Báo cáo hóa học: " CONVERGENCE THEOREMS FOR FIXED POINTS OF DEMICONTINUOUS PSEUDOCONTRACTIVE MAPPINGS" pdf

CONVERGENCE THEOREMS FOR FIXED POINTS OF DEMICONTINUOUS PSEUDOCONTRACTIVE MAPPINGS C. E. CHIDUME AND H. ZEGEYE Received 26 August 2004 Let D be an open subset of a real uniformly smooth Banach space E.SupposeT : ¯ D → E is a demicontinuous pseudocontractive mapping satisfying an appropriate condition, where ¯ D denotes the closure of D. Then, it is proved that (i) ¯ D ⊆᏾ (I + r(I −T)) for every r>0; (ii) for a given y 0 ∈ D, there exists a unique path t → y t ∈ ¯ D, t ∈ [0,1], satisfying y t := tT y t +(1−t)y 0 .Moreover,ifF(T) =∅or there exists y 0 ∈ D such that the set K :={y ∈ D : Ty= λy+(1−λ)y 0 for λ>1} is bounded, then it is proved that, as t →1 − , the path {y t } converges strongly to a fixed point of T. Furthermore, explicit iteration procedures with bounded error terms are proved to converge strongly to a fixed point of T. 1. Introduction Let D be a nonempty subset of a real linear space E.AmappingT : D → E is called a contraction mapping if there exists L ∈ [0,1)suchthatTx −Ty≤Lx − y for all x, y ∈D.IfL = 1thenT is called nonexpansive. T is called pseudocontractive if there exists j(x − y) ∈J(x − y)suchthat  Tx−Ty, j(x − y)  ≤x − y 2 , ∀x, y ∈K, (1.1) where J is the normalized duality mapping from E to 2 E ∗ defined by Jx :=  f ∗ ∈ E ∗ :  x, f ∗  = x 2 =   f ∗   2  . (1.2) T is called strongly pseudocontractive if there exists k ∈ (0,1) such that  Tx−Ty, j(x − y)  ≤ kx − y 2 , ∀x, y ∈K. (1.3) Clearlytheclassofnonexpansivemappingsisasubsetofclassofpseudocontractivemap- pings. T is said to be demicontinuous if {x n }⊆D and x n → x ∈ D together imply that Tx n  Tx,where→ and  denote the strong and weak convergences, respectively. We denote by F(T) the set of fixed points of T. Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:1 (2005) 67–77 DOI: 10.1155/FPTA.2005.67 68 Fixed points of demicontinuous pseudocontractive maps Closely related to the class of pseudocontractive mappings is the class of accretive map- pings. A mapping A : D(A) ⊆E → E is called accretive if T := (I −A) is pseudocontractive. If E is a Hilbert space, accretive operators are also called monotone.AnoperatorA is called m-accretive if it is accretive and ᏾(I + rA), the range of (I + rA), is E for all r>0; and A is said to satisfy the range condition if cl(D(A)) ⊆᏾( I + rA), for all r>0, where cl(D(A)) denotes the closure of the domain of A. Let z ∈D,thenforeacht ∈(0,1), and for a nonexpansive map T, there exists a unique point x t ∈ D satisfying the condition, x t = tTx t +(1−t)z (1.4) since the mapping x → tTx +(1−t)z is a contraction. When E is a Hilbert space and T is aself-map,Browder[1] showed that {x t } converges strongly to an element of F(T) which is nearest to u as t →1 − . This result was extended to various more general Banach spaces by Reich [10], Takahashi and Ueda [11], and a host of other authors. Recently, Morales and Jung [7] proved the existence and convergence of a continuous path to a fixed point of a continuous pseudocontractive mapping in reflexive Banach spaces. More precisely, they proved the following theorem. Theorem 1.1 [7, Proposition 2(iv), Theorem 1]. Suppose D is a nonempty closed con- vex subset of a reflexive Banach space E and T : D → E is a