A new continuous dependence result for impulsive retarded functional differential equations


 Linette Berry
 1 years ago
 Views:
Transcription
1 CADERNOS DE MATEMÁTICA 11, May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo  Campus de São Carlos, Caixa Postal 668, São Carlos SP, Brazil J. B. Godoy Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo  Campus de São Carlos, Caixa Postal 668, São Carlos SP, Brazil We consider a large class of impulsive retarded functional differential equations and we present a new continuous dependence result for this class of equations. May, 2010 ICMCUSP Key Words: Retarded functional differential equations, impulses, existence and uniqueness, continuous dependence. 1. INTRODUCTION We consider a class of impulsive retarded differential equations (we write impulsive RFDEs for short) with Lebesgue integrable righthand sides whose indefinite integrals satisfy Carathéodory and Lypschitztype conditions. We also consider that the impulse operators are Lypschitzian functions and that the initial conditions are regulated functions. Then we prove an existence and uniqueness theorem for this class of equations as well as a new continuous dependence type result. In the above setting, the following continuous dependence result for impulsive RFDEs is wellknown (see [1], Theorem 4.1). Consider a sequence of impulsive retarded initial value problems (we write IVPs for short) whose righthand sides converge to the righthand side of an impulsive RFDE and whose initial data also converge. Let the sequence of impulse operators be convergent as well. Suppose each element of the sequence of impulsive retarded IVPs admits a unique solution and that the sequence of solutions is uniformly convergent. * Supported by FAPESP grant 2008/ and by CNPq grant / Supported by FAPESP grant 2007/
2 38 M. FEDERSON AND J. B. GODOY Consider also the limiting IVP with limiting righthand side, limiting impulse operators and limiting initial condition. Then the limit of the sequence of solutions is a solution of the limiting IVP, provided certain conditions are fulfilled. In the present paper, we state and prove a certain reciprocal of this result. Roughly speaking, we consider a sequence of IVPs for impulsive RFDEs, in the above setting, with convergent righthand sides, convergent impulse operators and uniformly convergent initial data. We assume the limiting equation is an impulsive RFDE whose initial condition is the uniform limit of the sequence of initial data and whose solution exists and is unique. Then, under certain conditions, for sufficient large indexes, the elements of the sequence of impulsive retarded IVPs admit a unique solution and such a sequence of solutions converges to the solution of the limiting IVP. 2. THE SETTING OF IMPULSIVE RFDES Let X be a Banach space. A function f : [a, b] X is called regulated, if the following limits exist lim f(s) = f(t ) X, t (a, b], and lim f(s) = f(t+) X, s t s t+ t [a, b). In this case, we write f G([a, b], X) and we endow G([a, b], X) with the usual supremum norm f = sup a t b f(t). Then (G([a, b], X), ) is a Banach space. Also, any function in G([a, b], X) is the uniform limit of step functions (see [2]). Define G ([a, b], X) = {u G([a, b], X) : u is left continuous at every t (a, b]}. In G ([a, b], X), we consider the norm induced by G([a, b], X). Given a function y : [ r, + σ] R n, with r > 0 and σ > 0, we consider y t : [ r, 0] R n given by y t (θ) = y (t + θ), θ [ r, 0], t [, + σ]. Then it is clear that for a function y G ([ r, +σ], R n ), we have y t G ([ r, 0], R n ) for all t [, + σ]. We consider the following initial value problem for a retarded functional differential equation (RFDE) with impulses ẏ (t) = f (y t, t), t t k y (t k ) = I k (y (t k )), k = 0, 1,..., m, (1) y t0 = φ, where t k, k = 0, 1,..., m, with < t 1 <... < t k <... < t m + σ, σ > 0, are preassigned moments of impulse, for k = 0, 1,..., m, y I k (y) maps R n into itself and y (t k ) := y (t k +) y (t k ) = y (t k +) y (t k ), Sob a supervisão da CPq/ICMC
