Contracting Mapping on Normed Linear Space

Summary In this article, we described the contracting mapping on normed linear space. Furthermore, we applied that mapping to ordinary differential equations on real normed space. Our method is based on the one presented by Schwarz [29].

Let f be a function. We say that f has unique fixpoint if and only if: (Def. 1) There exists a set x such that x is a fixpoint of f and for every set y such that y is a fixpoint of f holds x = y. Next we state two propositions: (1) Every set x is a fixpoint of { x, x }.
(2) For all sets x, y, z such that x is a fixpoint of { y, z } holds x = y.
Let x be a set. Observe that { x, x } has unique fixpoint. Next we state three propositions: (3) Let X be a real normed space and x be a point of X. If for every real number e such that e > 0 holds x < e, then x = 0 X . (4) Let X be a real normed space and x, y be points of X. If for every real number e such that e > 0 holds x − y < e, then x = y. (5) For all real numbers K, L, e such that 0 < K < 1 and 0 < e there exists a natural number n such that |L · K n | < e.
Let X be a real normed space. Note that every function from X into X which is constant is also contraction.
Let X be a real Banach space. One can verify that every function from X into X which is contraction also has unique fixpoint.
One can prove the following three propositions: (6) Let X be a real Banach space and f be a function from X into X.
Suppose f is contraction. Then there exists a point x 1 of X such that f (x 1 ) = x 1 and for every point x of X such that f (x) = x holds x 1 = x. (7) Let X be a real Banach space and f be a function from X into X such that there exists a natural number n 0 such that f n 0 is contraction. Then f has unique fixpoint. (8) Let X be a real Banach space and f be a function from X into X.
Given an element n 0 of N such that f n 0 is contraction. Then there exists a point x 1 of X such that f (x 1 ) = x 1 and for every point x of X such that f (x) = x holds x 1 = x.

The Real Banach Space C([a,b],X)
We now state the proposition (9) Let X be a non empty closed interval subset of R, Y be a real normed space, and f be a continuous partial function from R to Y . If dom f = X, then f is a bounded function from X into Y .
Let X be a non empty closed interval subset of R and let Y be a real normed space. The continuous functions of X and Y yields a subset of the set of bounded real sequences from X into Y and is defined by the condition (Def. 2). Let X be a non empty closed interval subset of R and let Y be a real normed space. Note that the continuous functions of X and Y is non empty.
Let X be a non empty closed interval subset of R and let Y be a real normed space. Observe that the continuous functions of X and Y is linearly closed.
Let X be a non empty closed interval subset of R and let Y be a real normed space. The R-vector space of continuous functions of X and Y yielding a strict real linear space is defined by the condition (Def. 3). (Def. 3) The R-vector space of continuous functions of X and Y = the continuous functions of X and Y , Zero(the continuous functions of X and Y , the set of bounded real sequences from X into Y ), Add(the continuous functions of X and Y , the set of bounded real sequences from X into Y ), Mult(the continuous functions of X and Y , the set of bounded real sequences from X into Y ) .
Let X be a non empty closed interval subset of R and let Y be a real normed space. Observe that the R-vector space of continuous functions of X and Y is Abelian, add-associative, right zeroed, right complementable, vector distributive, scalar distributive, scalar associative, and scalar unital.
One can prove the following three propositions: (10) Let X be a non empty closed interval subset of R, Y be a real normed space, f , g, h be vectors of the R-vector space of continuous functions of X and Y , and f 9 , g 9 , h 9 be continuous partial functions from R to Y . Suppose f 9 = f and g 9 = g and h 9 = h and dom f 9 = X and dom g 9 = X and dom h 9 = X. Then h = f + g if and only if for every element x of X holds (h 9 ) x = (f 9 ) x + (g 9 ) x .
