(t) State and prove an existence theorem for the equation S-f- + f (x) = 0 with initial conditions x(0) = 0 and x'(0) = 0 under the assumption that f is continuous and I f j (2) Let J : (c2) be...

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(t) State and prove an existence theorem for the equation S-f- + f (x) = 0 with initial conditions x(0) = 0 and x'(0) = 0 under the assumption that f is continuous and I f j
(2) Let J : (c2) be defined by
11 VA U /G U 12 in 4 / ./0/) = in U 4 hu)dv
for h fixed in L2(Q) where 12 is a bounded region in W. Find the Frechet derivative of J. Show that inf J is attained. You may use the fact that H(1(12) is compactly embedded in Lt(Q)for t

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(I) State and prove an existence theorem for the equation ^ + f{x) = 0 with initial conditions 3;(0) = 0 and .•r'(O) = 0 under the assumption that / is continuous and | / |<><>

(I ) State and prove an existence theorem for the equation ^ + f{x) = 0 with initial conditions 3;(0) = 0 and .•r'(O) = 0 under the assumption that / is continuous and | / |< r="" .="" (you="" can="" assimie="" rothe's="" fixed="" point="" theorem.).="" (a)="" let="" j="" :="" hl{n)="" 3j="" be="" defined="" by="" ^="" •="" j(u)="/" (i="" vu="" p="" 12="" +="" u^/a="" -="" hu)dv="" for="" h="" fiixed="" in="" lp'{q)="" where="" q="" is="" a="" bounded="" region="" in="" 3?̂="" .="" find="" the="" prechet="" derivative="" of="" j.="" show="" that="" inf="" j="" is="" attained.="" you="" may="" use="" the="" fact="" that="" hl[^)="" is="" compactly="" embedded="" in="" i/="" '="" (0)for="" t="">< 6 and that / j ^ | v u p is a norm 6="" and="" that="" j="" ^="" |="" v="" u="" p="" is="" a="">

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Answered Same DayDec 22, 2021

Answer To: (t) State and prove an existence theorem for the equation S-f- + f (x) = 0 with initial conditions...

Robert answered on Dec 22 2021

131 Votes

Solution
From Rothe’s fixed point theorem
Let B denote the closed unit ball of a normed linear space X. If f(x) maps B continuously into a compact subset
of X and if f(dB) ⊂ B, then f(x) has a fixed point.

Let f denotes the radial projection into B defined by f(x) = x if ‖??‖ ≤ 1 and f(x) = x/‖??‖ is x >1 , then the given
map will be continous. Hence x 0 f maps B into acompact subset of B. then there will be a fixed point x in B. If
‖??‖ = 1, then ‖??(??)‖ = 1by hypothesis and we have x = f(x) hence our assumed assumption is true that f(x) is a
continous function.
Now since it is proved that the function is continous so from the given equation we have
d2x/dt2 +f(x) = 0 ---------------(1)
Existence Theorem: Assume that the function f : ?? → R is continuous and has continuous first order derivatives
with respect to the second and the third argument...

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(t) State and prove an existence theorem for the equation S-f- + f (x) = 0 with initial conditions x(0) = 0 and x'(0) = 0 under the assumption that f is continuous and I f j (2) Let J : (c2) be...

Answer To: (t) State and prove an existence theorem for the equation S-f- + f (x) = 0 with initial conditions...

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