NON-AUTONOMOUS INHOMOGENEOUS BOUNDARY CAUCHY PROBLEMS AND RETARDED EQUATIONS. M. Filali and M. Moussi
|
|
- Chrystal McDonald
- 5 years ago
- Views:
Transcription
1 Electronic Journal: Southwest Journal o Pure and Applied Mathematics Internet: ISSN Issue 2, December, 23, pp Submitted: December 24, 22. Published: December NON-AUTONOMOUS INHOMOGENEOUS BOUNDARY CAUCHY PROBLEMS AND RETARDED EQUATIONS M. Filali and M. Moussi Abstract. In this paper we prove the existence and the uniqueness o classical solution o non-autonomous inhomogeneous boundary Cauchy problems, this solution is given by a variation o constants ormula. Then, we apply this result to show the existence o solution o a retarded equation. A.M.S. MOS Subject Classiication Codes. 34G; 47D6 Key Words and Phrases. Boundary Cauchy problem, Evolution amilies, Classical solution, Wellposedness, Variation o constants ormula, Retarded equation Introduction Consider the ollowing Cauchy problem with boundary conditions d ut = Atut, s t T, dt IBCP Ltut = φtut + t, s t T, us = u. This type o problems presents an abstract ormulation o several natural equations such as retarded dierential equations, retarded dierence equations, dynamical population equations and neutral dierential equations. In the autonomous case At = A, Lt = L, φt = φ the Cauchy problem IBCP was studied by Greiner [2,3]. He used a perturbation o domain o generator o semigroups, and showed the existence o classical solutions o IBCP via variation o constants ormula. In the homogeneous case =, Kellermann [6] and Nguyen Lan [8] have showed the existence o an evolution amily Ut, s t s as the classical solution o the problem IBCP. Department o Mathematics, Faculty o Sciences, Oujda, Morocco Address: ilali@sciences.univ-oujda.ac.ma moussi@sciences.univ-oujda.ac.ma c 23 Cameron University 26 Typeset by AMS-TEX
2 NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS 27 The aim o this paper is to show well-posedness in the general case and apply this result to get a solution o a retarded equation. In Section 2 we prove the existence and uniqueness o the classical solution o IBCP. For that purpose, we transorm IBCP into an ordinary Cauchy problem and prove the equivalence o the two problems. Moreover, the solution o IBCP is explicitly given by a variation o constants ormula similar to the one given in [3] in the autonomous case. We note that the operator matrices method was also used in [4, 8, 9] or the investigation o inhomogeneous Cauchy problems without boundary conditions. Section 3 is devoted to an application to the retarded equation { vt = Ktvt + t, t s, RE v s = ϕ. We introduce now the ollowing basic deinitions which will be used in the sequel. A amily o linear unbounded operators At t T on a Banach space X is called a stable amily i there are constants M, ω R such that ]ω, [ ρat or all t T and R, At i M ω k or > ω i= or any inite sequence t... t k T. A amily o bounded linear operators Ut, s s t on X is said an evolution amily i Ut, t = I d and Ut, rur, s = Ut, s or all s r t, 2 the mapping { t, s R 2 + : t s} t, s Ut, s is strongly continuous. For evolution amilies and their applications to non-autonomous Cauchy problems we reer to [,5,]. 2 Well-posedness o Cauchy problem with boundary coditions Let D, X and Y be Banach spaces, D densely and continuously embedded in X, consider amilies o operators At LD, X, Lt LD, Y and φt LX, Y, t T. In this section we will use the operator matrices method in order to prove the existence o classical solution or the non-autonomous Cauchy problem with inhomogeneous boundary conditions d ut = Atut, s t T, dt IBCP Ltut = φtut + t, s t T, us = u, it means that we will transorm this Cauchy problem into an ordinary homogeneous one. In all this section we consider the ollowing hypotheses : H t Atx is continuously dierentiable or all x D; H 2 the amily A t t T, A t := At kerlt, is stable, with M, ω constants o stability; H 3 the operator Lt is surjective or every t [, T ] and t Ltx is continuously dierentiable or all x D;
3 28 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS H 4 t φtx is continuously dierentiable or all x X; H 5 there exist constants γ > and ω R such that Ltx Y γ ω x X or x ker At, > ω and t [, T ]. Deinition 2.. A unction u : [s, T ] X is called a classical solution o IBCP i it is continuously dierentiable, ut D, t [s, T ] and u satisies IBCP. I IBCP has a classical solution, we say that it is well-posed. We recall the ollowing results which will be used in the sequel. Lemma 2.. [6,7] For t [, T ] and ρa t we have the ollowing properties i D = DA t ker At. ii Lt ker At is an isomorphism rom ker At onto Y. iii t L,t := Lt ker At is strongly continuously dierentiable. As consequences o this lemma we have LtL,t = I dy, L,t Lt and I L,t Lt are the projections in D onto ker At and DA t respectively. In order to get the homogenization o IBCP, we introduce the Banach space E := X C [, T ], Y Y, where C [, T ], Y is the space o continuously dierentiable unctions rom [, T ] into Y equipped with the norm g := g + g, or g C [, T ], Y. Let A φ t be a matrix operator deined on E by A φ t :=, DA φ t := D C [, T ], Y {}, t [, T ], At Lt φt δ t here δ t : C[, T ], Y Y is such that δ t g = gt. To the amily A φ we associate the homogeneous Cauchy problem d NCP dt Ut = Aφ tut, s t T, u Us =. In the ollowing proposition we give an equivalence between solutions o IBCP and those o NCP. u Proposition 2.. Let D C [, T ], Y. ut u 2t i I the unction t Ut := is a classical solution o NCP with initial value. Then t u t is a classical solution o IBCP with initial u value u. ii Let u be a classical solution o IBCP with initial value u. Then, the unction t Ut = is a classical solution o NCP with initial value. ut u
4 NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS 29 Proo.. i Since U is a classical solution, then, rom Deinition 2., u is continuously dierentiable and u t D, or t [s, T ]. Moreover we have Thereore U t = u t u 2 t = A φ tut = Atu t Ltu t φtu t δ tu 2t u t = Atu t and u 2t =.. 2. This implies that u 2 t = u 2 s =, t [s, T ], hence the equation 2. yields to The initial value condition is obvious. The assertion ii is obvious. Ltu t = φtu t + t, s t T. Now we return to the study o the Cauchy problem NCP. For that aim, we recall the ollowing result. Theorem 2.. [], Theorem.3 Let At t T be a stable amily o linear operators on a Banach space X such that i the domain D := DAt, D is a Banach space independent o t, ii the mapping t Atx is continuously dierentiable in X or every x D. Then there is an evolution amily Ut, s s t T on D. Moreover Ut, s has the ollowing properties : Ut, sds Dt or all s t T, where Dr is deined by Dr := { x D : Arx D }, 2 the mapping t Ut, sx is continuously dierentiable in X on [s, T ] and d Ut, sx = AtUt, sx or all x Ds and t [s, T ]. dt In order to apply Theorem 2., we need the ollowing lemma. Lemma 2.2. The amily o operators A φ t t T is stable. Proo.. For t [, T ], we write A φ t as A φ t = At +, φt δ t where At = At, with domain DAt = DA φ t. Lt
5 3 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS Since A φ t is a perturbation o At by a linear bounded operator on E, hence, in view o a perturbation result [], Thm it is suicient to show the stability o At t T. x Let > ω and y, we have R, A t L,t At x y = AtR, A tx AtL,t y LtR, A tx + LtL,t y Since R, A tx DA t = kerlt, L,t y ker At and LtL,t = I dy, we obtain At R, A t L,t = I de. 2.2 x On the other hand, or DAt, we have R, A t L,t x At = R, A t Atx + L,t Ltx. From Lemma 2., let x DA t and x 2 ker At such that x = x +x 2. Then R, A t Atx + L,t Ltx = R, A t Atx + x 2 + L,t Ltx + x 2 = R, A t Atx + L,t Ltx 2 = x + x 2 = x. As a consequence, we get R, A t L,t At = I DAt. From 2.2 and 2.3, we obtain that the resolvent o At is given by R, At = R, A t L,t. Hence, by a direct computation one can obtain, or a inite sequence t... t k T, R, A t i R, A t i L,t R, At i = i= i=2 i=. k
6 NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS 3 From the hypothesis H 5, we conclude that L,t γ ω or all t [, T ] and > ω. Deine ω = maxω, ω. Thereore, by using H 2, we obtain or E x y i= R, At i x y = R, A t i x R, A t i L,t y + k i= i=2 M ω k x + M ω k γ ω y + ω k x M ω k, y where M := maxm, Mγ. Thus the lemma is proved. Now we are ready to state the main result. Theorem 2.2. Let be a continuously dierentiable unction on [, T ] onto Y. Then, or all initial value u D, such that Lsu = φsu + s, the Cauchy problem IBCP has a unique classical solution u. Moreover, u is given by the variation o constants ormula ut = Ut, si L,s Lsu + L,t t, ut t [ + Ut, r L,r r, ur L,r r, ur ] dr, 2.4 s where Ut, s t s is the evolution amily generated by A t and t, ut := φtut + t. Proo. First, or the existence o Ut, s we reer to [7]. Since A φ t t T is stable and rom assumptions H, H 3 and H 4, A φ t t T satisies all conditions o Theorem 2., then there exists an evolution amily U φ t, s such that, ut or all initial value DA φ s, the unction u 2t := U φ t, s is u a classical solution o NCP. Thereore, rom i o Proposition 2., u is a classical solution o IBCP. The uniqueness o u comes rom the uniqueness o the solution o NCP and Proposition 2.. Let u be a classical solution o IBCP, at irst, we assume that φt =, then u 2 t := L,t Ltut = L,t t, and u t := I L,t Ltut are dierentiable on t and we have u t = u t u 2 t = Atu t + u 2 t L,t t = A tu t + L,t t L,t t. u