(11) Let X be a non empty closed interval subset of R, Y be a real normed space, f , h be vectors of the R-vector space of continuous functions of X and Y , and f 9 , h 9 be continuous partial functions from R to Y . Suppose f 9 = f and h 9 = h and dom f 9 = X and dom h 9 = X. Then h = a · f if and only if for every element x of X holds (h 9 ) x = a · (f 9 ) x .
(12) Let X be a non empty closed interval subset of R and Y be a real normed space. Then 0 the R-vector space of continuous functions of X and Let X be a non empty closed interval subset of R and let Y be a real normed space. The continuous functions norm of X and Y yields a function from the continuous functions of X and Y into R and is defined as follows: space, and let f be a set. Let us assume that f ∈ the continuous functions of X and Y . The functor modetrans(f, X, Y ) yielding a continuous partial function from R to Y is defined by: Let X be a non empty closed interval subset of R and let Y be a real normed space. The R-norm space of continuous functions of X and Y yields a strict non empty normed structure and is defined by the condition (Def. 6).
(Def. 6) The R-norm space of continuous functions of X and Y = the continuous functions of X and Y , Zero(the continuous functions of X and Y , the set of bounded real sequences from X into Y ), Add(the continuous functions of X and Y , the set of bounded real sequences from X into Y ), Mult(the continuous functions of X and Y , the set of bounded real sequences from X into Y ), the continuous functions norm of X and Y .
We now state several propositions: (13) Let X be a non empty closed interval subset of R, Y be a real normed space, and f be a continuous partial Let X be a non empty closed interval subset of R and Y be a real normed space. Then X −→ 0 Y = 0 the R-norm space of continuous functions of X and Y .
(15) Let X be a non empty closed interval subset of R, Y be a real normed space, f , g, h be points of the R-norm space of continuous functions of X and Y , and f 9 , g 9 , h 9 be continuous partial functions from R to Y . Suppose f 9 = f and g 9 = g and h 9 = h and dom f 9 = X and dom g 9 = X and dom h 9 = X. Then h = f + g if and only if for every element x of X holds (h 9 ) x = (f 9 ) x + (g 9 ) x .
(16) Let X be a non empty closed interval subset of R, Y be a real normed space, f , h be points of the R-norm space of continuous functions of X and Y , and f 9 , h 9 be continuous partial functions from R to Y . Suppose f 9 = f and h 9 = h and dom f 9 = X and dom h 9 = X. Then h = a · f if and only if for every element (17) Let X be a non empty closed interval subset of R, Y be a real normed space, f be a point of the R-norm space of continuous functions of X and Y , and g be a point of the real normed space of bounded functions from X into Y . If f = g, then f = g .
(18) Let X be a non empty closed interval subset of R, Y be a real normed space, f , g be points of the R-norm space of continuous functions of X and Y , and f 1 , g 1 be points of the real normed space of bounded functions from X into Y . If f 1 = f and g 1 = g, then f + g = f 1 + g 1 . Y , and f 1 be a point of the real normed space of bounded functions from Let X be a non empty closed interval subset of R and let Y be a real normed space. Observe that the R-norm space of continuous functions of X and Y is reflexive, discernible, real normed space-like, vector distributive, scalar distributive, scalar associative, scalar unital, Abelian, add-associative, right zeroed, and right complementable.
One can prove the following propositions: (20) Let X be a non empty closed interval subset of R, Y be a real normed space, f , g, h be points of the R-norm space of continuous functions of X and Y , and f 9 , g 9 , h 9 be continuous partial functions from R to Y .
Suppose f 9 = f and g 9 = g and h 9 = h and dom f 9 = X and dom g 9 = X and dom h 9 = X. Then h = f − g if and only if for every element x of X holds (h 9 ) x = (f 9 ) x − (g 9 ) x .
(21) Let X be a non empty closed interval subset of R, Y be a real normed space, f , g be points of the R-norm space of continuous functions of X and Y , and f 1 , g 1 be points of the real normed space of bounded functions from X into Y . If f 1 = f and Let X be a non empty closed interval subset of R and let Y be a real normed space. Note that there exists a subset of the real normed space of bounded functions from X into Y which is closed.
