ON 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.
|
|
- Hester Maxwell
- 5 years ago
- Views:
Transcription
1 Bulletin of the Iranian Mathematical Society Vol. 36 No. 1 (2010), pp 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 by Mohammad Asadzadeh Abstract. Our main objective is to study some qualitative behavior of the solutions of the difference equation x n+1 γx n k + (ax n + bx n k ) / (cx n dx n k ), n 0, 1, 2,..., where the initial conditions x k,..., x 1, x 0 are arbitrary positive real numbers the coefficients γ, a, b, c d are positive constants, while k is a positive integer number. 1. Introduction Our goal is to investigate some qualitative behavior of the solutions of the difference equation, (1.1) x n+1 γx n k + ax n + bx n k cx n dx n k, n 0, 1, 2,..., where the initial conditions x k,..., x 1, x 0 are arbitrary positive real numbers the coefficients γ, a, b, c d are positive constants, while k is a positive integer number. The case where any of α, a, c, d is allowed to be zero gives different special cases of the equation (1) which are studied by many authors (see for example, 3, 6, 8, 12, 17, 29]). For MSC(2000): Primary: 39A10, 39A11, 39A99; Secondary: 34C99. Keywords: Difference equations, prime period two solution, locally asymptotically stable, global attractor, convergence. Received: 15 July 2008, Accepted: 07 March Corresponding author c 2010 Iranian Mathematical Society. 103
2 104 Zayed El-Moneam related work, see also 1, 2, 4, 5, 7, 9-11, 13-16, 18-28, 30 40]. The study of these equations is challenging rewarding still in its infancy. We believe that the nonlinear rational difference equations are of paramount importance in their own rights. Furthermore, the results about such equations offer prototypes for the development of the basic theory of the global behavior of nonlinear difference equations. Note that Eq. (1.1) can be considered as a generalization of that obtained in 8, 33]. Definition 1.1. A difference equation of order (k + 1) is of the form (1.2) x n+1 F (x n, x n 1,..., x n k ), n 0, 1, 2,..., where F is a continuous function which maps some set J k+1 into J J is a set of real numbers. An equilibrium point x of this equation is a point that satisfies the condition x F ( x, x,..., x). That is, the constant sequence {x n } n k with x n x, for all n k, is a solution of that equation. Definition 1.2. Let x (0, ) be an equilibrium point of the difference equation (1.2). Then, we have: (i) An equilibrium point x of the difference equation (1.2) is called locally stable if for every ε > 0 there exists δ > 0 such that, if x k,..., x 1, x 0 (0, ) with x k x x 1 x + x 0 x < δ, then x n x < ε, for all n k. (ii) An equilibrium point x of the difference equation (1.2) is called locally asymptotically stable if it is locally stable there exists γ > 0 such that, if x k,..., x 1, x 0 (0, ) with x k x x 1 x + x 0 x < γ, then lim n x n x. (iii) An equilibrium point x of the difference equation (1.2) is called a global attractor if x k,..., x 1, x 0 (0, ), then lim x n x. n (iv) An equilibrium point x of the equation (1.2) is called globally asymptotically stable if it is locally stable a global attractor. (v) An equilibrium point x, of the difference equation (2) is called unstable if it is not locally stable.
3 On the rational recursive sequence 105 Definition 1.3. A sequence {x n } n k is said to be periodic with period p if x n+p x n, for all n k. A sequence {x n } n k is said to be periodic with prime period p if p is the smallest positive integer having this property. The linearized equation of the difference equation (1.2) about the equilibrium point x is the linear difference equation, (1.3) y n+1 k i0 F ( x, x,..., x) x n i y n i. Now, assume that the characteristic equation associated with (1.3) is (1.4) p (λ) p 0 λ k + p 1 λ k p k 1 λ + p k 0, where, p i F ( x, x,..., x) / x n i. Theorem 1.1. (See 18]). Assume that p i R, i 1, 2,..., k {0, 1, 2,...}. Then, (1.5) k p i < 1 i1 is a sufficient condition for the asymptotic stability of the difference equation, (1.6) x n+k + p 1 x n+k p k x n 0, n 0, 1, 2,... Theorem 1.2. (The linearized stability theorem; see15,18,19]). Suppose F is a continuously differentiable function defined on an open neighbourhood of the equilibrium x. Then, the following statements are true. (i) If all roots of the characteristic equation (1.4) of the linearized equation (1.3) have absolute value less than one, then the equilibrium point x is locally asymptotically stable. (ii) If at least one root of Eq.(1.4) has absolute value greater than one, then the equilibrium point x is unstable. The following Theorem will be useful for the proof of our main results.
4 106 Zayed El-Moneam Theorem 1.3. (See15, p.18]). Let F : a, b] k+1 a, b] be a continuous function, where k is a positive integer, a, b] is an interval of real numbers consider the difference equation (1.2). Suppose that F satisfies the following conditions: (i) For every integer i with 1 i k+1, the function F (z 1, z 2,..., z k+1 ) is weakly monotonic in z i, for fixed z 1, z 2,..., z i 1, z i+1,..., z k+1. (ii) If (m, M) is a solution of the system (1.7) m F (m 1, m 2,..., m k+1 ) M F (M 1, M 2,..., M k+1 ), then m M, where for each i 1, 2,..., k + 1, we set m if F is nondecreasing in z i m i M if F is nonincreasing in z i M if F is nondecreasing in z i M i m if F is nonincreasing in z i. Then, there exists exactly one equilibrium point x of the difference equation (1.2), every solution of (1.2) converges to x. 2. Periodic solutions Theorem 2.1. If k is an even positive integer c d, then Eq. (1.1) has no positive solution of prime period two. Proof. Assume that there exists a distinctive positive solution..., P, Q, P, Q,... of prime period two of Eq. (1.1). If k is even, then x n x n k. It follows from Eq. (1.1) that P γq + aq + bq cq dq Q γp + ap + bp cp dp, provided that c d. Hence, we have (P Q) (γ + 1) 0. Thus, P Q, which is a contradiction. The proof of Theorem 2.1 is now complete.
