On a Fuzzy Logistic Difference Equation
|
|
- Moses Reynolds
- 5 years ago
- Views:
Transcription
1 On a Fuzzy Logistic Difference Euation QIANHONG ZHANG Guizhou University of Finance and Economics Guizhou Key Laboratory of Economics System Simulation Guiyang Guizhou CHINA zianhong68@163com JINGZHONG LIU Hunan Institute of Technology Deartment of Mathematics and Physics Hengyang Hunan CHINA hnhyls@126com WENZHUAN ZHANG Guizhou University of Finance and Economics Guizhou Key Laboratory of Economics System Simulation Guiyang Guizhou CHINA zhwenzh97@hotmailcom YUANFU SHAO Guilin University of Technology School of Science Guilin Guangxi CHINA shaoyuanfu@163com Abstract: This aer is concerned with the existence uniueness and asymtotic behavior of the ositive solutions of a fuzzy Logistic difference euation x n+1 = A + Bx n 1 e xn n = 0 1 where (x n ) is a seuence of ositive fuzzy number A B are ositive fuzzy numbers and the initial conditions x 1 x 0 are ositive fuzzy numbers Moreover an illustrative examle is given to demonstrate the effectiveness of the results obtained Key Words: Fuzzy Logistic difference euation Euilibrium oint Bounded Persistence 1 Introduction It is well known that difference euation aears naturally as discrete analogous and as numerical solutions of differential euations and delay differential euation having many alications in economics biology comuter science control engineering The study of asymtotic stability and oscillatory roerties of solutions of difference euations is extremely useful in the behavior of mathematical models of various biological systems and other alications This is due to the fact that difference euations are aroriate models for describing situations where the variable is assumed to take only a discrete set of values and they arise freuently in the study of biological models in the formulation and analysis of discrete time systems etc Recently there has been a lot of work concerning the asymtotic behavior the eriodicity and the boundedness of nonlinear difference euation and system of nonlinear difference euations(see for examle [ ] and the references therein) EI-Metwally et al[7] investigated the asymtotic behavior of oulation model: x n+1 = α + βx n 1 e x n n = 0 1 (1) where α is the immigration rate and β is the oulation growth rate Fuzzy set theory is a owerful tool for modeling uncertainty and for rocessing vague or subjective information in mathematical model Particularly the use of fuzzy difference euations is a natural way to model the dynamical systems with embedded uncertainty Fuzzy difference euation is a difference euation where constants and the initial values are fuzzy numbers and its solutions are seuences of fuzzy numbers Recently there is an increasing interest in the study of fuzzy difference euation (see for examle [ ? 21]) In this aer we study the fuzzy analogs of (1) x n+1 = A + Bx n 1 e x n n = 0 1 (2) where (x n ) is a seuence of ositive fuzzy numbers A B and the initial values x 1 x 0 are ositive fuzzy numbers E-ISSN: Volume
2 For readers convenience we need some definitions: A is said to be a fuzzy number if A : R [0 1] satisfies the below (i)-(iv) (i) A is normal ie there exists an x R such that A(x) = 1; (ii) A is fuzzy convex ie for all t [0 1] and x 1 x 2 R such that A(tx 1 + (1 t)x 2 ) min{a(x 1 ) A(x 2 )}; (iii) A is uer semi-continuous; (iv) The suort of A sua = [A] α = {x : A(x) > 0} is comact The α-cuts of A are denoted by [A] α = {x R : A(x) α} α [0 1] it is clear that the [A] α are closed interval We say that a fuzzy number is ositive if sua (0 ) It is obvious that if A is a ositive real number then A is a fuzzy numbers and [A] α = [A A] α (0 1] Then we say that A is a trivial fuzzy number Let A B be fuzzy numbers with [A] α = [A lα A rα ] [B] α = [B lα B rα ] α (0 1] We define a norm on fuzzy numbers sace as follows: A = We take the following metric : D(A B) = su max{ A lα A rα } su max{ A lα B lα A rα B rα } The fuzzy analog of the boundedness and ersistence (see [4 11]) as follows: we say that a seuence of ositive fuzzy numbers x n ersists (res is bounded) if there exists a ositive real number M (res N) such that sux n [M )(res sux n (0 N]) n N x n is bounded and ersists if there exist ositive real numbers M N > 0 such that sux n [M N] n = 1 2 x n n = 1 2 is an unbounded seuence if the norm x n n = 1 2 is an unbounded seuence x n is a ositive solution of (2) if x n is a seuence of ositive fuzzy numbers which satisfies (2) We say a ositive fuzzy number x is a ositive euilibrium for (2) if x = A + Bxe x Let (x n ) be a seuence of ositive fuzzy numbers and x is a ositive fuzzy number Suose that [x n ] α = [α R nα ] n = α (0 1] (3) and [x] α = [L α R α ] α (0 1] (4) The seuence (x n ) converges to x with resect to D as n if lim D(x n x) = 0 Suose that (2) has a uniue ositive euilibrium x We say that the ositive euilibrium x of (2) is stable if for every ε > 0 there exists a δ = δ(ε) > 0 such that for every ositive solution x n of (2) which satisfies D(x i x) δ i = 0 1 we have D(x n x) ε for all n > 0 Moreover we say that the ositive euilibrium x of (2) is asymtotically stable if it is stable and every ositive solution of (2) tends to the ositive euilibrium of (2) with resect to D as n The urose of this aer is to study existence of the ositive solutions of (2) Furthermore we give some conditions so that every ositive solution of (2) is boundedness and ersistence Finally under some conditions we rove that (2) has a uniue ositive euilibrium x which is asymtotic stable 2 Main results 21 Existence of the ositive solution Firstly we study the existence of the ositive solutions of (2) We need the following lemma Lemma 1 [12] Let f :: R + R + R + R + R + be continuous A B C D are fuzzy numbers Then for α (0 1] [f(a B C D)] α = f([a] α [B] α [C] α [D] α ) (5) Lemma 2 [19] Let u E write [u] α = [u (α) u + (α)] α (0 1] Then u (α) and u + (α) can be regarded as functions on (0 1] which satisfy (i) u (α) is nondecreasing and left continuous; (ii) u + (α) is nonincreasing and left continuous; (iii) u (1) u + (1) Conversely for any functions a(α) and b(α) defined on (0 1] which satisfy (i) (iii) in the above there exists a uniue u E such that [u] α = [a(α) b(α)] for any α (0 1] E-ISSN: Volume