7 32 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS I we deine t := L,t t L,t t, we obtain u t = Ut, su s + t Replacing u s by I L,s Lsu, we obtain consequently, u t = Ut, si L,s Lsu + ut = Ut, si L,s Lsu +L,t t+ t s s Ut, r r dr. t s Ut, r r dr, [ Ut, r L,r r L,r r ] dr. 2.5 Now in the case Φt, since, u is continuously dierentiable, similar arguments are used to obtain the ormula 2.5 or :=, u which is exactly Retarded equation On the Banach space CE := C [ r, ], E, where r > and E is a Banach space, we consider the retarded equation { vt = Ktvt + t, s t T, RE v s = ϕ C E. Where v t τ := vt + τ, or τ [ r, ], and : [, T ] E. Deinition 3.. A unction v : [s r, T ] E is said a solution o RE, i it is continuously dierentiable, Ktv t is well deined, t [, T ] and v satisies RE. In this section we are interested in the study o the retarded equation RE, we will apply the abstract result obtained in the previous section in order to get a solution o RE. As a irst step, we show that this problem can be written as a boundary Cauchy one. More precisely, we show in the ollowing theorem that solutions o RE are equivalent to those o the boundary Cauchy problem d ut = Atut, s t T, 3. IBCP dt Ltut = φtut + t, 3.2 us = ϕ. 3.3 Where At is deined by { Atu := u D := DAt = C [ r, ], E, Lt : D E : Ltϕ = ϕ and φt : C[ r, ], E E : φtϕ = Ktϕ. Note that the spaces D, X and Y in section 2, are given here by D := C [ r, ], E, X := C[ r, ], E and Y := E. We have the ollowing theorem
8 NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS 33 Theorem. i I u is a classical solution o IBCP, then the unction v deined by { ut, s t T, vt := ϕt s, r + s t s, is a solution o RE. ii I v is a solution o RE, then t ut := v t is a classical solution o IBCP. Proo. i Let u be a classical solution o IBCP, then rom Deinition 2., v is continuously dierentiable. On the other hand, 3. and 3.3 implies that u veriies the translation property utτ = { ut + τ, s t + τ T ϕt + τ s, r + s t + τ s, which implies that v t = ut. Thereore, rom 3.2, we obtain vt = ut = Ltut + t = Ktut + t = Ktv t + t. Hence v satisies RE. ii Now, let v be a solution o RE. From Deinition 3., ut C [ r, ], E = DAt, or s t T. Moreover, The equation 3. is obvious. Ltut = ut = vt = Ktvt + t = φtut + t. This theorem allows us to concentrate our sel on the problem IBCP. So, it remains to show that the hypotheses H H 5 are satisied. The hypotheses H, H 3 are obvious and H 4 can be obtained rom the assumptions on the operator Kt. For H 2, let ϕ DA t = { ϕ C [ r, ], E ; ϕ = } and C[ r, ], E such that A tϕ =, or >. Then Since ϕ =, we get ϕτ = e τ ϕ + R, A tτ = τ e τ σ σ dσ, τ [ r, ]. τ e τ σ σ dσ.
9 34 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS By induction, we can show that R, A t i τ = k! i= or a inite sequence t... t k T. Hence Thereore R, A t i τ k! i= τ σ τ k e τ σ σ dσ, = e τ i k τ i i! i=k = eτ τ i k i! i=k τ σ τ k e τ σ dσ, or τ [ r, ]. k R, A t i, >. k i= This proves the stability o A t t [,T ]. Now, i ϕ ker At, then ϕτ = e τ ϕ, or τ [ r, ]. Hence since Ltϕ = ϕ = e τ ϕτ, e lim + = +, in C E, we can take c > such that e Ltϕ c ϕ, t [, T ]. So H 5 holds. As a conclusion, we get the ollowing corollary c, thereore Corollary 3.. Let be a continuously dierentiable unction rom [, T ] onto E, then or all ϕ CE such that, ϕ = Ksϕ + s, the retarded equation RE has a solution v, moreover, v satisies v t = T t sϕ e ϕ + e t, v t + t where T t t is the c -semigroup generated by A t. s Reerences T t re [r, v r r, v r ] dr,. K.J. Engel and R. Nagel, One-Parameter Semigroups or Linear Evolution Equations, Graduate Texts in Mathematics 94, Springer-Verlag, G. Greiner, Perturbing the boundary conditions o a generator, Houston J. Math. vol 3,
10 NON-AUTONOMOUS BOUNDARY CAUCHY PROBLEMS G. Greiner, Semilinear boundary conditions or evolution equations o hyperbolic type, Lecture Notes in Pure and Appl. Math. vol 6 Dekker, New York, R. Grimmer and E. Sinestrari, Maximum norm in one-dimensional hyperbolic problems, Di. Integ. Equat. vol 5, T. Kato,, Linear evolution equations o hyperbolic type, J. Fac. Sci. Tokyo. vol 7, H. Kellermann, Linear evolution equations with time dependent domain, Tübingerbericht Funktionalanalysis, N. T. Lan, On nonautonomous Functional Dierential Equations, J. Math. Anal. Appl. vol 239, N. T. Lan, G. Nickel, Time-Dependent operator matrices and inhomogeneous Cauchy Problems, Rendiconti Del circolo matematico di Palermo. XLVII, R. Nagel and E. Sinestrari, Inhomogeneous Volterra integrodierential equations or Hille- Yosida operators, Marcel Dekker, Lecture Notes Pure Appl. Math. vol 5, A. Pazy, Semigroups o linear operators and applications to partial dierential equations, New york Springer, N. Tanaka, Quasilinear evolution equations with non-densely deined operators, Di. Integral Eq. vol 9,
NON-AUTONOMOUS INHOMOGENEOUS BOUNDARY CAUCHY PROBLEMS
25-Ouja International Conerence on Nonlinear Analysis. Electronic Journal o Dierential Equations, Conerence 14, 26, pp. 191 25. ISSN: 172-6691. URL: http://eje.math.tstate.eu or http://eje.math.unt.eu
More informationINTEGRAL MANIFOLDS OF NONAUTONOMOUS BOUNDARY CAUCHY PROBLEMS
Journal of Nonlinear Evolution Equations and Applications ISSN 2161-368 Volume 212, Number 1, pp. 1 15 (January 212) http://www.jneea.com INTEGRAL MANIFOLDS OF NONAUTONOMOUS BOUNDARY CAUCHY PROBLEMS T.