continuous pseudocontractive mapping satisfying the weakly inward condition. Then for z ∈ D, there exists a unique path t → y t ∈ D, t ∈[0, 1), satisfying the following condition, y t = tTy t +(1−t)z. (1.5) Further more, suppose E is assumed to have a uniformly G ˆ ateaux differentiable norm and is such that every closed convex and bounded subset of D has the fixed point propert y for nonexpansive self-mappings. If F(T) =∅or there exists x 0 ∈ D such that the set K :={x ∈ D : Tx =λx +(1−λ)x 0 for λ>1} is bounded, then as t →1 − , the path converges strongly to afixedpointofT. From Theorem 1.1, one question arises quite naturally. Question. Can the continuity of T be weakened to demicontinuity of T? In connection with this, Lan and Wu [3] proved the following theorem in the Hilbert space setting. Theorem 1.2 [3, Theorems 2.3 and 2.5]. Le t E be a Hilbert space. Suppose D is a nonempty closed convex subset of E and T : D → E is a demicontinuous pseudocontractive mapping satisfying the weakly inward condit ion. Then for z ∈ D, there exists a unique path t → y t ∈ D, t ∈(0,1), satisfying the following condition: y t = tTy t +(1−t)z. (1.6) Moreover, if (i) D is bounded then F(T) =∅and {y t } converges strongly to a fixed point of T as t → 1 − ; (ii) D is unbounded and F(T) =∅then {y t } converges strongly to a fixed point of T as t →1 − . C. E. Chidume and H. Zegeye 69 Let D be a nonempty open and convex subset of a real uniformly smooth Banach space E.SupposeT : ¯ D →E is a demicontinuous pseudocontractive mapping which satisfies for some z ∈D, Tx−z = λ(x −z)forx ∈∂D, λ>1, (1.7) where ¯ D is the closure of D. It is our purpose in this paper to give sufficient conditions to ensure that ¯ D ⊆ (I + r(I −T))( ¯ D)foreveryr>0 and to prove the existence and convergence of a path to a fixed point of a demicontinuous pseudocontractive mapping in spaces more general than Hilbert spaces. More precisely, we prove that for a given y 0 ∈ D, there exists a unique path t → y t ∈ ¯ D, t ∈ (0,1), satisfying y t := tTy t +(1−t)y 0 .Moreover,ifF(T) =∅or there ex- ists y 0 ∈ D such that the set K :={y ∈ D : Ty = λy +(1−λ)y 0 for λ>1} is bounded, then the path {y t } converges strongly to a fixed point of T. Furthermore, the sequence {x n } generated from x 1 ∈ K by x n+1 := (1 −λ n )x n + λ n Tx n −λ n θ n (x n −x 1 ), for all integers n ≥ 1, where {λ n } and {θ n } are real sequences satisfying appropriate conditions, con- verges strongly to a fixed point of T.Ourtheoremsprovideanaffirmativeanswertothe above question in uniformly smooth Banach spaces and extend Theorem 1.2 to uniformly smooth spaces provided that the interior of D,int(D), is nonempty. 2. Preliminaries Let E be a real normed linear space of dimension ≥ 2. The modulus of smoothness of E is defined by ρ E (τ):= sup  x + y+ x − y 2 −1:x=1, y=τ  , τ>0. (2.1) If there exist a constant c>0 and a real number 1 <q<∞,suchthatρ E (τ) ≤ cτ q ,then E is said to be q-uniformly smooth. Typical examples of such spaces are L p and the Sobolev spaces W m p for 1<p<∞.ABanachspaceE is called uniformly smooth if lim τ→0 (ρ E (τ)/τ) = 0. If E is a real uniformly smooth Banach space, then x + y 2 ≤x 2 +2  y, j(x)  +max  x,1  yb  y  (2.2) holds for every x, y ∈ E where b :[0,∞) → [0,∞) is a continuous strictly increasing func- tion satisfying