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

Question

(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 by11 VA U /G U 12 in 4 / ./0/) = in U 4 hu)dvfor 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 
	
Document Preview: (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 | / |

Robert · Accepted Answer

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 : ??

(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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