More informationALMOST PERIODIC SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS ON HILBERT SPACES
Electronic Journal of Differential Equations, Vol. 21(21, No. 72, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ALMOST PERIODIC SOLUTIONS
More informationarxiv: v1 [math.fa] 30 Nov 2012
SEMIGROUPS AND LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH DELAY arxiv:1211.7197v1 [math.fa] 3 Nov 212 ANDRÁS BÁTKAI AND SUSANNA PIAZZERA Abstract. We prove the equivalence o the well-posedness o a partial
More informationEXISTENCE OF SOLUTIONS TO SYSTEMS OF EQUATIONS MODELLING COMPRESSIBLE FLUID FLOW
Electronic Journal o Dierential Equations, Vol. 15 (15, No. 16, pp. 1 8. ISSN: 17-6691. URL: http://ejde.math.txstate.e or http://ejde.math.unt.e tp ejde.math.txstate.e EXISTENCE OF SOLUTIONS TO SYSTEMS
More informationSEMIGROUP APPROACH FOR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTION
SEMIGROUP APPROACH FOR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTION Istanbul Kemerburgaz University Istanbul Analysis Seminars 24 October 2014 Sabanc University Karaköy Communication Center 1 2 3 4 5 u(x,
More informationSolution of the Synthesis Problem in Hilbert Spaces
Solution o the Synthesis Problem in Hilbert Spaces Valery I. Korobov, Grigory M. Sklyar, Vasily A. Skoryk Kharkov National University 4, sqr. Svoboda 677 Kharkov, Ukraine Szczecin University 5, str. Wielkopolska
More informationSemigroups and Linear Partial Differential Equations with Delay
Journal of Mathematical Analysis and Applications 264, 1 2 (21 doi:1.16/jmaa.21.675, available online at http://www.idealibrary.com on Semigroups and Linear Partial Differential Equations with Delay András
More informationNON-ARCHIMEDEAN BANACH SPACE. ( ( x + y
J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. ISSN(Print 6-0657 https://doi.org/0.7468/jksmeb.08.5.3.9 ISSN(Online 87-608 Volume 5, Number 3 (August 08, Pages 9 7 ADDITIVE ρ-functional EQUATIONS
More informationSemigroup Generation
Semigroup Generation Yudi Soeharyadi Analysis & Geometry Research Division Faculty of Mathematics and Natural Sciences Institut Teknologi Bandung WIDE-Workshoop in Integral and Differensial Equations 2017
More informationTelescoping Decomposition Method for Solving First Order Nonlinear Differential Equations
Telescoping Decomposition Method or Solving First Order Nonlinear Dierential Equations 1 Mohammed Al-Reai 2 Maysem Abu-Dalu 3 Ahmed Al-Rawashdeh Abstract The Telescoping Decomposition Method TDM is a new
More informationWellposedness and inhomogeneous equations
LECTRE 6 Wellposedness and inhomogeneous equations In this lecture we complete the linear existence theory. In the introduction we have explained the concept of wellposedness and stressed that only wellposed
More informationA Simple Explanation of the Sobolev Gradient Method
A Simple Explanation o the Sobolev Gradient Method R. J. Renka July 3, 2006 Abstract We have observed that the term Sobolev gradient is used more oten than it is understood. Also, the term is oten used
More informationPerturbation theory of boundary value problems and approximate controllability of perturbed boundary control problems
Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 28 TuC7.6 Perturbation theory of boundary value problems and approximate controllability of perturbed boundary
More informationYURI LEVIN AND ADI BEN-ISRAEL
Pp. 1447-1457 in Progress in Analysis, Vol. Heinrich G W Begehr. Robert P Gilbert and Man Wah Wong, Editors, World Scientiic, Singapore, 003, ISBN 981-38-967-9 AN INVERSE-FREE DIRECTIONAL NEWTON METHOD
More informationJ. M. Almira A MONTEL-TYPE THEOREM FOR MIXED DIFFERENCES
Rendiconti Sem. Mat. Univ. Pol. Torino Vol. 75, 2 (2017), 5 10 J. M. Almira A MONTEL-TYPE THEOREM FOR MIXED DIFFERENCES Abstract. We prove a generalization o classical Montel s theorem or the mixed dierences
More informationNONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST. Genni Fragnelli. Dimitri Mugnai. (Communicated by the associate editor name)
Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 200X Website: http://aimsciences.org pp. X XX NONLINEAR DELAY EQUATIONS WITH NONAUTONOMOUS PAST Genni Fragnelli Dipartimento di Ingegneria
More informationSyed Abbas. June 19th, 2009
Almost Department of Mathematics and Statistics Indian Institute of Technology Kanpur, Kanpur, 208016, India June 19th, 2009 Harald Bohr Function Almost Bohr s early research was mainly concerned with
More informationVALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS
VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where
More informationTHE PERRON PROBLEM FOR C-SEMIGROUPS
Math. J. Okayama Univ. 46 (24), 141 151 THE PERRON PROBLEM FOR C-SEMIGROUPS Petre PREDA, Alin POGAN and Ciprian PREDA Abstract. Characterizations of Perron-type for the exponential stability of exponentially
More informationGlobal Weak Solution of Planetary Geostrophic Equations with Inviscid Geostrophic Balance
Global Weak Solution o Planetary Geostrophic Equations with Inviscid Geostrophic Balance Jian-Guo Liu 1, Roger Samelson 2, Cheng Wang 3 Communicated by R. Temam) Abstract. A reormulation o the planetary
More informationSome Hermite-Hadamard type integral inequalities for operator AG-preinvex functions
Acta Univ. Sapientiae, Mathematica, 8, (16 31 33 DOI: 1.1515/ausm-16-1 Some Hermite-Hadamard type integral inequalities or operator AG-preinvex unctions Ali Taghavi Department o Mathematics, Faculty o
