Minimum-Norm Fixed Point of Pseudocontractive Mappings

and Applied Analysis 3 where K and Q are nonempty closed convex subsets of the infinite-dimension real Hilbert spaces H1 and H2, respectively, and A is bounded linear mapping from H1 to H2. Equation 1.9 models many applied problems arising from image reconstructions and learning theory see, e.g., 4 . Someworks on the finite dimensional setting with relevant projectionmethods for solving image recovery problems can be found in 5–7 . Defining the proximity function f by f x : 1 2 ∥ ∥Ax − PQAx ∥ ∥ 2 , 1.10 we consider the convex optimization problem: min x∈K f x : min x∈K 1 2 ∥ ∥Ax − PQAx ∥ ∥ 2 . 1.11 It is clear that x∗ is a solution to the split feasibility problem 1.9 if and only if x∗ ∈ K and Ax∗ − PQAx∗ 0 which is the minimum-norm solution of the minimization problem 1.11 . Motivated by the above split feasibility problem, we study the general case of finding the minimum-norm fixed point of a pseudocontractive mapping T : K → K, that is, we find minimum norm fixed point of T which satisfies x∗ ∈ F T such that ‖x∗‖ min{‖x‖ : x ∈ F T }. 1.12 Let T : K → K be a nonexpansive self-mapping on closed convex subset K of a Banach space E. For a given u ∈ K and for a given t ∈ 0, 1 define a contraction Tt : K → K by Ttx 1 − t u tTx, x ∈ K. 1.13 By Banach contraction principle, it yields a fixed point zt ∈ K of Tt, that is, zt is the unique solution of the equation: zt 1 − t u tTzt. 1.14 Browder 8 proved that as t → 1, zt converges strongly to a fixed point of T which is closer to u, that is, the nearest point projection of u onto F T . In 1980, Reich 9 extended the result of Browder to a more general Banach spaces. Furthermore, Takahashi and Ueda 10 and Morales and Jung 11 improved results of Reich 9 to the class of continuous pseudocontractive mappings. For other results on pseudocontractive mappings, we refer to 12–15 . We note that the above methods can be used to find the minimum-norm fixed point x∗ of T if 0 ∈ K. However, if 0 / ∈ K neither Browder’s, Reich’s, Takahashi and Ueda’s, nor Morales and Jung’s method works to find minimum-norm fixed point of T . Our concern is now the following: is it possible to construct a scheme, implicit or explicit, which converges strongly to the minimum-norm fixed point of T for any closed convex domain K of T? 4 Abstract and Applied Analysis In this direction, Yang et al. 4 introduced an implicit and explicit iteration processes which converge strongly to the minimum-norm fixed point of nonexpansive self-mapping T , in real Hilbert spaces. In fact, they proved the following theorems. Theorem YLY1 see 4 . Let K be a nonempty closed convex subset of a real Hilbert space H and T : K → K a nonexpansive mapping with F T / ∅. For β ∈ 0, 1 and each t ∈ 0, 1 , let yt be defined as the unique solution of fixed point equation: yt βTyt ( 1 − β)PK [ 1 − t yt ] , t ∈ 0, 1 . 1.15 Then the net {yt} converges strongly, as t → 0, to the minimum-norm fixed point of T . Theorem YLY2 see 4 . Let K be a nonempty closed convex subset of a real Hilbert space H, and let T : K → K be a nonexpansive mapping with F T / ∅. For a given x0 ∈ K, define a sequence {xn} iteratively by xn 1 βTxn ( 1 − β)PK 1 − αn xn , n ≥ 1, 1.16 where β ∈ 0, 1 and αn ∈ 0, 1 , satisfying certain conditions. Then the sequence {xn} converges strongly to the minimum-norm fixed point of T . A natural question arises whether the above theorems can be extended to a more