Differential Equations on Functions from R into Real Banach Space

Abstract In this article, we describe the differential equations on functions from R into real Banach space. The descriptions are based on the article [20]. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article [21] and the article [32]. And applying the theorems of Riemann integral introduced in the article [22], we proved the ordinary differential equations on real Banach space. We referred to the methods of proof in [30].


Some Properties of Differentiable Functions on Real Normed Space
From now on Y denotes a real normed space.Now we state the propositions: keiko narita, noboru endou, and yasunari shidama (1) Let us consider a real normed space Y, a function J from E 1 , • into R, a point x 0 of E 1 , • , an element y 0 of R, a partial function g from R to Y, and a partial function f from E 1 , • to Y. Suppose (i) J = proj (1,1), and (ii) x 0 ∈ dom f , and (iii) y 0 ∈ dom g, and (iv) x 0 = y 0 , and Then f is continuous in x 0 if and only if g is continuous in y 0 .Proof: If f is continuous in x 0 , then g is continuous in y 0 by [14, (2)], [6, (39)], [37, (36)].(2) Let us consider a real normed space Y, a function I from R into E 1 , • , a point x 0 of E 1 , • , an element y 0 of R, a partial function g from R to Y, and a partial function

and
(ii) x 0 ∈ dom f , and (iii) y 0 ∈ dom g, and (iv) x 0 = y 0 , and Then (vii) g is differentiable in y 0 , and (ix) for every element The theorem is a consequence of (3).Proof: Consider N 1 being a neighbourhood of x 0 such that N 1 ⊆ dom f and there exists a point L of the real norm space of bounded linear operators from E 1 , • into Y and there exists a rest R of E 1 , • , Y such that for every point x of Consider e being a real number such that 0 < e and {z, where Consider L being a point of the real norm space of bounded linear operators from [28, (1)].For every real number y 1 such that [6, (12)], [7, (35)], [14, (3)].(6) Let us consider a function a real number y 0 , a partial function g from R to Y, and a partial function

and
(ii) x 0 ∈ dom f , and (iii) y 0 ∈ dom g, and (iv) x 0 = y 0 , and Then f is differentiable in x 0 if and only if g is differentiable in y 0 .The theorem is a consequence of ( 5) and ( 4).Proof: Reconsider J = proj(1, 1) as a function from E 1 , • into R. Consider N 0 being a neighbourhood of y 0 such that N 0 ⊆ dom(f • I) and there exists a linear L of Y and there exists a rest R of Y such that for every real number y such that y ∈ N 0 holds (f Consider e 0 being a real number such that 0 < e 0 and N 0 = ]y 0 −e 0 , y 0 +e 0 [.Reconsider e = e 0 as an element of R. Consider L being a linear of Y, R being a rest of Y such that for every real number [6, (13)], [7, (35)], [14, (4)].( 7) Let us consider a function an element y 0 of R, a partial function g from R to Y, and a partial function (ii) x 0 ∈ dom f , and (iii) y 0 ∈ dom g, and (iv) x 0 = y 0 , and Then f is differentiable in x 0 if and only if g is differentiable in y 0 .The theorem is a consequence of ( 6). ( 8) Let us consider a function an element y 0 of R, a partial function g from R to Y, and a partial function
Let us consider real numbers a, b, z and points p, q, x of E 1 , • .Now we state the propositions: (9) Suppose p = a and q = b and x = z .Then ) Suppose p = a and q = b and x = z .Then and Now we state the propositions: (11) Let us consider real numbers a, b, points p, q of E 1 , • , and a function (ii) q = b , and The theorem is a consequence of ( 10) and ( 9).Then The theorem is a consequence of ( 11), ( 10), (1), ( 9), (7), and (8).

Differential Equations
In the sequel X, Y denote real Banach spaces, Z denotes an open subset of R, a, b, c, d, e, r, x 0 denote real numbers, y 0 denotes a vector of X, and G denotes a function from X into X.Now we state the propositions: (13) Let us consider a real Banach space X, a partial function F from R to the carrier of X, and a continuous partial function f from R to the carrier of X. Suppose (15) Let us consider a continuous partial function f from R to the carrier of Let us consider a continuous partial function f from R to the carrier of X and a partial function g from R to the carrier of X.Now we state the propositions: Then y = g.The theorem is a consequence of ( 17), ( 16), (19), and (20).
Proof: Reconsider h = y − g as a continuous partial function from R to the carrier of X.For every real number x such that x ∈ dom h holds h x = 0 X by [35, (15)].For every element x of R such that x ∈ dom y holds y(x) = g(x) by [35, (21)].
Let X be a real Banach space, y 0 be a vector of X, G be a function from X into X, and a, b be real numbers.For every real number x such that x ∈ [a, t] holds G 4x r • u − v by [20, (26)], [6, (12)].(ii) y is differentiable on Z, and (iii) y a = y 0 , and (iv) for every real number t such that t ∈ Z holds y (t) = G(y t ).
The theorem is a consequence of ( 26) and ( 27).