5 On the rational recursive sequence 107 Theorem 2.2. If k is an odd positive integer, γ > 1, b > a, then the difference equation (1.1) has no positive solution of prime period two. Proof. Assume that there exists a distinctive positive solution..., P, Q, P, Q,... of prime period two of Eq. (1.1). If k is odd, then x n+1 x n k. It follows from the difference equation (1.1) that P γp + Consequently, we obtain: aq + bp cq dp Q γq + ap + bq cp dq. (2.1) cp Q dp 2 γcp Q γdp 2 + aq + bp, (2.2) cp Q dq 2 γcp Q γdq 2 + ap + bq. By subtracting, we deduce: (2.3) P + Q b a d (γ 1), while by adding we obtain: (2.4) P Q a (b a) d (c + d) (γ 1) 2. Since γ > 1 b > a, then P Q is negative. But P Q are both positive, we have a contradiction. Therefore, the proof of Theorem 2.2 is complete. Theorem 2.3. If k is an odd positive integer, then the necessary sufficient condition for the difference equation (1.1) to have a positive prime period two solution is that the inequality (2.5) (c + d) (a b) > 4ad
6 108 Zayed El-Moneam is valid, provided that a > b 0 < γ < 1. Proof. First, suppose that there exists a positive prime period two solution..., P, Q, P, Q,... of the difference equation (1.1). If k is odd, then x n+1 x n k. We shall prove that the condition (2.5) holds. It follows from the difference equation (1.1) that aq + bp P γp + cq dp Consequently, we have, Q γq + ap + bq cp dq. (2.6) cp Q dp 2 γcp Q γdp 2 + aq + bp, (2.7) cp Q dq 2 γcp Q γdq 2 + ap + bq. By subtracting (2.6) from (2.7), we deduce: (2.8) P + Q a b d (1 γ). while, by adding (2.6) (2.7), we have (2.9) P Q a (a b) d (c + d) (1 γ) 2, where a > b 0 < γ < 1. Assume that P Q are two positive distinct real roots of the quadratic equation, (2.10) t 2 ( P + Q) t + P Q 0. Thus, we deduce: (2.11) ( ) a b 2 > d (1 γ) 4a (a b) d (c + d) (1 γ) 2. From (2.11), we obtain the condition (2.5). Thus, the necessary condition is satisfied. Conversely, suppose that the condition (2.5) is valid. Then, we deduce immediately from (2.5) that the inequality (2.11) holds.
7 On the rational recursive sequence 109 Consequently, there exist two positive distinct real numbers P Q such that (2.12) P (a b) + β 2d (1 γ) Q (a b) β 2d (1 γ), where β (a b) 2 4ad (a b) / (c + d). Thus, P Q represent two positive distinct real roots of the quadratic equation (2.10). Now, we prove that P Q form a positive prime period two solution of the difference equation (1.1). To this end, we assume that x k P, x k+1 Q,..., x 1 P, x 0 Q. We wish to show that x 1 P x 2 Q. To this end, we deduce from the difference equation (1.1) that (2.13) x 1 γx k + ax 0 + bx k aq + bp γp + cx 0 dx k cq dp. Thus, we deduce from (2.12) (2.13) that (2.14) aq + bp x 1 P (1 γ) P cq dp aq + bp c (1 γ) P Q + d (1 γ) P 2 cq dp ] ] ] a (a b) β +b (a b)+β c (1 γ) +d (1 γ) d(c+d)(1 γ) ] ] 2 c d. (a b) β (a b)+β ] 2 (a b)+β
8 110 Zayed El-Moneam Multiplying the denominator numerator of (2.14) by 4d 2 (1 γ) 2, we get: (2.15) x 1 P 4d 2 (1 γ) 2 ( a ] (a b) β + b 4d 2 (1 γ) 2 ( c 4d 2 (1 γ) (d 2 (1 γ) + ( 4d 2 (1 γ) 2 c (a b) β ] (a b)+β ] (a b) β (a b)+β (a b)+β ] d d ] ) 2 c (1 γ) ]) (a b)+β 2ad (1 γ) (a b) β]+2bd (1 γ) (a b)+β] 4cd (1 γ) 2cd (1 γ) (a b) β] 2d 2 (1 γ) (a b)+β] d (1 γ) (a b) + β] 2 + 2cd (1 γ) (a b) β] 2d 2 (1 γ) (a b) + β] 2a (a b) β] + 2b (a b) + β] 4c (c+d) 2 c (a b) β] d (a b) + β]] 2 (a b) (a+b) 2 (a b) β 4c 2 ( a 2 b 2) 4c (c+d) 2 c (a b) β] d (a b) + β]] ] + (a b) 2 + β 2 (c+d) 2 c (a b) β] d (a b) + β]] 2 ( a 2 b 2) 4c (c+d) ]) d(c+d)(1 γ) 2 ]) ] (c+d) ] + (a b) + β] 2 ] +(a b) 2 +2 (a b) β+β 2 ] + (a b) 2 + (a b) 2 4ad(a b) c+d 2 c (a b) β] d (a b) + β]] Thus, x 1 P. Similarly, we can show, x 2 γx 1 k + ax 1 + bx 1 k ap + bq γq + cx 1 dx 1 k cp dq Q. Using the mathematical induction, we have, x n P x n+1 Q, for all n k. 0. Thus, the difference equation (1.1) has a positive prime period two solution,..., P, Q, P, Q,... Hence, the proof is now complete.
9 On the rational recursive sequence Local stability of the equilibrium point Here, we study the local stability character of the solutions of the difference equation (1.1). The equilibrium points of the difference equation (1.1) are given by the relation (3.1) x γ x + a x + b x c x d x. If (1 γ) (c d) > 0, then the only positive equilibrium point x of the difference equation (1.1) is given by (3.2) x a + b (1 γ) (c d), where 0 < γ < 1 c > d. Let F : (0, ) k+1 (0, ) be a continuous function defined by (3.3) F (u 0, u 1 ) γu 1 + au 0 + bu 1 cu 0 du 1, provided that cu 0 du 1. Therefore, F (u 0, u 1 ) (ad + bc) u 1 u 0 (cu 0 du 1 ) 2 F (u 0, u 1 ) γ+ (ad + bc) u 0 u 1 (cu 0 du 1 ) 2. (3.4) Then, we see that F ( x, x) u 0 (1 γ) (ad + bc) ρ 0 (a + b) (c d) F ( x, x) u 1 γ + (1 γ) (ad + bc) (a + b) (c d) ρ 1. Then, the linearized equation of the difference equation (1.1) about x is: (3.5) y n+1 ρ 0 y n ρ 1 y n k 0. Theorem 3.1. Assume that 0 < γ < 1, c > d (3.6) 2 (1 γ) (ad + bc) + γ (a + b) (c d) < (a + b) (c d). Then, the equilibrium point x of the difference equation (1.1) is locally asymptotically stable.
