ON A CLASS OF SPLIT EQUALITY FIXED POINT PROBLEMS IN HILBERT SPACES

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1 J. Noliear Var. Aal. (207), No. 2, pp Available olie at ON A CLASS OF SPLIT EQUALITY FIXED POINT PROBLEMS IN HILBERT SPACES SHIH-SEN CHANG,, LIN WANG 2, YUNHE ZHAO 2 Ceter for Geeral Educati, Chia Medical Uiversity, Taichug, 40402, Taiwa 2 College of Statistics ad Mathematics, Yua Uiversity of Fiace ad Ecoomics, Kumig 65002, Chia Abstract. The purpose of this paper is to study the split equality fixed poit problem for quasi-asymptotically pseudocotractive mappigs via a iterative algorithm. Weak ad strog covergece theorems are proved i the framework of ifiite dimesioal Hilbert spaces. The results preseted i the article geeralize ad improve some recet results. Keywords. Hilbert space; Split equality fixed poit problem; Quasi-asymptotically pseudo-cotractive mappig; Quasiasymptotically oexpasive mappig. 200 Mathematics Subject Classificatio. 47J20, 47H0, 49J25.. INTRODUCTION-PRELIMINARIES Let H be a real Hilbert space with ier product, ad orm, respectively. Let C be a oempty closed covex subset of H. Let T : C C be a operator. We use Fix(T ) to deote the fixed poit set of mappig T, i.e., Fix(T ) = {x C : x = T x}. Recall that T : C C is said to be oexpasive iff T x Ty x y, x,y C. T : C C is said to be quasi-oexpasive iff Fix(T ) /0 ad T x p x p, x C, p Fix(T ). T : C C is said to be asymptotically oexpasive iff there exists a sequece {l } [,+) with l such that T x T y l x y, x,y C. T : C C is said to be quasi-asymptotically oexpasive iff Fix(T ) /0 ad T x p l x p, x C, p Fix(T ). Correspodig author. address: chagss203@63.com (S.S. Chag). Received December, 206; Accepted April 28, 207. c 207 Joural of Noliear ad Variatioal Aalysis 20

2 202 S.S. CHANG, L. WANG, Y. ZHAO T : C C is said to be pseudo-cotractive if It is easy to kow that T is pseudo-cotractive iff T x Ty,x y x y 2, x,y C. T x Ty 2 x y 2 + (I T )x (I T )y 2, x,y C. T : C C is said to be quasi-pseudo-cotractive iff Fix(T ) /0 ad T x p 2 x p 2 + T x x 2, x C, p Fix(T ). T : C C is said to be asymptotically pseudo-cotractive iff there exists a sequece {l } [,+) with l such that for all x,y C ad for all. T x T y 2 l x y 2 + (I T )x (I T )y 2, It is well kow that T is asymptotically pseudo-cotractive iff for all x,y C ad. T x T y,x y l + 2 x y 2, T : C C is said to be quasi-asymptotically pseudo-cotractive iff Fix(T ) /0 ad there exists a sequece {l } [,+) with l such that for all x C, p Fix(T ) ad for all. T x p 2 l x p 2 + T x x 2, Next we give a example of quasi-asymptotically pseudo-cotractive mappig. Example.. [0] Let C be a uit ball i a real Hilbert space l 2 ad let S : C C be a mappig defied by It is proved i [0] that (i) Sx Sy x y 2, x,y C; S : (x,x 2, ) (0,x 2,a 2 x 2,a 3 x 3, ). (ii) S x S y 2 (2 j=2 a j) 2 x y 2, x,y C, 2. Takig a j = 2 2 j, j 2, it is easy to see that j=2 a j = 2. Lettig l = 4, l = (2 j=2 a j) 2, 2, the we have ad lim l = lim (2 j=2 2 2 j ) 2 =, S x S y 2 l x y 2 l x y 2 + (I S )x (I S )y 2, x,y, C. This shows that S : C C is a asymptotically pseudo-cotractive mappig with Fix(S) = {(0,0,0, )}, so it is also a quasi-asymptotically pseudo-cotractive mappig.