the following conditions: (i) b(ct) ≤ cb(t), ∀c ≥1, (ii) lim t→0 b(t) =0. (See, e.g., [8].) Let D beanonemptysubsetofaBanachspaceE.Forx ∈D,theinward set of x, I D (x), is defined by I D (x):={x + λ(u −x):u ∈D, λ ≥1}.AmappingT : D → E is called weakly inward if Tx ∈cl[I D (x)] for all x ∈ D,wherecl[I D (x)] denotes the closure of the inward set. Every self-map is trivially weakly inward. Let D ⊆ E be closed convex and let Q be a mapping of E onto D.AmappingQ of E into E is said to be a retraction if Q 2 = Q.IfamappingQ is a retraction, then Qz =z for every z ∈ R(Q), range of Q.AsubsetD of E is said to be a nonexpansive ret ract of E if there exists a nonexpansive retraction of E onto D. If E = H, the metric projection P D is a nonexpansive retraction from H to any closed convex subset D of H. 70 Fixed points of demicontinuous pseudocontractive maps In what follows, we will make use of the following lemma and theorems. Lemma 2.1 [2]. Let {λ n }, {γ n },and{α n } be sequences of nonnegative numbers satisfying  ∞ 1 α n =∞and γ n /α n → 0,asn →∞. Let the recursive inequality λ n+1 ≤ λ n −2α n ψ  λ n  + γ n , n =1,2, , (2.3) be given where ψ :[0,∞) → [0,∞) is a nondecreasing function such that it is positive on (0,∞) and ψ(0) =0. Then λ n → 0,asn →∞. Theorem 2.2 [6]. Let E be a uniformly smooth Banach space and let D be an open subset of E.SupposeT : ¯ D → E is a demicontinuous strongly pseudocontractive mapping which satisfies for some z ∈ D : Tx−z = λ(x −z) for x ∈∂D, λ>1. (2.4) Then T has a unique fixed point in ¯ D. Remark 2.3. We observe that, in Theorem 2.2, if, in addition, D is convex, then any weakly inward map satisfies condition (2.4). Theorem 2.4 (Reich [10]). Let E be uniformly smooth. Let A ⊂ E ×E be accretive with cl(D(A)) convex. Suppose A satisfies the range condition. Le t J t := (I + tA) −1 , t>0 be the resolvent of A and assume that A −1 (0) is nonempty. Then, for each x ∈ ᏾(I + rA)( ¯ D), lim t→∞ J t x = Px ∈A −1 (0),whereP is the sunny nonexpansive retraction of cl(D(A)) onto A −1 (0). Remark 2.5. Fr om the proof of Theorem 2.4, we observe that we may replace the as- sumption that A −1 (0) =∅with the assumption that x t = J t x is bounded, for each x ∈ ᏾(I + tA)andt>0. 3. Main results We first prove the following results which will be used in the sequel. Proposition 3.1. Let D be an open subset of a real uniformly smooth Banach space E and let T : ¯ D → E be a demicontinuous pseudocontractive mapping which satisfies condition (2.4). Let A T : ¯ D →E be defined by A T := I + r(I −T) for any r>0. Then ¯ D ⊆A T [ ¯ D]. Proof. Let z ∈ ¯ D.Thenitsuffices to show that there exists x ∈ ¯ D such that z = A T (x). Define g : ¯ D → E by g(x):= (1/(1 + r))(rT(x)+z)forsomer>0. Then clearly g is demi- continuous and for x, y ∈ ¯ D we have that g(x) −g(y), j(x − y)≤(r/(1 + r))x − y 2 . Thus, g is a strongly pseudocontractive mapping which satisfies condition (2.4). There- fore, by Theorem 2.2, there exists x ∈ ¯ D such that g(x) =x, that is, z = A T (x). The proof is complete.  Corollary 3.2. Le t E be a real uniformly smooth Banach space and let A : E →E be demi- continuous accretive mapping. Then A is m-accretive. C. E. Chidume and H. Zegeye 71 Proof. Set T := (I −A). Then, we obtain that T is a demicontinuous pseudocontractive self-map of E. Clearly, condition (2.4) is satisfied. The conclusion follows from Proposition 3.1.  