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• 1.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Compactness of embedding between Sobolev type spaces with multiweighted derivatives2009In: AIHT : Analysis, Inequalities and Homogenization Theory: Midnight sun conference in honor of Lars-Erik Persson, 2009Conference paper (Other academic)

We consider a new Sobolev type function space called the space with multiweighted derivatives. As basis for this space serves some differential operators containing weight functions. We establish necessary and sufficient conditions for the boundedness and compactness of the embedding between the spaces with multiweighted derivatives in different selections of weights.

• 2.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Embedding theorems for spaces with multiweighted derivatives2007Licentiate thesis, comprehensive summary (Other academic)

This Licentiate Thesis consists of four chapters, which deal with a new Sobolev type function space called the space with multiweighted derivatives. This space is a generalization of the usual one dimensional Sobolev space. Chapter 1 is an introduction, where, in particular, the importance to study function spaces with weights is discussed and motivated. In Chapter 2 we consider and analyze some results of L. D. Kudryavtsev, where he investigated one dimensional Sobolev spaces. Moreover, in this chapter we present and prove analogous results by B. L. Baidel'dinov for generalized Sobolev spaces. These results are crucially for the proofs of the main results of this Licentiate Thesis. In Chapter 3 we prove some embedding theorems for these new generalized Sobolev spaces. The main results of Kudryavtsev and Baidel'dinov about characterization of the behavior of functions at a singularity take place in weak degeneration of spaces. However, with the help of our new embedding theorems we can extend these results to the case of strong degeneration. In Chapter 4 we prove some new estimates for each function in a Tchebychev system. In order to be able to study also compactness of the embeddings from Chapter 3 such estimates are crucial. I plan to study this question in detail in my further PhD studies.

• 3.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new results concerning boundedness and compactness for embeddings between spaces with multiweighted derivatives2009Doctoral thesis, comprehensive summary (Other academic)

This Doctoral Thesis consists of five chapters, which deal with a new Sobolev type function space called the space with multiweighted derivatives. This space is a generalization of the usual one dimensional Sobolev space. As basis for this space serves some differential operators containing weight functions.Chapter 1 is an introduction, where, in particular, the importance to study function spaces with weights is discussed and motivated. In Chapter 2 we prove some new estimates for each function in a Tchebychev system. In order to be able to study compactness of the embeddings from Chapter 3 such estimates are crucial.In Chapter 3 we rewrite and present some results of L. D. Kudryavtsev, where he investigated one dimensional Sobolev spaces. Moreover, in this chapter we rewrite and discuss some analogous results by B. L. Baidel'dinov for generalized Sobolev spaces. These results are not available in the Western literatures in this way and they are crucial for the proofs of the main results in Chapter 4. In Chapter 4 we prove some embedding theorems for these new generalized Sobolev spaces. The main results of Kudryavtsev and Baidel'dinov about characterization of the behavior of functions at a singularity take place in weak degeneration of the spaces. However, with the help of our new embedding theorems we can extend theseresults to the case of strong degeneration.The main aim of Chapter 5 is to establish boundedness and compactness of the embedding considered in Chapter 4.In Chapter 4 basically only sufficient conditions for boundedness of this embedding were obtained. In Chapter 5 we obtain necessary and sufficient conditions for boundedness and compactness of this embedding and the main results are proved in a different way.

• 4. Abdikalikova, Zamira
Compactness of embedding between spaces with multiweighted derivatives: the case 1 ≤ p ≤ q2009Report (Other academic)
• 5.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Summability of a Tchebysheff system of functions2007Report (Other academic)
• 6.
L.N. Gumilyov Eurasian National University.
L.N. Gumilyov Eurasian National University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Boundedness and compactness of the embedding between spaces with multiweighted derivatives when 12011In: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 61, no 1, p. 7-26Article in journal (Refereed)

