The purpose of this short note is to present some corrections and clarifications concerning the proof of the main result given in the above mentioned paper.
Point TheoremInitial PointDifferential GeometryFixed Point TheoremPoint Theory
Concerning the text and the proof of the main result (Theorem 2.2), we would like to do the following corrections and clarifications:
(1) In Theorem 2.2, an additional hypothesis is needed, namely:
3. There exists such that
(2) On page 3, the fact that the sequence is bounded follows now from the above mentioned hypothesis, using the following remark: by 3., for each , we have that
Indeed, let and suppose that . Let (where ). Let us choose . Then
(3) On page 5, to prove that the Picard iteration converges to , we have to do as follows.
Let us show now that the Picard iteration converges to for any initial point . We know that f has a unique fixed point (denoted by ) and the sequence converges to a certain . We will show that y is also a fixed point of f. For this purpose, we have
This shows that y is a fixed point of f and, thus, .
Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, Jeddah, Saudi Arabia
Department of Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
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