 By P. S. Milojevic

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Extra resources for Nonlinear Functional Analysis

Example text

3) being, for a given condition, contradictory. 4). 6) which is now the unique actual and not the wrong solution of the problem. 4) are not equivalent. 4) but not vice versa. 4) were always equivalent and therefore the solutions obtained were correct. § 26. The Cauchy conditions and the question of their being equivalent to the general conditions. 3) amxw+ ... +aQx =/(A), in order to determine the function f(X) it is sufficient to know the form, in a given interval a < λ < β, of the functions § 26.

19. The case where the right side is a linear combination of two functions. 1) amx™ + ... + a0x = b^W+b&W, where bt and b2 are arbitrary operators, then it is sufficient to find the solutions of two simpler equations, amx™+ ... +a0x=f1(X)9 amx

2) we have {*A2ta(A, t)} = s2x"-sxX2(X9 0)-Λ:Λ2ί(Α, 0), {xM(A, 0 } = s4x-s3x(X9 0)-s2xt(X9 0)-sxt2(X9 0)-χί3(λ9 0). 11) is x^+2s2x"+s4x = s3x(X9 0)+s2xt(X, 0) + +s[2xX2(X, 0)+*„(λ, 0)]+ [2χλ2ί(λ, 0)+χ„(λ, 0)]. 13) x^+2s2x"+s4x χλ2ί(λ9 0) = 0, = ^3AsinA+2^(2cosA-AsinA). 13) and ordering, we have ( ^ - ^ [ ( ^ ^ O ^ +^ ^ - ^ s i n A + ^ - l i ^ - l i f e + ^^+^lcosA = (s3 -2s) λ sin λ+ 4s cos λ9 and by comparing the coefficients (s2-l)2c2 = s3-2s9 (s 2 -l) 2 c 4 = 0, Part IV. Chapter III 42 (s2-l)[(s2-l)Cl-4c4] (s2-l)[(s2-l)c3+4c2] = 0, = 4s.