3 Theorem 3 Consider euation (2) where A B are ositive fuzzy numbers Then for any ositive fuzzy numbers x 1 x 0 there exists a uniue ositive solution x n of (2) with initial conditions x 1 x 0 Proof: The roof is similar to Proosition 21 in [12] Suose that there exists a seuence of fuzzy numbers x n satisfying (2) with initial conditions x 1 x 0 Consider the α cuts α (0 1] n = [x n ] α = [α R nα ] [A] α = [A lα A rα ] [B] α = [B lα B rα ] It follows from (2) (6) and Lemma 1 that [x n+1 ] α = [+1α R n+1α ] = [A + Bx n 1 e x n ] α = [A] α + [B] α [x n 1 ] α [e x n ] α = [ A lα + B lα 1α e Rnα A rα + B rα R n 1α e α ] (6) from which we have that for n = α (0 1] +1α = A lα + B lα 1α e Rnα R n+1α = A rα + B rα R n 1α e Lnα (7) Then it is obvious that for any initial condition (L iα R iα ) i = 0 1 α (0 1] there exists a uniue solution (α R nα ) Now we rove that [α R nα ] α (0 1] where (α R nα ) is the solution of system (7) with initial conditions (L iα R iα ) i = 0 1 determines the solution x n of (2) with the initial conditions x i i = 0 1 such that for n = [x n ] α = [α R nα ] α (0 1] (8) From reference [20] and since A B x 1 x 0 are ositive fuzzy numbers for any α 1 α 2 (0 1] α 1 α 2 we have 0 < A lα1 A lα2 A rα2 A rα1 We claim that for n = α1 α2 R nα2 R nα1 (10) We rove it by induction It is obvious from (9) that (10) holds true for n = 1 0 Suose that (10) are true for n k k {1 2 } Then from (7) (9) and (10) for n k it follows that L k+1α1 = A lα1 + B lα1 L k 1α1 e R kα 1 A lα2 + B lα2 L kα2 e R kα 2 = L k+1α2 = A lα2 + B lα2 L kα2 e R kα 2 A rα2 + B rα2 L kα2 e R kα 2 = R k+1α2 = A rα2 + B rα2 L kα2 e R kα 2 A rα1 + B rα1 L kα1 e R kα 1 = R k+1α1 Therefore (10) are satisfied Moreover from (7) we have for α (0 1] L 1α = A lα + B lα L 1α e R 0α (11) R 1α = A rα + B rα R 1α e L 0α Since A B x 1 x 0 are ositive fuzzy numbers then we have that A lα A rα B lα B rα L lα R 1α L 0α R 0α are left continuous So from (11) we have that L 1α R 1α are also left continuous By induction we can get that α R nα n = 1 2 are left continuous Now we rove that the suort of x n sux n = [α R nα ] is comact It is sufficient to rove that [α R nα ] is bounded Let n = 1 since A B x 1 x 0 are ositive fuzzy numbers there exist constants M A > 0 N A > 0 M B > 0 N B > 0 M 1 > 0 N 1 > 0 M 0 > 0 N 0 > 0 such that for all α (0 1] [A lα A rα ] [M A N A ] [B lα B rα ] [M B N B ] [L lα R 1α ] [M 1 N 1 ] [L 0α R 0α ] [M 0 N 0 ] (12) Hence from (11) and (12) we can easily get for α (0 1] [L 1α R 1α ] [ M A + M B M 1 e N 0 N A + N B N 1 e M 0 ] (13) 0 < B lα1 B lα2 B rα2 B rα1 0 < L 1α1 L 1α2 R 1α2 R 1α1 0 < L 0α1 L 0α2 R 0α2 R 0α1 (9) From which it is obvious that [L 1α R 1α ] [ M A +M B M 1 e N 0 N A +N B N 1 e M 0 ] (14) E-ISSN: Volume
4 Therefore (14) imlies that [L 1α R 1α ] is comact and [L 1α R 1α ] (0 ) Deducing inductively we can easily follow that [α R nα ] is comact and [α R nα ] (0 ) n = 1 2 (15) Therefore (10) (15) and since α R nα are left continuous we have that [α R nα ] determines a seuence of ositive fuzzy numbers x n such that (8) holds We rove now that x n is the solution of (2) with initial condition x 1 x 0 Since for all α (0 1] [x n+1 ] α = [+1α R n+1α ] = [ A lα + B lα 1α e R nα A rα + B rα R n 1α e Lnα] = [ A + Bx n 1 e x n] α we have that x n is the solution of (2) with initial condition x 1 x 0 Suose that there exists another solution x n of (2) with initial conditions x 1 x 0 Then from arguing as above we can easily rove thatfor n = [x n ] α = [α R nα ] α (0 1] (16) Then from (8) and (16) we have [x n ] α = [x n ] α α (0 1] n = from which it follows that x n = x n n = 0 1 Thus the roof of the theorem 3 is comleted 22 Boundedness and Permanence In the following we will study the boundedness and ermanence of the fuzzy ositive solution of (2) We need the following lemma Lemma 4 (comarison results) Assume that α (0 ) β [0 ) Let {X n } n= 1 {Y n} n= 1 be seuences of real numbers such that X 1 Y 1 X 0 Y 0 and for n = 0 1 { Xn+1 αx n 1 + β Y n+1 = αy n 1 + β Then X n Y n for all n 1 Lemma 5 Consider the system of the difference euations for n = 0 1 y n+1 = + cy n 1 e z n (17) z n+1 = + dz n 1 e yn where c d are ositive real numbers and the initial values y i z i (i = 0 1) are ositive real numbers If c < e d < e (18) Then system (17) is bounded and ersists Proof: Let {(y n z n )} n= 1 be a ositive solution of system (17) Then it follows from system (17) that for n 0 y n+1 = + cy n 1 e z n > (19) z n+1 = + dz n 1 e y n > Thus y n+1 = + cy n 1 e zn < + ce y n 1 z n+1 = + dz n 1 e yn < + de z n 1 (20) Now we consider the initial value roblem for n = 1 2 Y n+1 = + ce Y n 1 Z n+1 = + de Z n 1 with initial conditions y i Y i z i Z i (i = 0 1) and so it follows from Lemma 4 that Observe that lim Y n = and then lim su y n y n Y n z n Z n n 1 lim 1 ce Z n = lim 1 ce su z n 1 de 1 de Therefore {(y n z n )} n= 1 is bounded and ersists and the roof is comleted Theorem 6 Consider fuzzy difference euation (2) where A B and initial values x 1 x 0 are ositive fuzzy numbers If for all α (0 1] B rα < e A lα (21) Then every ositive solution of (2) is bounded and ersists Proof: Let x n be ositive solution of E(2) such that (8) holds From (7) it is obvious that for n = 1 2 α (0 1] A lα α A rα R nα (22) E-ISSN: Volume