10 112 Zayed El-Moneam Proof. From (3.5), we get: ρ 0 + ρ 1 (1 γ) (ad + bc) (a + b) (c d) + (1 γ) (ad + bc) γ + (a + b) (c d) (1 γ) (ad + bc) (1 γ) (ad + bc) + γ + (a + b) (c d) (a + b) (c d) 2 (1 γ) (ad + bc) + γ (a + b) (c d) (a + b) (c d) (3.7) 2 (1 γ) (ad + bc) + γ (a + b) (c d). (a + b) (c d) From (3.6) (3.7), we deduce that (3.8) ρ 0 + ρ 1 < 1. It is followed by Theorem 1.1 that Eq. (1.1) is locally asymptotically stable. Thus, the proof of Theorem 3.1 is now complete. 4. Global attractor of the equilibrium point Here, we investigate the global attractivety character of the solutions of the difference equation (1.1). Theorem 4.1. The equilibrium point x of the difference equation (1.1) is a global attractor if 0 < γ < 1. Proof. By using (3.4), we can see that the function F (u 0, u 1 ) which is defined by (3.3) is decreasing in u 0 increasing in u 1. Suppose that (m, M) is a solution of the system (4.1) m F (M, m) M F (m, M). Then, we get: we have, (4.2) am + bm m F (M, m) γm + cm dm, M F (m, M) γm + am + bm cm dm, am + bm m (cm dm) (1 γ) am + bm (1 γ). M (cm dm)
11 On the rational recursive sequence 113 From (4.2) we deduce that M m. It follows by Theorem 1.3 that x is a global attractor of the difference equation (1.1). Thus, the proof of Theorem 4.1 is now complete. References 1] M.T. Aboutaleb, M.A. El-Sayed A. E. Hamza, Stability of the recursive sequence x n+1 (α βx n)/(γ + x n 1), J. Math. Anal. Appl. 261 (2001) ] R. Agarwal, Difference Equations Inequalities. Theory, Methods Applications, Marcel Dekker Inc., New York, ] A.M. Amleh, E.A. Grove, G.Ladas D.A. Georgiou, On the recursive sequence x n+1 α + (x n 1/x n), J. Math. Anal. Appl. 233 (1999) ] C.W. Clark, A delayed recruitment model of population dynamics with an application to baleen whale populations, J. Math. Biol. 3 (1976) ] R. Devault, W. Kosmala, G. Ladas S.W. Schultz, Global behavior of y n+1 (p + y n k )/(qy n + y n k ), Nonlinear Analysis, 47 (2001) ] R. Devault, G. Ladas S.W. Schultz, On the recursive sequence x n+1 α + (x n/x n 1), Proc. Amer. Math. Soc. 126(11) (1998) ] R. Devault S.W. Schultz, On the dynamics of x n+1 (βx n+γx n 1)/(Bx n+ Dx n 2), Comm. Appl. Nonlinear Analysis 12 (2005) ] E.M. Elabbasy, H.El-Metwally E.M. Elsayed, On the difference equation x n+1 ax n bx n/ (cx n dx n 1), Advances in Difference Equations, Volume 2006, Article ID 82579, pages 1-10, doi: /2006/ ] H. El-Metwally, E.A. Grove G. Ladas, A global convergence result with applications to periodic solutions, J. Math. Anal. Appl. 245 (2000) ] H. El-Metwally, G. Ladas, E.A. Grove H.D. Voulov, On the global attractivity the periodic character of some difference equations, J. Difference Equations Appl. 7 (2001) ] H.A. El-Morshedy, New explicit global asymptotic stability criteria for higher order difference equations, J. Math. Anal. Appl. 336 (2007), ] H.M. EL- Owaidy, A.M. Ahmed M.S. Mousa, On asymptotic behavior of the difference equation x n+1 α + (x p n 1 /xp n), J. Appl. Math. & Computing 12 (2003) ] H.M. EL- Owaidy, A.M. Ahmed Z. Elsady, Global attractivity of the recursive sequence x n+1 (α βx n k )/(γ + x n), J. Appl. Math. & Computing 16 (2004) ] C.H. Gibbons, M.R.S. Kulenovic G. Ladas, On the recursive sequence x n+1 (α + βx n 1)/(γ + x n), Math. Sci. Res. Hot-Line 4 (2) (2000) ] E.A. Grove G. Ladas, Periodicities in nonlinear difference equations, Vol. 4, Chapman & Hall / CRC, ] G. Karakostas, Convergence of a difference equation via the full limiting sequences method, Diff. Equations Dynamical. System 1 (1993) ] G. Karakostas S. Stevic, On the recursive sequences x n+1 A + f(x n,..., x n k+1 )/x n 1, Comm. Appl. Nonlinear Analysis 11 (2004)
12 114 Zayed El-Moneam 18] V.L. Kocic G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, ] M.R.S. Kulenovic G. Ladas, Dynamics of second order rational difference equations with open problems conjectures, Chapman & Hall / CRC, Florida, ] M.R.S. Kulenovic, G. Ladas W.S. Sizer, On the recursive sequence x n+1 (αx n + βx n 1)/(γx n + δx n 1), Math. Sci. Res. Hot-Line 2(5) (1998) ] S.A. Kuruklis, The asymptotic stability of x n+1 ax n + bx n k 0, J. Math. Anal. Appl. 188 (1994) ] G. Ladas, C.H. Gibbons, M.R.S. Kulenovic H.D. Voulov, On the trichotomy character of x n+1 (α + βx n + γx n 1)/(A + x n), J. Difference Equations Appl. 8 (2002) ] G. Ladas, C.H. Gibbons M.R.S. Kulenovic, On the dynamics of x n+1 (α + βx n + γx n 1)/(A + Bx n), Proceeding of the Fifth International Conference on Difference Equations Applications, Temuco, Chile, Jan. 3-7, 2000, Taylor Francis, London, (2002) ] G. Ladas, E. Camouzis H.D. Voulov, On the dynamic of x n+1 (α+γx n 1+ δx n 2)/(A + x n 2), J. Difference Equations Appl. 9 (2003) ] G. Ladas, On the rational recursive sequence x n+1 (α + βx n + γx n 1)/(A + Bx n + Cx n 1), J. Difference Equations Appl. 1 (1995) ] W. T. Li H.R. Sun, Global attractivity in a rational recursive sequence, Dynamical Systems. Appl. 11 (2002) ] S. Stevic, On the recursive sequences x n+1 x n 1/g(x n), Taiwanese J. Math. 6 (2002) ] S. Stevic, On the recursive sequences x n+1 g(x n, x n 1)/(A+x n), Appl. Math. Letter 15 (2002) ] S. Stevic, On the recursive sequences x n+1 α + (x p n 1 /xp n), J. Appl. Math.& Computing 18 (2005) ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 (D + αx n + βx n 1 + γx n 2)/(Ax n + Bx n 1 + Cx n 2), Comm. Appl. Nonlinear Analysis 12 (2005) ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 (αx n + βx n 1 + γx n 2 + δx n 3)/(Ax n + Bx n 1 + Cx n 2 + Dx n 3), J. Appl. Math. & Computing 22 (2006) ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 ( A + k ( B + k ) i0 αixn i /, Int. J. Math. & Math. Sci, Volume 2007, Article ID23618, 12 pages, doi: /2007/ ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 ax n bx n/ (cx n dx n k ), Comm. Appl. Nonlinear Analysis 15(2008) ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 ( A + k i0 αixn i ) i0 βixn i ) / k i0 βixn i, Mathematica Bohemica 133(2008) ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 (ax n + bx n k ) / (cx n dx n k ), Comm. Appl. Nonlinear Analysis 15(2008)