3 CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES 39 that is, y is left continuous at t = t k and the lateral limit y(t k +) exists, for k = 0, 1, 2,..., m. We also consider φ G ([ r, 0], R n ). It is known that the impulsive system (1) is equivalent to the integral equation y(t) = y( ) + f(y s, s)ds + y t0 = φ, t k < t I k (y(t k )) whenever the integral exists in some sense. We will consider Lebesgue integration in (2). Let P C 1 G ([ r, + σ], R n ) be an open set (in the topology of locally uniform convergence in G ([ r, + σ], R n ) with the following property: if y is an element of P C 1 and t [ r, + σ], then ȳ given by (2) ȳ (t) = { y (t), t0 r t t, y ( t), t < t, (3) is also an element of P C 1. In particular, any open ball in G ([ r, + σ], R n ) has this property. We assume that f : (ψ, t) G ([ r, 0], R n ) [, + σ] R n and that the mapping t f (y t, t) is Lebesgue integrable. We also assume the following conditions: (A) There is a Lebesgue integrable function M : [, +σ] R such that for all x P C 1 and all u 1, u 2 [, + σ], u2 u 1 u2 f (x s, s) ds M (s) ds; (B) There is a Lebesgue integrable function L : [, + σ] R such that for all x, y P C 1 and all u 1, u 2 [, + σ], u2 u 1 u2 [f (x s, s) f (y s, s)] ds L (s) x s y s ds. For the impulse operators I k : R n R n, k = 1, 2,..., we assume the following conditions: (A ) there is a constant K 1 > 0 such that for all k = 0, 1, 2,..., and all x R n, u 1 I k (x) K 1 ; (B ) there is a constant K 2 > 0 such that for all k = 0, 1, 2,..., and all x, y R n, u 1 I k (x) I k (y) K 2 x y.
4 40 M. FEDERSON AND J. B. GODOY Note that the Carathéodory and Lipschitztype conditions (A) and (B) are required for the indefinite integral of f only and not for the function f itself. Thus the standard requirement that f(ψ, t) is continuous in ψ does not need to be fulfilled. Also, the mapping t f (y t, t) does not need to be piecewise continuous. Let us recall the concept of a solution of problem (1). Definition 2.1. A function y P C 1 for which (y t, t) G ([ r, 0], R n ) [, + σ] for t < + σ and the conditions (i) ẏ (t) = f (y t, t), almost everywhere (in Lebesgue s sense), whenever t t k. (ii) y (t k +) = y (t k ) + I k (y (t k )), k = 0, 1, 2,..., m. (iii) y t0 = φ are satisfied is called a solution of (1) in [, + σ] (or sometimes also in [ r, + σ]) with initial condition (φ, ). In [1], it was proved that under the conditions (A), (B), (A ) and (B ), system (1) can be identified in a onetoone correspondence with a system of generalized ordinary differential equations taking values in a Banach space. Local existence and uniqueness of solutions are guaranteed by Theorems 2.15, 3.4 and 3.5 from [1]. In what follows, we give a direct proof of an existence and uniqueness theorem for the impulsive RFDE (1) without employing the theory of generalized ODEs. Nevertheless, our proof is inspired in the proof of [1], Theorem Theorem 2.1. Consider problem (1) and suppose conditions (A), (B), (A ) and (B ) are fulfilled. Then there is a > 0 such that on the interval [, + ] there exists a unique solution y : [ r, + ] R n of problem (1) for which y t0 = φ. Proof. For t [, + σ], define h 1 (t) = [M(s) + L(s)]ds and h 2 (t) = max(k 1, K 2 ) m H tk (t), k=0 where H tk denotes the left continuous Heaviside function concentrated at t k, that is, { 0, for t tk H tk (t) = 1, for t k > t. Let h = h 1 +h 2. Then h is nondecreasing and left continuous. Besides, for s [, +σ], we have f(y t, t)dt + I j (y(t j )) = s m f(y t, t)dt + I j (y(t j ))H tj (s) t j<s j=0 Sob a supervisão da CPq/ICMC