5 From Lemma 4 and (19) it follows that for n = 1 2 α R nα A lα 1 B lα e A rα (23) A rα 1 B rαe A lα Hence from (12) (20) and (21) it is obvious that for n 1 α (0 1] [ ] N A [α R nα ] M A 1 N B e M (24) A From which we get for n 1 [ N A [α R nα ] M A 1 N B e M A and so [α R nα ] [ M A N A 1 N B e M A ] ] ie the ositive solution of (2) is bounded and ersists 23 Dynamic of solution of E(2) Lemma 7 Consider the system of difference euation (17) where c d are ositive real numbers If (18) holds Then there exists a uniue ositive euilibrium (ȳ z) such that < ȳ < 1 ce < z < (25) 1 de Proof: Let (ȳ z) be the solution of the following systems y = + cye z z = + dze y (26) Set Then and f() = de f(z) = + dze 1 ce z z (27) ce 1 > 0 lim f(z) = (28) z + f (z) = de ce z 1 ce z dz (1 ce z ) 2 e 1 ce z 1 < 0 (29) It follows from (26) and (16) that (25) has exactly one solution z > On the other hand set g(y) = + cye 1 de y y (30) Similarly it can easily rove that (28) has exactly one solution ȳ > Noting (18) and (24) it follows that ȳ < z < The roof is comleted 1 ce 1 de Theorem 8 Consider the system (17) where c d are ositive real numbers and the initial values y i z i (i = 0 1) are ositive real numbers Assume that (18) and the following condition hold { ( ) 1 2 max (1 + )ce 1 ce } (1 + )de < 1 (31) 1 de Then the ositive euilibrium (ȳ z) of system (17) is globally asymtotically stable Proof: First we rove that the ositive euilibrium (ȳ z) of (17) is locally asymtotically stable We can easily obtain that the linearized system of (17) about the ositive euilibrium (ȳ z) is where D = Ψ n+1 = DΨ n 0 ce z cȳe z d ze ȳ 0 0 de ȳ Since (18) and (29) hold we can take a ositive number ε such that { ((1 ε = max + ȳ)ce z ) 1 2 (1 + z)de ȳ} Let < 1 (32) We consider the matrix S = diag(1 ε 1 ε 2 ε 3 ) C = (c ij ) = S 1 DS = 0 ce z ε 1 cȳe z ε 2 0 ε d ze ȳ ε de ȳ ε ε 0 with the norm 4 C = max c 1 i 4 ij j=1 Using (30) we can rove that C < 1 Therefore since λ i < C ( where λ i (i = ) are the eigenvalues of D) we have that all eigenvalues of D lie inside the unit disk This imlies that (ȳ z) is locally asymtotically stable E-ISSN: Volume
6 Let now (y n z n ) be a ositive solution of (17) We rove that Using Lemma 4 we have lim y n = ȳ lim z n = z (33) Λ 1 = lim su y n < Λ 2 = lim su z n < λ 1 = lim inf y n > 0 λ 2 = lim inf z n > 0 Then from (17) and (32) we take Λ 1 1 ce λ λ 2 1 Λ 2 1 de λ 1 λ 2 1 ce Λ 2 (34) (35) 1 de Λ 1 From (33) it follows that Λ 1 λ 2 λ 2 1 ce Λ λ 2 2λ 1 Λ 2 1 ce Λ 2 Λ 2 λ 1 λ 1 1 de λ 1 λ 2Λ 1 Λ 1 1 de Λ 1 from which we take Λ 1 1 de λ Λ ce λ 2 Λ 2 1 ce Λ 2 λ 1 1 de λ 1 Now we consider the functions for f(y) = y ( (36) y 1 de y g(z) = z (37) 1 ce z ) ( 1 ce z Then from (35) it follows that From (30) Noting ( y f (y) = (1 de y (1+y)) (1 de y ) 2 ) 1 de g (z) = (1 ce z (1+z)) (1 ce z ) 2 (38) 1 ce ) ( z ) 1 de we get 1 de y (1 + y) > 1 de (1 + ) > 0 1 de 1 ce z (1 + z) > 1 ce (1 + ) > 0 1 ce (39) Therefore from (36) and (37) we obtain for y ( f (y) > 0 g (z) > 0 ) ( 1 ce z ) 1 de Hence f g are increasing functions and this together with (34) imlies that λ 1 = Λ 1 Then from (34) again we see that λ 2 = Λ 2 This imlies that lim y n = ȳ lim z n = z This comletes the roof of the theorem Remark 9 The condition satisfying the uniue ositive euilibrium of (12) is globally asymtotically stable is different from that of [13] Theorem 10 Consider fuzzy difference euation (2) where A B and the initial values x i (i = 0 1) are ositive fuzzy numbers if (19) is satisfied Then the following statements are true (i) The euation (2) has a uniue ositive euilibrium (ii) The every ositive solution x n of (2) converges to the uniue ositive euilibrium x with resect to D as n Proof: (i) We consider the system L α = A lα + B lα L α e Rα R α = A rα + B rα R α e Lα (40) Obviously system (40) has a uniue solution (L α R α ) Let x n be a ositive solution of (2) such that [x n ] α = [α R nα ] α (0 1] n = 0 1 Then alying Theorem 8 to the system (7) we have lim α = L α lim R nα = R α (41) From (22) and (41) we have for 0 < α 1 α < L α1 L α2 R α2 R α1 (42) Since A lα A rα B lα B rα are left continuous it follows from (40) that L α R α are also left continuous From (40) and (12) we get L α A lα M A R α = A rα 1 B rαe Lα N A 1 N B e M A (43) E-ISSN: Volume
7 Therefore (43) imlies that [ N A [L α R α ] M A 1 N B e M A From which it is obvious that [L α R α ] is comact [L α R α ] (0 ) ] (44) So from Lemma 2 relations (40) (42) (44) and since L α R α α (0 1] determines a fuzzy number x such that x = A + Bxe x [x] α = [L α R α ] α (0 1] and 10x 5 05 x 06 B(x) = 5x x 08 (48) We take initial value x 1 x 0 such that 10x 2 02 x 03 x 1 (x) = (49) 10 3 x x 06 and 5x 5 01 x 03 x 0 (x) = 5x x 05 (50) and so x is a ositive euilibrium of (2) Provided that there exists another ositive euilibrium x for (2) then there exist functions L α : (0 1] (0 ) R α : (0 1] (0 ) such that x = A + B xe x [ x] α = [L α R α ] α (0 1] From which we have L α = A lα +B lα L α e R α R α = A rα +B rα R α e L α So L α = L α R α = R α α (0 1] namely x = x This comletes the roof of (i) (ii) From (41) we have lim D(x n x) = lim su {max { α L α R nα R α }} = 0 (45) From which it is obvious that every ositive solution x n of E(2) converges the uniue euilibrium x with resect to D as n 3 Numerical examle To illustrate our results we give some examles in which the conditions of our roositions hold Examle 1 Consider the following fuzzy difference euation x n+1 = A + Bx n 1 e x n n = 0 1 (46) where A B are fuzzy numbers such that 10x 3 03 x 04 A(x) = 5x x 06 (47) From (47) and (48) we get for α (0 1] [A] α = [ α α] And so [B] α = [ α α] [A] α = [03 06] [B] α = [05 08] (51) (52) Moreover from (49) and (50) we get for α (0 1] [x 1 ] α = [ α α] [x 0 ] α = [ α α] It follows that [x 1 ] α = [02 06] (53) [x 0 ] α = [01 05] (54) From (46) it results in a couled system of difference euation with arameter α (0 1] +1α = α + ( α) 1α e Rnα R n+1α = α + ( α) R n 1α e α (55) Therefore B rα < e A lα for any α (0 1] namely the condition (19) is satisfied so from Theorem 6 we have that every ositive solution x n of E(46) is bounded and ersists In addition E(46) has a uniue ositive euilibrium x = ( ) Moreover every ositive solution x n of E(46) converges the uniue euilibrium x with resect to D as n (see Fig1-Fig4) E-ISSN: Volume