13 On the rational recursive sequence ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 (α + βx n k ) / (γ x n), J. Appl. Math. & Computing, /s , in press. 37] E.M.E. Zayed M.A. El-Moneam, On the global attractivity of two nonlinear difference equations, J. Math. Sci., to appear. 38] E.M.E. Zayed M.A. El-Moneam, On the rational recursive two sequences x n+1 ax n k + bx n k / (cx n + δdx n k ), Acta Math. Vietnamica, to appear. 39] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 Ax n + (βx n + γx n k ) / (Cx n + Dx n k ), Comm. Appl. Nonlinear Analysis 16 (2009) ] E.M.E. Zayed M.A. El-Moneam, On the rational recursive sequence x n+1 Ax n + Bx n k + (βx n + γx n k ) / (Cx n + Dx n k ), Acta Appl. Math. doi /s y, in press. E. M. E. Zayed Mathematics Department, Faculty of Science, Taif University, El-Taif, EL-Hawiyah, P.O. Box 888, Kingdom of Saudi Arabia emezayed@hotmail.com M. A. El-Moneam Mathematics Department, Faculty of Science Arts, Jazan University, Farasan, Jazan, Kingdom of Saudi Arabia mabdelmeneam2004@yahoo.com
ON THE GLOBAL ATTRACTIVITY AND THE PERIODIC CHARACTER OF A RECURSIVE SEQUENCE. E.M. Elsayed
Opuscula Mathematica Vol. 30 No. 4 2010 http://dx.doi.org/10.7494/opmath.2010.30.4.431 ON THE GLOBAL ATTRACTIVITY AND THE PERIODIC CHARACTER OF A RECURSIVE SEQUENCE E.M. Elsayed Abstract. In this paper
More informationResearch Article Global Attractivity and Periodic Character of Difference Equation of Order Four
Discrete Dynamics in Nature and Society Volume 2012, Article ID 746738, 20 pages doi:10.1155/2012/746738 Research Article Global Attractivity and Periodic Character of Difference Equation of Order Four
More informationReview Article Solution and Attractivity for a Rational Recursive Sequence
Discrete Dynamics in Nature and Society Volume 2011, Article ID 982309, 17 pages doi:10.1155/2011/982309 Review Article Solution and Attractivity for a Rational Recursive Sequence E. M. Elsayed 1, 2 1
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 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 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 informationDynamics and behavior of a higher order rational recursive sequence
RESEARCH Open Access Dynamics and behavior of a higher order rational recursive sequence Elsayed M Elsayed 1,2* and Mohammed M El-Dessoky 1,2 * Correspondence: emelsayed@mansedueg 1 Mathematics Department,
More informationDynamics of higher order rational difference equation x n+1 = (α + βx n )/(A + Bx n + Cx n k )
Int. J. Nonlinear Anal. Appl. 8 (2017) No. 2, 363-379 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2017.10822.1526 Dynamics of higher order rational difference equation x n+1 = (α + βx
More informationGlobal Attractivity of a Rational Difference Equation
Math. Sci. Lett. 2, No. 3, 161-165 (2013) 161 Mathematical Sciences Letters An International Journal http://dx.doi.org/10.12785/msl/020302 Global Attractivity of a Rational Difference Equation Nouressadat
More informationOn the Dynamics of the Higher Order Nonlinear Rational Difference Equation
Math. Sci. Lett. 3, No. 2, 121-129 (214) 121 Mathematical Sciences Letters An International Journal http://dx.doi.org/1.12785/msl/328 On the Dynamics of the Higher Order Nonlinear Rational Difference Equation
More informationOn the Dynamics of a Rational Difference Equation, Part 1
International Journal of Difference Equations (IJDE). ISSN 0973-6069 Volume 3 Number 1 (2008), pp. 1 35 Research India Publications http://www.ripublication.com/ijde.htm On the Dynamics of a Rational Difference
More informationDYNAMICS AND BEHAVIOR OF HIGHER ORDER AND NONLINEAR RATIONAL DIFFERENCE EQUATION
DYNAMICS AND BEHAVIOR OF HIGHER ORDER AND NONLINEAR RATIONAL DIFFERENCE EQUATION Nirmaladevi.S 1, Karthikeyan.N 2 1 Research scholar in mathematics Vivekanha college of Arts Science for Women (Autonomous)
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 informationConvergence and Oscillation in a Rational Fourth Order Difference Equation
Australian Journal of Basic and Applied Sciences, 5(7): 771-777, 011 ISSN 1991-8178 Convergence and Oscillation in a Rational Fourth Order Difference Equation Hamid Gazor, Azadeh Memar, Tahereh Gazor and
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 informationResearch Article Global Attractivity of a Higher-Order Difference Equation
Discrete Dynamics in Nature and Society Volume 2012, Article ID 930410, 11 pages doi:10.1155/2012/930410 Research Article Global Attractivity of a Higher-Order Difference Equation R. Abo-Zeid Department
More informationMonotone and oscillatory solutions of a rational difference equation containing quadratic terms
Monotone and oscillatory solutions of a rational difference equation containing quadratic terms M. Dehghan, C.M. Kent, R. Mazrooei-Sebdani N.L. Ortiz, H. Sedaghat * Department of Mathematics, Virginia
More informationOn the Third Order Rational Difference Equation
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 27, 32-334 On the Third Order Rational Difference Equation x n x n 2 x (a+x n x n 2 ) T. F. Irahim Department of Mathematics, Faculty of Science Mansoura
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 A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE
IJMMS 2004:55, 295 2926 PII. S067204402270 http://ijmms.hindawi.com Hindawi Publishing Corp. ON A DIFFERENCE EQUATION WITH MIN-MAX RESPONSE GEORGE L. KARAKOSTAS and STEVO STEVIĆ Received 25 February 2004
