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Some new results concerning a class of third-order differential equations
Department of Pure Mathematics, Eurasian National University.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper. Department of Mathematics, Narvik University College.
Department of Pure Mathematics, Eurasian National University.
Luleå tekniska universitet, Institutionen för teknikvetenskap och matematik, Matematiska vetenskaper.ORCID-id: 0000-0001-8211-3671
2015 (Engelska)Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, nr 2, s. 419-434Artikel i tidskrift (Refereegranskat) Published
##### Abstract [en]

We consider the following third-order differential equation with unbounded coefficients:Some new existence and uniqueness results are proved, and precise estimates of the norms of the solutions are given. The obtained results may be regarded as a unification and extension of all other results of this type

##### Ort, förlag, år, upplaga, sidor
Taylor & Francis, 2015. Vol. 94, nr 2, s. 419-434
##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
ISI: 000347395000013Scopus ID: 2-s2.0-84920697521Lokalt ID: 1959cbfa-5e19-4ea4-9c4d-93b09eb1a578OAI: oai:DiVA.org:ltu-3756DiVA, id: diva2:976617
##### Anmärkning

Validerad; 2015; Nivå 2; 20140415 (andbra)

##### Ingår i avhandling
1. Maximal regularity of the solutions for some degenerate differential equations and their applications
Öppna denna publikation i ny flik eller fönster >>Maximal regularity of the solutions for some degenerate differential equations and their applications
2018 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
##### Alternativ titel[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).

##### Ort, förlag, år, upplaga, sidor
Luleå: Luleå University of Technology, 2018
##### Serie
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544
##### Nationell ämneskategori
Matematisk analys
Matematik
##### Identifikatorer
urn:nbn:se:ltu:diva-68293 (URN)978-91-7790-100-6 (ISBN)978-91-7790-101-3 (ISBN)
##### Disputation
2018-06-07, E243, Luleå, 10:00 (Engelska)

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