8 α=0 α=05 α= R n R n 1 08 &R n n Figure 1: The dynamics of system (55) Figure 3: The solution of system (55) at α = R n R n &R n &R n n n Figure 2: The solution of system (55) at α = 0 4 Conclusion In this work fuzzy Logistic difference x n+1 = A + Bx n 1 e x n n = 0 1 is discussed Firstly the existence of ositive solution to this euation is roved Secondly we find that under condition B rα < e A lα the ositive solutions of fuzzy Logistic difference are bounded and ersistence and there exists uniue ositive euilibrium x such that every ositive solution converges it Finally an examle is given to illustrate our results obtained Acknowledgements: This research was financially suorted by the National Natural Science Foundation of China (Grant No ) and suorted by the Scientific Research Foundation of Guizhou Science and Technology Deartment ([2009]2061[2013]2083) References: [1] D Benest and C Froeschle Analysis and Modelling of Discrete Dynamical Systems Gordon Figure 4: The solution of system (55) at α = 1 and Breach Science Publishers The Netherland 1998 [2] Mukminah Darus and Abd Fatah Wahab Review on fuzzy difference euation Malaysian Journal of Fundamental & Alied Sciences 8 No [3] E Y Deeba and A De Korvin Analysis by fuzzy difference euations of a model of CO 2 level in the blood Al Math Lett [4] E Y Deeba and A De Korvin and E L Koh A fuzzy difference euation with an alication J Difference Euation Al [5] R DeVault G Ladas and S W Schultz Necessary and sufficient conditions the boundedness of x n+1 = A/x n + B/x n 1 JDifference Euations Al [6] R DeVault G Ladas and S W Schultz On the recursive seuence x n+1 = A/x n +1/x n 2 Proc Amer Math Soc [7] E El-Metwally E A Grove G Ladas R Levins and M Radin On the difference eua- E-ISSN: Volume
9 tion x n+1 = α + βx n 1 e x n Nonlinear Anal [8] T F Ibrahim and Q Zhang Stability of an anticometitive system of rational difference euations Archives Des Sciences [9] V L Kocic and G LadasGlobal behavior of nolinear difference euations of higher order with alication Kluwer Academic Dordrecht1993 [10] W T Li and H R Sun Dynamics of a rational difference euation Al Math Comut [11] G Paaschinooulos and B K Paadooulos On the fuzzy difference euation x n+1 = A + B/x n Soft Comut [12] G Paaschinooulos and B K Paadooulos On the fuzzy difference euation x n+1 = A + x n /x n m Fuzzy Sets and Systems [13] G Paaschinooulos M A Radin and CJ Schinas On the system of two difference euations of exonential form: x n+1 = a + bx n 1 e yn y n+1 = c+dy n 1 e xn Math Comut Model [14] G Paaschinooulos and C J Schinas On a systems of two nonlinear difference euation J Math Anal Al [15] G Paaschinooulos and C J Schinas On the fuzzy difference euation x n+1 = k 1 i=0 A i/x i n i + 1/x k n k J Difference Euation Al [16] G Paaschinooulos G Stefanidou Boundedness and asymtotic behavior of the solutions of a fuzzy difference euation Fuzzy Sets and Systems [17] G Stefanidou and G Paaschinooulos A fuzzy difference euation of a rational form J Nonlin Math Phys 12 Sulement [18] G Stefanidou G Paaschinooulos and C J Schinas On an exonential-tye fuzzy difference euation Advances in Difference Euations : doi:101155/2010/ [19] C Wu and B Zhang Embedding roblem of noncomact fuzzy number sace E Fuzzy Sets and Systems [20] Q Zhang L Yang and D Liao On the fuzzy difference euation x n+1 = A + k B i=0 x n i World Acad Sci Eng Technol [21] Q Zhang L Yang and D Liao Behovior of solutions to a fuzzy nonlinear difference eution Iranian Journal of Fuzzy Systems Vol 9 No E-ISSN: Volume
Global Behavior of a Higher Order Rational Difference Equation
International Journal of Difference Euations ISSN 0973-6069, Volume 10, Number 1,. 1 11 (2015) htt://camus.mst.edu/ijde Global Behavior of a Higher Order Rational Difference Euation Raafat Abo-Zeid The
More informationAnna Andruch-Sobi lo, Ma lgorzata Migda. FURTHER PROPERTIES OF THE RATIONAL RECURSIVE SEQUENCE x n+1 =
Ouscula Mathematica Vol. 26 No. 3 2006 Anna Andruch-Sobi lo, Ma lgorzata Migda FURTHER PROPERTIES OF THE RATIONAL RECURSIVE SEQUENCE x n+1 = Abstract. In this aer we consider the difference equation x
More informationA Note on the Positive Nonoscillatory Solutions of the Difference Equation
Int. Journal of Math. Analysis, Vol. 4, 1, no. 36, 1787-1798 A Note on the Positive Nonoscillatory Solutions of the Difference Equation x n+1 = α c ix n i + x n k c ix n i ) Vu Van Khuong 1 and Mai Nam
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationAn Existence Theorem for a Class of Nonuniformly Nonlinear Systems
Australian Journal of Basic and Alied Sciences, 5(7): 1313-1317, 11 ISSN 1991-8178 An Existence Theorem for a Class of Nonuniformly Nonlinear Systems G.A. Afrouzi and Z. Naghizadeh Deartment of Mathematics,
More informationElementary Analysis in Q p
Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some
More informationHermite-Hadamard Inequalities Involving Riemann-Liouville Fractional Integrals via s-convex Functions and Applications to Special Means
Filomat 3:5 6), 43 5 DOI.98/FIL6543W Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt://www.mf.ni.ac.rs/filomat Hermite-Hadamard Ineualities Involving Riemann-Liouville
More informationExistence and nonexistence of positive solutions for quasilinear elliptic systems
ISSN 1746-7233, England, UK World Journal of Modelling and Simulation Vol. 4 (2008) No. 1,. 44-48 Existence and nonexistence of ositive solutions for uasilinear ellitic systems G. A. Afrouzi, H. Ghorbani
More informationResearch Article An iterative Algorithm for Hemicontractive Mappings in Banach Spaces
Abstract and Alied Analysis Volume 2012, Article ID 264103, 11 ages doi:10.1155/2012/264103 Research Article An iterative Algorithm for Hemicontractive Maings in Banach Saces Youli Yu, 1 Zhitao Wu, 2 and
More informationMultiplicity of weak solutions for a class of nonuniformly elliptic equations of p-laplacian type