5 CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES 41 M(t)dt + K 1 m H tj (s) h(s) h( ), j=0 by conditions (A) and (A ). We will consider two cases: when is a point of continuity of h : [, + σ] R n and otherwise. At first, let be a point of continuity of h. Assume that > 0 is such that [, + ] [, + σ) and h( + ) h( ) < 1 2. Let Q be the set of functions z : [, + ] R n such that z BV ([, + ], R n ) and z(t) y( ) h(t) h( ) for t [, + ]. It is easy to show that the set Q BV ([, + ], R n ) is closed. For s [, + ] and z Q, define T z(s) = y( ) + f(z t, t)dt + <t j s I j (z(t j )). Then, T z(s) y( ) = f(z t, t)dt + t j<s I j (z(t j )) h(s) h(), s [, + ]. Also, the fact that T z belongs to BV ([, + ], R n ) is not difficult to prove. Thus, it follows that T maps Q into itself. Take s 1 < s 2 + and z 1, z 2 Q. Using conditions (B) and (B ), we obtain T z 2 (s 2 ) T z 1 (s 2 ) [T z 2 (s 1 ) T z 1 (s 1 )] = 2 = [f(z 2, t) f(z 1, t)]dt + [I j (z 2 (t j )) I j (z 1 (t j ))] s 1 s 1 t j<s 2 2 m L(t) z 2 (t) z 1 (t) dt + K 2 z 2 (t j ) z 1 (t j ) [ H tj (s 2 ) H tj (s 1 ) ] s 1 j=0 2 m [ sup z 2 (s) z 1 (s) L(t)dt + K 2 Htj (s 2 ) H tj (s 1 ) ] s [,+ ] s 1 z 2 z 1 BV ([t0,+ ])[h(s 2 ) h(s 1 )]. Recall that z BV ([t0,+ ]) = z( ) +vart t0+ 0 (z) defines a norm in BV ([, + ], R n ), where vart t0+ 0 (z) denotes the variation of z on the interval [, + ]. Therefore T z 2 T z 1 BV ([t0,+ ]) z 2 z 1 BV ([t0,+ ])[h( + ) h( )] j=0 < 1 2 z 2 z 1 BV ([t0,+ ])
6 42 M. FEDERSON AND J. B. GODOY and hence T is a contraction. Then, by the Banach fixedpoint theorem, the result follows. Now, we consider the case where is not a point of continuity of h (or of h 2 ). Define { h2 (t), t = t h 2 (t) = 0 h 2 (t) h 2 ( +) + h 2 ( ) = h 2 (t) h 2 ( +), t + σ, and h = h 1 + h 2. Then h is continuous at t =, left continuous and nondecreasing. Define an impulse operator I 0 from R n to R n such that, for y G ([, + σ], R n ), I 0 (y(t)) = { I0 (y(t)), t = I 0 (y(t)) I 0 (y( +)) + I 0 (y( )), t + σ and consider the impulsive RFDE ż (t) = f (z t, t), t t k z (t k ) = I k (z (t k )), k = 1,..., m, z ( ) = I 0 (z ( )), z t0 = φ, (4) where φ(θ) = { φ(θ), r θ < 0 y( +), θ = 0. Then, for s [, + σ], we have f(z t, t)dt + I 0 (z( )) + <t j<s I j (z(t j )) h(s) h(). As in the previous case, let > 0 be such that [, + ] [, + σ) and h( + ) h( ) < 1 2. Then, define Q as the set of functions z : [, + ] R n such that z BV ([, + ], R n ) and z(t) y( +) h(t) h( ) for t [, + ] and consider the operator T defined on Q and given by T z(s) = y( +) + f(z t, t)dt + I 0 (z( )) + <t j s I j (z(t j )). Following the procedure of the previous case, it can be shown that equation (4) admits a unique solution on [, + ]. Then, defining y t0 = φ and y(t) = z(t), for t >, we obtain a unique solution of (1), for which y t0 = φ in [, + ]. For the next theorem, we consider the following sequence of initial value problems: ẏ (t) = f p (y t, t), t t k y (t k ) = I p k (y (t k)), k = 0, 1,..., m, y t0 = φ p, (5) Sob a supervisão da CPq/ICMC
7 CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES 43 where < t 1 <... < t k <... < t m + σ, and for each p = 1, 2,..., x I p k (x) maps R n into itself and y (t k ) := y (t k +) y (t k ) = y (t k +) y (t k ), k = 0, 1, 2,..., m. We will show that, under conditions (A), (B), (A ) and (B ) for f p and I p k, k = 0, 1, 2,..., m, the sequence {y p } p 1 of solutions of (5) is equibounded and of uniformly bounded variation on some closed subinterval of [, + σ]. Theorem 2.2. Assume that for p = 0, 1,..., φ p G ([ r, 0], R n ) and moreover f p : G ([ r, 0], R n ) [, + σ] R n and I p k : Rn R n satisfy conditions (A), (B), (A ) and (B ) for the same functions M, L and the same constants K 1, K 2. Then there is a > 0 such that y p : [ r, + ] R n is a solution of (5), for each p, and the sequence {y p } p 1 is equibounded and