advertisement

International Journal of Civil Engineering and Technology (IJCIET) Volume 10, Issue 04, April 2019, pp. 54-61. Article ID: IJCIET_10_04_007 Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=04 ISSN Print: 0976-6308 and ISSN Online: 0976-6316 © IAEME Publication Scopus Indexed SOME PROPERTIES ON THE GLOBAL BEHAVIOUR OF FIRST ORDER NEUTRAL DELAY DIFFERENCE EQUATION WITH POSITIVE AND NEGATIVE COEFFICIENT IN THE NEUTRAL TERM G. Gomathi Jawahar Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore, India ABSTRACT In this paper some criteria for oscillatory behavior of first order Neutral Delay Difference equation with the positive coefficient and the negative coefficient in the neutral term is obtained, where k,l>0, {pn}, {qn} are positive sequences. Keywords: Neutral, Delay, Difference Equations, Oscillatory, Behavior, Difference Inequality,Positive Coefficient, Negative Coefficient Cite this Article: G. Gomathi Jawahar, Some Properties on the Global Behaviour of First Order Neutral Delay Difference Equation with Positive and Negative Coefficient in the Neutral Term, International Journal of Civil Engineering and Technology, 10(4), 2019, pp. 54-61. http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=04 1. INTRODUCTION First order Neutral Delay Difference Equation is gaining interest because they are the discrete analogue of differential Equations. In recent years, several papers on oscillation of solutions of Neutral Delay Difference Equations have appeared. [3] S.S. Cheng and Y.Z. Lin (Lin, 1998) have provided a complete characterization of oscillation solutions of first order Neutral Delay Difference Equations with positive coefficient in the Neutral term. [6] John R.Graef, R.Savithri, E.Thandapani (John R. Graef, 2004) have analyzed the non-oscillatory solutions of first order Neutral Delay Differential Equations with positive coefficient in the Neutral term. [4]Elmetwally M.Elabbasy, Taher S.Hassan, Samir H.Saker (Elmetwally M. Elabbasy, 2005) have provided oscillatory solutions of first order Neutral Delay Differential Equations with negative coefficient in the Neutral term. http://www.iaeme.com/IJCIET/index.asp 54 [email protected] G. Gomathi Jawahar [8]Ozkan Ocalan extensively discuss the problem of Oscillation of neutral differential equation with positive and negative coefficients, J.Math.Anal.Appl. 33(1), 2007, 644-654. [9]Tanaka, S. (2002) discussed the various Solutions of Oscillation First order Neutral Delay Differential Equations. Here some oscillation results in difference equations based on the existence results of differential equations are provided. Examples are provided to illustrate the results. Section I In this section some criteria for oscillatory behavior of first order Neutral Delay Difference Equation, ( xn + pn xn−k ) + qn f ( xn−l ) = 0 (1.1) Is obtained where k,l > 0 , {pn}, {qn} are positive sequences. The following assumptions has been made to prove the results: H1: f (u) is an increasing function and u f (u) > 0 H2: There exists a function w such that w (u) > 0 for u > 0 and w (u) f (v) > f(u v). H3: φ (u) is an increasing function such that u φ(u) > 0 and |φ(u+v)|<|f(u)+f(v)| 2. EXISTENCE OF OSCILLATORY SOLUTIONS Theorem 1.1 Every solution of the equation (1.1) is oscillatory if there exists a constant λn such that 0 < λn < 1 for some n ≥ n0 and the difference inequality Δzn+Qn φ(zn-l+k) ≤ 0 Satisfies where ( (1.2) ) 1− q n−k n−l Q = min q , n n n wp n−l z = n and ( m Qn yn − l n +k 0 ) Proof Suppose to the contrary that there is a non-oscillatory solution {xn} of (1.1). Suppose that xn > 0 for all n ≥ n0. Let yn= xn + pn xn-k. Then by the equation (1.1), Δyn = -qnf(xn-l) < 0 yn+1 < yn, yn is a decreasing function, yn+1-yn = -qn f (xn-l) yn+1+ qn f (xn-l) = yn yn > qn f (xn-l) for some n ≥ n0, taking