Nonlinear Analysis 7 29 536 546 www.elsevier.com/locate/na Multilicity of weak solutions for a class of nonuniformly ellitic equations of -Lalacian tye Hoang Quoc Toan, Quô c-anh Ngô Deartment of Mathematics,
More informationOn the minimax inequality for a special class of functionals
ISSN 1 746-7233, Engl, UK World Journal of Modelling Simulation Vol. 3 (2007) No. 3,. 220-224 On the minimax inequality for a secial class of functionals G. A. Afrouzi, S. Heidarkhani, S. H. Rasouli Deartment
More informationMULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS INVOLVING CONCAVE AND CONVEX NONLINEARITIES IN R 3
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 301,. 1 16. ISSN: 1072-6691. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu MULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF TYPE
More informationOn the minimax inequality and its application to existence of three solutions for elliptic equations with Dirichlet boundary condition
ISSN 1 746-7233 England UK World Journal of Modelling and Simulation Vol. 3 (2007) No. 2. 83-89 On the minimax inequality and its alication to existence of three solutions for ellitic equations with Dirichlet
More informationResearch Article Positive Solutions of Sturm-Liouville Boundary Value Problems in Presence of Upper and Lower Solutions
International Differential Equations Volume 11, Article ID 38394, 11 ages doi:1.1155/11/38394 Research Article Positive Solutions of Sturm-Liouville Boundary Value Problems in Presence of Uer and Lower
More informationMollifiers and its applications in L p (Ω) space
Mollifiers and its alications in L () sace MA Shiqi Deartment of Mathematics, Hong Kong Batist University November 19, 2016 Abstract This note gives definition of mollifier and mollification. We illustrate
More informationKIRCHHOFF TYPE PROBLEMS INVOLVING P -BIHARMONIC OPERATORS AND CRITICAL EXPONENTS
Journal of Alied Analysis and Comutation Volume 7, Number 2, May 2017, 659 669 Website:htt://jaac-online.com/ DOI:10.11948/2017041 KIRCHHOFF TYPE PROBLEMS INVOLVING P -BIHARMONIC OPERATORS AND CRITICAL
More informationGlobal Attractivity of a Higher-Order Nonlinear Difference Equation
International Journal of Difference Equations ISSN 0973-6069, Volume 5, Number 1, pp. 95 101 (010) http://campus.mst.edu/ijde Global Attractivity of a Higher-Order Nonlinear Difference Equation Xiu-Mei
More informationRIEMANN-STIELTJES OPERATORS BETWEEN WEIGHTED BERGMAN SPACES
RIEMANN-STIELTJES OPERATORS BETWEEN WEIGHTED BERGMAN SPACES JIE XIAO This aer is dedicated to the memory of Nikolaos Danikas 1947-2004) Abstract. This note comletely describes the bounded or comact Riemann-
More informationTwo Modifications of the Beverton Holt Equation
International Journal of Difference Equations ISSN 0973-5321 Volume 4 Number 1 pp. 115 136 2009 http://campus.mst.edu/ijde Two Modifications of the Beverton Holt Equation G. Papaschinopoulos. J. Schinas
More information1 Riesz Potential and Enbeddings Theorems
Riesz Potential and Enbeddings Theorems Given 0 < < and a function u L loc R, the Riesz otential of u is defined by u y I u x := R x y dy, x R We begin by finding an exonent such that I u L R c u L R for
More informationSharp gradient estimate and spectral rigidity for p-laplacian
Shar gradient estimate and sectral rigidity for -Lalacian Chiung-Jue Anna Sung and Jiaing Wang To aear in ath. Research Letters. Abstract We derive a shar gradient estimate for ositive eigenfunctions of
More informationStability In A Nonlinear Four-Term Recurrence Equation
Applied Mathematics E-Notes, 4(2004), 68-73 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Stability In A Nonlinear Four-Term Recurrence Equation Shu-rong Sun, Zhenlai
More informationOn the statistical and σ-cores
STUDIA MATHEMATICA 154 (1) (2003) On the statistical and σ-cores by Hüsamett in Çoşun (Malatya), Celal Çaan (Malatya) and Mursaleen (Aligarh) Abstract. In [11] and [7], the concets of σ-core and statistical
More informationLocation of solutions for quasi-linear elliptic equations with general gradient dependence
Electronic Journal of Qualitative Theory of Differential Equations 217, No. 87, 1 1; htts://doi.org/1.14232/ejqtde.217.1.87 www.math.u-szeged.hu/ejqtde/ Location of solutions for quasi-linear ellitic equations
More informationExtremal Polynomials with Varying Measures
International Mathematical Forum, 2, 2007, no. 39, 1927-1934 Extremal Polynomials with Varying Measures Rabah Khaldi Deartment of Mathematics, Annaba University B.P. 12, 23000 Annaba, Algeria rkhadi@yahoo.fr
More informationProducts of Composition, Multiplication and Differentiation between Hardy Spaces and Weighted Growth Spaces of the Upper-Half Plane
Global Journal of Pure and Alied Mathematics. ISSN 0973-768 Volume 3, Number 9 (207),. 6303-636 Research India Publications htt://www.riublication.com Products of Comosition, Multilication and Differentiation
More informationGlobal Attractivity in a Nonlinear Difference Equation and Applications to a Biological Model
International Journal of Difference Equations ISSN 0973-6069, Volume 9, Number 2, pp. 233 242 (204) http://campus.mst.edu/ijde Global Attractivity in a Nonlinear Difference Equation and Applications to
More informationOn the continuity property of L p balls and an application
J. Math. Anal. Al. 335 007) 347 359 www.elsevier.com/locate/jmaa On the continuity roerty of L balls and an alication Kh.G. Guseinov, A.S. Nazliinar Anadolu University, Science Faculty, Deartment of Mathematics,
More informationAnalysis of some entrance probabilities for killed birth-death processes
Analysis of some entrance robabilities for killed birth-death rocesses Master s Thesis O.J.G. van der Velde Suervisor: Dr. F.M. Sieksma July 5, 207 Mathematical Institute, Leiden University Contents Introduction
More informationIntroduction to Banach Spaces
CHAPTER 8 Introduction to Banach Saces 1. Uniform and Absolute Convergence As a rearation we begin by reviewing some familiar roerties of Cauchy sequences and uniform limits in the setting of metric saces.