of uniformly bounded variation on [, + ]. Proof. From the proof of Theorem 2.1, it is clear that a unique > 0 can be obtained such that a solution y p : [ r, + ] R n of (5) exists and is unique, independently of p. Thus, for p = 0, 1,..., we have y p (s) = y p ( ) + f p ((y p ) t, t)dt + I p j (y p(t j )), < t j s s [, + ]. Consider a partition = s 0 < s 1 <... < s n = + of [, + ]. Then y p (s i ) y p (s i 1 ) = i s i 1 f p ((y p ) t, t)dt + Therefore, conditions (A) and (A ) imply n y p (s i ) y p (s i 1 ) i=1 i=1 0+ n i f p ((y p ) t, t)dt s i 1 + M(s)ds + K 1 l, s i 1< t j s i I j (y p (t j )), i = 1, 2,..., n. < t j + where l is the number of impulse moments in the interval [, + ]. Then I j (y p (t j )) 0+ vart t0+ 0 (y p ) M(s)ds + K 1 l, for all p N. (6) where vart t0+ 0 (y p ) denotes the variation of y p [, + ], and hence {y p } p 1 is of uniformly bounded variation on [, + ]. Now, we are going to show that {y p } p 1 is equibounded. Again, since y p (t) = y p ( ) + f((y p ) s, s)ds + <t j t I j (y p (t j ))
8 44 M. FEDERSON AND J. B. GODOY for each t [, + ] and each p = 1, 2, 3,..., we have y p (t) y p ( ) + f p ((y p ) s, s)ds + y p ( ) + M(s)ds + K 1 l y p ( ) + 0+σ M(s)ds + K 1 l < t j + I j (y p (t j )) Therefore {y p } p 1 is equibounded on [, + ] and the result follows. 3. CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES In general, one cannot expect that an impulsive RFDE depends on the initial data. We mention [3] for an elucidative discussion on the continuous dependence of solutions of an impulsive RFDE whose impulse operators also involve delays. The next theorem is a continuous dependence result which, together with Theorem 2.2, are important to prove the theorem following it. A proof of it can be found in [1], Theorem 4.1. Theorem 3.1. Assume that for p = 0, 1,..., φ p G ([ r, 0], R n ) and moreover f p : G ([ r, 0], R n ) [, + σ] R n and I p k : Rn R n, k = 0, 1, 2,..., satisfy conditions (A), (B), (A ) and (B ) for the same functions M, L and the same constants K 1, K 2. Let the relations ϑ lim sup [f p ϑ [,+σ] p (y s, s) f 0 (y s, s)]ds = 0 (7) for every y P C 1 and lim p Ip k (x) = I0 k(x) (8) for every x R n, k = 0, 1,..., m be satisfied. Assume that y p : [ r, + σ] R n, p = 1, 2,..., is a solution on [ r, + σ] of the problem such that ẏ (t) = f p (y t, t), t t k y (t k ) = I p k (y (t k)), k = 1,..., m y t0 = φ p, lim y p(s) = y(s) uniformly on [ r, + σ]. (10) p (9) Sob a supervisão da CPq/ICMC
9 CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES 45 Then y : [ r, + σ] R n is a solution on [ r, + σ] of the problem ẏ (t) = f 0 (y t, t), t t k y (t k ) = Ik 0 (y (t k)), k = 1,..., m y t0 = φ 0. (11) The assumptions (7) and (8) in Theorem 2.2 ensure that if the sequence {y p } p 1, y p : [ r, + ] R n, p = 1, 2,..., of solutions of (5) converges uniformly to a function y : [ r, + ] R n, then this limit is a solution of (11). The next result says that adding an uniqueness condition to the limiting equation, then for sufficient large p N, y p : [ r, + ] R n is a solution of (5), provided the sequence of initial data {φ p } p 1 converges uniformly on [ r, 0]. Theorem 3.2. Assume that f p (φ, t) : G ([ r, 0], R n ) [, + σ] R n, p = 0, 1, 2,..., satisfies conditions (A) and (B) for the same functions M and L. Let I k p : R n R n k = 1, 2,..., p = 0, 1, 2,..., be impulse operators which satisfy conditions (A ) and (B ) for the same constants K 1 and K 2. Assume that for every y P C 1, and lim p [f p (y s, s) f 0 (y s, s)]ds = 0, t [, + σ] (12) lim p Ip k (x) = I0 k(x) (13) for every x R n and k = 1,..., m. Let y : [ r, + σ] R n be a solution of ẏ (t) = f 0 (y t, t), t t k y (t k ) = Ik 0 (y (t k)), k = 1,..., m y t0 = φ 0, (14) on [ r, + σ]. Assume that if there exists ρ > 0 such that sup θ [ r,0] u(θ) φ 0 (θ) < ρ, then u G ([ r, 0], R n ). Assume further that φ p φ 0 uniformly on [ r, 0] as p. Then, for sufficiently large p N, there exists a solution y p of on [ r, + σ] and ẏ (t) = f p (y t, t), t t k y (t k ) = I p k (y (t k)), k = 1,..., m y t0 = φ p, (15) lim y p(s) = y(s), s [ r, + σ] (16) p