More information1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
More informationOn Picard value problem of some difference polynomials
Arab J Math 018 7:7 37 https://doiorg/101007/s40065-017-0189-x Arabian Journal o Mathematics Zinelâabidine Latreuch Benharrat Belaïdi On Picard value problem o some dierence polynomials Received: 4 April
More informationON THE WELL-POSEDNESS OF THE HEAT EQUATION ON UNBOUNDED DOMAINS. = ϕ(t), t [0, τ] u(0) = u 0,
24-Fez conference on Differential Equations and Mechanics Electronic Journal of Differential Equations, Conference 11, 24, pp. 23 32. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
More informationSOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR PREINVEX FUNCTIONS
Banach J. Math. Anal. 9 15), no., uncorrected galleyproo DOI: 1.1535/bjma/9-- ISSN: 1735-8787 electronic) www.emis.de/journals/bjma/ SOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR
More informationClassification of effective GKM graphs with combinatorial type K 4
Classiication o eective GKM graphs with combinatorial type K 4 Shintarô Kuroki Department o Applied Mathematics, Faculty o Science, Okayama Uniervsity o Science, 1-1 Ridai-cho Kita-ku, Okayama 700-0005,
More informationSOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS
International Journal o Analysis and Applications ISSN 2291-8639 Volume 15, Number 2 2017, 179-187 DOI: 10.28924/2291-8639-15-2017-179 SOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM
More informationMath 216A. A gluing construction of Proj(S)
Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does
More informationAnalysis of Age-Structured Population Models with an Additional Structure
Analysis of Age-Structured Population Models with an Additional Structure Horst R. Thieme Department of Mathematics Arizona State University Tempe, AZ 85287, USA Mathematical Population Dynamics Proceedings
More informationVALUATIVE CRITERIA BRIAN OSSERMAN
VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not
More informationANALYTIC SEMIGROUPS AND APPLICATIONS. 1. Introduction
ANALYTIC SEMIGROUPS AND APPLICATIONS KELLER VANDEBOGERT. Introduction Consider a Banach space X and let f : D X and u : G X, where D and G are real intervals. A is a bounded or unbounded linear operator
More informationOn Semigroups Of Linear Operators
On Semigroups Of Linear Operators Elona Fetahu Submitted to Central European University Department of Mathematics and its Applications In partial fulfillment of the requirements for the degree of Master
More informationExistence Results for Multivalued Semilinear Functional Differential Equations
E extracta mathematicae Vol. 18, Núm. 1, 1 12 (23) Existence Results for Multivalued Semilinear Functional Differential Equations M. Benchohra, S.K. Ntouyas Department of Mathematics, University of Sidi
More informationA NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages 3185 3189 S 0002-9939(97)04112-9 A NOTE ON HENSEL S LEMMA IN SEVERAL VARIABLES BENJI FISHER (Communicated by
More informationFluctuationlessness Theorem and its Application to Boundary Value Problems of ODEs
Fluctuationlessness Theorem and its Application to Boundary Value Problems o ODEs NEJLA ALTAY İstanbul Technical University Inormatics Institute Maslak, 34469, İstanbul TÜRKİYE TURKEY) nejla@be.itu.edu.tr
More informationMathematical Journal of Okayama University
Mathematical Journal of Okayama University Volume 46, Issue 1 24 Article 29 JANUARY 24 The Perron Problem for C-Semigroups Petre Prada Alin Pogan Ciprian Preda West University of Timisoara University of
More informationMassera-type theorem for the existence of C (n) -almost-periodic solutions for partial functional differential equations with infinite delay
Nonlinear Analysis 69 (2008) 1413 1424 www.elsevier.com/locate/na Massera-type theorem for the existence of C (n) -almost-periodic solutions for partial functional differential equations with infinite
More informationThe fundamental properties of quasi-semigroups
Journal of Physics: Conference Series PAPER OPEN ACCESS The fundamental properties of quasi-semigroups To cite this article: Sutrima et al 2017 J. Phys.: Conf. Ser. 855 012052 View the article online for
More informationExample: When describing where a function is increasing, decreasing or constant we use the x- axis values.
Business Calculus Lecture Notes (also Calculus With Applications or Business Math II) Chapter 3 Applications o Derivatives 31 Increasing and Decreasing Functions Inormal Deinition: A unction is increasing
More informationCHOW S LEMMA. Matthew Emerton
CHOW LEMMA Matthew Emerton The aim o this note is to prove the ollowing orm o Chow s Lemma: uppose that : is a separated inite type morphism o Noetherian schemes. Then (or some suiciently large n) there
More informationGENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS
GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS CHRIS HENDERSON Abstract. This paper will move through the basics o category theory, eventually deining natural transormations and adjunctions
More informationOn Controllability of Linear Systems 1
On Controllability of Linear Systems 1 M.T.Nair Department of Mathematics, IIT Madras Abstract In this article we discuss some issues related to the observability and controllability of linear systems.