We consider a new Sobolev type function space called the space with multiweighted derivatives W-p(n),(alpha) over bar, where (alpha) over bar = (alpha(0), alpha(1), ......, alpha(n)), alpha(i) is an element of R, i = 0, 1,......,n, and parallel to f parallel to W-p(n),((alpha) over bar) = parallel to D((alpha) over bar)(n)f parallel to(p) + Sigma(n-1) (i=0) vertical bar D((alpha) over bar)(i)f(1)vertical bar, D((alpha) over bar)(0)f(t) = t(alpha 0) f(t), d((alpha) over bar)(i)f(t) = t(alpha i) d/dt D-(alpha) over bar(i-1) f(t), i = 1, 2, ....., n. We establish necessary and sufficient conditions for the boundedness and compactness of the embedding W-p,(alpha) over bar(n) -> W-q,(beta) over bar,(m) when 1 <= q < p < infinity, 0 <= m < n

• 7.
L.N. Gumilyov Eurasian National University.
L.N. Gumilyov Eurasian National University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Boundedness and compactness of the embedding between spaces with multiweighted derivatives when 1≤ q2009Report (Other academic)
• 8.
University of Haifa.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. University of Tromsø ; The Arctic University of Norway, Narvik.
Fejer and Hermite–Hadamard Type Inequalitiesfor N-Quasiconvex Functions2017In: Mathematical notes of the Academy of Sciences of the USSR, ISSN 0001-4346, E-ISSN 1573-8876, Vol. 102, no 5-6, p. 599-609Article in journal (Refereed)

Abstract—Some new extensions and refinements of Hermite–Hadamard and Fejer type inequali-ties for functions which are N-quasiconvex are derived and discussed.

• 9.
Department of Mathematics, University of Haifa, Haifa, Israel.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. UIT The Arctic University of Norway, Narvik, Norway.
Extensions and Refinements of Fejer and Hermite–Hadamard Type Inequalities2018In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 21, no 3, p. 759-772Article in journal (Refereed)

In this paper extensions and refinements of Hermite-Hadamard and Fejer type inequalities are derived including monotonicity of some functions related to the Fejer inequality and extensions for functions, which are 1-quasiconvex and for function with bounded second derivative. We deal also with Fejer inequalities in cases that p, the weight function in Fejer inequality, is not symmetric but monotone on [a, b] .

• 10.
Department of Mathematics, University of Haifa.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some New Refined Hardy Type Inequalities with Breaking Points p = 2 or p = 32014In: Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation: 22nd International Workshop in Operator Theory and its Applications, Sevilla, July 2011 / [ed] Manuel Cepedello Boiso; Håkan Hedenmalm; Marinus A. Kaashoek; Alfonso Montes Rodríguez; Sergei Treil, Basel: Encyclopedia of Global Archaeology/Springer Verlag, 2014, p. 1-10Conference paper (Refereed)

For usual Hardy type inequalities the natural “breaking point” (the parameter value where the inequality reverses) is p = 1. Recently, J. Oguntuase and L.-E. Persson proved a refined Hardy type inequality with breaking point at p = 2. In this paper we show that this refinement is not unique and can be replaced by another refined Hardy type inequality with breaking point at p = 2. Moreover, a new refined Hardy type inequality with breaking point at p = 3 is obtained. One key idea is to prove some new Jensen type inequalities related to convex or superquadratic funcions, which are also of independent interest

• 11.
Department of Mathematics, University of Haifa.
Faculty of Textile Technology, University of Zagreb. Faculty of Textile Technology, University of Zagreb. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new refined Hardy type inequalities with general kernels and measures2010In: Aequationes Mathematicae, ISSN 0001-9054, E-ISSN 1420-8903, Vol. 79, no 1-2, p. 157-172Article in journal (Refereed)

We state and prove some new refined Hardy type inequalities using the notation of superquadratic and subquadratic functions with an integral operator Ak defined by, where k: Ω1 × Ω2 is a general nonnegative kernel, (Ω1, μ1) and (Ω2, μ2) are measure spaces and,. The relations to other results of this type are discussed and, in particular, some new integral identities of independent interest are obtained.

