|
Browder Fixed-point Theorem
The Browder fixed-point theorem is a refinement of the Banach fixed-point theorem for uniformly convex Banach spaces. It asserts that if K is a nonempty convex closed bounded set in uniformly convex Banach space and f is a mapping of K into itself such that \, f(x)-f(y)\, \leq\, x-y\, (i.e. f is ''non-expansive''), then f has a fixed point. History Following the publication in 1965 of two independent versions of the theorem by Felix Browder and by William Kirk, a new proof of Michael Edelstein showed that, in a uniformly convex Banach space, every iterative sequence f^nx_0 of a non-expansive map f has a unique asymptotic center, which is a fixed point of f. (An ''asymptotic center'' of a sequence (x_k)_, if it exists, is a limit of the Chebyshev centers c_n for truncated sequences (x_k)_.) A stronger property than asymptotic center is Delta-limit of Teck-Cheong Lim, which in the uniformly convex space coincides with the weak limit if the space has the Opial property. See ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Fixed-point Theorem
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922. Statement ''Definition.'' Let (X, d) be a complete metric space. Then a map T : X \to X is called a contraction mapping on ''X'' if there exists q \in non-empty complete metric space with a contraction mapping T : X \to X. Then ''T'' admits a unique Fixed point (mathematics)">fixed-point x^* in ''X'' (i.e. T(x^*) = x^*). Furthermore, x^* can be found as follows: start with an arbitrary element x_0 \in X and define a sequence (x_n)_ by x_n = T(x_) for n \geq 1. Then \li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Convex Space
In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. Definition A uniformly convex space is a normed vector space such that, for every 00 such that for any two vectors with \, x\, = 1 and \, y\, = 1, the condition :\, x-y\, \geq\varepsilon implies that: :\left\, \frac\right\, \leq 1-\delta. Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short. Properties * The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space X is uniformly convex if and only if for every 00 so that, for any two vectors x and y in the closed unit ball (i.e. \, x\, \le 1 and \, y\, \le 1 ) with \, x-y\, \ge \varepsilon , one has \left\, \right\, \le 1-\delta (note that, given \varepsilon , the corresponding value of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function. In physics, the term fixed point can refer to a temperature that can be used as a reproducible reference point, usually defined by a phase change or triple point. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain and the codomain of , and . For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, , has no fixed points, since is never equal to for any real number. In graphical terms, a fixed point means the point is on the line , or in other words the graph of has a point in common with that line. Fixed-point iteration In numerical analysis, ''fixed-point iter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Browder
Felix Earl Browder (; July 31, 1927 – December 10, 2016) was an American mathematician known for his work in nonlinear functional analysis. He received the National Medal of Science in 1999 and was President of the American Mathematical Society until 2000. His two younger brothers also became notable mathematicians, William Browder (an algebraic topologist) and Andrew Browder (a specialist in function algebras). Early life and education Felix Earl Browder was born in 1927 in Moscow, Russia, while his American father Earl Browder, born in Wichita, Kansas, was living and working there. He had gone to the Soviet Union in 1927. His mother was Raissa Berkmann, a Russian Jewish woman from St. Petersburg whom Browder met and married while living in the Soviet Union. As a child, Felix Browder moved with his family to the United States, where his father Earl Browder for a time was head of the American Communist Party. The father ran for US president in 1936 and 1940. A 1999 book by Al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Arthur Kirk
William Arthur ("Art") Kirk was an American mathematician. His research interests include nonlinear functional analysis, the geometry of Banach spaces and metric spaces. In particular, he has made notable contributions to the fixed point theory of metric spaces; for example, he is one of the two namesakes of the Caristi-Kirk fixed point theorem of 1976. He is also known for the Kirk theorem of 1964. He completed his PhD, entitled "Metrization of Surface Curvature", at the University of Missouri in August 1962 under the supervision of Leonard Blumenthal. He then became an assistant professor of mathematics at the University of California, Riverside from 1962 to 1967. Since 1967 he has worked from the University of Iowa, as a full professor of mathematics since 1971 and as department chair from 1985 to 1991. He holds an honorary doctorate from Maria Curie-Skłodowska University Maria Curie-Skłodowska University (MCSU) ( pl, Uniwersytet Marii Curie-Skłodowskiej w Lublinie, UMCS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Center
In geometry, the Chebyshev center of a bounded set Q having non-empty interior is the center of the minimal-radius ball enclosing the entire set Q, or alternatively (and non-equivalently) the center of largest inscribed ball of Q. In the field of parameter estimation, the Chebyshev center approach tries to find an estimator \hat x for x given the feasibility set Q , such that \hat x minimizes the worst possible estimation error for x (e.g. best worst case). Mathematical representation There exist several alternative representations for the Chebyshev center. Consider the set Q and denote its Chebyshev center by \hat. \hat can be computed by solving: : \min_ \left\ with respect to the Euclidean norm \, \cdot\, , or alternatively by solving: : \operatorname \max_ \left\, x - \hat x \right\, ^2. Despite these properties, finding the Chebyshev center may be a hard numerical optimization problem. For example, in the second representation above, the inner maximization i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delta-convergence
In mathematics, Delta-convergence, or Δ-convergence, is a mode of convergence in metric spaces, weaker than the usual metric convergence, and similar to (but distinct from) the weak convergence in Banach spaces. In Hilbert space, Delta-convergence and weak convergence coincide. For a general class of spaces, similarly to weak convergence, every bounded sequence has a Delta-convergent subsequence. Delta convergence was first introduced by Teck-Cheong Lim,T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182. and, soon after, under the name of ''almost convergence,'' by Tadeusz Kuczumow. Definition A sequence (x_k) in a metric space (X,d) is said to be Δ-convergent to x\in X if for every y\in X, \limsup(d(x_k,x)-d(x_k,y))\le 0. Characterization in Banach spaces If X is a uniformly convex and uniformly smooth Banach space, with the duality mapping x\mapsto x^* given by \, x\, =\, x^*\, , \langle x^*,x\rangle=\, x\, ^2, then a sequenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Opial Property
In mathematics, the Opial property is an abstract property of Banach spaces that plays an important role in the study of weak convergence of iterates of mappings of Banach spaces, and of the asymptotic behaviour of nonlinear semigroups. The property is named after the Polish mathematician ZdzisÅ‚aw Opial. Definitions Let (''X'', , , , , ) be a Banach space. ''X'' is said to have the Opial property if, whenever (''x''''n'')''n''∈N is a sequence in ''X'' converging weakly to some ''x''0 ∈ ''X'' and ''x'' ≠ ''x''0, it follows that :\liminf_ \, x_ - x_ \, < \liminf_ \, x_ - x \, . Alternatively, using the contrapositive, this condition may be written as : If ''X'' is the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed-point Theorem
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors claim that results of this kind are amongst the most generally useful in mathematics. In mathematical analysis The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed-point theorem (1911) is a non- constructive result: it says that any continuous function from the closed unit ball in ''n''-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma). For example, the cosine function is continuous in ˆ’1,1and maps it into ˆ’1, 1 and thus must have a fixed point. This is clear when examining a sketched graph of the cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