general class of pseudocontractive mappings or not. Let K be a closed convex subset a real Hilbert space H and let T : K → K be continuous pseudocontractive mapping. It is our purpose in this paper to prove that for β ∈ 0, 1 and each t ∈ 0, 1 , there exists a sequence {yt} ⊂ K satisfying yt βPK 1 − t yt 1 − β T yt which converges strongly, as t → 0 , to the minimum-norm fixed point of T . Moreover, we provide an explicit iteration process which converges strongly to the minimum-norm fixed point of T provided that T is Lipschitz. Our theorems improve Theorem YLY1 and Theorem YLY2 of Yang et al. 4 and Theorems 3.1, and 3.2 of Cai et al. 16 . 2. Preliminaries In what follows, we shall make use of the following lemmas. Lemma 2.1 see 11 . Let H be a real Hilbert space. Then, for any given x, y ∈ H, the following inequality holds: ∥ ∥x y ∥ ∥ 2 ≤ ‖x‖ 2〈y, x y〉. 2.1 Lemma 2.2 see 17 . Let K be a closed and convex subset of a real Hilbert space H. Let x ∈ H. Then x0 PKx if and only if 〈z − x0, x − x0〉 ≤ 0, ∀z ∈ K. 2.2 Abstract and Applied Analysis 5 Lemma 2.3 see 18 . Let {λn}, {αn}, and {γn} be sequences of nonnegative numbers satisfying the conditions: limn→∞αn 0, ∑∞ n 1 αn ∞, and γn/αn → 0, as n → ∞. Let the recursive inequality: λn 1 ≤ λn − αnψ λn 1 γn, n 1, 2, . . . , 2.3and Applied Analysis 5 Lemma 2.3 see 18 . Let {λn}, {αn}, and {γn} be sequences of nonnegative numbers satisfying the conditions: limn→∞αn 0, ∑∞ n 1 αn ∞, and γn/αn → 0, as n → ∞. Let the recursive inequality: λn 1 ≤ λn − αnψ λn 1 γn, n 1, 2, . . . , 2.3 be given where ψ : 0,∞ → 0,∞ is a strictly increasing function such that it is positive on 0,∞ and ψ 0 0. Then λn → 0, as n → ∞. Lemma 2.4 see 3 . Let H be a real Hilbert space, K be a closed convex subset of H and T : K → K be a continuous pseudocontractive mapping, then i F T is closed convex subset of K; ii I − T is demiclosed at zero, that is, if {xn} is a sequence in K such that xn ⇀ x and Txn − xn → 0, as n → ∞, then x T x . Lemma 2.5 see 19 . Let H be a real Hilbert space. Then for all x, y ∈ H and α ∈ 0, 1 , the following equality holds: ‖αx 1 − α x‖ α‖x‖ 1 − α ∥∥y∥∥2 − α 1 − α ∥∥x − y∥∥2. 2.4 3. Main Results Theorem 3.1. LetK be a nonempty closed and convex subset of a real Hilbert spaceH. Let T : K → K be a continuous pseudocontractive mapping with F T / ∅. Then for β ∈ 0, 1 and each t ∈ 0, 1 , there exists a sequence {yt} ⊂ K satisfying the following condition: yt βPK [ 1 − t yt ] ( 1 − β)T(yt ) 3.1 and the net {yt} converges strongly, as t → 0 , to the minimum-norm fixed point of T . Proof. For β ∈ 0, 1 and each t ∈ 0, 1 let Tt y : βPK 1 − t y 1 − β T y . Then using nonexpansiveness of PK and pseudocontractivity of T , for x, y ∈ K, we have that 〈 Ttx − Tty, x − y 〉 β 〈 PK 1 − t x − PK [ 1 − t y], x − y〉 ( 1 − β)〈T x − T(y), x − y〉 ≤ β 1 − t ∥∥x − y∥∥2 (1 − β)∥∥x − y∥∥2 ≤ (1 − tβ)∥∥x − y∥∥2. 3.2 This implies that Tt is strongly pseudocontractive on K. Thus, by Corollary 1 of 20 Tt has a unique fixed point, yt, in K. This means that the equation: yt βPK [ 1 − t yt ] ( 1 − β)T(yt ) 3.3 6 Abstract and Applied Analysis has a unique solution for each t ∈ 0, 1 . Furthermore, since F T / ∅, for y∗ ∈ F T , we have that ∥ yt − y∗ ∥ ∥ 2 〈 βPK [ 1 − t yt ] ( 1 − β)Tyt − y∗, yt − y∗ 〉 β 〈 PK [ 1 − t yt ] − PKy∗, yt − y∗ 〉 ( 1 − β)〈Tyt − Ty∗, yt − y∗ 〉 ≤ β∥∥ 1 − t yt − y∗ ∥ ∥ · ∥∥yt − y∗ ∥ ∥ ( 1 − β)∥∥yt − y∗ ∥ ∥ 2 ≤ β[ 1 − t ∥∥yt − y∗ ∥ ∥ t ∥ ∥y∗ ∥ ∥ ]∥ yt − y∗ ∥ ∥ ( 1 − β)∥∥yt − y∗ ∥ ∥ 2 , 3.4