This PhD thesis deals with the study of existence and uniqueness together with coercive estimates for solutions of certain differential equations.
The thesis consists of six papers (papers A, B, C, D, E and F), two appendices and an introduction, which put these papers and appendices into a more general frame and which also serves as an overview of this interesting field of mathematics.
In the text below the functionsr = r(x), q = q(x), m = m(x) etc. are functions on (−∞,+∞), which are different but well defined in each paper. Paper A deals with the study of separation and approximation properties for the differential operator
y'+s(x)%5Cbar%7By%7D'%0A$$)
in the Hilbert space ,%5C%20%5Cmathbb%7BR%7D=(-%5Cinfty,%20+%5Cinfty),)
(here 
is the complex conjugate of 
). A coercive estimate for the solution of the second order differential equation 
is obtained and its applications to spectral problems for the corresponding differential operator 
is demonstrated. Some sufficient conditions for the existence of the solutions of a class of nonlinear second order differential equations on the real axis are obtained.
In paper B necessary and sufficient conditions for the compactness of the resolvent of the second order degenerate differential operator in 
is obtained. We also discuss the two-sided estimates for the radius of fredholmness of this operator.
In paper C we consider the minimal closed differential operator
(%5Crho(x)y')'+%20r(x)y'%20+%20q(x)y%0A)
in )
, where ,%20r=r(x))
are continuously differentiable functions, and )
is a continuous function. In this paper we show that the operator 
is continuously invertible when these coefficients satisfy some suitable conditions and obtain the following estimate for )
:
'%5C%7C%7C_2+%5C%7C%7Cr%20y'%5C%7C%7C_2+%5C%7C%7Cq%20y%5C%7C%7C_2%5Cleq%20c%20%5C%7C%7CL%20y%5C%7C%7C_2)
,
where $)
is the domain of 
.
In papers D, E, and F various differential equations of the third order of the form
%5Cleft(m_2(x)%5Cleft(m_3(x)y'%5Cright)'%5Cright)'+%5Bq(x)+ir(x)+%5Clambda%5Dy=f(x)%20%20%5C%20%20%20%20%20(0.1))
are studied in the space )
.
In paper D we investigate the case when 
and 
.
Moreover, in paper E the equation (0.1) is studied when 
. Finally, in paper F the equation (0.1) is investigated under certain additional conditions on %20(j=1,2,3)$)
.
For these equations we establish sufficient conditions for the existence and uniqueness of the solution, and also prove an estimate of the form
(m_2(x)%5Cleft(m_3(x)y')'%5Cright)'%5Cright%5C%7C%5Ep_p+%5Cleft%5C%7C(q(x)+ir(x)+%5Clambda)y%5Cright%5C%7C%5Ep_p%20%5Cleq%20c%20%5Cleft%5C%7Cf(x)%5Cright%5C%7C%5Ep_p)
for the solution of equation (0.1).
Luleå: Luleå University of Technology, 2018.