• 12.
Department of Mathematics, University of Haifa.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Inequalities for averages of quasiconvex and superquadratic functions2016In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 19, no 2, p. 535-550Article in journal (Refereed)

For n ε ℤ+ we consider the difference Bn-1 (f)-Bn(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1 nηi=0f(ai/an) where the sequences{ai} and {ai-ai-1} are increasing. Some lower bounds are derived when f is 1-quasiconvex and when f is a closely related superquadratic function. In particular, by using some fairly new results concerning the so called "Jensen gap", these bounds can be compared. Some applications and related results about An-1 (f)-An(f):= 1/an n-1/ηi=0 f(ai/an-1)-1/an+1 nηi=0f(ai/an) are also included.

• 13.
Department of Mathematics, University of Haifa.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new estimates of the ‘Jensen gap’2016In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2016, article id 39Article in journal (Refereed)

Let (μ,Ω) be a probability measure space. We consider the so-called ‘Jensen gap’ J(φ,μ,f)=∫ Ω φ(f(s))dμ(s)−φ(∫ Ω f(s)dμ(s)) for some classes of functions φ. Several new estimates and equalities are derived and compared with other results of this type. Especially the case when φ has a Taylor expansion is treated and the corresponding discrete results are pointed out.

• 14.
University of Haifa.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new scales of refined Hardy type inequalities via functions related to superquadracity2013In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 16, no 3, p. 679-695Article in journal (Refereed)

For the Hardy type inequalities the "breaking point" (=the point where the inequality reverses) is p = 1. Recently, J. Oguntoase and L. E. Persson proved a refined Hardy type inequality with a breaking point at p = 2. In this paper we prove a new scale of refined Hardy type inequality which can have a breaking point at any p ≥ 2. The technique is to first make some further investigations for superquadratic and superterzatic functions of independent interest, among which, a new Jensen type inequality is proved

• 15.
University of Haifa, Department of Mathematics.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. University of Zagreb. University of Zagreb.
General inequalities via isotonic subadditive functionals2007In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 10, no 1, p. 15-28Article in journal (Refereed)

In this manuscript a number of general inequalities for isotonic subadditive functionals on a set of positive mappings are proved and applied. In particular, it is pointed out that these inequalities both unify and generalize some general forms of the Holder, Popoviciu, Minkowski, Bellman and Power mean inequalities. Also some refinements of some of these results are proved.

• 16.
University of Haifa.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On some new developments of Hardy-type inequalities2012In: 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2012 / [ed] Seenith Sivasundaram, Melville, NY: American Institute of Physics (AIP), 2012, p. 739-746Conference paper (Refereed)

In this paper we present and discuss some new developments of Hardy-type inequalities, namely to derive (a) Hardy-type inequalities via a convexity approach, (b) refined scales of Hardy-type inequalities with other “breaking points” than p = 1 via superquadratic and superterzatic functions, (c) scales of conditions to characterize modern forms of weighted Hardy-type inequalities.

• 17.
Department of Mathematics, University of Haifa.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On γ-quasiconvexity, superquadracity and two-sided reversed Jensen type inequalities2015In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 18, no 2, p. 615-627Article in journal (Refereed)

In this paper we deal with γ -quasiconvex functions when −1γ 0, to derive sometwo-sided Jensen type inequalities. We also discuss some Jensen-Steffensen type inequalitiesfor 1-quasiconvex functions. We compare Jensen type inequalities for 1-quasiconvex functionswith Jensen type inequalities for superquadratic functions and we extend the result obtained forγ -quasiconvex functions to more general classes of functions.

• 18.
Department of Mathematics, University of Haifa.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new scales of refined Jensen and Hardy type inequalities2014In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 17, no 3, p. 1105-1114Article in journal (Refereed)

Some scales of refined Jensen and Hardy type inequalities are derived and discussed. The key object in our technique is ? -quasiconvex functions K(x) defined by K(x)x-? =? (x) , where Φ is convex on [0,b) , 0 < b > ∞ and γ > 0.

