C. E. Chidume

Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces

Let be a nonempty, closed, and convex subset of a real Hilbert space . Suppose that is a multivalued strictly pseudocontractive mapping such that . A Krasnoselskii-type iteration sequence is constructed and shown to be an approximate fixed point sequence of ; that is, holds. Convergence theorems are also proved under appropriate additional conditions.

Approximation of Solutions of Nonlinear Integral Equations of Hammerstein Type

Suppose that 𝐻 is a real Hilbert space and 𝐹,𝐾∶𝐻→𝐻 are bounded monotone maps with 𝐷(𝐾)=𝐷(𝐹)=𝐻. Let 𝑢∗ denote a solution of the Hammerstein equation 𝑢

Strong Convergence Theorems for Zeros of Bounded Maximal Monotone Nonlinear Operators

An iteration process studied by Chidume and Zegeye 2002 is proved to converge strongly to a solution of the equation where A is a bounded m-accretive operator on certain real Banach spaces E that include spaces The iteration process does not involve the computation of the resolvent at any step of the process and does not involve the projection of an initial vector onto the intersection of two convex subsets of E, setbacks associated with the classical proximal point algorithm of Martinet 1970, Rockafellar 1976 and its modifications by various authors for approximating of a solution of this equation. The ideas of the iteration process are applied to approximate fixed points of uniformly continuous pseudocontractive maps.

Krasnoselskii-type algorithm for family of multi-valued strictly pseudo-contractive mappings

AbstractA Krasnoselskii-type algorithm is constructed and the sequence of the algorithm is proved to be an approximate fixed point sequence for a common fixed point of a suitable finite family of multi-valued strictly pseudo-contractive mappings in a real Hilbert space. Under some mild additional compactness-type condition on the operators, the sequence is proved to converge strongly to a common fixed point of the family. MSC:47H04, 47H06, 47H15, 47H17, 47J25.

An iterative method for solving nonlinear integral equations of Hammerstein type

Suppose H is a real Hilbert space and F,K:H->H are continuous bounded monotone maps with D(K)=D(F)=H. Assume that the Hammerstein equation u+KFu=0 has a solution. An explicit iteration process is proved to converge strongly to a solution of this equation. No invertibility assumption is imposed on K and the operator F is not restricted to be angle-bounded. Our theorem complements the Galerkin method of Brezis and Browder to provide methods for approximating solutions of nonlinear integral equations of Hammerstein type.

Strong and -Convergence Theorems for Common Fixed Points of a Finite Family of Multivalued Demicontractive Mappings in CAT Spaces

Let K be a nonempty closed and convex subset of a complete CAT(0) space. Let , be a family of multivalued demicontractive mappings such that . A Krasnoselskii-type iterative sequence is shown to -converge to a common fixed point of the family . Strong convergence theorems are also proved under some additional conditions. Our theorems complement and extend several recent important results on approximation of fixed points of certain nonlinear mappings in CAT spaces. Furthermore, our method of the proof is of special interest.

A New Iterative Algorithm for Zeros of Generalized Phi-strongly Monotone and Bounded Maps with Application

Let E be a uniformly smooth and uniformly convex real Banach space and A : E → E∗ be a generalized Φ-strongly monotone and bounded map with A−1(0) 6= ∅. A new iterative process is constructed and proved to converge strongly to the unique solution of the equation Au = 0. An application to convex minimization problem is given. Furthermore, the technique of proof is of independent interest.

Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings

Let q>1 and let K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Let T:K→CB(K) be a multi-valued strictly pseudo-contractive map with a nonempty fixed point set. A Krasnoselskii-type iteration sequence {xn} is constructed and proved to be an approximate fixed point sequence of T, i.e., limn→∞d(xn,Txn)=0. This result is then applied to prove strong convergence theorems for a fixed point of T under additional appropriate conditions. Our theorems improve several important well-known results.MSC:47H04, 47H06, 47H15, 47H17, 47J25.

An algorithm for computing zeros of generalized phi-strongly monotone and bounded maps in classical Banach spaces

Let , , and be a generalized -strongly monotone and bounded map with . An iterative process is constructed and proved to converge strongly to the unique solution of the equation . In the special case in which is the subdifferential of a proper convex function , a solution of corresponds to a minimizer of . Furthermore, our technique of proof is of independent interest.

Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical banach spaces

AbstractLet