10 46 M. FEDERSON AND J. B. GODOY Proof. The present proof is inspired in the proof of [4], Theorem 8.6, for generalized ODEs. We strongly use the fact that the functions f p, p = 0, 1, 2,..., take values in a finite dimensional space so that we can apply Helly s choice principle. Because φ p φ 0 uniformly on [ r, 0] as p, there exists ρ > 0 and k N such that sup φ p (θ) φ 0 (θ) < ρ, for p > k. (17) θ [ r,0] Thus, by the assumption, φ p G ([ r, 0], R n ), for p > k. Then this implies by the existence theorem (Theorem 2.1) that for p > k, there exists a > 0 such that y p is a solution of (15) on [ r, + ]. We assert that lim y p(s) = y(s), for s [ r, + σ]. (18) p By Theorem 3.1, if the sequence {y p } p 1 admits a convergent subsequence, then since φ p φ 0, it will follow by the uniqueness of solutions that there is a > 0 such that lim p y p (s) = y(s), for s [ r, + ], where y t0 = φ 0. We will use Helly s choice principle to prove that in fact {y p } p 1 admits a convergent subsequence. By Theorem 2.2 the sequence {y p }, p > k, of functions on [, + ] is equibounded and of uniformly bounded variation. Thus, by the Helly s choice principle, the sequence {y p }, p > k, contains a pointwise convergent subsequence and hence y(t) is the only accumulation point of the sequence y k (t) for every t [, + ]. Therefore the theorem holds on [, + ], > 0. It also holds on [ r, ], since φ p φ 0. Now, let us assume that the convergence result does not hold on the whole interval [ r, +σ]. Thus there exist a, 0 < < σ, such that for every < and for p > k, there is a solution y p of (15) on [ r, + ], with (y p ) t0 = φ 0, and lim p y p (t) = y(t) for t [ r, + ], but this does not hold on [ r, + ], whenever >. By the proof of Theorem 2.1, y k (s 2 ) y k (s 1 ) < h(s 2 ) h(s 1 ) for every s 2, s 1 [ r, + ) and every p > k. Hence the limits y p (( + ) ) = lim ε 0 y p ( + + ε), p > k, exist and y p (( + ) ) = y( + ), for p > k, since y is left continuous. Defining y p ( + ) = y p (( + ) ), for p > k, then lim p y p ( + ) = y( + ). Therefore the theorem holds on [ r, + ] as well. Then, using + < + σ as the starting point, it can be proved analogously that the theorem holds on the interval [ +, + +η], for some η > 0, and this contradicts our assumption. Thus the theorem holds on the whole interval [ r, + σ]. REFERENCES Sob a supervisão da CPq/ICMC
11 CONTINUOUS DEPENDENCE FOR IMPULSIVE RFDES M. Federson; S. Schwabik, Generalized ODEs approach to impulsive retarded differential equations, Differential and Integral Equations 19(11) (2006), MR (2007i:34092) 2. C. S. Hönig, Volterra Stieltjes Integral Equations. Functional analytic methods; linear constraints. Mathematics Studies, No. 16. Notas de Matemtica, No. 56. [Notes on Mathematics, No. 56] North Holland Publishing Co., AmsterdamOxford; American Elsevier Publishing Co., Inc., New York, MR (58 #17705) 3. Xinzhi Liu and G. Ballinger, Continuous dependence on initial values for impulsive delay differential equations, Appl. Math. Letters, 17(4), (2004), MR (2004m:34020) 4. Š. Schwabik, Generalized Ordinary Differential Equations, Series in Real Analysis, 5. World Scientific Publishing Co., Inc., River Edge, NJ, MR (94g:34004)
On the existence of Gequivariant maps
CADERNOS DE MATEMÁTICA 12, 69 76 May (2011) ARTIGO NÚMERO SMA# 345 On the existence of Gequivariant maps Denise de Mattos * Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação,
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More information{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...