More informationThe Clifford algebra and the Chevalley map - a computational approach (detailed version 1 ) Darij Grinberg Version 0.6 (3 June 2016). Not proofread!
The Cliord algebra and the Chevalley map - a computational approach detailed version 1 Darij Grinberg Version 0.6 3 June 2016. Not prooread! 1. Introduction: the Cliord algebra The theory o the Cliord
More informationELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables
Department o Electrical Engineering University o Arkansas ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Two discrete random variables
More informationDerivation, f-derivation and generalized derivation of KUS-algebras
PURE MATHEMATICS RESEARCH ARTICLE Derivation, -derivation and generalized derivation o KUS-algebras Chiranjibe Jana 1 *, Tapan Senapati 2 and Madhumangal Pal 1 Received: 08 February 2015 Accepted: 10 June
More informationA NOTE ON SHEAVES WITHOUT SELF-EXTENSIONS ON THE PROJECTIVE n-space.
A NOTE ON SHEAVES WITHOUT SELF-EXTENSIONS ON THE PROJECTIVE n-space. DIETER HAPPEL AND DAN ZACHARIA Abstract. Let P n be the projective n space over the complex numbers. In this note we show that an indecomposable
More informationPartial Averaging of Fuzzy Differential Equations with Maxima
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 6, Number 2, pp. 199 27 211 http://campus.mst.edu/adsa Partial Averaging o Fuzzy Dierential Equations with Maxima Olga Kichmarenko and
More informationBounded and periodic solutions of infinite delay evolution equations
J. Math. Anal. Appl. 286 (2003) 705 712 www.elsevier.com/locate/jmaa Bounded and periodic solutions of infinite delay evolution equations James Liu, a,,1 Toshiki Naito, b and Nguyen Van Minh c a Department
More informationAbstract. 1. Introduction
Chin. Ann. o Math. 19B: 4(1998),401-408. THE GROWTH THEOREM FOR STARLIKE MAPPINGS ON BOUNDED STARLIKE CIRCULAR DOMAINS** Liu Taishun* Ren Guangbin* Abstract 1 4 The authors obtain the growth and covering
More informationFeedback Linearization
Feedback Linearization Peter Al Hokayem and Eduardo Gallestey May 14, 2015 1 Introduction Consider a class o single-input-single-output (SISO) nonlinear systems o the orm ẋ = (x) + g(x)u (1) y = h(x) (2)
More informationMidterm Exam Solutions February 27, 2009
Midterm Exam Solutions February 27, 2009 (24. Deine each o the ollowing statements. You may assume that everyone knows the deinition o a topological space and a linear space. (4 a. X is a compact topological
More informationTheorem Let J and f be as in the previous theorem. Then for any w 0 Int(J), f(z) (z w 0 ) n+1
(w) Second, since lim z w z w z w δ. Thus, i r δ, then z w =r (w) z w = (w), there exist δ, M > 0 such that (w) z w M i dz ML({ z w = r}) = M2πr, which tends to 0 as r 0. This shows that g = 2πi(w), which
More informationUniversidad de Chile, Casilla 653, Santiago, Chile. Universidad de Santiago de Chile. Casilla 307 Correo 2, Santiago, Chile. May 10, 2017.
A Laplace transorm approach to linear equations with ininitely many derivatives and zeta-nonlocal ield equations arxiv:1705.01525v2 [math-ph] 9 May 2017 A. Chávez 1, H. Prado 2, and E.G. Reyes 3 1 Departamento
More informationOn Convexity of Reachable Sets for Nonlinear Control Systems
Proceedings o the European Control Conerence 27 Kos, Greece, July 2-5, 27 WeC5.2 On Convexity o Reachable Sets or Nonlinear Control Systems Vadim Azhmyakov, Dietrich Flockerzi and Jörg Raisch Abstract
More informationNon-stationary Friedrichs systems
Department of Mathematics, University of Osijek BCAM, Bilbao, November 2013 Joint work with Marko Erceg 1 Stationary Friedrichs systems Classical theory Abstract theory 2 3 Motivation Stationary Friedrichs
More informationFractional Evolution Integro-Differential Systems with Nonlocal Conditions
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 5, Number 1, pp. 49 6 (21) http://campus.mst.edu/adsa Fractional Evolution Integro-Differential Systems with Nonlocal Conditions Amar
More informationAPPROXIMATE CONTROLLABILITY OF DISTRIBUTED SYSTEMS BY DISTRIBUTED CONTROLLERS
2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada. Electronic Journal of Differential Equations, Conference 12, 2005, pp. 159 169. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
More informationPERSISTENCE AND EXTINCTION OF SINGLE POPULATION IN A POLLUTED ENVIRONMENT
Electronic Journal o Dierential Equations, Vol 2004(2004), No 108, pp 1 5 ISSN: 1072-6691 URL: http://ejdemathtxstateedu or http://ejdemathuntedu tp ejdemathtxstateedu (login: tp) PERSISTENCE AND EXTINCTION
More informationControllability of non-densely defined functional differential systems in abstract space
Applied Mathematics Letters 19 (26) 369 377 www.elsevier.com/locate/aml Controllability of non-densely defined functional differential systems in abstract space Xianlong Fu Department of Mathematics, East
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the
More informationBasic mathematics of economic models. 3. Maximization