More informationOn the difference equation. x n+1 = ax n l + bx n k + f(x n l, x n k ),
Abdelrahman et al. Advances in Difference Equations (2018) 2018:431 https://doi.org/10.1186/s13662-018-1880-8 R E S E A R C H Open Access On the difference equation x n+1 = ax n l + bx n k + f (x n l,
More informationOn the Solutions of the Recursive Sequence. i=0. Saniye Ergin and Ramazan Karataş 1
Thai Journal of Mathematics Volume 14 (2016) Number 2 : 391 397 http://thaijmathincmuacth ISSN 1686-0209 On the Solutions of the Recursive Sequence ax n k a k x n i Saniye Ergin and Ramazan Karataş 1 Akdeniz
More informationON THE RATIONAL RECURSIVE SEQUENCE. 1. Introduction
Tatra Mt. Math. Publ. 43 (2009), 1 9 DOI: 10.2478/v10127-009-0020-y t m Mathematical Publications ON THE RATIONAL RECURSIVE SEQUENCE ax b+cx n x Anna Andruch-Sobi lo Ma lgorzata Migda ABSTRACT. In this
More informationNonhyperbolic Dynamics for Competitive Systems in the Plane and Global Period-doubling Bifurcations
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 3, Number 2, pp. 229 249 (2008) http://campus.mst.edu/adsa Nonhyperbolic Dynamics for Competitive Systems in the Plane and Global Period-doubling
More informationAsymptotic Behavior of a Higher-Order Recursive Sequence
International Journal of Difference Equations ISSN 0973-6069, Volume 7, Number 2, pp. 75 80 (202) http://campus.mst.edu/ijde Asymptotic Behavior of a Higher-Order Recursive Sequence Özkan Öcalan Afyon
More informationGlobal Attractivity in a Higher Order Difference Equation with Applications
International Jr, of Qualitative Theory of Differential Equations and Applications International Vol. 3 No. 1 Jr, (January-June, of Qualitative 2017) Theory of Differential Equations and Applications Vol.
More informationThe Fibonacci sequence modulo π, chaos and some rational recursive equations
J. Math. Anal. Appl. 310 (2005) 506 517 www.elsevier.com/locate/jmaa The Fibonacci sequence modulo π chaos and some rational recursive equations Mohamed Ben H. Rhouma Department of Mathematics and Statistics
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 informationTwo Rational Recursive Sequences
ELSEVIER An Intemational Journal Available online at www.sciencedirectcom computers &.c,=.c= ~--~=,,,==T. mathematics with applications Computers Mathematics with Applications 47 (2004) 487-494 www.elsevier.eom/locat
More informationGlobal 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 informationResearch Article On Boundedness of Solutions of the Difference Equation x n 1 px n qx n 1 / 1 x n for q>1 p>1
Hindawi Publishing Corporation Advances in Difference Equations Volume 2009, Article ID 463169, 11 pages doi:10.1155/2009/463169 Research Article On Boundedness of Solutions of the Difference Equation
More informationGlobal Asymptotic Stability of a Nonlinear Recursive Sequence
International Mathematical Forum, 5, 200, no. 22, 083-089 Global Asymptotic Stability of a Nonlinear Recursive Sequence Mustafa Bayram Department of Mathematics, Faculty of Arts and Sciences Fatih University,
More informationPeriodicity and Solution of Rational Recurrence Relation of Order Six
Applied Mathematics 3 79-733 http://d.doi.org/.436/am..377 Published Online July (http://www.scirp.org/journal/am) Periodicity and Solution of Rational Recurrence Relation of Order Si Tarek F. Ibrahim
More informationGlobal Stability and Periodic Solutions for a Second-Order Quadratic Rational Difference Equation
International Journal of Difference Equations ISSN 0973-6069, Volume 11, Number 1, pp. 79 103 (2016) http://campus.mst.edu/ijde Global Stability and Periodic Solutions for a Second-Order Quadratic Rational
More informationM.ARCIERO, G.LADAS AND S.W.SCHULTZ
Georgian Mathematical Journal 1(1994), No. 3, 229-233 SOME OPEN PROBLEMS ABOUT THE SOLUTIONS OF THE DELAY DIFFERENCE EQUATION x n+1 = A/x 2 n + 1/x p n k M.ARCIERO, G.LADAS AND S.W.SCHULTZ Abstract. We
More informationON THE STABILITY OF SOME SYSTEMS OF EXPONENTIAL DIFFERENCE EQUATIONS. N. Psarros, G. Papaschinopoulos, and C.J. Schinas
Opuscula Math. 38, no. 1 2018, 95 115 https://doi.org/10.7494/opmath.2018.38.1.95 Opuscula Mathematica ON THE STABILITY OF SOME SYSTEMS OF EXPONENTIAL DIFFERENCE EQUATIONS N. Psarros, G. Papaschinopoulos,
More informationUnboundedness Results for Rational Difference Equations
University of Rhode Island DigitalCommons@URI Open Access Dissertations 203 Unboundedness Results for Rational Difference Equations Gabriel Lugo University of Rhode Island, lugogabrielmath@gmailcom Follow
More informationA System of Difference Equations with Solutions Associated to Fibonacci Numbers
International Journal of Difference Equations ISSN 0973-6069 Volume Number pp 6 77 06) http://campusmstedu/ijde A System of Difference Equations with Solutions Associated to Fibonacci Numbers Yacine Halim
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 informationSTABILITY OF THE kth ORDER LYNESS EQUATION WITH A PERIOD-k COEFFICIENT
International Journal of Bifurcation Chaos Vol 7 No 2007 43 52 c World Scientific Publishing Company STABILITY OF THE kth ORDER LYNESS EQUATION WITH A PERIOD-k COEFFICIENT E J JANOWSKI M R S KULENOVIĆ
More informationDynamics of rational difference equations containing quadratic terms
Dynamics of rational difference equations containing quadratic terms M. Dehghan, C.M. Kent, R. Mazrooei-Sebdani N.L. Ortiz, H. Sedaghat * Department of Mathematics, Virginia Commonwealth University, Richmond,
More informationGLOBAL ATTRACTIVITY IN A NONLINEAR DIFFERENCE EQUATION
Sixth Mississippi State Conference on ifferential Equations and Computational Simulations, Electronic Journal of ifferential Equations, Conference 15 (2007), pp. 229 238. ISSN: 1072-6691. URL: http://ejde.mathmississippi