More informationTranspose of the Weighted Mean Matrix on Weighted Sequence Spaces
Transose of the Weighted Mean Matri on Weighted Sequence Saces Rahmatollah Lashkariour Deartment of Mathematics, Faculty of Sciences, Sistan and Baluchestan University, Zahedan, Iran Lashkari@hamoon.usb.ac.ir,
More informationAttractivity of the Recursive Sequence x n+1 = (α βx n 1 )F (x n )
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.7(2009) No.2,pp.201-206 Attractivity of the Recursive Sequence (α βx n 1 )F (x n ) A. M. Ahmed 1,, Alaa E. Hamza
More informationElementary theory of L p spaces
CHAPTER 3 Elementary theory of L saces 3.1 Convexity. Jensen, Hölder, Minkowski inequality. We begin with two definitions. A set A R d is said to be convex if, for any x 0, x 1 2 A x = x 0 + (x 1 x 0 )
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 informationA sharp generalization on cone b-metric space over Banach algebra
Available online at www.isr-ublications.com/jnsa J. Nonlinear Sci. Al., 10 2017), 429 435 Research Article Journal Homeage: www.tjnsa.com - www.isr-ublications.com/jnsa A shar generalization on cone b-metric
More informationarxiv: v2 [math.ap] 19 Jan 2010
GENERALIZED ELLIPTIC FUNCTIONS AND THEIR APPLICATION TO A NONLINEAR EIGENVALUE PROBLEM WITH -LAPLACIAN arxiv:11.377v2 [math.ap] 19 Jan 21 SHINGO TAKEUCHI Dedicated to Professor Yoshio Yamada on occasion
More informationAn Estimate For Heilbronn s Exponential Sum
An Estimate For Heilbronn s Exonential Sum D.R. Heath-Brown Magdalen College, Oxford For Heini Halberstam, on his retirement Let be a rime, and set e(x) = ex(2πix). Heilbronn s exonential sum is defined
More informationDifferential Sandwich Theorem for Multivalent Meromorphic Functions associated with the Liu-Srivastava Operator
KYUNGPOOK Math. J. 512011, 217-232 DOI 10.5666/KMJ.2011.51.2.217 Differential Sandwich Theorem for Multivalent Meromorhic Functions associated with the Liu-Srivastava Oerator Rosihan M. Ali, R. Chandrashekar
More informationGlobal dynamics of two systems of exponential difference equations by Lyapunov function
Khan Advances in Difference Equations 2014 2014:297 R E S E A R C H Open Access Global dynamics of two systems of exponential difference equations by Lyapunov function Abdul Qadeer Khan * * Correspondence:
More informationGROUND STATES OF LINEARLY COUPLED SCHRÖDINGER SYSTEMS
Electronic Journal of Differential Equations, Vol. 2017 (2017), o. 05,. 1 10. ISS: 1072-6691. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu GROUD STATES OF LIEARLY COUPLED SCHRÖDIGER SYSTEMS
More information#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS
#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS Ramy F. Taki ElDin Physics and Engineering Mathematics Deartment, Faculty of Engineering, Ain Shams University, Cairo, Egyt
More informationJournal of Mathematical Analysis and Applications
J. Math. Anal. Al. 44 (3) 3 38 Contents lists available at SciVerse ScienceDirect Journal of Mathematical Analysis and Alications journal homeage: www.elsevier.com/locate/jmaa Maximal surface area of a
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 informationOn some nonlinear elliptic systems with coercive perturbations in R N
On some nonlinear ellitic systems with coercive erturbations in R Said EL MAOUI and Abdelfattah TOUZAI Déartement de Mathématiues et Informatiue Faculté des Sciences Dhar-Mahraz B.P. 1796 Atlas-Fès Fès
More informationArithmetic and Metric Properties of p-adic Alternating Engel Series Expansions
International Journal of Algebra, Vol 2, 2008, no 8, 383-393 Arithmetic and Metric Proerties of -Adic Alternating Engel Series Exansions Yue-Hua Liu and Lu-Ming Shen Science College of Hunan Agriculture
More informationGENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS
GENERICITY OF INFINITE-ORDER ELEMENTS IN HYPERBOLIC GROUPS PALLAVI DANI 1. Introduction Let Γ be a finitely generated grou and let S be a finite set of generators for Γ. This determines a word metric on
More informationDynamics of a Rational Recursive Sequence
International Journal of Difference Equations ISSN 0973-6069, Volume 4, Number 2, pp 185 200 (2009) http://campusmstedu/ijde Dynamics of a Rational Recursive Sequence E M Elsayed Mansoura University Department
More informationHAUSDORFF MEASURE OF p-cantor SETS
Real Analysis Exchange Vol. 302), 2004/2005,. 20 C. Cabrelli, U. Molter, Deartamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and CONICET, Pabellón I - Ciudad Universitaria,
More informationINFINITELY MANY SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS WITH NONLINEAR NEUMANN BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 2016 2016, No. 188,. 1 9. ISSN: 1072-6691. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu INFINITELY MANY SOLUTIONS FOR KIRCHHOFF TYPE PROBLEMS
More informationGENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS
International Journal of Analysis Alications ISSN 9-8639 Volume 5, Number (04), -9 htt://www.etamaths.com GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS ILYAS ALI, HU YANG, ABDUL SHAKOOR Abstract.
More informationNEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS
Kragujevac Journal of Mathematics Volume 42(1) (2018), Pages 83 95. NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS M. ALBEHBAH 1 AND M. DARUS 2 Abstract. In this aer, we introduce a new
More informationIteration with Stepsize Parameter and Condition Numbers for a Nonlinear Matrix Equation
Electronic Journal of Linear Algebra Volume 34 Volume 34 2018) Article 16 2018 Iteration with Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation Syed M Raza Shah Naqvi Pusan National
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Alied Mathematics htt://jiam.vu.edu.au/ Volume 3, Issue 5, Article 8, 22 REVERSE CONVOLUTION INEQUALITIES AND APPLICATIONS TO INVERSE HEAT SOURCE PROBLEMS SABUROU SAITOH,
More information1. Introduction In this note we prove the following result which seems to have been informally conjectured by Semmes [Sem01, p. 17].
A REMARK ON POINCARÉ INEQUALITIES ON METRIC MEASURE SPACES STEPHEN KEITH AND KAI RAJALA Abstract. We show that, in a comlete metric measure sace equied with a doubling Borel regular measure, the Poincaré
More informationAdvanced Calculus I. Part A, for both Section 200 and Section 501
Sring 2 Instructions Please write your solutions on your own aer. These roblems should be treated as essay questions. A roblem that says give an examle requires a suorting exlanation. In all roblems, you
More informationOn the irreducibility of a polynomial associated with the Strong Factorial Conjecture
On the irreducibility of a olynomial associated with the Strong Factorial Conecture Michael Filaseta Mathematics Deartment University of South Carolina Columbia, SC 29208 USA E-mail: filaseta@math.sc.edu
More informationON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS
#A13 INTEGERS 14 (014) ON THE LEAST SIGNIFICANT ADIC DIGITS OF CERTAIN LUCAS NUMBERS Tamás Lengyel Deartment of Mathematics, Occidental College, Los Angeles, California lengyel@oxy.edu Received: 6/13/13,
More informationA PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL
A PRIORI ESTIMATES AND APPLICATION TO THE SYMMETRY OF SOLUTIONS FOR CRITICAL LAPLACE EQUATIONS Abstract. We establish ointwise a riori estimates for solutions in D 1, of equations of tye u = f x, u, where
More informationANNALES MATHÉMATIQUES BLAISE PASCAL. Tibor Šalát, Vladimír Toma A Classical Olivier s Theorem and Statistical Convergence
ANNALES MATHÉMATIQUES BLAISE PASCAL Tibor Šalát, Vladimír Toma A Classical Olivier s Theorem and Statistical Convergence Volume 10, n o 2 (2003),. 305-313.