44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationThe HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line
The HenstockKurzweilStieltjes type integral for real functions on a fractal subset of the real line D. Bongiorno, G. Corrao Dipartimento di Ingegneria lettrica, lettronica e delle Telecomunicazioni,
More informationUNIVERSIDADE DE SÃO PAULO
UNIVERSIDADE DE SÃO PAULO Instituto de Ciências Matemáticas e de Computação ISSN 01032569 Comments on On minimizing the lengths of checking sequences Adenilso da Silva Simão N ō 307 RELATÓRIOS TÉCNICOS
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationBOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS. Muhammad N. Islam
Opuscula Math. 35, no. 2 (215), 181 19 http://dx.doi.org/1.7494/opmath.215.35.2.181 Opuscula Mathematica BOUNDED, ASYMPTOTICALLY STABLE, AND L 1 SOLUTIONS OF CAPUTO FRACTIONAL DIFFERENTIAL EQUATIONS Muhammad
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More information1. Let X and Y be normed spaces and let T B(X, Y ).
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 20050314 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationx a x 2 (1 + x 2 ) n.
Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number
More informationON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS
ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWOPOINT BOUNDARY VALUE PROBLEMS FOR TWODIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationAsian Option Pricing Formula for Uncertain Financial Market
Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s446715357 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei
More informationTHE CONTRACTION MAPPING THEOREM
THE CONTRACTION MAPPING THEOREM KEITH CONRAD 1. Introduction Let f : X X be a mapping from a set X to itself. We call a point x X a fixed point of f if f(x) = x. For example, if [a, b] is a closed interval
More informationA Simple Observation Concerning Contraction Mappings
Revista Colombiana de Matemáticas Volumen 46202)2, páginas 229233 A Simple Observation Concerning Contraction Mappings Una simple observación acerca de las contracciones German LozadaCruz a Universidade
More informationTRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh
Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp . TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI*
More informationGeneral Theory of Differential Equations Sections 2.8, 3.13.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.13.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationKnown Results and Open Problems on C 1 linearization in Banach Spaces
São Paulo Journal of Mathematical Sciences 6, 2 (2012), 375 384 Known Results and Open Problems on C 1 linearization in Banach Spaces Hildebrando M. Rodrigues Instituto de Ciências Matemáticas e de Computaçao
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationI. Pointwise convergence
MATH 40  NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationStochastic Inventory Control
Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationand s n (x) f(x) for all x and s.t. s n is measurable if f is. REAL ANALYSIS Measures. A (positive) measure on a measurable space
RAL ANALYSIS A survey of MA 641643, UAB 19992000 M. Griesemer Throughout these notes m denotes Lebesgue measure. 1. Abstract Integration σalgebras. A σalgebra in X is a nonempty collection of subsets
More informationOn the Fractional Derivatives at Extreme Points
Electronic Journal of Qualitative Theory of Differential Equations 212, No. 55, 15; http://www.math.uszeged.hu/ejqtde/ On the Fractional Derivatives at Extreme Points Mohammed AlRefai Department of
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationSome stability results of parameter identification in a jump diffusion model
Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss
More informationMathematics for Econometrics, Fourth Edition
Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationThe integrating factor method (Sect. 2.1).