3 ON A CLASS OF SPLIT EQUALITY FIXED POINT PROBLEMS IN HILBERT SPACES 203 Recall that T : C C is said to be uiformly L-Lipschtzia iff there exists some L > 0 such that T x T y L x y, for all x,y C ad for all. The split equality fixed poit problem, which is a geeralizatio of the split feasibility problem ad of the covex feasibility problem, has bee extesively ivestigated recetly due to its extraordiary utility ad broad applicability i may areas of applied mathematics. Let C ad Q be oempty closed ad covex subsets of real Hilbert spaces H ad H 2, respectively, ad A : H H 2 be a bouded liear operator. Recall that the split feasibility problem (SFP) is formulated as to fid a poit q H such that: q C ad Aq Q. (.) Let P C ad P Q be the metric projectios from H ad H 2 to C ad Q, respectively. It is easy to see that q C solves equatio (.) iff it solves the followig fixed poit equatio q = P C (I γa (I P Q )A)q, where γ > 0 is a real costat ad A is the adjoit of A. I 994, Cesor ad Elfvig [5] first itroduced the (SFP) i fiite-dimesioal Hilbert spaces for modelig iverse problems which arise from phase retrievals ad i medical image recostructio []. It has bee foud that the (SFP) ca also be used i various disciplies such as image restoratio, computer tomograph ad radiatio therapy treatmet plaig [6, 7, 8]. The (SFP) i a ifiite dimesioal Hilbert space ca be foud i [2, 3, 4, 5, 6]. Recetly, Moudafi [], Moudafi ad Al-Shemas [2] ad Moudafi [3] itroduced the followig split equality feasibility problem (SEFP): to f id x C, y Q such that Ax = By, (.2) where A : H H 3 ad B : H 2 H 3 are two bouded liear operators. Obviously, if B = I ( idetity mappig o H 2 ) ad H 3 = H 2, the (.2) reduces to (.). I order to solve split equality feasibility problem (.2), Moudafi [] itroduced the followig simultaeous iterative algorithm: x k+ = P C (x k γa (Ax k By k )), (.3) y k+ = P Q (y k + βb (Ax k+ By k )), ad uder suitable coditios some weak covergece of the sequece {(x,y )} to a solutio of (.2) i Hilbert spaces is proved. I order to avoid usig the projectio, recetly, Moudafi [2] itroduced ad studied the followig problem: Let T : H H ad S : H 2 H 2 be oliear operators such that Fix(T ) /0 ad Fix(S) /0. If C = Fix(T ) ad Q = Fix(S), the the split equality feasibility problem (.2) reduces to: f id x Fix(T ) ad y Fix(S) such that Ax = By, (.4)

4 204 S.S. CHANG, L. WANG, Y. ZHAO which is called split equality fixed poit problem (i short, (SEFPP)). I the sequel, we deote by Γ the solutio set of split equality fixed poit problem (.4). Recetly Moudafi [3] proposed the followig iterative algorithm for fidig a solutio of (SEFPP) (.4): x + = T (x γ A (Ax By )), (.5) y + = S(y + β B (Ax + By )). He also studied the weak covergece of the sequeces geerated by scheme (.5) uder the coditio that T ad S are firmly quasi-oexpasive mappigs. I 205, Che ad Li [9] proposed the followig iterative algorithm for fidig a solutio of (SEFPP) (.4): u = x γ A (Ax By ), x + = α x + ( α )Tu, (.6) v = y + γ B (Ax By ), y + = α y + ( α )Sv. They also established the weak covergece of scheme (.6) uder the coditio that the operators T ad S are quasi-oexpasive mappigs. Recetly, Chag, Wag ad Qi [4] proposed the followig iterative algorithm for fidig a solutio of (SEFPP) (.4): u = x γ A (Ax By ), x + = α x + ( α )(( ξ )I + ξ T (( η )I + η T ))u, (.7) v = y + γ B (Ax By ), y + = α y + ( α )(( ξ )I + ξ S(( η )I + η S))v. They established the weak covergece of scheme (.7) uder the coditio that operators T ad S are quasi-pseudo-cotractive mappigs which is more geeral tha the classes of quasi-oexpasive mappigs, directed mappigs ad demi-cotractive mappigs. Motived by above results, the purpose of this paper is to cosider split equality fixed poit problem (.4) for a class of quasi-asymptotically pseudo-cotractive mappigs which is more geeral tha the classes of quasi-oexpasive mappigs ad quasi-pseudo-cotractive mappigs. Uder suitable coditios, some weak ad strog covergece theorems are proved. To prove our mai results, the followig defiitios ad tools are essetial. Recall that a operator T : C C is said to be demiclosed at 0 if, for ay sequece {x } C which coverges weakly to x ad x T (x ) 0, the T x = x. T : H H is said to be semi-compact if, for ay bouded sequece {x } H with x T x 0, there exists a subsequece {x i } {x } such that {x i } coverges strogly to some poit x H. Lemma.. Let H be a real Hilbert space. For ay x,y H, the followig coclusios hold: tx + ( t)y 2 = t x 2 + ( t) y 2 t( t) x y 2, t [0,]; (.8)