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 informationGLOBAL DYNAMICS OF THE SYSTEM OF TWO EXPONENTIAL DIFFERENCE EQUATIONS
Electronic Journal of Mathematical Analysis an Applications Vol. 7(2) July 209, pp. 256-266 ISSN: 2090-729X(online) http://math-frac.org/journals/ejmaa/ GLOBAL DYNAMICS OF THE SYSTEM OF TWO EXPONENTIAL
More informationTrichotomy of a system of two difference equations
J. Math. Anal. Appl. 289 (2004) 216 230 www.elsevier.com/locate/jmaa Trichotomy of a system of two difference equations G. Papaschinopoulos and G. Stefanidou 1 Democritus University of Thrace, Department
More informationGlobal Dynamics and Bifurcations of Two Quadratic Fractional Second Order Difference Equations
University of Rhode Island DigitalCommons@URI Mathematics Faculty Publications Mathematics 2016 Global Dynamics and Bifurcations of Two Quadratic Fractional Second Order Difference Equations Senada Kalabušić
More informationResearch Article On the Difference Equation x n 1 x n x n k / x n k 1 a bx n x n k
Abstract and Applied Analysis Volume 2012, Article ID 108047, 9 pages doi:10.1155/2012/108047 Research Article On the Difference Equation x n 1 x n x n k / x n k 1 a bx n x n k Stevo Stević, 1 Josef Diblík,
More informationInvariants for Some Rational Recursive Sequences with Periodic Coefficients
Invariants for Some Rational Recursive Sequences with Periodic Coefficients By J.FEUER, E.J.JANOWSKI, and G.LADAS Department of Mathematics University of Rhode Island Kingston, R.I. 02881-0816, USA Abstract
More informationOscillationofNonlinearFirstOrderNeutral Di erenceequations
AppliedMathematics E-Notes, 1(2001), 5-10 c Availablefreeatmirrorsites ofhttp://math2.math.nthu.edu.tw/»amen/ OscillationofNonlinearFirstOrderNeutral Di erenceequations YingGaoandGuangZhang yz Received1June2000
More informationGlobal Stability for Mixed Monotone Systems
uly 1, 007 9:44 Journal of Difference Equations and Applications mixedmonotonesystemsfinal Journal of Difference Equations and Applications, Vol. 00, No. 00, Month 00x, 1 6 Global Stability for Mixed Monotone
More informationGlobal stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures
Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing-Henson conjectures Saber Elaydi Department of Mathematics Trinity University San Antonio,
More informationGENERAL QUARTIC-CUBIC-QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN NORMED SPACES
U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 3, 200 ISSN 223-7027 GENERAL QUARTIC-CUBIC-QUADRATIC FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN NORMED SPACES M. Eshaghi Gordji, H. Khodaei 2, R. Khodabakhsh 3 The
More informationGlobal attractivity in a rational delay difference equation with quadratic terms
Journal of Difference Equations and Applications ISSN: 1023-6198 (Print) 1563-5120 (Online) Journal homepage: http://www.tandfonline.com/loi/gdea20 Global attractivity in a rational delay difference equation
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 informationSome Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph
J o u r n a l of Mathematics and Applications JMA No 39, pp 81-90 (2016) Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph Hamid
More informationOscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments
Journal of Mathematical Analysis Applications 6, 601 6 001) doi:10.1006/jmaa.001.7571, available online at http://www.idealibrary.com on Oscillation Criteria for Certain nth Order Differential Equations
More informationarxiv: v1 [math.ds] 29 Feb 2016
Computational Complex Dynamics of f α,β,γ,δ (z) = αz + β γz 2 + δz arxiv:1603.00007v1 [math.ds] 29 Feb 2016 Sk. Sarif Hassan Department of Mathematics College of Engineering Studies University of Petroleum
More informationBulletin of the Iranian Mathematical Society Vol. 39 No.6 (2013), pp
Bulletin of the Iranian Mathematical Society Vol. 39 No.6 (2013), pp 1125-1135. COMMON FIXED POINTS OF A FINITE FAMILY OF MULTIVALUED QUASI-NONEXPANSIVE MAPPINGS IN UNIFORMLY CONVEX BANACH SPACES A. BUNYAWAT
More informationResearch Article Asymptotic Behavior of the Solutions of System of Difference Equations of Exponential Form
Difference Equations Article ID 936302 6 pages http://dx.doi.org/10.1155/2014/936302 Research Article Asymptotic Behavior of the Solutions of System of Difference Equations of Exponential Form Vu Van Khuong
More informationA fixed point problem under two constraint inequalities
Jleli and Samet Fixed Point Theory and Applications 2016) 2016:18 DOI 10.1186/s13663-016-0504-9 RESEARCH Open Access A fixed point problem under two constraint inequalities Mohamed Jleli and Bessem Samet*
More informationThe main objective of this work is to establish necessary and sufficient conditions for oscillations of (1.1), under the assumptions
Journal of Applied Mathematics and Computation (JAMC), 2018, 2(3), 100-106 http://www.hillpublisher.org/journal/jamc ISSN Online:2576-0645 ISSN Print:2576-0653 Necessary and Sufficient Conditions for Oscillation
More informationDynamics of the equation complex plane
APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Dynamics of the equation in the complex plane Sk. Sarif Hassan 1 and Esha Chatterjee * Received: 0 April 015 Accepted: 08 November 015 Published:
More informationSome unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces
An. Şt. Univ. Ovidius Constanţa Vol. 19(1), 211, 331 346 Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces Yonghong Yao, Yeong-Cheng Liou Abstract
More informationOscillation of second-order nonlinear difference equations with sublinear neutral term
Mathematica Moravica Vol. 23, No. (209), 0 Oscillation of second-order nonlinear difference equations with sublinear neutral term Martin Bohner, Hassan A. El-Morshedy, Said R. Grace and Ilgin Sağer Abstract.
