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  • 1.
    Akhmetkaliyeva, Raya
    Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
    Coercive estimates for the solutions of some singular differential equations and their applications2013Licentiatavhandling, med artikler (Annet vitenskapelig)
    Abstract [en]

    This Licentiate thesis deals with the study of existence and uniqueness together with coercive estimates for solutions of certain differential equations. The thesis consists of four papers (papers A, B, C and D) and an introduction, which put these papers into a more general frame and which also serves as an overview of this interesting field of mathematics. In the text below the functions r(x), q(x), m(x) etc. are functions on (-∞,+∞), which are different but well defined in each paper. In paper A we study the separation and approximation properties for the differential operator ly=-y″+r(x)y′+q(x)y in the Hilbert space L2 :=L2(R), R=(-∞,+∞), as well as the existence problem for a second order nonlinear differential equation in L2 . Paper B deals with the study of separation and approximation properties for the differential operator ly=-y″+r(x)y′+s(x)‾y′ in the Hilbert spaceL2:=L2(R), R=(-∞,+∞), (here ¯y is the complex conjugate of y). A coercive estimate for the solution of the second order differential equation ly =f is obtained and its applications to spectral problems for the corresponding differential operatorlis 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 C we study questions of the existence and uniqueness of solutions of the third order differential equation (L+λE)y:=-m(x)(m(x)y′)″+[q(x)+ir(x)+λ]y=f(x), (0.1) and conditions, which provide the following estimate: ||m(x)(m(x)y′)″||pp+||(q(x)+ir(x)+λ)y||pp≤ c||f(x)||pp for a solution y of (0.1). Paper D is devoted to the study of the existence and uniqueness for the solutions of the following more general third order differential equation with unbounded coefficients: -μ1(x)(μ2(x)(μ1(x)y′)′)′+(q(x)+ir(x)+λ)y=f(x). Some new existence and uniqueness results are proved and some normestimates of the solutions are given.

  • 2.
    Akhmetkaliyeva, Raya
    Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.
    Maximal regularity of the solutions for some degenerate differential equations and their applications2018Doktoravhandling, med artikler (Annet vitenskapelig)
    Abstract [en]

    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                                                                                       

                                                       

    in the Hilbert space (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

                                          

    in , where 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 :

                                                ,

    where is the domain of .

    In papers D, E, and F various differential equations of the third order of the form

          

    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 .

    For these equations we establish sufficient conditions for the existence and uniqueness of the solution, and also prove an estimate of the form

         

    for the solution of equation (0.1).

                             

                           

                                 

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