Halfspace-Relaxation Projection Methods for Solving the Split Feasibility Problem

Research Article Modified Halfspace-Relaxation Projection Methods for Solving the Split Feasibility Problem Advances in Operations Research Volume 01, Article ID 483479, 17 pages doi:10.1155/01/483479 Research Article Modified Halfspace-Relaxation Projection Methods for Solving the Split Feasibility Problem

More information

STRONG CONVERGENCE OF A MODIFIED ISHIKAWA ITERATIVE ALGORITHM FOR LIPSCHITZ PSEUDOCONTRACTIVE MAPPINGS

STRONG CONVERGENCE OF A MODIFIED ISHIKAWA ITERATIVE ALGORITHM FOR LIPSCHITZ PSEUDOCONTRACTIVE MAPPINGS J. Appl. Math. & Informatics Vol. 3(203), No. 3-4, pp. 565-575 Website: http://www.kcam.biz STRONG CONVERGENCE OF A MODIFIED ISHIKAWA ITERATIVE ALGORITHM FOR LIPSCHITZ PSEUDOCONTRACTIVE MAPPINGS M.O. OSILIKE,

More information

A NEW COMPOSITE IMPLICIT ITERATIVE PROCESS FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS IN BANACH SPACES

A NEW COMPOSITE IMPLICIT ITERATIVE PROCESS FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS IN BANACH SPACES A NEW COMPOSITE IMPLICIT ITERATIVE PROCESS FOR A FINITE FAMILY OF NONEXPANSIVE MAPPINGS IN BANACH SPACES FENG GU AND JING LU Received 18 January 2006; Revised 22 August 2006; Accepted 23 August 2006 The

More information

SHRINKING PROJECTION METHOD FOR A SEQUENCE OF RELATIVELY QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM IN BANACH SPACES

SHRINKING PROJECTION METHOD FOR A SEQUENCE OF RELATIVELY QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM IN BANACH SPACES U.P.B. Sci. Bull., Series A, Vol. 76, Iss. 2, 2014 ISSN 1223-7027 SHRINKING PROJECTION METHOD FOR A SEQUENCE OF RELATIVELY QUASI-NONEXPANSIVE MULTIVALUED MAPPINGS AND EQUILIBRIUM PROBLEM IN BANACH SPACES

More information

CONVERGENCE THEOREMS FOR MULTI-VALUED MAPPINGS. 1. Introduction

CONVERGENCE THEOREMS FOR MULTI-VALUED MAPPINGS. 1. Introduction CONVERGENCE THEOREMS FOR MULTI-VALUED MAPPINGS YEKINI SHEHU, G. C. UGWUNNADI Abstract. In this paper, we introduce a new iterative process to approximate a common fixed point of an infinite family of multi-valued

More information

CONVERGENCE OF THE STEEPEST DESCENT METHOD FOR ACCRETIVE OPERATORS

CONVERGENCE OF THE STEEPEST DESCENT METHOD FOR ACCRETIVE OPERATORS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 12, Pages 3677 3683 S 0002-9939(99)04975-8 Article electronically published on May 11, 1999 CONVERGENCE OF THE STEEPEST DESCENT METHOD

More information

Split equality monotone variational inclusions and fixed point problem of set-valued operator

Split equality monotone variational inclusions and fixed point problem of set-valued operator Acta Univ. Sapientiae, Mathematica, 9, 1 (2017) 94 121 DOI: 10.1515/ausm-2017-0007 Split equality monotone variational inclusions and fixed point problem of set-valued operator Mohammad Eslamian Department

More information

FIXED POINT ITERATION FOR PSEUDOCONTRACTIVE MAPS

FIXED POINT ITERATION FOR PSEUDOCONTRACTIVE MAPS PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 4, April 1999, Pages 1163 1170 S 0002-9939(99)05050-9 FIXED POINT ITERATION FOR PSEUDOCONTRACTIVE MAPS C. E. CHIDUME AND CHIKA MOORE

More information

KKM-Type Theorems for Best Proximal Points in Normed Linear Space

KKM-Type Theorems for Best Proximal Points in Normed Linear Space International Journal of Mathematical Analysis Vol. 12, 2018, no. 12, 603-609 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.81069 KKM-Type Theorems for Best Proximal Points in Normed

More information

International Journal of Scientific & Engineering Research, Volume 7, Issue 12, December-2016 ISSN

International Journal of Scientific & Engineering Research, Volume 7, Issue 12, December-2016 ISSN 1750 Approximation of Fixed Points of Multivalued Demicontractive and Multivalued Hemicontractive Mappings in Hilbert Spaces B. G. Akuchu Department of Mathematics University of Nigeria Nsukka e-mail:

More information

New Nonlinear Conditions for Approximate Sequences and New Best Proximity Point Theorems

New Nonlinear Conditions for Approximate Sequences and New Best Proximity Point Theorems Applied Mathematical Sciences, Vol., 207, no. 49, 2447-2457 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ams.207.7928 New Nonlinear Conditions for Approximate Sequences and New Best Proximity Point

More information

STRONG CONVERGENCE RESULTS FOR NEARLY WEAK UNIFORMLY L-LIPSCHITZIAN MAPPINGS

STRONG CONVERGENCE RESULTS FOR NEARLY WEAK UNIFORMLY L-LIPSCHITZIAN MAPPINGS BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org / JOURNALS / BULLETIN Vol. 6(2016), 199-208 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS

More information

Available online at J. Nonlinear Sci. Appl., 10 (2017), Research Article

Available online at   J. Nonlinear Sci. Appl., 10 (2017), Research Article Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2719 2726 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa An affirmative answer to

More information

APPROXIMATING SOLUTIONS FOR THE SYSTEM OF REFLEXIVE BANACH SPACE

APPROXIMATING SOLUTIONS FOR THE SYSTEM OF REFLEXIVE BANACH SPACE Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 2 Issue 3(2010), Pages 32-39. APPROXIMATING SOLUTIONS FOR THE SYSTEM OF φ-strongly ACCRETIVE OPERATOR

More information

Caristi-type Fixed Point Theorem of Set-Valued Maps in Metric Spaces

Caristi-type Fixed Point Theorem of Set-Valued Maps in Metric Spaces International Journal of Mathematical Analysis Vol. 11, 2017, no. 6, 267-275 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.717 Caristi-type Fixed Point Theorem of Set-Valued Maps in Metric

More information

A GENERALIZATION OF THE REGULARIZATION PROXIMAL POINT METHOD

A GENERALIZATION OF THE REGULARIZATION PROXIMAL POINT METHOD A GENERALIZATION OF THE REGULARIZATION PROXIMAL POINT METHOD OGANEDITSE A. BOIKANYO AND GHEORGHE MOROŞANU Abstract. This paper deals with the generalized regularization proximal point method which was

More information

STRONG CONVERGENCE THEOREMS BY A HYBRID STEEPEST DESCENT METHOD FOR COUNTABLE NONEXPANSIVE MAPPINGS IN HILBERT SPACES

STRONG CONVERGENCE THEOREMS BY A HYBRID STEEPEST DESCENT METHOD FOR COUNTABLE NONEXPANSIVE MAPPINGS IN HILBERT SPACES Scientiae Mathematicae Japonicae Online, e-2008, 557 570 557 STRONG CONVERGENCE THEOREMS BY A HYBRID STEEPEST DESCENT METHOD FOR COUNTABLE NONEXPANSIVE MAPPINGS IN HILBERT SPACES SHIGERU IEMOTO AND WATARU

More information

Strong convergence theorems for asymptotically nonexpansive nonself-mappings with applications

Strong convergence theorems for asymptotically nonexpansive nonself-mappings with applications Guo et al. Fixed Point Theory and Applications (2015) 2015:212 DOI 10.1186/s13663-015-0463-6 R E S E A R C H Open Access Strong convergence theorems for asymptotically nonexpansive nonself-mappings with

More information

ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES

ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES TJMM 6 (2014), No. 1, 45-51 ON THE CONVERGENCE OF MODIFIED NOOR ITERATION METHOD FOR NEARLY LIPSCHITZIAN MAPPINGS IN ARBITRARY REAL BANACH SPACES ADESANMI ALAO MOGBADEMU Abstract. In this present paper,

More information

ON WEAK AND STRONG CONVERGENCE THEOREMS FOR TWO NONEXPANSIVE MAPPINGS IN BANACH SPACES. Pankaj Kumar Jhade and A. S. Saluja

ON WEAK AND STRONG CONVERGENCE THEOREMS FOR TWO NONEXPANSIVE MAPPINGS IN BANACH SPACES. Pankaj Kumar Jhade and A. S. Saluja MATEMATIQKI VESNIK 66, 1 (2014), 1 8 March 2014 originalni nauqni rad research paper ON WEAK AND STRONG CONVERGENCE THEOREMS FOR TWO NONEXPANSIVE MAPPINGS IN BANACH SPACES Pankaj Kumar Jhade and A. S.

More information