5 ON A CLASS OF SPLIT EQUALITY FIXED POINT PROBLEMS IN HILBERT SPACES 205 x + y 2 x y,x + y. (.9) Recall that a Baach space X is said to satisfy Opial s coditio, if for ay sequece {x } i X which coverges weakly to x, we have which is equivalet to lim if x x < limif x y, y Xwith y x, lim sup x x < limsup x y, y Xwith y x. Lemma.2. Let {p },{q } ad {r } be the oegative real sequeces satisfyig the followig coditios p + ( + q )p + r, 0, q <, ad r <. = = The (i) lim p exists; (ii) if limif p = 0, the lim p = 0. Lemma.3. Let H be a real Hilbert space, T : H H be a uiformly L-Lipschitzia ad {l } quasiasymptotically pseudocotractive mappig with L, {l } [,) ad lim l =. Let {K : H H} be a sequece of mappigs defied by: K := ξ T (ηt + ( η)i) + ( ξ )I. (.0) If 0 < ξ < η <, where M = sup M+ (M+) l, the the followig coclusios hold: 4 +L 2 (i) Fix(T ) = Fix(T (( η)i + ηt )) = Fix(K ) for all ; (ii) If T is demiclosed at 0, the K is also demiclosed at 0; (iii) For all ad for all x H,u Fix(T ) = Fix(K ), K x u k x u, where k = + ξ (l )( + ηl ),{k } [,+) ad lim k =. Proof. (i) If x Fix(T ), i.e., x = T x, we have T (( η)i + ηt )x = T (( η)x + ηt x ) = T x = x. This shows that x Fix(T (( η)i +ηt )). Coversely, if x Fix(T (( η)i +ηt )) for all, i.e., x = T (( η)i + ηt )x, lettig U = ( η)i + ηt, we have T U x = x. Put U x = y, the T y = x. Now we prove that x = y. I fact, we have x y = x U x = x (( η)i + ηt )x =η x T x = η T y T x Lη x y. Sice 0 < Lη <, we have x = y, i.e.,x Fix(T ). This shows that Fix(T ) = Fix(T (( η)i +ηt )) for all. It is obvious that x Fix(K ) if ad oly if x Fix(T (( η)i + ηt )). The coclusio (i) is proved.