John Riley 1 January 16 Basic mathematics o economic models 3 Maimization 31 Single variable maimization 1 3 Multi variable maimization 6 33 Concave unctions 9 34 Maimization with non-negativity constraints
More informationNumerical Solution of Ordinary Differential Equations in Fluctuationlessness Theorem Perspective
Numerical Solution o Ordinary Dierential Equations in Fluctuationlessness Theorem Perspective NEJLA ALTAY Bahçeşehir University Faculty o Arts and Sciences Beşiktaş, İstanbul TÜRKİYE TURKEY METİN DEMİRALP
More informationSlowly Changing Function Oriented Growth Analysis of Differential Monomials and Differential Polynomials
Slowly Chanin Function Oriented Growth Analysis o Dierential Monomials Dierential Polynomials SANJIB KUMAR DATTA Department o Mathematics University o kalyani Kalyani Dist-NadiaPIN- 7235 West Benal India
More informationProblem Set. Problems on Unordered Summation. Math 5323, Fall Februray 15, 2001 ANSWERS
Problem Set Problems on Unordered Summation Math 5323, Fall 2001 Februray 15, 2001 ANSWERS i 1 Unordered Sums o Real Terms In calculus and real analysis, one deines the convergence o an ininite series
More information1.4 The Jacobian of a map
1.4 The Jacobian of a map Derivative of a differentiable map Let F : M n N m be a differentiable map between two C 1 manifolds. Given a point p M we define the derivative of F at p by df p df (p) : T p
More informationMath 754 Chapter III: Fiber bundles. Classifying spaces. Applications
Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
More informationGeneral solution of the Inhomogeneous Div-Curl system and Consequences
General solution o the Inhomogeneous Div-Curl system and Consequences Briceyda Berenice Delgado López Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional March 07 2017 CINVESTAV
More informationWEAK AND STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR NONEXPANSIVE MAPPINGS IN HILBERT SPACES
Applicable Analysis and Discrete Mathematics available online at http://pemath.et.b.ac.yu Appl. Anal. Discrete Math. 2 (2008), 197 204. doi:10.2298/aadm0802197m WEAK AND STRONG CONVERGENCE OF AN ITERATIVE
More informationAN EXTENSION OF THE LAX-MILGRAM THEOREM AND ITS APPLICATION TO FRACTIONAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 215 (215), No. 95, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu AN EXTENSION OF THE
More informationConservative Control Systems Described by the Schrödinger Equation
Conservative Control Systems Described by the Schrödinger Equation Salah E. Rebiai Abstract An important subclass of well-posed linear systems is formed by the conservative systems. A conservative system
More informationIn the index (pages ), reduce all page numbers by 2.
Errata or Nilpotence and periodicity in stable homotopy theory (Annals O Mathematics Study No. 28, Princeton University Press, 992) by Douglas C. Ravenel, July 2, 997, edition. Most o these were ound by
More informationDISCRETE METHODS AND EXPONENTIAL DICHOTOMY OF SEMIGROUPS. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXIII, 2(2004), pp. 97 205 97 DISCRETE METHODS AND EXPONENTIAL DICHOTOMY OF SEMIGROUPS A. L. SASU Abstract. The aim of this paper is to characterize the uniform exponential
More informationRESTRICTED UNIFORM BOUNDEDNESS IN BANACH SPACES
RESTRICTED UNIFORM BOUNDEDNESS IN BANACH SPACES OLAV NYGAARD AND MÄRT PÕLDVERE Abstract. Precise conditions for a subset A of a Banach space X are known in order that pointwise bounded on A sequences of
More informationA CAUCHY PROBLEM OF SINE-GORDON EQUATIONS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS 1. INTRODUCTION
Trends in Mathematics Information Center for Mathematical Sciences Volume 4, Number 2, December 21, Pages 39 44 A CAUCHY PROBLEM OF SINE-GORDON EQUATIONS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS
More informationStrong Lyapunov Functions for Systems Satisfying the Conditions of La Salle
06 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 004 Strong Lyapunov Functions or Systems Satisying the Conditions o La Salle Frédéric Mazenc and Dragan Ne sić Abstract We present a construction
More informationBoundedly complete weak-cauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informationLINEAR MAPS ON M n (C) PRESERVING INNER LOCAL SPECTRAL RADIUS ZERO
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 2018 (757 763) 757 LINEAR MAPS ON M n (C) PRESERVING INNER LOCAL SPECTRAL RADIUS ZERO Hassane Benbouziane Mustapha Ech-Chérif Elkettani Ahmedou Mohamed
More informationASYMPTOTIC PROPERTIES OF SOLUTIONS TO LINEAR NONAUTONOMOUS DELAY DIFFERENTIAL EQUATIONS THROUGH GENERALIZED CHARACTERISTIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 2(2), No. 95, pp. 5. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ASYMPTOTIC PROPERTIES OF SOLUTIONS
More informationDIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS
Journal o Applied Analysis Vol. 14, No. 2 2008, pp. 259 271 DIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS B. BELAÏDI and A. EL FARISSI Received December 5, 2007 and,
