Convergence theorems for fixed points of demicontinuous pseudocontractive mappings
- PDF / 596,222 Bytes
- 11 Pages / 468 x 680 pts Page_size
- 102 Downloads / 211 Views
Let D be an open subset of a real uniformly smooth Banach space E. Suppose T : 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 ¯ t ∈ [0,1], satisfying r > 0; (ii) for a given y0 ∈ D, there exists a unique path t → yt ∈ D, yt := tT yt + (1 − t)y0 . Moreover, if F(T) = ∅ or there exists y0 ∈ D such that the set K := { y ∈ D : T y = λy + (1 − λ)y0 for λ > 1} is bounded, then it is proved that, as t → 1− , the path { yt } 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. A mapping T : D → E is called a contraction mapping if there exists L ∈ [0,1) such that Tx − T y ≤ Lx − y for all x, y ∈ D. If L = 1 then T is called nonexpansive. T is called pseudocontractive if there exists j(x − y) ∈ J(x − y) such that
Tx − T y, j(x − y) ≤ x − y 2 ,
∀x, y ∈ K,
(1.1)
∗
where J is the normalized duality mapping from E to 2E defined by
2
Jx := f ∗ ∈ E∗ : x, f ∗ = x2 = f ∗ .
(1.2)
T is called strongly pseudocontractive if there exists k ∈ (0,1) such that
Tx − T y, j(x − y) ≤ kx − y 2 ,
∀x, y ∈ K.
(1.3)
Clearly the class of nonexpansive mappings is a subset of class of pseudocontractive mappings. T is said to be demicontinuous if {xn } ⊆ D and xn → x ∈ D together imply that Txn 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 mappings. 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. An operator A 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, then for each t ∈ (0,1), and for a nonexpansive map T, there exists a unique point xt ∈ D satisfying the condition, xt = tTxt + (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 a self-map, Browder [1] showed that {xt } 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
Data Loading...