summation from n0 to m, m > n0 ( m m y n q n f xn − l n=n n=n 0 0 ) ( ( ( )( ) ( m n q n f x n − l + 1 − n q n f x n − l n=n 0 ) ( ) ( m m 1− q f x Qn f xn − l + n−k n−k n−k −l n=n n=n +k 0 0 http://www.iaeme.com/IJCIET/index.asp 55 )) ) [email protected] Some Properties on the Global Behaviour of First Order Neutral Delay Difference Equation with Positive and Negative Coefficient in the Neutral Term ( ) ( )( m m Q wp f x Qn f xn − l + n n−l n−k −l n=n n=n +k 0 0 ( ) ( m m Q f p x Qn f xn − l + n n−l n−k −l n=n n=n +k 0 0 ( ( m Q f x +f p x n n−l n−l n−k −l n=n +k 0 ( m Qn xn − l + pn − l xn − k − l n +k 0 = m Qn yn − l n +k 0 ) ) )) ) m Qn yn − l 0 n +k 0 Let z = n ( m Q y n n−l n=n +k 0 ) Also, Δzn = zn+1 - zn = ( m Qn + 1 yn + 1 − l − Qn yn − l n +k 0 ) +Q =Q y +Q y y + ....... + n + k + 1 n + k + 1 − l n + k + 2 n + k + 2 − l n + k + 3 n + k − l 0 0 0 0 0 0 ( ) −Q Q y −Q y y m +1 m +1− l n + k n + k +1 n + k + 1 n + k + 1 − l 0 0 0 0 −Q y + .... − Q y n + k + 2 n + k − l + 2 m m−l 0 0 =Q y −Q y m +1 m +1− l n + k n + k − l 0 0 z − Q y n n +k n + k −l 0 0 > -Qn φ (yn-l), n≥ n0+k > -Qn φ (zn-l+k) Δ zn + Qn φ (zn-l+k) > 0 (1.3) Condition (1.3) holds when zn is eventually positive solution. Contradiction to the equation (1.2). Since http://www.iaeme.com/IJCIET/index.asp 56 [email protected] G. Gomathi Jawahar z = n ( m Qn yn − l n +k 0 ) and y =x + p x n n n n−k , Hence we say that {xn }is an oscillatory solution of the equation (1.1). Theorem 1.2 Δyn + qn f Every solution of the equation (1.1) is oscillatory, if it satisfies the condition (1 − n ) P }, and (xn-l ) ≤ 0, where yn = xn + pn f(xn-k) and Qn = min{ λn pn , if there exists n a function such that (u ) > 0, for u > 0 and f(u)=u, f (uv ) n −l ≤ (u ) f (v ) and f (uv) ≥ uv, λ n is positive constant. Proof Suppose {xn} is a non-oscillatory solution. Let xn > 0 and let us assume {xn} is eventually positive. Let yn = xn + pn f(xn-k) yn = xn + pn f(xn-k) > pn f(xn-k) Hence yn > pn f(xn-k) = λn pn f(xn-k) + (1-λn ) pn f(xn-k) > Qn f(xn-k) + Qn f(xn-k) n-l+k > Qn f(xn-k) + Qn f(xn-k xn-l+k) > Qn f(xn-k) + xn-k xn-l+k > 0 Since yn > 0, Δyn > 0 i.e, Δyn + qn f(xn-l) > 0, yn > 0 This contradicts our assumption. Hence {xn} is an oscillatory solution. Theorem 1.3 If lim ∑qn > 0, and {xn}is an eventually positive solution of n → Δ(xn + pn xn-k) + qn f(xn-l) = 0, then Δzn + Mqn pn 2 (1.4) ( p n − 1) zn+k-l ≤ 0 Where zn = xn+ pn xn-k http://www.iaeme.com/IJCIET/index.asp 57 (1.5) [email protected] Some Properties on the Global Behaviour of First Order Neutral Delay Difference Equation with Positive and Negative Coefficient in the Neutral Term Proof: Equation (1.4) becomes, Δzn + qn f(xn-l) = 0 Δzn = -qn f(xn-l) < 0 zn+1-zn < 0 zn+1 < zn Hence zn is decreasing. From the equation (1.5) , pn xn-k = zn - xn (1.6) zn+k = xn+k + pn xn (1.7) From the equation (1.5), Since zn is decreasing zn > zn+k ≥ pn xn From the equation(1.6), pn2 xn-k= pn zn - pn xn (multiplying by pn) (1.8) pn xn = zn+k - xn+k p x z n n n+k - pn xn ≥ -zn+k Substituting inthe equation (1.8) , pn2 xn-k = pn zn - pn xn pn2 xn-k ≥ pn zn – zn+k (1.9) pn2 xn-k ≥ pn zn+k - zn+k xn-k ≥ xn-l ≥ pn − 1 pn 2 pn − 1 pn 2 zn+k (2.0) zn+k-l (2.1) Δzn + qn f(xn-l) = 0 = > Δzn = -qn f(xn-l) Δzn ≤ -qn M http://www.iaeme.com/IJCIET/index.asp pn − 1 pn 58 2 zn+k-l [email protected] G. Gomathi Jawahar Δzn + M q n ( p n − 1) pn zn+k-l ≤ 0 2 (2.2) Hence the theorem. Section II In this section some criteria for oscillatory behavior of first order Neutral Delay Difference Equation, ( ) x −q x +p x =0 n n n−k n n−l (2.3) is obtained where k,l > 0 and {pn} and {qn} are positive sequences. 