Corollary 3.2 was proved by Minty [5] in a Hilbert space setting for continuous accre- tive mappings and this was extended to general Banach spaces by Martin [4]. We now prove the following theorems. Theorem 3.3. Let D be an open and convex subset of a real uniformly smooth Banach space E.LetT : ¯ D →E be a demicontinuous pseudocontractive mapping satisfying condition (2.4). Then for a given y 0 ∈ D, there exists a unique path t → y t ∈ ¯ D, t ∈(0,1), satisfy ing y t = tTy t +(1−t)y 0 . (3.1) Furthermore, if F(T) =∅or there exists z ∈ D such that the set K :={y ∈ D : Ty= λy + (1 −λ)z for λ>1} is bounded, then the path {y t } described by (3.1) converges strongly to a fixed point of T as t →1 − . Proof. For each t ∈ (0,1) the mapping T t defined by T t x := tT(t n )x +(1−t)y 0 is demi- continuous and strongly pseudocontractive. By Theorem 2.2, it has a unique fixed point y t in ¯ D, that is, for each t ∈(0,1) there exists y t ∈ ¯ D satisfying (3.1). Continuity of y t fol- lows as in [7]. Now we show the convergence of {y t } to a fixed point of T.LetA :=I −T. Then A is accretive and by Proposition 3.1, ¯ D ⊆(I + rA)( ¯ D)forallr>0 and hence A sat- isfies the range condition. Moreover, from (3.1), y t +(t/(1 −t))Ay t = y 0 . But this implies that y t = (I +(t/(1 −t))A) −1 y 0 = J (t/(1−t)) y 0 . Furthermore, since A −1 (0) =∅or the fact that K is bounded implies that {y t } is bounded (see, e.g., [7]), we have by Theorem 2.4 that y t → y ∗ ∈ A −1 (0) and hence y t → y ∗ ∈ F(T)ast → 1 − . This completes the proof of the theorem.  Remark 3.4. We note that, in Theorem 3.3, the requirement that T satisfies condition (2.4) may be replaced with the weakly inward condition. Fur thermore, Theorem 3.3 extends [3, Theorems 2.3 and 2.5] to the more general Banach spaces which include l p ,L p ,W m p ,1 <p<∞, spaces, provided that int(D)isnonempty. For our next theorem and corollary, {λ n }, {θ n },and{c n } are real sequences in [0,1] satisfying the following conditions: (i) lim n→∞ θ n = 0; (ii)  ∞ n=1 λ n θ n =∞,lim n→∞ (b(λ n )/θ n ) =0; (iii) lim n→∞ ((θ n−1 /θ n −1)/λ n θ n ) =0, c n = o(λ n θ n ). Theorem 3.5. Let D be an open and convex subset of a real uniformly smooth Banach space E.SupposeT : ¯ D → E is a bounded demicontinuous pseudocontractive mapping satisfying condition (2.4). Suppose ¯ D is a nonexpansive re tract of E with Q as the nonexpansive retrac- tion. Let a sequence {x n } be generated from x 0 ∈ E by x n+1 = Q  1 −λ n  x n + λ n Tx n −λ n θ n  x n −x 0  −c n  x n −u n  , (3.2) 72 Fixed points of demicontinuous pseudocontractive maps for all positive integers n,where{u n }is a sequence of bounded error terms. If either F(T) =∅ or the set K :={x ∈ D : Tx = λx +(1−λ)x 0 for λ>1} is bounded, then there exists d>0 such that whenever λ n ≤ d and c n /λ n θ n ,b(λ n )/θ n ≤ d 2 for all n ≥0, {x n } converges strongly to a fixed point of T. Proof. By Theorem 3.3, F(T) =∅.Letx ∗ ∈ F(T). Let r>1besufficiently large such that x 0 ∈ B r/2 (x ∗ ). Claim 3.6. {x n } is bounded. It suffices to show by induction that {x n } belongs to B = B r (x ∗ ) for all positive inte- gers. Now, x 0 ∈ B by assumption. Hence we may assume that x n ∈ B and set M := 2r + sup{(I −T)x i + x i −u i ,fori ≤n}.Weprovethatx