The following two propositions are true: (22) Let X be a non empty closed interval subset of R and Y be a real normed space. Then the continuous functions of X and Y is a closed subset of the real normed space of bounded functions from X into Y .
(23) Let X be a non empty closed interval subset of R, Y be a real normed space, and s 1 be a sequence of the R-norm space of continuous functions of X and Y . Suppose Y is complete and s 1 is Cauchy sequence by norm. Then s 1 is convergent.
Let X be a non empty closed interval subset of R and let Y be a real Banach space. One can check that the R-norm space of continuous functions of X and Y is complete.
We now state four propositions: (24) Let X be a non empty closed interval subset of R, Y be a real normed space, v be a point of the R-norm space of continuous functions of X and Y , and g be a partial function from R to Y . If g = v, then for every real number t such that t ∈ X holds g t ≤ v .
(25) Let X be a non empty closed interval subset of R, Y be a real normed space, K be a real number, v be a point of the R-norm space of continuous functions of X and Y , and g be a partial function from R to Y . Suppose via Shinshu U Lib g = v and for every real number t such that t ∈ X holds g t ≤ K. Then v ≤ K.
(26) Let X be a non empty closed interval subset of R, Y be a real normed space, v 1 , v 2 be points of the R-norm space of continuous functions of X and Y , and g 1 , g 2 be partial functions from R to Y . Suppose g 1 = v 1 and (27) Let X be a non empty closed interval subset of R, Y be a real normed space, K be a real number, v 1 , v 2 be points of the R-norm space of continuous functions of X and Y , and g 1 , g 2 be partial functions from R to Y . Suppose g 1 = v 1 and g 2 = v 2 and for every real number t such that t ∈ X holds (g

Differential Equations
The following propositions are true: (28) Let n, i be natural numbers, f be a partial function from R to R n , and A be a subset of R. In the sequel Z denotes an open subset of R, y 0 denotes a vector of E n , · , and G denotes a function from E n , · into E n , · .
One can prove the following propositions: (36) Let f be a continuous partial function from R to E n , · and g be a partial function from R to E n , · . and g is differentiable on Z and for every real number t such that t ∈ Z holds g (t) = f t .
(39) Let g be a partial function from R to E n , · , x 0 be a real number, and i be a natural number. Suppose 1 ≤ i ≤ n and g is differentiable in x 0 . Then proj(i, n) · g is differentiable in x 0 and (proj(i, n))(g (x 0 )) = (proj(i, n) · g) (x 0 ).
(40) Let f be a partial function from R to E n , · and X be a set. Suppose that for every natural number i such that 1 ≤ i ≤ n holds (proj(i, n)·f ) X is constant. Then f X is constant. (43) Let y, G 1 be continuous partial functions from R to E n , · and g be a partial function from R to E n , · . Suppose that a < b and Z = ]a, b[ and dom y = [a, b] and dom g = [a, b] and dom G 1 = [a, b] and y is differentiable on Z and y a = y 0 and for every real number t such that t ∈ Z holds y (t) = (G 1 ) t and for every real number t such that t ∈ [a, b] holds g(t) = y 0 + t a G 1 (x)dx. Then y = g.
(44) Let a, b, c, d be real numbers and f be a partial function from R to E n , · . Suppose that a ≤ b and f is integrable on [a, b] and f is integrable on [a, b] and f [a, b] is bounded and [a, b] ⊆ dom f and c, d ∈ [a, b] (46) Let n be a natural number and g be a function from R into R. Suppose that for every real number x holds g(x) = (x − a) n+1 . Let x be a real number. Then g is differentiable in x and g (x) = (n + 1) · (x − a) n .
