<P>THIS BOOK SHOWS THE IMPORTANCE OF STUDYING SEMILOCAL CONVERGENCE IN ITERATIVE METHODS THROUGH NEWTON'S METHOD AND ADDRESSES THE MOST IMPORTANT ASPECTS OF THE KANTOROVICH'S THEORY INCLUDING IMPLICATED STUDIES. KANTOROVICH'S THEORY FOR NEWTON'S METHOD USED TECHNIQUES OF FUNCTIONAL ANALYSIS TO PROVE THE SEMILOCAL CONVERGENCE OF THE METHOD BY MEANS OF THE WELL-KNOWN MAJORANT PRINCIPLE. TO GAIN A DEEPER UNDERSTANDING OF THESE TECHNIQUES THE AUTHORS RETURN TO THE BEGINNING AND PRESENT A DEEP-DETAILED APPROACH OF KANTOROVICH'S THEORY FOR NEWTON'S METHOD, WHERE THEY INCLUDE OLD RESULTS, FOR A HISTORICAL PERSPECTIVE AND FOR COMPARISONS WITH NEW RESULTS, REFINE OLD RESULTS, AND PROVE THEIR MOST RELEVANT RESULTS, WHERE ALTERNATIVE APPROACHES LEADING TO NEW SUFFICIENT SEMILOCAL CONVERGENCE CRITERIA FOR NEWTON'S METHOD ARE GIVEN. THE BOOK CONTAINS MANY NUMERICAL EXAMPLES INVOLVING NONLINEAR INTEGRAL EQUATIONS, TWO BOUNDARY VALUE PROBLEMS AND SYSTEMS OF NONLINEAR EQUATIONS RELATED TO NUMEROUS PHYSICAL PHENOMENA. THE BOOK IS ADDRESSED TO RESEARCHERS IN COMPUTATIONAL SCIENCES, IN GENERAL, AND IN APPROXIMATION OF SOLUTIONS OF NONLINEAR PROBLEMS, IN PARTICULAR.</P>

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“The text is easy to follow with full technical details given. Historical remarks are given throughout, which makes the reading especially interesting. The book also contains some numerical examples illustrating the theoretical analysis. It is a useful reference for researchers working on Newton method in Banach spaces.” (Bangti Jin, zbMATH 1376.65088, 2018)
“This book is well written and will be useful to researchers interested in the theory of Newton’s method in Banach spaces. Two of its merits have to be mentioned explicitly: the authors offer all details for the proofs of all the results presented in the book, and, moreover, they also include significant material from their own results on the theory of Newton's method which were carried out over many years of research work.” (Vasile Berinde, Mathematical Reviews, March, 2018)

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