By Mohamed A. Khamsi, Wojciech M. Kozlowski
Presents cutting-edge developments within the box of modular functionality theory
Provides a self-contained review of the topic
Includes open difficulties, broad bibliographic references, and proposals for extra improvement
This monograph presents a concise advent to the most effects and strategies of the mounted aspect thought in modular functionality areas. Modular functionality areas are average generalizations of either functionality and series editions of many very important areas like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii areas, and others. regularly, fairly in functions to vital operators, approximation and stuck aspect effects, modular sort stipulations are even more ordinary and will be extra simply established than their metric or norm opposite numbers. There also are very important effects that may be proved merely utilizing the equipment of modular functionality areas. the cloth is gifted in a scientific and rigorous demeanour that enables readers to understand the main principles and to realize a operating wisdom of the idea. although the paintings is basically self-contained, wide bibliographic references are integrated, and open difficulties and additional improvement instructions are recommended whilst applicable.
The monograph is focused quite often on the mathematical examine neighborhood however it is additionally obtainable to graduate scholars attracted to practical research and its functions. it could possibly additionally function a textual content for a complicated path in mounted element idea of mappings appearing in modular functionality spaces.
Content point » Research
Keywords » fastened aspect - Iterative tactics - Metric mounted element thought - Modular functionality area - Modular Metric area - Orlicz area
Read or Download Fixed Point Theory in Modular Function Spaces PDF
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Additional resources for Fixed Point Theory in Modular Function Spaces
Sample text
Then the mapping f : K → K defined by setting f (x) = (x + T (x))/2 is asymptotically regular. Proof. Select x0 ∈ K, let y0 = f (x0 ), and having defined xn take yn = f (xn ) and set xn+1 = (xn + yn )/2, n = 1, 2, · · ·. 2. Thus for each i, n ∈ N, 1+ n 2 yi − xi ≤ 2n yi − xi − yi+n − xi+n + yi+n − xi . Since { xn − yn } = { xn − f (xn ) } is monotone nonincreasing, there exists a number r ≥ 0 such that lim xn+1 − f (xn+1 ) = r. Now let i → ∞ in the above inequality. n→∞ Then n r ≤ diam(K). 1+ 2 Clearly this implies r = 0.
26. Assume that (M, d) is complete and uniformly convex. Let C ⊂ M be nonempty, convex, and closed. Let x ∈ M be such that d(x,C) < ∞. , there exists a unique x0 ∈ C such that d(x, x0 ) = d(x,C). The following result gives the analog result to the well-known theorem that states that any uniformly convex Banach space is reflexive. 1 in [82]. 27. If (M, d) is complete and uniformly convex, then (M, d) has the property (R). 7. Note that any hyperbolic metric space M which satisfies the property (R) is complete.
113] Let (M, d) be hyperbolic metric space which is 2-uniformly convex. Let C be a nonempty, closed, convex, and bounded subset of M. Let T : C → C be uniformly Lipschitzian with λ (T ) < 1+ 1 + 8cM N(M)2 2 1/2 . Then T has a fixed point in C. 9 More on Convexity Structures Kirk’s fixed point theorem involves some kind of compactness and the normal structure property. Both properties are easy to define in metric spaces. The first attempt to extend Kirk’s theorem to metric spaces was done by Takahashi [197].