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Document Type |
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Article In Journal |
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Document Title |
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On Fixed point theorem related to banach s contraction principal على نظرية النقطة الثابتة المتعلقة الرئيسية الانكماش باناخ ق |
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Subject |
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On Fixed point theorem related to banach s contraction principal |
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Document Language |
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English |
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Abstract |
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fixed-point x* in X (i.e. T(x*) = x*). Furthermore, x* can be found as follows: start with an arbitrary element x0 in X and define a sequence {xn} by xn = T(xn−1), then xn → x*.
Remark 1. The following inequalities are equivalent and describes the speed of convergence:
Any such value of q is called a Lipschitz constant for T, and the smallest one is sometimes called "the best Lipschitz constant" of T.
Remark 2. d(T(x), T(y)) < d(x, y) for all x ≠ y is in general not enough to ensure the existence of a fixed point, as is shown by the map T : [1, ∞) → [1, ∞), T(x) = x + 1/x, which lacks a fixed point. However, if X is compact, then this weaker assumption does imply the existence and uniqueness of a fixed point, that can be easily found as a minimizer of d(x, T(x)), indeed, a minimizer exists by compactness, and has to be a fixed point of T. It then easily follows that the fixed point is the limit of any sequence of iterations of T.
Remark 3. When using the theorem in practice, the most difficult part is typically to define X properly so that T(X) ⊆ X. |
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ISSN |
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0918-5402 |
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Journal Name |
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journal of fractional calculus |
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Volume |
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23 |
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Issue Number |
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1 |
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Publishing Year |
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1423 AH
2003 AD |
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Article Type |
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Article |
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Added Date |
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Wednesday, March 20, 2013 |
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Researchers
| سهام جلال الصياد | alsayad, Siham Jalal | Researcher | | |
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