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis. Volume 9,. 29-36, 25. Coyright 25,. ISSN 68-963. ETNA ASYMPTOTICS FOR EXTREMAL POLYNOMIALS WITH VARYING MEASURES M. BELLO HERNÁNDEZ AND J. MíNGUEZ CENICEROS
More informationON THE RATIONAL RECURSIVE SEQUENCE X N+1 = γx N K + (AX N + BX N K ) / (CX N DX N K ) Communicated by Mohammad Asadzadeh. 1.
Bulletin of the Iranian Mathematical Society Vol. 36 No. 1 (2010), pp 103-115. ON THE RATIONAL RECURSIVE SEQUENCE X N+1 γx N K + (AX N + BX N K ) / (CX N DX N K ) E.M.E. ZAYED AND M.A. EL-MONEAM* Communicated
More informationHEAT AND LAPLACE TYPE EQUATIONS WITH COMPLEX SPATIAL VARIABLES IN WEIGHTED BERGMAN SPACES
Electronic Journal of ifferential Equations, Vol. 207 (207), No. 236,. 8. ISSN: 072-669. URL: htt://ejde.math.txstate.edu or htt://ejde.math.unt.edu HEAT AN LAPLACE TYPE EQUATIONS WITH COMPLEX SPATIAL
More informationSOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES. 1. Introduction
SOME INEQUALITIES FOR (α, β)-normal OPERATORS IN HILBERT SPACES SEVER S. DRAGOMIR 1 AND MOHAMMAD SAL MOSLEHIAN Abstract. An oerator T is called (α, β)-normal (0 α 1 β) if α T T T T β T T. In this aer,
More informationGlobal Attractivity in a Higher Order Nonlinear Difference Equation
Applied Mathematics E-Notes, (00), 51-58 c ISSN 1607-510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Global Attractivity in a Higher Order Nonlinear Difference Equation Xing-Xue
More informationZhi-Wei Sun Department of Mathematics, Nanjing University Nanjing , People s Republic of China
Ramanuan J. 40(2016, no. 3, 511-533. CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x Zhi-Wei Sun Deartment of Mathematics, Naning University Naning 210093, Peole s Reublic of China zwsun@nu.edu.cn htt://math.nu.edu.cn/
More informationSpectral Properties of Schrödinger-type Operators and Large-time Behavior of the Solutions to the Corresponding Wave Equation
Math. Model. Nat. Phenom. Vol. 8, No., 23,. 27 24 DOI:.5/mmn/2386 Sectral Proerties of Schrödinger-tye Oerators and Large-time Behavior of the Solutions to the Corresonding Wave Equation A.G. Ramm Deartment
More informationExistence of solutions of infinite systems of integral equations in the Fréchet spaces
Int. J. Nonlinear Anal. Al. 7 (216) No. 2, 25-216 ISSN: 28-6822 (electronic) htt://dx.doi.org/1.2275/ijnaa.217.174.1222 Existence of solutions of infinite systems of integral equations in the Fréchet saces
More informationSOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES
Kragujevac Journal of Mathematics Volume 411) 017), Pages 33 55. SOME TRACE INEQUALITIES FOR OPERATORS IN HILBERT SPACES SILVESTRU SEVER DRAGOMIR 1, Abstract. Some new trace ineualities for oerators in
More informationSolvability and Number of Roots of Bi-Quadratic Equations over p adic Fields
Malaysian Journal of Mathematical Sciences 10(S February: 15-35 (016 Secial Issue: The 3 rd International Conference on Mathematical Alications in Engineering 014 (ICMAE 14 MALAYSIAN JOURNAL OF MATHEMATICAL
More informationA viability result for second-order differential inclusions
Electronic Journal of Differential Equations Vol. 00(00) No. 76. 1 1. ISSN: 107-6691. URL: htt://ejde.math.swt.edu or htt://ejde.math.unt.edu ft ejde.math.swt.edu (login: ft) A viability result for second-order
More informationExistence of solutions to a superlinear p-laplacian equation
Electronic Journal of Differential Equations, Vol. 2001(2001), No. 66,. 1 6. ISSN: 1072-6691. URL: htt://ejde.math.swt.edu or htt://ejde.math.unt.edu ft ejde.math.swt.edu (login: ft) Existence of solutions
More informationON THE SET a x + b g x (mod p) 1 Introduction
PORTUGALIAE MATHEMATICA Vol 59 Fasc 00 Nova Série ON THE SET a x + b g x (mod ) Cristian Cobeli, Marian Vâjâitu and Alexandru Zaharescu Abstract: Given nonzero integers a, b we rove an asymtotic result
More informationAlaa Kamal and Taha Ibrahim Yassen
Korean J. Math. 26 2018), No. 1,. 87 101 htts://doi.org/10.11568/kjm.2018.26.1.87 ON HYPERHOLOMORPHIC Fω,G α, q, s) SPACES OF QUATERNION VALUED FUNCTIONS Alaa Kamal and Taha Ibrahim Yassen Abstract. The
More informationNONLOCAL p-laplace EQUATIONS DEPENDING ON THE L p NORM OF THE GRADIENT MICHEL CHIPOT AND TETIANA SAVITSKA
NONLOCAL -LAPLACE EQUATIONS DEPENDING ON THE L NORM OF THE GRADIENT MICHEL CHIPOT AND TETIANA SAVITSKA Abstract. We are studying a class of nonlinear nonlocal diffusion roblems associated with a -Lalace-tye
More informationFactorizations Of Functions In H p (T n ) Takahiko Nakazi
Factorizations Of Functions In H (T n ) By Takahiko Nakazi * This research was artially suorted by Grant-in-Aid for Scientific Research, Ministry of Education of Jaan 2000 Mathematics Subject Classification
More informationt 0 Xt sup X t p c p inf t 0
SHARP MAXIMAL L -ESTIMATES FOR MARTINGALES RODRIGO BAÑUELOS AND ADAM OSȨKOWSKI ABSTRACT. Let X be a suermartingale starting from 0 which has only nonnegative jums. For each 0 < < we determine the best
More informationPETER J. GRABNER AND ARNOLD KNOPFMACHER
ARITHMETIC AND METRIC PROPERTIES OF -ADIC ENGEL SERIES EXPANSIONS PETER J. GRABNER AND ARNOLD KNOPFMACHER Abstract. We derive a characterization of rational numbers in terms of their unique -adic Engel
More informationNew weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function - a review
University of Wollongong Research Online Faculty of Informatics - Paers (Archive) Faculty of Engineering and Information Sciences 1999 New weighing matrices and orthogonal designs constructed using two
More informationSome nonlinear dynamic inequalities on time scales
Proc. Indian Acad. Sci. Math. Sci.) Vol. 117, No. 4, November 2007,. 545 554. Printed in India Some nonlinear dynamic inequalities on time scales WEI NIAN LI 1,2 and WEIHONG SHENG 1 1 Deartment of Mathematics,
More informationOn generalizing happy numbers to fractional base number systems