The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable
More informationINTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES
INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the
More information1 Introduction, Notation and Statement of the main results
XX Congreso de Ecuaciones Diferenciales y Aplicaciones X Congreso de Matemática Aplicada Sevilla, 2428 septiembre 2007 (pp. 1 6) Skewproduct maps with base having closed set of periodic points. Juan
More informationEXISTENCE AND NONEXISTENCE RESULTS FOR A NONLINEAR HEAT EQUATION
Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 5 (7), pp. 5 65. ISSN: 7669. UL: http://ejde.math.txstate.edu
More informationAn example of a computable
An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept
More informationFIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.
FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is
More informationGambling Systems and MultiplicationInvariant Measures
Gambling Systems and MultiplicationInvariant Measures by Jeffrey S. Rosenthal* and Peter O. Schwartz** (May 28, 997.. Introduction. This short paper describes a surprising connection between two previously
More informationON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction
ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity
More informationThe Fourier Series of a Periodic Function
1 Chapter 1 he Fourier Series of a Periodic Function 1.1 Introduction Notation 1.1. We use the letter with a double meaning: a) [, 1) b) In the notations L p (), C(), C n () and C () we use the letter
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationMetric Spaces Joseph Muscat 2003 (Last revised May 2009)
1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of
More informationRolle s Theorem. q( x) = 1
Lecture 1 :The Mean Value Theorem We know that constant functions have derivative zero. Is it possible for a more complicated function to have derivative zero? In this section we will answer this question
More informationSeparation Properties for Locally Convex Cones
Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam
More informationWorking Paper Series
RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos HervésBeloso, Emma Moreno García and
More informationChapter 04  More General Annuities
Chapter 04  More General Annuities 41 Section 4.3  Annuities Payable Less Frequently Than Interest Conversion Payment 0 1 0 1.. k.. 2k... n Time k = interest conversion periods before each payment n
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationSOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction
italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationEssential Topic: Continuous cash flows
Essential Topic: Continuous cash flows Chapters 2 and 3 The Mathematics of Finance: A Deterministic Approach by S. J. Garrett CONTENTS PAGE MATERIAL Continuous payment streams Example Continuously paid
More informationLecture notes on Ordinary Differential Equations. Plan of lectures
1 Lecture notes on Ordinary Differential Equations Annual Foundation School, IIT Kanpur, Dec.328, 2007. by Dept. of Mathematics, IIT Bombay, Mumbai76. email: sivaji.ganesh@gmail.com References Plan
More information16.3 Fredholm Operators
Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this
More informationOPTIMAl PREMIUM CONTROl IN A NONliFE INSURANCE BUSINESS
ONDERZOEKSRAPPORT NR 8904 OPTIMAl PREMIUM CONTROl IN A NONliFE INSURANCE BUSINESS BY M. VANDEBROEK & J. DHAENE D/1989/2376/5 1 IN A OPTIMAl PREMIUM CONTROl NONliFE INSURANCE BUSINESS By Martina Vandebroek
More informationMath212a1010 Lebesgue measure.
Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.
More informationON THE EXISTENCE AND LIMIT BEHAVIOR OF THE OPTIMAL BANDWIDTH FOR KERNEL DENSITY ESTIMATION
Statistica Sinica 17(27), 2893 ON THE EXISTENCE AND LIMIT BEHAVIOR OF THE OPTIMAL BANDWIDTH FOR KERNEL DENSITY ESTIMATION J. E. Chacón, J. Montanero, A. G. Nogales and P. Pérez Universidad de Extremadura
More informationIMPLEMENTING ARROWDEBREU EQUILIBRIA BY TRADING INFINITELYLIVED SECURITIES
IMPLEMENTING ARROWDEBREU EQUILIBRIA BY TRADING INFINITELYLIVED SECURITIES Kevin X.D. Huang and Jan Werner DECEMBER 2002 RWP 0208 Research Division Federal Reserve Bank of Kansas City Kevin X.D. Huang
More informationA PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS. In memory of RouHuai Wang
A PRIORI ESTIMATES FOR SEMISTABLE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS XAVIER CABRÉ, MANEL SANCHÓN, AND JOEL SPRUCK In memory of RouHuai Wang 1. Introduction In this note we consider semistable
More informationALMOST COMMON PRIORS 1. INTRODUCTION
ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type
More informationON A GLOBALIZATION PROPERTY
APPLICATIONES MATHEMATICAE 22,1 (1993), pp. 69 73 S. ROLEWICZ (Warszawa) ON A GLOBALIZATION PROPERTY Abstract. Let (X, τ) be a topological space. Let Φ be a class of realvalued functions defined on X.