More informationMath Circles: Number Theory III
Math Circles: Number Theory III Centre for Education in Mathematics and Computing University of Waterloo March 9, 2011 A prime-generating polynomial The polynomial f (n) = n 2 n + 41 generates a lot of
More informationCore Mathematics 3 Algebra
http://kumarmathsweeblycom/ Core Mathematics 3 Algebra Edited by K V Kumaran Core Maths 3 Algebra Page Algebra fractions C3 The specifications suggest that you should be able to do the following: Simplify
More informationPREVALENCE OF SOME KNOWN TYPICAL PROPERTIES. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXX, 2(2001), pp. 185 192 185 PREVALENCE OF SOME KNOWN TYPICAL PROPERTIES H. SHI Abstract. In this paper, some known typical properties of function spaces are shown to
More informationx 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?
1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number
More informationOn coupled generalised Banach and Kannan type contractions
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 5 (01, 59 70 Research Article On coupled generalised Banach Kannan type contractions B.S.Choudhury a,, Amaresh Kundu b a Department of Mathematics,
More informationStudent: Date: Instructor: kumnit nong Course: MATH 105 by Nong https://xlitemprodpearsoncmgcom/api/v1/print/math Assignment: CH test review 1 Find the transformation form of the quadratic function graphed
More informationMultiplicative metric spaces and contractions of rational type
Advances in the Theory of Nonlinear Analysis and its Applications 2 (2018) No. 4, 195 201. https://doi.org/10.31197/atnaa.481995 Available online at www.atnaa.org Research Article Multiplicative metric
More informationH2 MATHS SET D PAPER 1
H Maths Set D Paper H MATHS Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e b The curve y ax c x 3 points, and, H Maths Set D Paper has a stationary point at x 3. It also
More informationOscillation Criteria for Delay and Advanced Difference Equations with General Arguments
Advances in Dynamical Systems Applications ISSN 0973-531, Volume 8, Number, pp. 349 364 (013) http://campus.mst.edu/adsa Oscillation Criteria for Delay Advanced Difference Equations with General Arguments
More informationCommon fixed points of two generalized asymptotically quasi-nonexpansive mappings
An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 2 Common fixed points of two generalized asymptotically quasi-nonexpansive mappings Safeer Hussain Khan Isa Yildirim Received: 5.VIII.2013
More informationSOME FIXED POINT RESULTS FOR ADMISSIBLE GERAGHTY CONTRACTION TYPE MAPPINGS IN FUZZY METRIC SPACES
Iranian Journal of Fuzzy Systems Vol. 4, No. 3, 207 pp. 6-77 6 SOME FIXED POINT RESULTS FOR ADMISSIBLE GERAGHTY CONTRACTION TYPE MAPPINGS IN FUZZY METRIC SPACES M. DINARVAND Abstract. In this paper, we
More informationSTABILITY OF A GENERALIZED MIXED TYPE ADDITIVE, QUADRATIC, CUBIC AND QUARTIC FUNCTIONAL EQUATION
Volume 0 009), Issue 4, Article 4, 9 pp. STABILITY OF A GENERALIZED MIXED TYPE ADDITIVE, QUADRATIC, CUBIC AND QUARTIC FUNCTIONAL EQUATION K. RAVI, J.M. RASSIAS, M. ARUNKUMAR, AND R. KODANDAN DEPARTMENT
More informationAvailable online at Advances in Fixed Point Theory, 2 (2012), No. 4, ISSN:
Available online at http://scik.org Advances in Fixed Point Theory, 2 (2012), No. 4, 442-451 ISSN: 1927-6303 FIXED POINTS AND SOLUTIONS OF NONLINEAR FUNCTIONAL EQUATIONS IN BANACH SPACES A. EL-SAYED AHMED
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics ISOMETRIES ON LINEAR n-normed SPACES CHUN-GIL PARK AND THEMISTOCLES M. RASSIAS Department of Mathematics Hanyang University Seoul 133-791 Republic
More informationLECTURE 22, WEDNESDAY in lowest terms by H(x) = max{ p, q } and proved. Last time, we defined the height of a rational number x = p q
LECTURE 22, WEDNESDAY 27.04.04 FRANZ LEMMERMEYER Last time, we defined the height of a rational number x = p q in lowest terms by H(x) = max{ p, q } and proved Proposition 1. Let f, g Z[X] be coprime,
More informationNontrivial solutions for fractional q-difference boundary value problems
Electronic Journal of Qualitative Theory of Differential Equations 21, No. 7, 1-1; http://www.math.u-szeged.hu/ejqtde/ Nontrivial solutions for fractional q-difference boundary value problems Rui A. C.