n+1 ∈ B.Supposex n+1 is not in B. Then x n+1 −x ∗  >rand thus from (3.2)wehavethatx n+1 −x ∗ ≤x n −x ∗ −λ n ((I − T)x n + θ n (x n −x 0 ))−c n (x n −u n )≤x n −x ∗ + λ n (I −T)x n + θ n (x n −x 0 )+(c n /λ n )(x n − u n )≤r + M. Moreover, from (3.2) and inequality (2.2), and using the fact that θ n ≤ 1, we get that   x n+1 −x ∗   2 =   Q  1 −λ n  x n + λ n Tx n −λ n θ n  x n −x 0  −c n  x n −u n  −x ∗   ≤   x n −x ∗ −λ n  (I −T)x n + θ n  x n −x 0  −c n (x n −u n )   2 ≤   x n −x ∗   2 −2λ n  (I −T)x n , j  x n −x ∗  −2λ n θ n  x n −x 0 , j  x n −x ∗  −2c n  x n −u n , j  x n −x ∗  +max    x n −x ∗   ,1  λ n     (I −T)x n + θ n  x n −x 0  + c n λ n  x n −u n      ×b  λ n     (I −T)x n + θ n  x n −x 0  + c n λ n  x n −u n       ≤   x n −x ∗   2 −2λ n  (I −T)x n , j  x n −x ∗  −2λ n θ n  x n −x 0 , j  x n −x ∗  −2c n  x n −u n , j  x n −x ∗  +(r +1)λ n Mb  λ n M  . (3.3) Since T is pseudocontractive and x ∗ ∈ F(T), we have (I −T)x n , j(x n −x ∗ )≥0. Hence (3.3)gives   x n+1 −x ∗   2 ≤   x n −x ∗   2 −2λ n θ n  x n −x 0 , j  x n −x ∗  +2c n   x n −u n   ·   x n −x ∗   +(r +1)λ n M 2 b  λ n  . (3.4) Choose L>0sufficiently small such that L ≤ r 2 /(2 √ D ∗ +2M) 2 ,whereD ∗ = (r +1)M 2 . Set d := √ L. Then since x n+1 −x ∗  > x n −x ∗  by our assumption, from (3.4)weget that 2λ n θ n x n −x 0 , j(x n −x ∗ )≤(r +1)M 2 λ n b(λ n )+2c n Mr which gives x n −x 0 , j(x n − x ∗ )≤D ∗ L, since c n /λ n θ n ,b(λ n )/θ n ≤ L = d 2 ,foralln ≥ 1 by our assumption. C. E. Chidume and H. Zegeye 73 Now adding x 0 −x ∗ , j(x n −x ∗ ) to both sides of this inequality, we get that   x n −x ∗   2 ≤ LD ∗ +  x 0 −x ∗ , j  x n −x ∗  ≤ LD ∗ +   x 0 −x ∗     x n −x ∗   ≤ LD ∗ + r 2   x n −x ∗   . (3.5) Solving this quadratic inequality for x n −x ∗  and using the estimate √ r 2 /16 + LD ∗ ≤ r/4+ √ LD ∗ ,weobtainthatx n −x ∗ ≤r/2+ √ LD ∗ . But in any case, x n+1 −x ∗ ≤ x n −x ∗ + λ n (I −T)x n + θ n (x n −x 0 )+(c n /λ n )(x n −u n ) so that x n+1 −x ∗ ≤r/2+ √ LD ∗ + λ n M ≤ r, by the original choices of L and λ n , and this contradicts the assumption that x n+1 is not in B. Therefore, x n ∈ B for all positive integers n.Thus{x n } is bounded. Now we show that x n → x ∗ .Let{y n } be a subsequence of {y t : t ∈[0,1)},suchthaty n := y t n , t n = 1/(1 + θ n ). Then from (3.2) and inequality (2.2) and using the fact that y n ∈ ¯ D for all n ≥0, we get   x n+1 − y n   2 =   Q  1 −λ n  x n + λ n Tx n −λ n θ n  x n −x 0  −c n  x n −u n  − y n   2 ≤   x n − y n −λ n  (I −T)x n + θ n  x n −x 0  −c n  x n −u n    2 ≤   x n − y n   2 −2λ n  (I −T)x n + θ n  x n −x 0  , j  x n − y n  −2c n  x n −u n , j  x n − y n  +max    x n − y n   ,1  λ n     (I −T)x n + θ n  x n −x 0  + c n λ n  x n −u n      ×b  λ n     (I −T)x n + θ n  x n −x 0  + c n λ n  x n −u n       ≤  1 −2λ n θ n    x n − y n   2 −2λ n  (I −T)x n + θ n  y n −x 0  , j  x n − y n  +2c n   x n −u n   ·   x n − y n   +max    x n − y n   ,1  λ n     (I −T)x n + θ n  x n −x 0  + c n λ n  x n −u n      × b  λ n     (I −T)x n + θ n  x n −x 0  + c n λ n  x n −u n       . (3.6) Since Ty n = y n + θ n (y n −x 0 )andT is pseudocontractive, we get that (I −T)x n + θ n (y n − x 0 ), j(x n − y n )≥0. Moreover, since {x n }, {y n }, and hence {Tx n }, are bounded, there ex- ists M 0 > 0 such that max {x n − y n ,1,x n − y n ·x n −u n ,(I −T)x n + θ n (x