More information1 The Brownian bridge construction
The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof
More informationFuzzy Differential Systems and the New Concept of Stability
Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More information1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 0050615 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationTRIPLE FIXED POINTS IN ORDERED METRIC SPACES
Bulletin of Mathematical Analysis and Applications ISSN: 18211291, URL: http://www.bmathaa.org Volume 4 Issue 1 (2012., Pages 197207 TRIPLE FIXED POINTS IN ORDERED METRIC SPACES (COMMUNICATED BY SIMEON
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationRend. Istit. Mat. Univ. Trieste Suppl. Vol. XXX, 111{121 (1999) Fuzziness in Chang's Fuzzy Topological Spaces Valentn Gregori and Anna Vidal () Summary.  It is known that fuzziness within the concept
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationSensitivity analysis of European options in jumpdiffusion models via the Malliavin calculus on the Wiener space
Sensitivity analysis of European options in jumpdiffusion models via the Malliavin calculus on the Wiener space Virginie Debelley and Nicolas Privault Département de Mathématiques Université de La Rochelle
More informationManual for SOA Exam MLC.
Chapter 5 Life annuities Extract from: Arcones Manual for the SOA Exam MLC Fall 2009 Edition available at http://wwwactexmadrivercom/ 1/94 Due n year temporary annuity Definition 1 A due n year term annuity
More informationFunctional analysis and its applications
Department of Mathematics, London School of Economics Functional analysis and its applications Amol Sasane ii Introduction Functional analysis plays an important role in the applied sciences as well as
More informationThesis Title. A. U. Thor. A.A.S., University of Southern Swampland, 1988 M.S., Art Therapy, University of New Mexico, 1991 THESIS
Thesis Title by A. U. Thor A.A.S., University of Southern Swampland, 1988 M.S., Art Therapy, University of New Mexico, 1991 THESIS Submitted in Partial Fulllment of the Requirements for the Degree of Master
More informationA Transmission Problem for EulerBernoulli beam with KelvinVoigt. Damping
Applied Mathematics & Information Sciences 5(1) (211), 1728 An International Journal c 211 NSP A Transmission Problem for EulerBernoulli beam with KelvinVoigt Damping C. A Raposo 1, W. D. Bastos 2 and
More informationA First Course in Elementary Differential Equations. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A First Course in Elementary Differential Equations Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 1 Basic Terminology 4 2 Qualitative Analysis: Direction Field of y = f(t, y)
More informationA Note on Di erential Calculus in R n by James Hebda August 2010.
A Note on Di erential Calculus in n by James Hebda August 2010 I. Partial Derivatives o Functions Let : U! be a real valued unction deined in an open neighborhood U o the point a =(a 1,...,a n ) in the
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationON A MIXED SUMDIFFERENCE EQUATION OF VOLTERRAFREDHOLM TYPE. 1. Introduction
SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17) (2009), 55 62 ON A MIXED SUMDIFFERENCE EQUATION OF VOLTERRAFREDHOLM TYPE B.. PACHPATTE Abstract. The main objective of this paper is to study some basic properties
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationOPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov
DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationA Second Course in Elementary Differential Equations: Problems and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved
A Second Course in Elementary Differential Equations: Problems and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved Contents 8 Calculus of MatrixValued Functions of a Real Variable
More informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. GacovskaBarandovska
Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. GacovskaBarandovska Smirnov (1949) derived four
More informationOur goal first will be to define a product measure on A 1 A 2.
1. Tensor product of measures and Fubini theorem. Let (A j, Ω j, µ j ), j = 1, 2, be two measure spaces. Recall that the new σ algebra A 1 A 2 with the unit element is the σ algebra generated by the
More information