More informationJournal of Number Theory
Journal of Number Theory 130 (2010) 1737 1749 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt A binary linear recurrence sequence of composite numbers Artūras
More informationON THE RECURSIVE SEQUENCE x n+1 = A x n. 1. Introduction Our aim in this paper is to establish that every positive solution of the equation
PROCEEDINGS OF THE MERICN MTHEMTICL SOCIETY Volume 26, Number, November 998, Pages 3257 326 S 0002-9939(98)04626-7 ON THE RECURSIVE SEQUENCE x n+ = x n R. DEVULT, G. LDS, ND S. W. SCHULTZ (Communicated
More informationFIXED POINTS AND CYCLIC CONTRACTION MAPPINGS UNDER IMPLICIT RELATIONS AND APPLICATIONS TO INTEGRAL EQUATIONS
SARAJEVO JOURNAL OF MATHEMATICS Vol.1 (23) (214), 257 27 DOI: 1.5644/SJM.1.2.11 FIXED POINTS AND CYCLIC CONTRACTION MAPPINGS UNDER IMPLICIT RELATIONS AND APPLICATIONS TO INTEGRAL EQUATIONS HEMANT KUMAR
More informationA new approach to the realization problem for fractional discrete-time linear systems
BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES Vol 64 No 26 DOI: 55/bpasts-26-2 A new approach to the realization problem for fractional discrete-time linear systems T KACZOREK Faculty of
More informationDiscrete Halanay-type inequalities and applications
Nonlinear Analysis 55 (2003) 669 678 www.elsevier.com/locate/na Discrete Halanay-type inequalities and applications Eduardo Liz a;, Anatoli Ivanov b, Juan Bosco Ferreiro c a Departamento de Matematica
More information1. Suppose that a, b, c and d are four different integers. Explain why. (a b)(a c)(a d)(b c)(b d)(c d) a 2 + ab b = 2018.
New Zealand Mathematical Olympiad Committee Camp Selection Problems 2018 Solutions Due: 28th September 2018 1. Suppose that a, b, c and d are four different integers. Explain why must be a multiple of
More informationASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE VOLTERRA EQUATIONS. Janusz Migda and Małgorzata Migda
Opuscula Math. 36, no. 2 (2016), 265 278 http://dx.doi.org/10.7494/opmath.2016.36.2.265 Opuscula Mathematica ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE VOLTERRA EQUATIONS Janusz Migda and Małgorzata
More informationON A THREE TERM LINEAR DIFFERENCE EQUATION WITH COMPLEX COEFFICIENTS. 1. Introduction
t m Mathematical Publications DOI: 10.1515/tmmp-2015-0027 Tatra Mt. Math. Publ. 63 2015), 153 161 ON A THREE TERM LINEAR DIFFERENCE EQUATION WITH COMPLEX COEFFICIENTS Jiří Jánsý ABSTRACT. The paper discusses
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationA GENERALIZATION OF CONTRACTION PRINCIPLE IN QUASI-METRIC SPACES
Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 9 Issue 1(2017), Pages 92-108. A GENERALIZATION OF CONTRACTION PRINCIPLE IN QUASI-METRIC SPACES HAMZA
More informationCONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS
CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas
More informationh(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote
Real Variables, Fall 4 Problem set 4 Solution suggestions Exercise. Let f be of bounded variation on [a, b]. Show that for each c (a, b), lim x c f(x) and lim x c f(x) exist. Prove that a monotone function
More informationMath 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2
Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is
More informationFIXED POINT AND COMMON FIXED POINT THEOREMS IN CONE BALL-METRIC SPACES. 1. Introduction and preliminaries
Asia Pacific Journal of Mathematics, Vol. 1, No. 1 (2014), 67-85 ISSN 2357-2205 FIXED POINT AND COMMON FIXED POINT THEOREMS IN CONE BALL-METRIC SPACES ANIMESH GUPTA Department of Applied Mathematics, Vidhyapeeth
More informationCOUPLED FIXED AND COINCIDENCE POINT THEOREMS FOR GENERALIZED CONTRACTIONS IN METRIC SPACES WITH A PARTIAL ORDER
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 2018 (434 450) 434 COUPLED FIXED AND COINCIDENCE POINT THEOREMS FOR GENERALIZED CONTRACTIONS IN METRIC SPACES WITH A PARTIAL ORDER K. Ravibabu Department
More informationApproximation exponents for algebraic functions in positive characteristic
ACTA ARITHMETICA LX.4 (1992) Approximation exponents for algebraic functions in positive characteristic by Bernard de Mathan (Talence) In this paper, we study rational approximations for algebraic functions
More informationPeriodic difference equations, population biology and the Cushing-Henson conjectures
Periodic difference equations, population biology and the Cushing-Henson conjectures Saber Elaydi Department of Mathematics Trinity University San Antonio, Texas 78212, USA E-mail: selaydi@trinity.edu
More informationHYBRID DHAGE S FIXED POINT THEOREM FOR ABSTRACT MEASURE INTEGRO-DIFFERENTIAL EQUATIONS
HYBRID DHAGE S FIXED POINT THEOREM FOR ABSTRACT MEASURE INTEGRO-DIFFERENTIAL EQUATIONS Dr. Sidheshw. S. Bellale Dept. of Mathematics, Dayanand Science College, Latur, Maharashtra (IND) Email: sidhesh.bellale@gmail.com
More informationON THE STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION
Bull. Korean Math. Soc. 45 (2008), No. 2, pp. 397 403 ON THE STABILITY OF THE MONOMIAL FUNCTIONAL EQUATION Yang-Hi Lee Reprinted from the Bulletin of the Korean Mathematical Society Vol. 45, No. 2, May
More informationSPACES ENDOWED WITH A GRAPH AND APPLICATIONS. Mina Dinarvand. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69, 1 (2017), 23 38 March 2017 research paper originalni nauqni rad FIXED POINT RESULTS FOR (ϕ, ψ)-contractions IN METRIC SPACES ENDOWED WITH A GRAPH AND APPLICATIONS
More information