• 19.
Luleå University of Technology, Department of Engineering Sciences and Mathematics.
Inequalities for some classes of Hardy type operators and compactness in weighted Lebesgue spaces2016Doctoral thesis, comprehensive summary (Other academic)

This PhD thesis is devoted to investigate weighted differential Hardy inequalities and Hardy-type inequalities with the kernel when the kernel has an integrable singularity, and also the additivity of the estimate of a Hardy type operator with a kernel.The thesis consists of seven papers (Papers 1, 2, 3, 4, 5, 6, 7) and an introduction where a review on the subject of the thesis is given. In Paper 1 weighted differential Hardy type inequalities are investigated on the set of compactly supported smooth functions, where necessary and sufficient conditions on the weight functions are established for which this inequality and two-sided estimates for the best constant hold. In Papers 2, 3, 4 a more general class of $\alpha$-order fractional integrationoperators are considered including the well-known classical Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard operators. Here 0 < $\alpha$ < 1. In Papers 2 and 3 the boundedness and compactness of two classes of such operators are investigated namely of Weyl and Riemann-Liouville type, respectively, in weighted Lebesgue spaces for 1 < p ≤ q < 1 and 0 < q < p < ∞. As applications some new results for the fractional integration operators of Weyl, Riemann-Liouville, Erdelyi-Kober and Hadamard are given and discussed.In Paper 4 the Riemann-Liouville type operator with variable upper limit is considered. The main results are proved by using a localization method equipped with the upper limit function and the kernel of the operator. In Papers 5 and 6 the Hardy operator with kernel is considered, where the kernel has a logarithmic singularity. The criteria of the boundedness and compactness of the operator in weighted Lebesgue spaces are given for 1 < p ≤ q < ∞ and 0 < q < p < ∞, respectively. In Paper 7 we investigated the weighted additive estimates for integral operators K+ and K¯ defined by

K+ ƒ(x) := ∫ k(x,s) ƒ(s)ds,  K¯ ƒ(x) := ∫ k(x,s)ƒ(s)ds.

It is assumed that the kernel k of the operators K+and K- belongs to the general Oinarov class. We derived the criteria for the validity of these addittive estimates when 1 ≤ p≤ q < ∞

• 20.
Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana .
Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Boundedness and compactness of a class of Hardy type operators2016In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, no 1, article id 324Article in journal (Refereed)

We establish characterizations of both boundedness and of compactness of a general class of fractional integral operators involving the Riemann-Liouville, Hadamard, and Erdelyi-Kober operators. In particular, these results imply new results in the theory of Hardy type inequalities. As applications both new and well-known results are pointed out.

• 21.
Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan .
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. UiT, Tromso, Norway. RUDN University, Moscow, Russia.
Hardy type inequalities and compactness of a class of integral operators with logarithmic singularities2018In: Mathematical Inequalities & Applications, ISSN 1331-4343, E-ISSN 1848-9966, Vol. 21, no 1, p. 201-215, article id 21-16Article in journal (Refereed)

We establish criteria for both boundedness and compactness for some classes of integraloperators with logarithmic singularities in weighted Lebesgue spaces for cases 1 < p 6 q <¥ and 1 < q < p < ¥. As corollaries some corresponding new Hardy inequalities are pointedout.1

• 22.
L. N.Gumilev Eurasian National University, Khazakstan.
L. N.Gumilev Eurasian National University, Khazakstan. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Additive weighted Lp estimates of some classes of integral operators involving generalized Oinarov Kernels2017In: Journal of Mathematical Inequalities, ISSN 1846-579X, E-ISSN 1848-9575, Vol. 11, no 3, p. 683-694Article in journal (Refereed)

