Picard-like iterations for nonlinear equations involving <inline-formula><graphic file="1029-242X-2002-703904-i1.gif"/></inline-formula>-accretive operators

• Main Author: Moore Chika
• Format: Article
• Language: English
• Published: SpringerOpen 2002-01-01
• Series: Journal of Inequalities and Applications
• Subjects: <it>K</it>-accretive; Normed linear spaces; Picard-like iterations; Strong convergence; Nonlinear equations
• Online Access: http://www.journalofinequalitiesandapplications.com/content/7/703904
• ISSN: 1025-5834; 1029-242X

<p/> <p>Let <inline-formula><graphic file="1029-242X-2002-703904-i2.gif"/></inline-formula> be an arbitrary real normed linear space and let <inline-formula><graphic file="1029-242X-2002-703904-i3.gif"/></inline-formula> be a <inline-formula><graphic file="1029-242X-2002-703904-i4.gif"/></inline-formula>-Lipschitz strongly <inline-formula><graphic file="1029-242X-2002-703904-i5.gif"/></inline-formula>-accretive operator. It is proved that Picard-like iteration processes converge strongly to the unique solutions of the operator equations <inline-formula><graphic file="1029-242X-2002-703904-i6.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-2002-703904-i7.gif"/></inline-formula> where <inline-formula><graphic file="1029-242X-2002-703904-i8.gif"/></inline-formula> is an arbitrary but fixed vector. Related results deal with the strong convergence of Picard-like iteration processes to the unique solution of equations involving linear <inline-formula><graphic file="1029-242X-2002-703904-i9.gif"/></inline-formula>-positive definite ( <inline-formula><graphic file="1029-242X-2002-703904-i10.gif"/></inline-formula>-p.d) operators. Nontrivial examples, indicating that this class of mappings properly contains the classes of nonlinear accretive, dissipative and linear <inline-formula><graphic file="1029-242X-2002-703904-i11.gif"/></inline-formula>-p.d. operators, are also given.</p>