n −x 0 )+ (c n /λ n )(x n −u n )} ≤M 0 . Therefore, (3.6)withpropertyofb gives   x n+1 − y n   2 ≤  1 −2λ n θ n    x n − y n   2 + M 0 λ n b  λ n  + c n M 0 . (3.7) On the other hand, by the pseudocontractivity of T and the fact that θ n (y n −x 0 )+(y n − Ty n ) =0, we have that   y n−1 − y n   ≤     y n−1 − y n + 1 θ n  (I −T)y n−1 −(I −T)y n      ≤ θ n−1 −θ n θ n    y n−1   + z  =  θ n−1 θ n −1     y n−1   + z  . (3.8) 74 Fixed points of demicontinuous pseudocontractive maps However,   x n − y n   2 ≤   x n − y n−1   2 +   y n−1 − y n      y n−1 − y n   +2   y n−1 −x n    . (3.9) Therefore, these estimates with (3.7)givethat   x n+1 − y n   2 ≤  1 −2λ n θ n    x n − y n−1   2 + M 1  θ n−1 θ n −1  + M 1 λ n b  λ n  + c n M 1 , (3.10) for some M 1 > 0. Thus, by Lemma 2.1, x n+1 − y n → 0. Hence, since y n → x ∗ by Theorem 3.3,wehavethatx n → x ∗ , this completes the proof of the theorem.  With the help of Remark 2.3 and Theorem 3.5 we obtain the following corollary. Corollary 3.7. Let D be an ope n and convex subset of a real uniformly smooth Banach space E.SupposeT : ¯ D → E is a bounded demicontinuous pseudocontractive mapping satis- fy ing the weakly inward condition. Suppose ¯ D is a nonexpansive retract of E with Q as the nonexpansive retraction. Let a sequence {x n } be generated from x 0 ∈E by x n+1 = Q  1 −λ n  x n + λ n Tx n −λ n θ n  x n −x 0  −c n  x n −u n  , (3.11) for all positive integers n,where{u n } isasequenceoferrorterms.IfeitherF(T) =∅or the set K :={x ∈D : Tx = λx +(1−λ)x 0 for λ>1} is bounded then, there exists d>0 such that whenever λ n ≤ d and c n /λ n θ n ,b(λ n )/θ n ≤ d 2 for all n ≥0, {x n } converges strongly to a fixed point of T. Remark 3.8. For the case where E is q-uniformly smooth, where q>1, and t ≤ M for some M>0, the function b in (2.2)isestimatedbyb(t) ≤ct q−1 for some c>0 (see [9]). Thus, we have the following corollary. Corollary 3.9. Let D be an open and convex subs et of a real q-uniformly smooth Banach space E.SupposeT : ¯ D → E is a bounded demicontinuous pseudocontractive mapping satis- fy ing condition (2.4). Suppose ¯ D is a nonexpansive retract of E with Q as the nonexpansive retraction and let {λ n }, {θ n },and{c n } be real sequences in (0,1] satisfying the following conditions: (i) lim n→∞ θ n = 0; (ii)  ∞ n=1 λ n θ n =∞, lim n→∞ (λ (q−1) n /θ n ) =0; (iii) lim n→∞ ((θ n−1 /θ n −1)/λ n θ n ) =0, c n = o(λ n θ n ). Let a sequence {x n } be generated from x 0 ∈ E by x n+1 = Q  1 −λ n  x n + λ n Tx n −λ n θ n  x n −x 0  −c n  x n −u n  , (3.12) for all positive integers n,where{u n }is a bounded sequence of error terms. If either F(T) =∅ or the set K :={x ∈ D : Tx = λx +(1−λ)x 0 for λ>1} is bounded, then there exists d>0 such that whenever λ n ≤ d and c n /λ n θ n ,λ (q−1) n /θ n ≤ d 2 for all n ≥0, {x n } converges strongly to a fixed point of T. C. E. Chidume and H. Zegeye 75 Remark 3.10. Examples of sequences {λ n } and {θ n } satisfying conditions of Corollary 3.9 are as follows: λ n = 2(n +1) −a , θ n = 2(n +1) −b ,andc n = 2(n +1) −1 with 0 <b<aand a + b<1if2≤q<∞, and with 0 <b<a(q −1) and a + b(q −1) < 1if1<q<2. If in Theorem 3.5, T is a self-map of ¯ D, then the projection operator Q is replaced with I, the identity map on E.Moreover,T satisfies condition (2.4). As a consequence, we have the following