Abstract. Inequalities of the formkuK f kq 6C(kr f kp +kvH f kp) , f > 0,are considered, where K is an integral operator of Volterra type and H is the Hardy operator.Under some assumptions on the kernel K we give necessary and sufficient conditions for suchan inequality to hold.1

Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
On a new class of Hardy-type inequalities2012In: Journal of inequalities and applications (Print), ISSN 1025-5834, E-ISSN 1029-242X, Vol. 2012, no 259Article in journal (Refereed)

In this paper, we generalize a Hardy-type inequality to the class of arbitrary non-negative functions bounded from below and above with a convex function multiplied with positive real constants. This enables us to obtain new generalizations of the classical integral Hardy's, Hardy-Hilbert's, Hardy-Littlewood-P\'{o}lya's and P\'{o}lya-Knopp's inequalities as well as of Godunova's and of some recently obtained inequalities in multidimensional settings. Finally, we apply a similar idea to functions bounded from below and above with a superquadratic function.

• 24.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Coercive estimates for the solutions of some singular differential equations and their applications2013Licentiate thesis, comprehensive summary (Other academic)

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.

• 25.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Maximal regularity of the solutions for some degenerate differential equations and their applications2018Doctoral thesis, comprehensive summary (Other academic)

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).

• 26. Akhmetkaliyeva, Raya D.
Coersive solvability of the differential equation of the third order with complex valued coefficients2013In: Vestnik ENU, Vol. 95, no 4, p. 355-361Article in journal (Refereed)
• 27.
L.N. Gumilyov Eurasian National University, Astana, Kazakhstan.
On Solvability of Third-Order Singular Differential Equation2017In: FAIA 2017: Functional Analysis in Interdisciplinary Applications, Springer, 2017, Vol. 216, p. 106-112Conference paper (Refereed)

In this paper some new existence and uniqueness results are proved and maximal regularity estimates of solutions of third-order differential equation with unbounded coefficients are given.

• 28. Akhmetkaliyeva, Raya D.
On separation of a degenerate differential operator in Hilbert space2011Conference paper (Refereed)

A coercive estimate for a solution of a degenerate second order di fferential equation is installed, and its applications to spectral problems for the corresponding dif ferential operator is demonstrated. The suffi cient conditions for existence of the solutions of one class of the nonlinear second order diff erential equations on the real axis are obtained.

• 29.
Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University.
Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University. Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University.
Compactness of the resolvent of one second order differential operator2014In: 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences (ICNPAA 2014) / [ed] Sivasundaram, Seenith, American Institute of Physics (AIP), 2014, Vol. 1637, no 1, p. 13-17, article id 13Conference paper (Refereed)

In this work a necessary and sufficient condition for the compactness of the resolvent of one second order degenerate differential operator in L-2 is obtained. We also discuss the two-sided estimates for the radius of fredholmness of this operator.

• 30.
Department of Pure Mathematics, Eurasian National University.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Department of Mathematics, Narvik University College. Department of Pure Mathematics, Eurasian National University. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Some new results concerning a class of third-order differential equations2015In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, no 2, p. 419-434Article in journal (Refereed)

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

• 31.
Department of Mathematics, Claremont McKenna College.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Real interpolation and measure of weak noncompactness1995In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 175, no 1, p. 5-12Article in journal (Refereed)

Behavior of weak measures of noncompactness under real interpolation is investigated. It is shown that "convexity type" theorems hold true for weak measures of noncompactness.

