
Research Article


Resolvent of Fourth Order Differential Equation in Half Axis 

Kevser Koklu



ABSTRACT

Let H be a separable Hilbert space and H_{1}=L_{2}
(0, ∞ ;H). The all functions are defined in range[0, ∞) , their
values belongs to space H, they are measurable in the meaning of Bochner and
provides the condition of
If the scalar product is defined in H_{1} by the formula
f(x), g(x) ∈ H_{1}, H_{1} forms a separable Hilbert space.
In this study, in space H_{1}, it is investigated that Green`s function
(resolvant) of the operator formed by the differential expression.
(1)^{n }y^{ (2n) }+Q(x)y, 
0 ≤ x ≤ ∞ 
And boundary conditions
Y^{(j)} (0)h _{j}y_{
}^{(jI)}(0)=0, 
j = 1,3,......,2n1 
Where Q(x) is a normal operator that has pure degree spectrum for every x ∈
[0, ∞) ) in H. Assumed that domain of Q(x) is independent from x and resolvent
set of Q(x) belongs to
(0 < ∈ ≤ π) of Complex plane λ. In addition assumed that
the operator function Q(x) satisfies the TitchmarsLevitan complex conditions.
h _{j} are arbitrary complex numbers. The obtained result has been applied
to an example.






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