On generalizing hay numbers to fractional base number systems Enriue Treviño, Mikita Zhylinski October 17, 018 Abstract Let n be a ositive integer and S (n) be the sum of the suares of its digits. It is
More informationOn the approximation of a polytope by its dual L p -centroid bodies
On the aroximation of a olytoe by its dual L -centroid bodies Grigoris Paouris and Elisabeth M. Werner Abstract We show that the rate of convergence on the aroximation of volumes of a convex symmetric
More informationTopic 30 Notes Jeremy Orloff
Toic 30 Notes Jeremy Orloff 30 Alications to oulation biology 30.1 Modeling examles 30.1.1 Volterra redator-rey model The Volterra redator-rey system models the oulations of two secies with a redatorrey
More informationResearch Article A New Method to Study Analytic Inequalities
Hindawi Publishing Cororation Journal of Inequalities and Alications Volume 200, Article ID 69802, 3 ages doi:0.55/200/69802 Research Article A New Method to Study Analytic Inequalities Xiao-Ming Zhang
More informationResearch Article An Iterative Algorithm for the Reflexive Solution of the General Coupled Matrix Equations
he Scientific World Journal Volume 013 Article ID 95974 15 ages htt://dxdoiorg/101155/013/95974 Research Article An Iterative Algorithm for the Reflexive Solution of the General Couled Matrix Euations
More informationOn Maximum Principle and Existence of Solutions for Nonlinear Cooperative Systems on R N
ISS: 2350-0328 On Maximum Princile and Existence of Solutions for onlinear Cooerative Systems on R M.Kasturi Associate Professor, Deartment of Mathematics, P.K.R. Arts College for Women, Gobi, Tamilnadu.
More information216 S. Chandrasearan and I.C.F. Isen Our results dier from those of Sun [14] in two asects: we assume that comuted eigenvalues or singular values are
Numer. Math. 68: 215{223 (1994) Numerische Mathemati c Sringer-Verlag 1994 Electronic Edition Bacward errors for eigenvalue and singular value decomositions? S. Chandrasearan??, I.C.F. Isen??? Deartment
More informationBOUNDS FOR THE SIZE OF SETS WITH THE PROPERTY D(n) Andrej Dujella University of Zagreb, Croatia
GLASNIK MATMATIČKI Vol. 39(59(2004, 199 205 BOUNDS FOR TH SIZ OF STS WITH TH PROPRTY D(n Andrej Dujella University of Zagreb, Croatia Abstract. Let n be a nonzero integer and a 1 < a 2 < < a m ositive
More informationDiscrete Calderón s Identity, Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces
J Geom Anal (010) 0: 670 689 DOI 10.1007/s10-010-913-6 Discrete Calderón s Identity, Atomic Decomosition and Boundedness Criterion of Oerators on Multiarameter Hardy Saces Y. Han G. Lu K. Zhao Received:
More informationarxiv: v1 [math.ap] 19 Mar 2011
Life-San of Solutions to Critical Semilinear Wave Equations Yi Zhou Wei Han. Abstract arxiv:113.3758v1 [math.ap] 19 Mar 11 The final oen art of the famous Strauss conjecture on semilinear wave equations
More informationarxiv: v1 [math.ap] 17 May 2018
Brezis-Gallouet-Wainger tye inequality with critical fractional Sobolev sace and BMO Nguyen-Anh Dao, Quoc-Hung Nguyen arxiv:1805.06672v1 [math.ap] 17 May 2018 May 18, 2018 Abstract. In this aer, we rove
More informationBoundary regularity for elliptic problems with continuous coefficients
Boundary regularity for ellitic roblems with continuous coefficients Lisa Beck Abstract: We consider weak solutions of second order nonlinear ellitic systems in divergence form or of quasi-convex variational
More informationConvolution Properties for Certain Meromorphically Multivalent Functions
Filomat 31:1 (2017), 113 123 DOI 10.2298/FIL1701113L Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: htt://www.mf.ni.ac.rs/filomat Convolution Proerties for Certain
More informationAn extended Hilbert s integral inequality in the whole plane with parameters
He et al. Journal of Ineualities and Alications 88:6 htts://doi.org/.86/s366-8-8-z R E S E A R C H Oen Access An extended Hilbert s integral ineuality in the whole lane with arameters Leing He *,YinLi
More informationSums of independent random variables
3 Sums of indeendent random variables This lecture collects a number of estimates for sums of indeendent random variables with values in a Banach sace E. We concentrate on sums of the form N γ nx n, where
More informationarxiv:math/ v2 [math.nt] 21 Oct 2004
SUMS OF THE FORM 1/x k 1 + +1/x k n MODULO A PRIME arxiv:math/0403360v2 [math.nt] 21 Oct 2004 Ernie Croot 1 Deartment of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 ecroot@math.gatech.edu
More informationSCHUR m-power CONVEXITY OF GEOMETRIC BONFERRONI MEAN
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 38 207 (769 776 769 SCHUR m-power CONVEXITY OF GEOMETRIC BONFERRONI MEAN Huan-Nan Shi Deartment of Mathematics Longyan University Longyan Fujian 36402
More informationOn the capacity of the general trapdoor channel with feedback
On the caacity of the general tradoor channel with feedback Jui Wu and Achilleas Anastasooulos Electrical Engineering and Comuter Science Deartment University of Michigan Ann Arbor, MI, 48109-1 email:
More information#A47 INTEGERS 15 (2015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS
#A47 INTEGERS 15 (015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS Mihai Ciu Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 5,
More informationHolder Continuity of Local Minimizers. Giovanni Cupini, Nicola Fusco, and Raffaella Petti
Journal of Mathematical Analysis and Alications 35, 578597 1999 Article ID jmaa199964 available online at htt:wwwidealibrarycom on older Continuity of Local Minimizers Giovanni Cuini, icola Fusco, and
More informationMarcinkiewicz-Zygmund Type Law of Large Numbers for Double Arrays of Random Elements in Banach Spaces
ISSN 995-0802, Lobachevskii Journal of Mathematics, 2009, Vol. 30, No. 4,. 337 346. c Pleiades Publishing, Ltd., 2009. Marcinkiewicz-Zygmund Tye Law of Large Numbers for Double Arrays of Random Elements
More information