• 32.
Department of Mathematics, Claremont McKenna College.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Lipschitz-Orlicz spaces and the Laplace equation1996In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, no 178, p. 81-101Article in journal (Refereed)
• 33.
Institute of Mathematics, Ruhr-Universität, D-44780 Bochum.
SISSA. Luleå tekniska universitet.
Symmetries of Schrödinger operator with point interactions1998In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 45, no 1, p. 33-47Article in journal (Refereed)

The transformations of all the Schrödinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling Wλ are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon

• 34.
Institut für Angewandte Mathematik, Universität Bonn.
Institut für Angewandte Mathematik, Universität Bonn. Luleå tekniska universitet.
Many body problems with "spin"-related contact interactions2001In: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, Vol. 47, no 2, p. 157-166Article in journal (Refereed)

We study quantum mechanical systems with "spin"-related contact interactions in one dimension. The boundary conditions describing the contact interactions are dependent on the spin states of the particles. In particular we investigate the integrability of N-body systems with δ-interactions and point spin couplings. Bethe ansatz solutions, bound states and scattering matrices are explicitly given. The cases of generalized separated boundary condition and some Hamiltonian operators corresponding to special spin related boundary conditions are also discussed.

• 35.
BiBoS Research Center.
Fachbereich Physik, Universität-GH Essen. Luleå tekniska universitet. Fachbereich Physik, Universität-GH Essen. Nuclear Physics Institute, Czech Academy of Sciences.
S-matrix, resonances, and wave functions for transport through billiards with leads1996In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 37, no 10, p. 4888-4903Article in journal (Refereed)

For a simple model describing the S-matrices of open resonators the statistical properties of the resonances are investigated, as well as the wave functions inside the resonator

• 36.
Department of Mathematics, Ruhr-University, Bochum.
Luleå tekniska universitet.
Finite rank perturbations and distribution theory1999In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 127, no 4, p. 1151-1161Article in journal (Refereed)
• 37.
Department of Mathematics, Ruhr-University, Bochum.
Luleå tekniska universitet.
Rank one perturbations of not semibounded operators1997In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 27, no 4, p. 379-400Article in journal (Refereed)

Rank one perturbations of selfadjoint operators which are not necessarily semibounded are studied in the present paper. It is proven that such perturbations are uniquely defined, if they are bounded in the sense of forms. We also show that form unbounded rank one perturbations can be uniquely defined if the original operator and the perturbation are homogeneous with respect to a certain one parameter semigroup. The perturbed operator is defined using the extension theory for symmetric operators. The resolvent of the perturbed operator is calculated using Krein's formula. It is proven that every rank one perturbation can be approximated in the operator norm. We prove that some form unbounded perturbations can be approximated in the strong resolvent sense without renormalization of the coupling constant only if the original operator is not semibounded. The present approach is applied to study first derivative and Dirac operators with point interaction, in one dimension

• 38. Albeverio, Sergio
Luleå tekniska universitet.
Sur le comportement asymptotique du noyau associé à une diffusion dégénérée2000In: Comptes Rendus Mathematiques de l'Academie des Sciences = Mathematical reports of the academy of science, ISSN 0706-1994, Vol. 22, no 4, p. 151-159Article in journal (Refereed)
• 39.
Department of Mathematics, Ruhr-University, Bochum.
Luleå tekniska universitet.
Pseudo-Differential Operators with Point Interactions1997In: Letters in Mathematical Physics, ISSN 0377-9017, E-ISSN 1573-0530, Vol. 41, no 1, p. 79-92Article in journal (Refereed)

Point interactions for pseudo-differential operators are studied. Necessary and sufficient conditions for a pseudo-differential operator to have nontrivial point perturbations are given. The results are applied to the construction of relativistic spin zero Hamiltonians with point interactions

• 40.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
New insights on lubrication theory for compressible fluids2019In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 145, article id 103170Article in journal (Refereed)

The fact that the film is thin is in lubrication theory utilised to simplify the full Navier–Stokes system of equations. For incompressible and iso-viscous fluids, it turns out that the inertial terms are small enough to be neglected. However, for a compressible fluid, we show that the influence of inertia depends on the (constitutive) density-pressure relationship and may not always be neglected. We consider a class of iso-viscous fluids obeying a power-law type of compressibility, which in particular includes both incompressible fluids and ideal gases. We show by scaling and asymptotic analysis, that the degree of compressibility determines whether the terms governing inertia may or may not be neglected. For instance, for an ideal gas, the inertial terms remain regardless of the film height-to-length ratio. However, by means of a specific modified Reynolds number that we define we show that the magnitudes of the inertial terms rarely are large enough to be influential. In addition, we consider fluids obeying the well-known Dowson and Higginson density-pressure relationship and show that the inertial terms can be neglected, which allows for obtaining a Reynolds type of equation. Finally, some numerical examples are presented in order to illustrate our theoretical results.