corollaries. Corollary 3.11. Let D be an open and convex subset of a real uniformly smooth Ba- nach space E.SupposeT : ¯ D → ¯ D is a bounded demicontinuous pseudocontractive mapping. Suppose {λ n }, {θ n },and{c n } are real seque nces in (0, 1] satisfying conditions (i)–(iii) of Theorem 3.5 and λ n (1 + θ n )+c n ≤ 1, ∀n ≥0.Letasequence{x n } be generated from x 0 ∈E by x n+1 =  1 −λ n  x n + λ n Tx n −λ n θ n  x n −x 0  −c n  x n −u n  , (3.13) for all positive integers n,where{u n }is a sequence of bounded error terms. If either F(T) =∅ or the set K :={x ∈ D : Tx = λx +(1−λ)x 0 for λ>1} is bounded, then there exists d>0 such that whenever λ n ≤ d and c n /λ n θ n ,b(λ n )/θ n ≤ d 2 for all n ≥0, {x n } converges strongly to a fixed point of T. Proof. The conditions on λ n , θ n ,andc n imply that the sequence {x n } is well defined. Thus, the proof follows from Theorem 3.5.  If in Theorem 3.5, D is assumed to be b ounded, then the conditions λ n ≤ d and c n /λ n θ n ,b(λ n )/θ n ≤ d 2 for some d>0 which guarantee the boundedness of the sequence {x n } are not needed. In fact, we have the following corollary. Corollary 3.12. Let D be an open convex and bounded subset of a real uniformly smooth Banach space E.SupposeT : ¯ D → E is a bounded demicontinuous pseudocontractive map- ping satisfying the weakly inward condition. Suppose ¯ D is a nonexpansive retract of E with Q as the nonexpansive retraction and let {λ n }, {θ n },and{c n } be real sequences in (0,1) satisfying conditions (i)–(iii) of Theorem 3.5.Letasequence{x n } be generated from x 0 ∈ E by x n+1 = Q  1 −λ n  x n + λ n Tx n −λ n θ n  x n −x 0  −c n  x n −u n  , (3.14) for all positive integers n,where{u n } isasequenceoferrorterms.Then{x n } converges strongly to a fixed point of T. Proof. Since D, and hence ¯ D, is bounded we have that {x n } is bounded. Thus the conclu- sion follows from Theorem 3.5.  Corollary 3.13. Let D be an open convex and bounded subset of a real uniformly smooth Banach space E.SupposeT : ¯ D → ¯ D is a bounded demicontinuous pseudocontractive map- ping. Let {λ n }, {θ n },and{c n } be real sequences in (0, 1] satisfying conditions (i)–(iii) of Theorem 3.5 and λ n (1 + θ n )+c n ≤ 1, ∀n ≥0.Letasequence{x n } be generated from x 0 ∈E 76 Fixed points of demicontinuous pseudocontractive maps by x n+1 =  1 −λ n  x n + λ n Tx n −λ n θ n  x n −x 0  −c n  x n −u n  , (3.15) for all positive integers n,where{u n } isasequenceoferrorterms.Then{x n } converges strongly to a fixed point of T. Remark 3.14. If in Theorem 3.5, D is bounded, T is a self-map, and c n ≡ 1foralln ≥1, that is, the error term is ignored, then the following corollary holds. Corollary 3.15. Let D be an open convex and bounded subset of a real uniformly smooth Banach space E.SupposeT : ¯ D → ¯ D is a bounded demicontinuous pseudocontractive map- ping. Let {λ n } and {θ n } be real sequences in (0,1] satisfying conditions (i)–(iii) of Theorem 3.5 with c n ≡ 0 for all n ≥1 and λ n (1 + θ n ) ≤1,foralln ≥ 0.Letasequence{x n } be gener- ated from x 0 ∈ E by x n+1 =  1 −λ n  x n + λ n Tx n −λ n θ n  x n −x 0  , (3.16) for all positive integers n. Then {x n } converges strongly to a fixed point of T. The following convergence theorem is for the approximation of solution of demicon- tinuous accretive mappings. Theorem 3.16. Let