Iterative Solution of Nonlinear Equations Involving Strongly Accretive Operators without the Lipschitz Assumption

AbstractLetEbe a real Banach space with a uniformly convex dual spaceE*. SupposeT:E→Eis a continuous (not necessarily Lipschitzian) strongly accretive map such that (I−T) has bounded range, whereIdenotes the identity operator. It is proved that the Ishikawa iterative sequence converges strongly to the unique solution of equationTx=f,f∈E. Our results extend and complement the recent results obtained by Chidume

Iterative approximation of common fixed points for two quasi-phiphi-nonexpansive mappings in Banach spaces

In this paper, we introduce a new type of a projective algorithm for a pair of quasi-$phi$-nonexpansive mappings. We establish strong convergence theorems of common fixed points in uniformly smooth and strictly convex Banach spaces with the property(K). Our results improve and extend the corresponding results announced by many others

Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings

AbstractIn this paper, a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real Banach space using an implicit iteration scheme recently introduced by Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Fuct. Anal. Optim. 22 (2001) 767–773] in condition αn∈(0,1], and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of Browder–Petryshyn type is obtained. The results presented extend and improve the corresponding results of M.O. Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81]

A modified projective algorithm of common elements for equilibrium problems and fixed point problems in Banach spaces

In this paper, we consider a modified projective algorithm for finding common elements of the set of common fixed points of a finite family of quasi-$phi$-nonexpansive mappings and the set of solutions of an equilibrium problem in uniformly smooth and strictly convex Banach spaces with the property(K). Our results improve and extend the corresponding results announced by many others

Strong convergence and control condition of modified Halpern iterations in Banach spaces

Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gâteaux differentiable norm. Let T∈ΓC and f∈ΠC. Assume that {xt} converges strongly to a fixed point z of T as t→0, where xt is the unique element of C which satisfies xt=tf(xt)+(1−t)Txt. Let {αn} and {βn} be two real sequences in (0,1) which satisfy the following conditions: (C1)lim⁡n→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1. For arbitrary x0∈C, let the sequence {xn} be defined iteratively by yn=αnf(xn)+(1−αn)Txn, n≥0, xn+1=βnxn+(1−βn)yn, n≥0. Then {xn} converges strongly to a fixed point of T

Solution of Nonlinear Elliptic Boundary Value Problems and Its Iterative Construction

We study a kind of nonlinear elliptic boundary value problems with generalized p-Laplacian operator. The unique solution is proved to be existing and the relationship between this solution and the zero point of a suitably defined nonlinear maximal monotone operator is investigated. Moreover, an iterative scheme is constructed to be strongly convergent to the unique solution. The work done in this paper is meaningful since it combines the knowledge of ranges for nonlinear operators, zero point of nonlinear operators, iterative schemes, and boundary value problems together. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work

Doppler Wind Lidar From UV to NIR: A Review With Case Study Examples

Doppler wind lidar (DWL) uses the optical Doppler effect to measure atmospheric wind speed with high spatial-temporal resolution and long detection range and has been widely applied in scientific research and engineering applications. With the development of related technology, especially laser and detector technology, the performance of the DWL has significantly improved for the past few decades. DWL utilizes different principles and different tracers to sense the wind speed from the ground to the mesosphere, which leads to the difference in choosing the laser working wavelength. This article will review the working wavelength consideration of DWL, and typical DWLs will present from ultraviolet to near-infrared, after which three typical applications will be shown

The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

Let E be an arbitrary uniformly smooth real Banach space, let D be a nonempty closed convex subset of E, and let T:D→D be a uniformly generalized Lipschitz generalized asymptotically Φ-strongly pseudocontractive mapping with q∈F(T)≠∅. Let {an},{bn},{cn},{dn} be four real sequences in [0,1] and satisfy the conditions: (i) an+cn≤1, bn+dn≤1; (ii) an,bn,dn→0 as n→∞ and cn=o(an); (iii) Σn=0∞an=∞. For some x0,z0∈D, let {un},{vn},{wn} be any bounded sequences in D, and let {xn},{zn} be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of {xn} is equivalent to that of {zn}