6 206 S.S. CHANG, L. WANG, Y. ZHAO (ii) For ay sequece {x } H satisfyig x x ad x Kx 0. Next we show that x Fix(K). From coclusio (i), it suffices to prove x Fix(T ). I fact, sice T is L-Lipschizia, we have Simplifyig it, we have ξ x T x ξ T (( η)i + ηt )x T x + ξ x T (( η)i + ηt )x ξ Lη x T x + x ( ξ )x ξ T (( η)i + ηt )x =ξ Lη x T x + x K x. x T x ξ ( Lη) x K x 0. (.) Sice T is demiclosed at 0, we have x F(T ) = F(K). The coclusio (ii) is proved. (iii) For all u Fix(T ), we have T (( η)i + ηt )x u 2 l ( η)x + ηt x u 2 + (( η)i + ηt )x T (( η)i + ηt )x 2 (.2) =ηt )x T (( η)i + ηt )x 2 + l ( η)(x u ) + η(t x u ) 2 + (( η)i ad T x u 2 l x u 2 + x T x 2. (.3) Sice T is L-Lip ad x (( η)x + ηt x) = η(x T x) we have T x T (( η)x + ηt x) L x (( η)x + ηt x) = Lη x T x. (.4) From (.8) ad (.4), we have ( η)(x u ) + η(t x u ) 2 =( η) x u 2 + η T x u 2 η( η) x T x 2 ( η) x u 2 + η(l x u 2 + x T x 2 ) η( η) x T x 2 (.5) =( + η(l )) x u 2 + η 2 x T x 2. From (.8) ad (.4), we have (( η)i + ηt )x T (( η)i + ηt )x 2 =( η) x T (( η)x + ηt x) 2 + η T x T (( η)x + ηt x) 2 η( η) x T x 2 (.6) ( η) x T (( η)x + ηt x) 2 η( η η 2 L 2 ) x T x 2.

7 ON A CLASS OF SPLIT EQUALITY FIXED POINT PROBLEMS IN HILBERT SPACES 207 Substitutig (.5) ad (.6) ito (.2), we obtai T (( η)i + ηt )x u 2 l ( + η(l )) x u 2 + l η 2 T x x 2 + ( η) x T (( η)x + ηt x) 2 η( η η 2 L 2 ) T x x 2 =l ( + η(l )) x u 2 + ( η) x T (( η)x + ηt x) 2 (.7) + η(η + η 2 L 2 + l η ) T x x 2. Sice η < M+ 2 + (M+) 2 4 +L 2, we fid from (.7) that T (( η)x + ηt x) u 2 l ( + η(l )) x u 2 + ( η) x T (( η)x + ηt x) 2. (.8) Combie (.8) ad (.8) we have that K x u 2 =( ξ ) x u 2 + ξ T (( η)x + ηt x) u 2 ξ ( ξ ) x T (( η)x + ηt x) 2 ( ξ ) x u 2 + ξ l ( + η(l )) x u 2 + (ξ ( η) ξ ( ξ )) x T (( η)x + ηt x) 2 =( + ξ (l )( + ηl )) x u 2 ξ (η ξ ) x T (( η)x + ηt x) 2. This together with ξ < η implies that K x u 2 k x u 2 for all x H,u Fix(K ) ad, where k = ξ (l )( + ηl ) +. I view of that {l } [,+) ad lim l = we have {k } [,+) ad lim k =. The coclusio (iii) is proved. 2. MAIN RESULTS Now we are i a positio to give the followig mai result. Theorem 2.. Let H,H 2 ad H 3 be three real Hilbert spaces, A : H H 3, B : H 2 H 3 be two bouded liear operators, A ad B be the adjoit operators of A ad B, respectively. Let T : H H ad S : H 2 H 2 be two uiformly L-Lipschitzia ad {l }-quasi-asymptotically pseudo-cotractive mappigs with L, l [,), l ad Σ = (l2 ) <, Fix(T ) /0, ad Fix(S) /0. Let {α,i } be the sequece i (0,) satisfyig the followig coditios: (i) i=0 α,i =, f or each ; limif α,i > 0, α,i <, f or each i 0. =