3. SOME OSCILLATORY RESULTS Theorem 2.1 Every solution of the equation (2.3) is oscillatory if k, l, p n , q n > 0 Proof Case 1 Suppose { x n }is a non oscillatory solution of equation (2.3) , let x n >0 and { xn } is eventually positive, xn −1 < x n ( ) x −q x =− p x n n n−k n n−l ( = −p z +q x n n−l n−l n−k −l z + p z +p q x =0 Hence n n n − l n n − l n − k − l z n +1 z ( n +1 ( ) −z + p z +q x =0 n n n−l n−l n−k −l p p ) ( ) +p z +q x =z n n−l n−l n−k −l n ( ) z +q x 0 n n−l n−l n−k −l ) x −q x +q x 0 n n−l n−l n−k −l n−l n−k −l p x 0 n n−l Which is a contradiction. Since p n is positive and x n −l is eventually positive. Hence equation (2.3) has a oscillatory solution. Case 2 Let ( us assume ) xn < 0 and { x n }is eventually negative, then x n −1 x x −q x =− p x n n n−k n n−l http://www.iaeme.com/IJCIET/index.asp 59 [email protected] n Some Properties on the Global Behaviour of First Order Neutral Delay Difference Equation with Positive and Negative Coefficient in the Neutral Term = ( −p z +q x n n−l n−l n−k −l ) z + p z +p q x =0 n n n−l n n−l n−k −l z n +1 −z + p z +p q x =0 n n n−l n n−l n−k −l p p ( ( ) z +q x 0 n n−l n−l n−k −l ) x −q x +q x 0 n n−l n−l n−k −l n−l n−k −l p x 0 n n−l Which is a contradiction, since p n is positive and x n −l is eventually negative solution. Hence equation (2.3) has an oscillatory solution. 4. Some Non oscillatory results Theorem 2.2 ( ) Let {xn} be an eventually positive solution of x − q x +p x =0 , n n n−k n n−l xn > 0 and let z n = x n -q n x n− k (2.4) Assume k,l > 0, p n ,q n > 0. Then z n is eventually non increasing positive function. Proof From the equation (2.3), Δz n = -p n x n− l < 0 z n +1 - z n < 0 z n +1 < z n Hence z n is eventually non increasing positive function. Theorem 2.3 Every non oscillatory solution of the equation (2.3) converges to zero monotonically for large n as n→∞ if k, l > 0, q n > 0, p n > 0. Proof Suppose {x n }is a non-oscillatory solution of (2.3). Let us assume {x n } is eventually positive (if x n is eventually negative, the proof is similar). Hence from the theorem 2.2, z n is eventually non increasing positive function. From equation (2.4), http://www.iaeme.com/IJCIET/index.asp 60 [email protected] G. Gomathi Jawahar xn ≥ zn Here zn = α ≥ 0 lim x n ≥ lim n → n → If α > 0, then from the equation (2.3), pn xn−l < 0 z n →-∞, which contradicts that z n is positive function. Δz n = - Hence lim n → Hence we take α = 0, therefore lim x n = 0. n→ Hence every non oscillatory solution of the equation (2.3) converges to zero monotonically as n→∞. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] R.P. Agarwal, S.R Grace and D.O’ Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Drdrechet, 2000. D.D Bainov and D.P Mishev, Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, New York, 1991. S. S. Cheng and Y. Z. Lin(1998), Complete Characterization of an oscillatory Neutral Difference Equations. J. Math. Anal. Appl. 221 , 73-91. Elmetwally M. Elabbasy, T. S. (2005),Oscillation criteria for first-order Nonlinear Neutral Delay Differential Equations, Electric Journal of Qualitative theory of differential Equation, vol. 2005, No. 134 , 1-18. I.Gyori and G.Ladas, Oscillation Theory of Delay Differential Equations With Applications, Clarendon Press, Oxford,1991. John R Graef, R. Savithiri , E. Thandapani, (John R Graef, 2004) Oscillation of First order Neutral Delay Differential Equations, Electronic Journal of Qualitative theory of differential Equation, Proc. 7th coll. QTDE, No. 12 , 1-11. W.T.Li, Classification and existence of nonoscillatory solutions of second order nonlinear neutral differential equations, Ann.Polon Math.LXV.3(1997),283-302. Ozkan Ocalan, Oscillation of neutral differential equation with positive and negative coefficients, J.Math.Anal.Appl. 33(1), 2007, 644-654. Tanaka, S. (2002),Oscillation of Solutions of First order Neutral Delay Differential Equations, Hiloshima Mathematical Journal(32) , 79-85. http://www.iaeme.com/IJCIET/index.asp 61 [email protected]