(47) Let n be a natural number and g be a function from R into R. Suppose that for every real number x holds g(x) = (x−a) n+1 (n+1)! . Let x be a real number. Then g is differentiable in x and g (x) = (x−a) n n! . Let n be a non empty element of N, let y 0 be a vector of E n , · , let G be a function from E n , · into E n , · , and let a, b be real numbers. Let us assume that a ≤ b and G is continuous on dom G. The functor Fredholm(G, a, b, y 0 ) yielding a function from the R-norm space of continuous functions of [a, b] and E n , · into the R-norm space of continuous functions of [a, b] and E n , · is defined by the condition (Def. 7).
(Def. 7) Let x be a vector of the R-norm space of continuous functions of [a, b] and E n , · . Then there exist continuous partial functions f , g, G 1 from R to E n , · such that x = f and (Fredholm(G, a, b, y 0 ))(x) = g and dom f = [a, b] and dom g = [a, b] and G 1 = G · f and for every real number We now state several propositions: (49) Suppose a ≤ b and 0 < r and for all vectors y 1 , y 2 of E n , · holds G y 1 − G y 2 ≤ r · y 1 − y 2 . Let u, v be vectors of the R-norm space of continuous functions of [a, b] and E n , · and g, h be continuous partial functions from R to E n , · . Suppose g = (Fredholm(G, a, b, y 0 ))(u) and h = (Fredholm(G, a, b, y 0 ) (50) Suppose a ≤ b and 0 < r and for all vectors y 1 , y 2 of E n , · holds G y 1 − G y 2 ≤ r · y 1 − y 2 . Let u, v be vectors of the R-norm space of continuous functions of [a, b] and E n , · , m be an element of N, and g, h be continuous partial functions from R to E n , · . Suppose g = (Fredholm(G, a, b, y 0 )) m+1 (u) and h = (Fredholm(G, a, b, y 0 (51) Let m be a natural number. Suppose a ≤ b and 0 < r and for all vectors y 1 , y 2 of E n , · holds G y 1 − G y 2 ≤ r · y 1 − y 2 . Let u, v be vectors of the R-norm space of continuous functions of [a, b] and E n , · .
(52) Suppose a < b and G is Lipschitzian on the carrier of E n , · . Then there exists a natural number m such that (Fredholm(G, a, b, y 0 )) m+1 is contraction.
(53) If a < b and G is Lipschitzian on the carrier of E n , · , then Fredholm(G, a, b, y 0 ) has unique fixpoint.
(54) Let f , g be continuous partial functions from R to E n , · . Suppose dom f = [a, b] and dom g = [a, b] and Z = ]a, b[ and a < b and G is Lipschitzian on the carrier of E n , · and g = (Fredholm(G, a, b, y 0 ))(f ). Then g a = y 0 and g is differentiable on Z and for every real number t such that t ∈ Z holds g (t) = (G · f ) t .
(55) Let y be a continuous partial function from R to E n , · . Suppose that a < b and Z = ]a, b[ and G is Lipschitzian on the carrier of E n , · and dom y = [a, b] and y is differentiable on Z and y a = y 0 and for every real number t such that t ∈ Z holds y (t) = G(y t ). Then y is a fixpoint of Fredholm(G, a, b, y 0 ).
(56) Let y 1 , y 2 be continuous partial functions from R to E n , · . Suppose that a < b and Z = ]a, b[ and G is Lipschitzian on the carrier of E n , · and dom y 1 = [a, b] and y 1 is differentiable on Z and (y 1 ) a = y 0 and for every real number t such that t ∈ Z holds y 1 (t) = G((y 1 ) t ) and dom y 2 = [a, b] and y 2 is differentiable on Z and (y 2 ) a = y 0 and for every real number t such that t ∈ Z holds y 2 (t) = G((y 2 ) t ). Then y 1 = y 2 .
(57) Suppose a < b and Z = ]a, b[ and G is Lipschitzian on the carrier of E n , · . Then there exists a continuous partial function y from R to E n , · such that dom y = [a, b] and y is differentiable on Z and y a = y 0 and for every real number t such that t ∈ Z holds y (t) = G(y t ).