8 208 S.S. CHANG, L. WANG, Y. ZHAO Suppose further that for ay give x 0 H,y 0 H 2, {(x,y )} is the sequece geerated by: (a) u = x γ A (Ax By ), where (b) x + = α,0 x + (d) y + = α,0 y + α,i K i u, (c) v = y + γ B (Ax By ), α,i G i v, (2.) K i = ( ξ )I + ξ T i (( η)i + ηt i ); ad G i = ( ξ )I + ξ S i (( η)i + ηs i ). (2.2) If T ad S are demiclosed at 0, ad the solutio set Γ of problem (.4) Γ = {(x,y ) Fix(T ) Fix(S) such that Ax = By } (2.3) is oempty, ad the followig coditios are satisfied: (ii) γ (0,mi( A 2, B 2 )), with limifγ > 0; (iii) 0 < a < ξ < η < b <,, M+ (M+) L 2 where M = sup l, the the followig coclusios hold: (I) the sequece {(x,y )} coverges weakly to a solutio of problem (.4); (II) I additio, if S,T both are semi-compact, the {(x,y )} coverges strogly to a solutio of problem (.4). Proof. First we prove the coclusio (I). For ay give (p,q) Γ, the p Fix(T ),q Fix(S) ad Ap = Bq. From (2.) (a) ad Lemma., we have u p 2 = x γ A (Ax By ) p 2 Similarly, from (2.) (c) ad Lemma., we have =γ 2 A (Ax By ) 2 2γ x p,a (Ax By ) + x p 2 γ 2 A 2 Ax By 2 2γ Ax Ap,Ax By + x p 2. (2.4) v q 2 γ 2 B 2 Ax By 2 + 2γ By Bq,Ax By + y q 2. (2.5) By coditio (iii) ad Lemma.3, the mappigs {K i } ad {G i } have the followig properties: (a) Fix(T ) = Fix(K i ) ad Fix(S) = Fix(G i ) for each i ; (b) K ad G are demiclosed at 0; (c) For each i ad for all x H, y H 2,u Fix(T ) = Fix(K i ),v Fix(S) = Fix(G i ), K i x u k i x u, G i y v k i y v,

9 ON A CLASS OF SPLIT EQUALITY FIXED POINT PROBLEMS IN HILBERT SPACES 209 where k i = +ξ (l i )(+ηl i ),{k i } [,+) ad lim i k i =. By the assumptio that (l2 i ) <, we have (k i ) ξ (l i )(l i + ) (l 2 i ) <. (2.6) O the other had, it follows from (2.) (b), (2.4) ad a well-kow result that for ay positive iteger l we have x + p 2 α,0 x p 2 + α,0 x p 2 + α,0 x p 2 + α,i K i u p 2 α,0 α,l x K l u 2 α,i k 2 i u p 2 α,0 α,l x K l u 2 α,i k 2 i { x p 2 + γ 2 A 2 Ax By 2 2γ Ax Ap,Ax By } α,0 α,l x K l u 2. (2.7) Similarly, it follows from (2.) (d) ad (2.5) that for ay positive iteger l y + q 2 α,0 y p 2 + α,i k 2 i { y q 2 + γ 2 B 2 Ax By 2 + 2γ By Bq,Ax By } α,0 α,l y G l v 2. (2.8) Addig up (2.7) ad (2.8) ad otig Ap = Bq, we have that y + p 2 + x + q 2 (α,0 + + α,i k 2 i ){ x p 2 + y q 2 } α,i k 2 i γ 2 ( A 2 + B 2 ) Ax By 2 2γ α,i k 2 i Ax Ap By + Bq,Ax By α,0 α,l { x K l u 2 + y G l v 2 } = ( + + α,i (k 2 i ){ x p 2 + y q 2 } α,i k 2 i γ (γ ( A 2 + B 2 ) 2) Ax By 2 α,0 α,l { x K l u 2 + y G l v 2 }. (2.9) Sice γ (0,mi{ A 2, B 2 }), γ A 2 < ad γ B 2 <. This implies that γ ( A 2 + B 2 ) 2 < 0. Therefore (2.9) ca be writte as y + q 2 + x + p 2 ( + = ( + σ )( x p 2 + y q 2 ), α,i (k 2 i ))( x p 2 + y q 2 ). (2.0)