• 41.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
The homogenization process of the Reynolds equation describing compressible liquid flow2006In: Tribology International, ISSN 0301-679X, E-ISSN 1879-2464, Vol. 39, no 9, p. 994-1002Article in journal (Refereed)

This paper summarizes the homogenization process of rough, hydrodynamic lubrication problems governed by the Reynolds equation used to describe compressible liquid flow. Here, the homogenized equation describes the limiting result when the wavelength of a modeled surface roughness goes to zero. The lubricant film thickness is modeled by one part describing the geometry/shape of the bearing and a periodic part describing the surface topography/roughness. By varying the periodic part as well as its wavelength, we can try to systematically investigate the applicability of homogenization on this type of problem. The load carrying capacity is the target parameter; deterministic solutions are compared to homogenized by this measure. We show that the load carrying capacity rapidly converges to the homogenized results as the wavelength decreases, proving that the homogenized solution gives a very accurate representation of the problem when real surface topographies are considered

• 42.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of the Reynolds equation2005Report (Other academic)
• 43.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Department of Mathematics and Statistics, University of Cape Coast. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Multiscale homogenization of a class of nonlinear equations with applications in lubrication theory and applications2011In: Journal of Function Spaces and Applications, ISSN 0972-6802, E-ISSN 1758-4965, Vol. 9, no 1, p. 17-40Article in journal (Refereed)

We prove a homogenization result for monotone operators by using the method of multiscale convergence. More precisely, we study the asymptotic behavior as epsilon -> 0 of the solutions u(epsilon) of the nonlinear equation div a(epsilon)(x, del u(epsilon)) = div b(epsilon), where both a(epsilon) and b(epsilon) oscillate rapidly on several microscopic scales and a(epsilon) satisfies certain continuity, monotonicity and boundedness conditions. This kind of problem has applications in hydrodynamic thin film lubrication where the bounding surfaces have roughness on several length scales. The homogenization result is obtained by extending the multiscale convergence method to the setting of Sobolev spaces W-0(1,p)(Omega), where 1 < p < infinity. In particular we give new proofs of some fundamental theorems concerning this convergence that were first obtained by Allaire and Briane for the case p = 2.

• 44.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reiterated homogenization applied in hydrodynamic lubrication2008In: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology, ISSN 1350-6501, E-ISSN 2041-305X, Vol. 222, no 7, p. 827-841Article in journal (Refereed)

This work is devoted to studying the combined effect that arises due to surface texture and surface roughness in hydrodynamic lubrication. An effective approach in tackling this problem is by using the theory of reiterated homogenization with three scales. In the numerical analysis of such problems, a very fine mesh is needed, suggesting some type of averaging. To this end, a general class of problems is studied that, e.g. includes the incompressible Reynolds problem in both artesian and cylindrical coordinate forms. To demonstrate the effectiveness of the method several numerical results are presented that clearly show the convergence of the deterministic solutions towards the homogenized solution.Moreover, the convergence of the friction force and the load carrying capacity of the lubricant film is also addressed in this paper. In conclusion, reiterated homogenization is a feasible mathematical tool that facilitates the analysis of this type of problem.