More informationEikonal slant helices and eikonal Darboux helices in 3-dimensional pseudo-riemannian manifolds
Eikonal slant helices and eikonal Darboux helices in -dimensional pseudo-riemannian maniolds Mehmet Önder a, Evren Zıplar b a Celal Bayar University, Faculty o Arts and Sciences, Department o Mathematics,
More informationGENERATORS WITH INTERIOR DEGENERACY ON SPACES OF L 2 TYPE
Electronic Journal of Differential Equations, Vol. 22 (22), No. 89, pp. 3. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GENERATORS WITH INTERIOR
More informationPROPERTIES OF L P (K)-SOLUTIONS OF LINEAR NONHOMOGENEOUS IMPULSIVE DIFFERENTIAL EQUATIONS WITH UNBOUNDED LINEAR OPERATOR
PROPERTIES OF L P (K)-SOLUTIONS OF LINEAR NONHOMOGENEOUS IMPULSIVE DIFFERENTIAL EQUATIONS WITH UNBOUNDED LINEAR OPERATOR Atanaska Georgieva Abstract. Sufficient conditions for the existence of L (k)-solutions
More informationL -uniqueness of Schrödinger operators on a Riemannian manifold
L -uniqueness of Schrödinger operators on a Riemannian manifold Ludovic Dan Lemle Abstract. The main purpose of this paper is to study L -uniqueness of Schrödinger operators and generalized Schrödinger
More informationTHE SNAIL LEMMA ENRICO M. VITALE
THE SNIL LEMM ENRICO M. VITLE STRCT. The classical snake lemma produces a six terms exact sequence starting rom a commutative square with one o the edge being a regular epimorphism. We establish a new
More informationMath 248B. Base change morphisms
Math 248B. Base change morphisms 1. Motivation A basic operation with shea cohomology is pullback. For a continuous map o topological spaces : X X and an abelian shea F on X with (topological) pullback
More informationBIHARMONIC WAVE MAPS INTO SPHERES
BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.
More informationDependence on Initial Conditions of Attainable Sets of Control Systems with p-integrable Controls
Nonlinear Analysis: Modelling and Control, 2007, Vol. 12, No. 3, 293 306 Deendence on Initial Conditions o Attainable Sets o Control Systems with -Integrable Controls E. Akyar Anadolu University, Deartment
More information(C) The rationals and the reals as linearly ordered sets. Contents. 1 The characterizing results
(C) The rationals and the reals as linearly ordered sets We know that both Q and R are something special. When we think about about either o these we usually view it as a ield, or at least some kind o
More informationOn a Closed Formula for the Derivatives of e f(x) and Related Financial Applications
International Mathematical Forum, 4, 9, no. 9, 41-47 On a Closed Formula or the Derivatives o e x) and Related Financial Applications Konstantinos Draais 1 UCD CASL, University College Dublin, Ireland
More informationON MÜNTZ RATIONAL APPROXIMATION IN MULTIVARIABLES
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXVIII 1995 FASC. 1 O MÜTZ RATIOAL APPROXIMATIO I MULTIVARIABLES BY S. P. Z H O U EDMOTO ALBERTA The present paper shows that or any s sequences o real
More informationDiscrete Mathematics. On the number of graphs with a given endomorphism monoid
Discrete Mathematics 30 00 376 384 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On the number o graphs with a given endomorphism monoid
More informationON PERIODIC SOLUTIONS TO SOME LAGRANGIAN SYSTEM WITH TWO DEGREES OF FREEDOM
ON PERIODIC SOLUTIONS TO SOME LAGRANGIAN SYSTEM WITH TWO DEGREES OF FREEDOM OLEG ZUBELEVICH DEPT. OF THEORETICAL MECHANICS, MECHANICS AND MATHEMATICS FACULTY, M. V. LOMONOSOV MOSCOW STATE UNIVERSITY RUSSIA,
More informationON SEMILINEAR CAUCHY PROBLEMS WITH NON-DENSE DOMAIN
Advances in Differential Equations Volume 14, Numbers 11-12 (29), 141 184 ON SEMILINEAR CAUCHY PROBLEMS WITH NON-DENSE DOMAIN Pierre Magal UMR CNRS 5251, IMB & INRIA sud-ouest Anubis University of Bordeaux,
More informationOn m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry
On m-accretive Schrödinger operators in L p -spaces on manifolds of bounded geometry Ognjen Milatovic Department of Mathematics and Statistics University of North Florida Jacksonville, FL 32224 USA. Abstract
More informationAdditive functional inequalities in Banach spaces
Lu and Park Journal o Inequalities and Applications 01, 01:94 http://www.journaloinequalitiesandapplications.com/content/01/1/94 R E S E A R C H Open Access Additive unctional inequalities in Banach spaces
More informationOBSERVATION OPERATORS FOR EVOLUTIONARY INTEGRAL EQUATIONS
OBSERVATION OPERATORS FOR EVOLUTIONARY INTEGRAL EQUATIONS MICHAEL JUNG Abstract. We analyze admissibility and exactness of observation operators arising in control theory for Volterra integral equations.
More informationCategories and Natural Transformations
Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical
More informationTHE GAMMA FUNCTION THU NGỌC DƯƠNG
THE GAMMA FUNCTION THU NGỌC DƯƠNG The Gamma unction was discovered during the search or a actorial analog deined on real numbers. This paper will explore the properties o the actorial unction and use them
More information