10 20 S.S. CHANG, L. WANG, Y. ZHAO where σ = α,i(k 2 i ). Sice k ad by (2.6) (k i ) <, this implies that (k2 i ) <. Agai sice = σ = = α,i (k 2 i ) (k 2 i ) = α,i (k 2 i ) < σ 0. By virtue of Lemma.2 ad (2.0), it gets that the limit lim ( x p 2 + y q 2 ) exists, so the limits lim x p ad lim y q exist for all (p,q) Γ. Now rewrite (2.9) as α,i k 2 i γ (2 γ ( A 2 + B 2 )) Ax By 2 + α,0 α,l { x K l u 2 + y G l v 2 } ( + σ )( x p 2 + y q 2 ) ( x + p 2 (2.) ( x + p 2 + y + q 2 ) 0 (as ) (sice σ 0). It follows from coditios (i), (ii) ad (2.) that for each l =,2,3, lim Ax By = 0, lim K l u x = 0, lim G l v y = 0. (2.2) This together with (2.) ad the coditio (i) shows that lim u x = 0; lim x + x = lim lim v y = 0. lim lim y + y = lim sup i lim sup i α,i K i u x K i u x = 0. α,i G i v y = 0 G i v y = 0. From (2.2) ad (2.3) we have K u u K u x + x u 0; G v v G v y + y v 0. (2.3) (2.4) Sice {x } ad {y } are bouded sequeces, there exist some weakly coverget subsequeces, say {x i } {x } ad {y i } {y } such that x i x ad y i y. Sice every Hilbert space has the Opial s property which guaratees that the weak limit of {(x,y )} is uique. Therefore we have x x, ad y y. O the other had, it follows from (2.3) that u x ad v y. Sice K ad G both are demiclosed at 0 (by Lemma.3 (ii), from (2.4) it gets K x = x ad G y = y. This implies that x Fix(T ) ad y Fix(S).

11 ON A CLASS OF SPLIT EQUALITY FIXED POINT PROBLEMS IN HILBERT SPACES 2 Now we show that Ax = By. I fact, sice Ax By Ax By, by usig the weakly lower semi-cotiuity of orm, we have Ax By 2 limif Ax By 2 = lim Ax By 2 = 0. Thus Ax = By. This completes the proof of the coclusio (I). Now we prove the coclusio (II). I fact, sice K is uiformly cotiuous, we have lim K u K x = 0. Hece from (2.3), we have K x x x K u + K u K x. It follows that Similarly, we ca also prove that lim K x x = 0. (2.5) lim G y y = 0. (2.6) By virtue of (.), (2.4), (2.5) ad (2.6), we have x T x ξ ( Lη) x K x 0 ( ); y Sy ξ ( Lη) y G y 0 ( ). (2.7) Sice S,T are semi-compact, it follows from (2.7) that there exist subsequeces {x i } {x } ad {y j } {y } such that x i x (some poit i Fix(T )) ad y j y (some poit i Fix(S)). Sice x x ad y y ad the limits lim x p ad lim y q exist for all (p,q) Γ, it follows from (2.2) that x x ad y y ad Ax = By. This completes the proof of Theorem 2.. Remark 2.. Theorem 2. is a improvemet ad geeralizatio of the correspodig results i Chag, Wag ad Qi [4], Che ad Li [9], Moudafi [], Moudafi ad Al-Shemas [2], Moudafi [3]. Ackowledgemet This work was supported by the Natural Sciece Foudatio of Ceter for Geeral Educatio, Chia Medical Uiversity, Taichug, Taiwa. REFERENCES [] C. Byre, Iterative oblique projectio oto covex subsets ad the split feasibility problem. Iverse Probl. 8 (2002), [2] S.S. Chag, R.P. Agarwal, Strog covergece theorems of geeral split equality problems for quasi-oexpasive mappigs, J. Iequal. Appl. 204 (204), Articl ID 367. [3] S.S. Chag, L. Wag, Y.K. Tag, G. Wag, Moudafi s ope questio ad simultaeous iterative algorithm for geeral split equality variatioal iclusio problems ad geeral split equality optimizatio problems, Fixed Poit Theory Appl. 204 (204), Article ID 25. [4] S.S. Chag, L. Wag, L.J. Qi, Split equality fixed poit problem for quasi-pseudo-cotractive mappigs with applicatios, Fixed Poit Theory Appl. 205 (205), Article ID 208. [5] Y. Cesor, T. Elfvig, A multiprojectio algorithm usig Bregma projectios i a product space, Numer. Algorithms, 8 (994),

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