Problem (6.6.7). Prove Part (2) of Theorem 6.36: Let f S-T with C C T. Then f(f (C)) CC. Also, give an example where f(f (C)) C; that is, where f(f(C)) is a proper subset of C Problem (6.6.7). P...
Prove that the following two-point boundary-value problem has a
UNIQUE solution.
Thank you
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00<s< 00. Assume that on this domain THEOREM4 11. Prove that the following two-point boundary-value problem has a unique solution: "(t3 5)x +sin t
Theorem on Unique Solution, Boundary-Value Problem Let f be a continuous function of (t, s), where 0stSl and-00
Example: Let x, y ∈ Rn, where n ∈ N. The line segment joining x to y is the subset {(1 − t)x + ty : 0 ≤ t ≤ 1 } of R n . A subset A of Rn, where n ∈ N, is called convex if it contains the line segment joining any two of its points. It is easy to check that any convex set is path-connected. (a) Let f : X → Y be an...
3-2. Prove Theorem 3.2. Theorem 3.2 Let I S R be an open interval, xe I, and let f. 8:1\{x} → R be functions. If there is a number 8 > 0 so that f and g are equal on the subset 12 € 7\(x): 13-X1 < 8 of I\(x), then f converges at x iff g converges at x and in this case the equality lim f(x) = lim g(z) holds.
Theorem 2. Let E be an open subset of R² and suppose that fe C'(E). Let y(t) be a periodic solution of (1) of period T. Then the derivative of the Poincaré map P(8) along a straight line normal to r = {x E R x = y(t) - (0),O SE ST} at x = 0 is given by T P(0) = exp V. f(y(t)) dt. 4. Show that the system • = -y + (1 – 22 - y2)2...
* SUBSET-SUM-kS, t> I S -[xi Xk] and for some lyı yn)cIxi.... xk) the sum of the yi's equals t. For example, <S-2, 3, 5, 7, 11, 14], t-21> is in SUBSET-SUM because 3+7 11-21. xk) can be partitioned into two parts A and -A where -A * SET-PARTITION <S> S-Ixi S-A and the sum of the elements in A is equal to the sum of the elements in A. For example, 〈 S-12, 3, 4, 7, 8/> works because...
please answer both
questions
Let ג: ] 9,00)-C be a continuous and bounded function such that the limit d exists f(t) dt = Let ริ่ be the LapIde transform oss, î.e.. Prove that km s1(s) = d 〈this is a version of the final vahe theorem Give an exa mple of f as in 3 which does not have a limit as t-> oo
Let ג: ] 9,00)-C be a continuous and bounded function such that the limit d exists...
I need the answer to problem 6
Clear and step by step please
Problem 4. Let V be a vector space and let T : V → V and U : V → V be two linear transforinations 1. Show that. TU is also a linear transformation. 2. Show that aT is a linear transformation for any scalar a. 3. Suppose that T is invertible. Show that T-1 is also a linear transformation. Problem 5. Let T : R3 →...
·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3. In this problem, we prove the Intermedinte Value Theorem. Let Intermediate Value Theorem. Let f : [a → R be continuous, and suppose f(a) < 0 and f(b) >0. Define S = {t E [a, b] : f(z) < 0 for allェE [a,t)) (a) Prove that s is nonempty and bounded above. Deduce that c= sup S exists, and that astst (b) Use Problem...
Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) Prove that if the image of any linearly in depen dent subset of V is linearly indepen dent then T is injective (3) Suppose that {,... ,b,b^1,...,5} is Prove that T(b1), .. . , T(b,)} is a basis of im(T) (4) Let v1,. Vk} be T(v1),..,T(vk) span W lin ear transform ation between vector...
. Let C be a collection of open subsets of R. Thus, C is a set whose elements are open subsets of R. Note that C need not be finite, or even countable. (a) Prove that the union U S is also an open subset of R. SEC (b) Assuming C is finite, prove that the intersection n S is an open subset of R. SEC (c) Give an example where C is infinite and n S is not open....