D be an open and convex subset of a real uniformly smooth Banach space E.SupposeA : ¯ D → E is a bounded demicontinuous accretive mapping which satisfies, for some x 0 ∈ D, Ax =t(x −x 0 ) for all x ∈ ∂D and t<0.Suppose ¯ D is a nonexpansive retract of E with Q as the nonexpansive retraction and let {λ n }, {θ n },and{c n } be real sequences in (0,1] satisfying conditions (i)–(iii) of Theorem 3.5.Letasequence{x n } be generated from x 0 ∈ E by x n+1 = Q  x n −λ n  Ax n + θ n  x n −x 0  −c n  x n −u n  , (3.17) for all positive integers n,where{u n } is a s equence of bounded error ter ms. Suppose ei- ther N(A) =∅(N(A) is the null space of A)or the set K :={x ∈ D :(I − A)x = λx + (1 −λ)x 0 for λ>1} is bounded. Then there exists d>0 such that whenever λ n ≤ d and c n /λ n θ n ,b(λ n )/θ n ≤ d 2 for all n ≥0, {x n } converges strongly to a zero of A. Proof. Set T := (I −A). Then, we have that for some x 0 ∈ D,(I − T)x = t(x −x 0 )for x ∈∂D and t<0. This implies that Tx−x 0 = λ(x −x 0 )forallx ∈∂D and λ>1. Moreover, F(T) =∅or the set K ={x ∈ D : Tx = λx +(1−λ)x 0 ,forλ = (1 −t) > 1} is bounded. Therefore, by Theorem 3.5, {x n } converges strongly to x ∗ ∈ F(T). But F(T) = N(A). Hence, {x n } converges strongly to x ∗ ∈ N(A). The proof of the theorem is complete.  The following corollary follows from Theorem 3.16. Corollary 3.17. Let E be a real uniformly smooth Banach space and suppose A : E → E is a bounded demicontinuous accretive mapping. Let {λ n }, {θ n },and{c n } be real sequences in (0,1] satisfying conditions (i)–(iii) of Theorem 3.5.Letasequence{x n } be generated from [...]... 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Wu, Convergence of approximants for demicontinuous pseudo-contractive maps in Hilbert spaces, Nonlinear Anal Ser A: Theory Methods 49 (2002), no 6, 737–746 R H Martin Jr., A global existence theorem for autonomous differential equations in a Banach space, Proc Amer Math Soc 26 (1970), 307–314 G J Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math J 29 (1962), 341–346 C H Morales, Zeros for. .. = xn − λn Axn + θn xn − x0 − cn xn − un , (3.18) for all positive integers n, where {un } is a sequence of bounded error terms If either N(A) = ∅ or the set K := {x ∈ E : (I − A)x = λx + (1 − λ)x0 for λ > 1} is bounded, then there exists d > 0 such that whenever λn ≤ d and cn /λn θn ,b(λn )/θn ≤ d2 for all n ≥ 0, {xn } converges strongly to a point of N(A) Acknowledgment The second author undertook... (1984), no 2, 546–553 C E Chidume: The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy E-mail address: chidume@ictp.trieste.it H Zegeye: The Abdus Salam International Centre for Theoretical Physics, 34014 Trieste, Italy E-mail address: habtuzh@yahoo.com . CONVERGENCE THEOREMS FOR FIXED POINTS OF DEMICONTINUOUS PSEUDOCONTRACTIVE MAPPINGS C. E. CHIDUME AND H. ZEGEYE Received 26 August 2004 Let D be an open subset of a real uniformly smooth. Corporation Fixed Point Theory and Applications 2005:1 (2005) 67–77 DOI: 10.1155/FPTA.2005.67 68 Fixed points of demicontinuous pseudocontractive maps Closely related to the class of pseudocontractive. from H to any closed convex subset D of H. 70 Fixed points of demicontinuous pseudocontractive maps In what follows, we will make use of the following lemma and theorems. Lemma 2.1 [2]. Let {λ n },