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Akhmetkaliyeva, Raya
##### Publications (2 of 2) Show all publications
Akhmetkaliyeva, R. (2018). Maximal regularity of the solutions for some degenerate differential equations and their applications. (Doctoral dissertation). Luleå: Luleå University of Technology
Open this publication in new window or tab >>Maximal regularity of the solutions for some degenerate differential equations and their applications
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Alternative title[sv]
Maximal regularitet av lösningarna till några degenererade differentialekvationer och deras tillämpningar
##### 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

$ly=-y''+r(x)y'+s(x)\bar{y}'$

in the Hilbert space $L_2:= L_2 (\mathbb{R}),\ \mathbb{R}=(-\infty, +\infty),$(here $\bar y$ 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 operator $l$ 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 $l$ in $L_2$ 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

$Ly = -\rho(x)(\rho(x)y')'+ r(x)y' + q(x)y$

in $L_2(\mathbb{R})$, where $\rho=\rho (x), r=r(x)$ are continuously differentiable functions, and $q=q(x)$ is a continuous function. In this paper we show that the operator $L$ is continuously invertible when these coefficients satisfy some suitable conditions and obtain the following estimate for $y \in D(L)$:

$\||-\rho(\rho y')'\||_2+\||r y'\||_2+\||q y\||_2\leq c \||L y\||_2$,

where $D(L)$ is the domain of $L$.

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

$-m_1(x)\left(m_2(x)\left(m_3(x)y'\right)'\right)'+[q(x)+ir(x)+\lambda]y=f(x) \ (0.1)$

are studied in the space $L_p(\mathbb{R})$.

In paper D we investigate the case when $m_1=m_3=m$ and $m_2=1$.

Moreover, in paper E the equation (0.1) is studied when $m_3=1$. Finally, in paper F the equation (0.1) is investigated under certain additional conditions on $m_j(x) (j=1,2,3)$.

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

$\left\|m_1(x)(m_2(x)\left(m_3(x)y')'\right)'\right\|^p_p+\left\|(q(x)+ir(x)+\lambda)y\right\|^p_p \leq c \left\|f(x)\right\|^p_p$

for the solution $y$ of equation (0.1).

##### Place, publisher, year, edition, pages
Luleå: Luleå University of Technology, 2018
##### Series
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-68293 (URN)978-91-7790-100-6 (ISBN)978-91-7790-101-3 (ISBN)
##### Public defence
2018-06-07, E243, Luleå, 10:00 (English)
##### Supervisors
Available from: 2018-04-11 Created: 2018-04-11 Last updated: 2018-05-29Bibliographically approved
Akhmetkaliyeva, R. (2013). Coercive estimates for the solutions of some singular differential equations and their applications (ed.). (Licentiate dissertation). Paper presented at . Luleå: Luleå tekniska universitet
Open this publication in new window or tab >>Coercive estimates for the solutions of some singular differential equations and their applications
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
##### 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.

##### Place, publisher, year, edition, pages
Luleå: Luleå tekniska universitet, 2013. p. 105
##### Series
Licentiate thesis / Luleå University of Technology, ISSN 1402-1757
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-16970 (URN)0f123f49-ab51-4486-b8e3-53e81e7156c8 (Local ID)978-91-7439-560-0 (ISBN)0f123f49-ab51-4486-b8e3-53e81e7156c8 (Archive number)0f123f49-ab51-4486-b8e3-53e81e7156c8 (OAI)
##### Note

Godkänd; 2013; 20130208 (rayakh); Tillkännagivande licentiatseminarium 2013-02-26 Nedanstående person kommer att hålla licentiatseminarium för avläggande av teknologie licentiatexamen. Namn: Raya Akhmetkaliyeva Ämne: Matematik/Mathematics Uppsats: Coercive Estimates for the Solutions of some Singular Differential Equations and their Applications Examinator: Professor Lars-Erik Persson, Institutionen för teknikvetenskap och matematik, Luleå tekniska universitet Diskutant: Associate Professor Gabriela Holubová, Department of Mathematics, University of West Bohemia, Czech Republic Tid: Onsdag den 20 mars 2013 kl 10.00 Plats: E246, Luleå tekniska universitet

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-04-19Bibliographically approved

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