( 12 )
Let us consider a real normed space Y, a partial function g from R to the carrier of Y, and real numbers a, b, M .Suppose (i) a b, and (ii) [a, b] ⊆ dom g, and (iii) for every real number x such that x ∈ [a, b] holds g is continuous in x, and (iv) for every real number x such that x ∈ ]a, b[ holds g is differentiable in x, and (v) for every real number x such that x ∈ ]a, b[ holds g (x) M .
(i) [a, b] ⊆ dom f , and (ii) ]a, b[ ⊆ dom F , and (iii) for every real number x such that x ∈ ]a, b[ holds F x = x a f (x)dx, and (iv) x 0 ∈ ]a, b[, and (v) f is continuous in x 0 .Then (vi) F is differentiable in x 0 , and (vii) F (x 0 ) = f x 0 .(14) Let us consider a partial function F from R to the carrier of X and a continuous partial function f from R to the carrier of X. Suppose (i) dom f = [a, b], and (ii) dom F = [a, b], and (iii) for every real number t such that t ∈ [a, b] holds F t = t a f (x)dx.Let us consider a real number x.If x ∈ [a, b], then F is continuous in x.

( 16 )G 1
Suppose a b and dom f = [a, b] and for every real number t such that t ∈ [a, b] holds g t = y 0 + t a f (x)dx.Then g a = y 0 .(17) Suppose dom f = [a, b] and dom g = [a, b] and Z = ]a, b[ and for every real number t such that t ∈ [a, b] holds g t = y 0 + t a f (x)dx.Then (i) g is continuous and differentiable on Z, and (ii) for every real number t such that t ∈ Z holds g (t) = f t .Let us consider a partial function f from R to the carrier of X.Now we state the propositions: (18) Suppose a b and [a, b] ⊆ dom f and for every real number x such that x ∈ [a, b] holds f is continuous in x and f is differentiable on ]a, b[ and for every real number x such that x ∈ ]a, b[ holds f (x) = 0 X .Then f b = f a .(19) Suppose [a, b] ⊆ dom f and for every real number x such that x ∈ [a, b] holds f is continuous in x and f is differentiable on ]a, b[ and for every real number x such that x ∈ ]a, b[ holds f (x) = 0 X .Then f ]a, b[ is constant.Now we state the propositions: (20) Let us consider a continuous partial function f from R to the carrier of X. Suppose (i) [a, b] = dom f , and (ii) f ]a, b[ is constant.Let us consider a real number x.If x ∈ [a, b], then f x = f a .(21) Let us consider continuous partial functions y, G 1 from R to the carrier of X and a partial function g from R to the carrier of X. Suppose (i) a b, and (ii) Z = ]a, b[, and (iii) dom y = [a, b], and (iv) dom g = [a, b], and (v) dom G 1 = [a, b], and (vi) y is differentiable on Z, and (vii) y a = y 0 , and (viii) for every real number t such that t ∈ Z holds y (t) = G 1t , and (ix) for every real number t such that t ∈ [a, b] holds g t = y 0 + t a (x)dx.

G 1 G 3 G 5
Assume 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 X into the R-norm space of continuous functions of [a, b] and X is defined by (Def. 1) Let us consider a vector x of the R-norm space of continuous functions of [a, b] and X.Then there exist continuous partial functions f , g, G 1 from R to the carrier of X such that (i) x = f , and (ii) it(x) = g, and (iii) dom f [a, b], and (iv) dom g = [a, b], and (v) G 1 = G • f , and (vi) for every real number t such that t ∈ [a, b] holds g t = y 0 + t a (x)dx.Now we state the propositions: (22) Suppose a b and 0 < r and for every vectors y 1 , y 2 of X, G y 1 −G y 2 r • y 1 −y 2 .Let us consider vectors u, v of the R-norm space of continuous functions of [a, b] and X and continuous partial functions g, h from R to the carrier of X. Suppose (i) g = (Fredholm(G, a, b, y 0 ))(u), and (ii) h = (Fredholm(G, a, b, y 0 ))(v).Let us consider a real number t.Suppose t ∈ [a, b].Then g t − h t (r • (t − a)) • u − v .Proof: Set F = Fredholm(G, a, b, y 0 ).Consider f 1 , g 1 , G 3 beingcontinuous partial functions from R to the carrier of X such that u = f 1 and F (u) = g 1 and dom f 1 = [a, b] and dom g 1 = [a, b] and G 3 = G • f 1 and for every real number t such that t ∈ [a, b] holds g 1t = y 0 + t a (x)dx.Consider f 2 , g 2 , G 5 being continuous partial functions from R to the carrier of X such that v = f 2 and F (v) = g 2 and dom f 2 = [a, b] and dom g 2 = [a, b] and G 5 = G • f 2 and for every real number t such that t ∈ [a, b] holds g 2t = y 0 + t a (x)dx.Set G 4 = G 3 −G 5 .
(23) Suppose a b and 0 < r and for every vectorsy 1 , y 2 of X, G y 1 − G y 2 r • y 1 − y 2 .Let us consider vectors u, v of the R-norm space of (i) dom y = [a, b], and