• 45.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reiterated homogenization of a nonlinear Reynolds-type equation2008Report (Other academic)
• 46.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Department of Mathematics and Statistics, University of Cape Coast. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Variational bounds applied to unstationary hydrodynamic lubrication2008In: International Journal of Engineering Science, ISSN 0020-7225, E-ISSN 1879-2197, Vol. 46, no 9, p. 891-906Article in journal (Refereed)

This paper is devoted to the effects of surface roughness in hydrodynamic lubrication. The numerical analysis of such problems requires a very fine mesh to resolve the surface roughness, hence it is often necessary to do some type of averaging. Previously, homogenization (a rigorous form of averaging) has been successfully applied to Reynolds type differential equations. More recently, the idea of finding upper and lower bounds on the effective behavior, obtained by homogenization, was applied for the first time in tribology. In these pioneering works, it has been assumed that only one surface is rough. In this paper we develop these results to include the unstationary case where both surfaces may be rough. More precisely, we first use multiple-scale expansion to obtain a homogenization result for a class of variational problems including the variational formulation associated with the unstationary Reynolds equation. Thereafter, we derive lower and upper bounds corresponding to the homogenized (averaged) variational problem. The bounds reduce the numerical analysis, in that one only needs to solve two smooth problems, i.e. no local scale has to be considered. Finally, we present several examples, where it is shown that the bounds can be used to estimate the effects of surface roughness with very high accuracy.

• 47.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Homogenization of the unstationary incompressible Reynolds equation2007In: Tribology International, ISSN 0301-679X, E-ISSN 1879-2464, Vol. 40, no 9, p. 1344-1350Article in journal (Refereed)

This paper is devoted to the effects of surface roughness during hydrodynamic lubrication. In the numerical analysis a very fine mesh is needed to resolve the surface roughness, suggesting some type of averaging. A rigorous way to do this is to use the general theory of homogenization. In most works about the influence of surface roughness, it is assumed that only the stationary surface is rough. This means that the governing Reynolds type equation does not involve time. However, recently, homogenization was successfully applied to analyze a situation where both surfaces are rough and the lubricant is assumed to have constant bulk modulus. In this paper we will consider a case where both surfaces are assumed to be rough, but the lubricant is incompressible. It is also clearly demonstrated, in this case that homogenization is an efficient approach. Moreover, several numerical results are presented and compared with those corresponding to where a constant bulk modulus is assumed to govern the lubricant compressibility. In particular, the result shows a significant difference in the asymptotic behavior between the incompressible case and that with constant bulk modulus.

• 48.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
A new approach for studying cavitation in lubrication2014In: Journal of tribology, ISSN 0742-4787, E-ISSN 1528-8897, Vol. 136, no 1, article id 11706Article in journal (Refereed)

The underlying theory, in this paper, is based on clear physical arguments related to conservation of mass flow and considers both incompressible and compressible fluids. The result of the mathematical modeling is a system of equations with two unknowns, which are related to the hydrodynamic pressure and the degree of saturation of the fluid. Discretization of the system leads to a linear complementarity problem (LCP), which easily can be solved numerically with readily available standard methods and an implementation of a model problem in matlab code is made available for the reader of the paper. The model and the associated numerical solution method have significant advantages over today's most frequently used cavitation algorithms, which are based on Elrod-Adams pioneering work

• 49.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Reynolds equation flow factor estimates by means of homogenization2010In: ASIATRIB 2010: Frontiers in tribology - knowledge & friendship . proceedings of the fourth Asia International Conference on Tribology, 5-9 December 2010, Perth, Western Australia, 2010, p. 185-Conference paper (Refereed)
• 50.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Machine Elements.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Fluid and Experimental Mechanics. Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
Flow in thin domains with a microstructure: Lubrication and thin porous media2017In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1798, article id 020172Article in journal (Refereed)

This paper is devoted to homogenization of different models of flow in thin domains with a microstructure. The focus is on applications connected to the effect of surface roughness in full film lubrication, but a parallel to flow in thin porous media is also discussed. Mathematical models of such flows naturally include two small parameters. One is connected to the fluid film thickness and the other to the microstructure. The corresponding asymptotic analysis is a delicate problem, since the result depends on how fast the two small parameters tend to zero relative to each other. We give a review of the current status in this area and point out some future challenges.

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