In this paper, we discuss the existence of nonoscillatory solutions of first order nonlinear neutral difference equations of the form
We use the Knaster-Tarski xed point theorem to obtain some sucient conditions for the existence of nonoscillatory solutions of above equations. Example are given to illustrate the main results

Alagesan, K

Arul, R

Ayyappan, G

English

KHALSA PUBLICATIONS

  ISSN 2347-1921                                                           5230 | P a g e                                                O c t o b e r  1 2 ,  2 0 1 5   EXISTENCE OF NONOSCILLATORY SOLUTIONS OF FIRST ORDER NONLINEAR NEUTRAL DIFFERENCE EQUATIONS R.Arul1, K.Alagesan2 and G.Ayyappan3 1&2Department of Mathematics, Kandaswami Kandar’s College, Velur – 638 182, Namakkal Dt. Tamil Nadu, India. 3Department of Mathematics, Periyar University College of Arts and Science College, Pappireddipatti-636 905, Tamil Nadu, India. Abstract In this paper, we discuss the existence of nonoscillatory solutions of first order nonlinear neutral  difference equations of the from           ( ) ( ) ( ) 0,x n p n x n Q n G x n       .           ( ) ( , ) ( ) 0,ds cx n p n x n Q n s G x n s       and        ( ) , ( , ) ( ) 0.b ds a s cx n p n s x n s Q n s G x n s                We use the Knaster-Tarski fixed point theorem to obtain some sufficient conditions for the existence of nonoscillatory solutions of above equations. Examples are provided to illustrate the main results. 2010 AMS Subject Classification: 39A10 Keywords and Phrases: First order; Nonoscillation;  neutral difference equations.                      Council for Innovative Research Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN MATHEMATICS Vol .11, No. 5 www.cirjam.com , editorjam@gmail.com   ISSN 2347-1921                                                           5231 | P a g e                                                O c t o b e r  1 2 ,  2 0 1 5   1. INTRODUCTION  In this paper, we discuss the existence of nonoscillatory solutions of first order nonlinear neutral difference equations of the from            0( ) ( ) ( ) 0,x n p n x n Q n G x n n       N .    (1.1)           0( ) ( , ) ( ) 0,ds cx n p n x n Q n s G x n s n       N    (1.2) and         0( ) , ( , ) ( ) 0,b ds a s cx n p n s x n s Q n s G x n s n                N   (1.3) where   is the forward difference operator defined by ( ) ( 1) ( )x n x n x n     and    0 0 0 0, 1, 2,...n n n n  N  and 0n  is a nonnegative integer subject to the following conditions:  1C   is a ratio of odd positive integers;  2 , , ,C a b c    and d  are nonnegative integer with a b  and c d ;      3 ( ) , ( )C p n Q n  and  ( , )Q n s  are nonnegative real sequences;  4 ( )C G x  is a positive continuous real valued function with  ( ) 0xG x  for 0x  . Let  max ,   . By a solution of equations (1.1)-(1.3), we mean a real sequence  ( )x n defined and satisfying equations (1.1)-(1.3) for all 0n n   . Such a solution is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise. In recent years, there has been much research concerning the oscillation of first order neutral delay difference equations, see for example [1-4, 9, 11-13, 16] and the references cited therein. In [2, 5 ,7, 8, 10, 14, 15], the authors investigated the existence of nonoscilatory solutions of first order difference equations. Following this trend, we obtain some new sufficient conditions for the existence of nonoscillatory solutions of equations (1.1)-(1.3). In Section 2, we establish some sufficient conditions for the existence of nonoscillatory  solutions of equations (1.1)-(1.3). In Section 3, we present some examples to illustrate the main results. The results established in this paper are discrete analogue of that in [6]. 2. Nonoscillation Theorems  In this section, we present some sufficient conditions for the existence of bounded nonoscillatory solutions of equations (1.1)-(1.3). We begin with the following lemma. Lemma 2.1. (Knaster-Tarski Fixed Point Theorem)   Let B  be a partially ordered Banach space with ordering  . Let M  be a subset of B with the following properties: the infimum of M  belongs to M  and every nonempty subset ofM  has a supremum which belongs to M . Let :T M M  be an increasing mapping, that is, x y   implies Tx Ty  .Then T  has a fixed point inM .  The proof of Lemma 2.1 can be found in [3]. Theorem 2.1. Assume that  0 ( ) 1,p n p G    is nondecreasing and      0( ) ,n nQ n           (2.1) then equation (1.1) has a bounded nonoscillatory solution. Proof: Let B  be the set of all bounded real valued sequence with the supremum norm,   ISSN 2347-1921                                                           5232 | P a g e                                                O c t o b e r  1 2 ,  2 0 1 5       sup | | .nnx Bx x     Then clearly B  is a Banach space. We can define a partial ordering as follows: for given 1 2 1 2, ,x x B x x   means that 1 2( ) ( )x n x n for 0 0n n N . Define      1 2 0: ( ) , ,S x B C x n C n n       where 1C  and 2C  are positive constants such that    1 2(1 ) .C p C    If  1 1 0( ) , ,x n C n n   then 1x S  and 1 infx S . In addition, if *S S   , then      1 2 0* : ( ) , , , .S x B x n C C n n           Let  2 0 1 2 0( ) sup : , .x n C C n n        Then 2x S   and  2 sup *x S . From the condition (2.1) there exists 1 0n n  with     1 0 max{ , }n n            (2.2) sufficiently large that    212[(1 ) ]( ) , .( )s np CQ s n nG C          (2.3) For x S , we define    1/11 0 1( ) ( ) ( ) ( ( )) ,( )( )( )( ), .s np n x n Q s G x s n nTx nTx n n n n               For 1n n  and x S ,  by making use of (2.3), we obtain     1/2 21/22 221/2 22( )( ) ( ) ( )[(1 ) ]( )( )[(1 ) ],s nTx n pC G C Q sp CpC G CG CpC p CC               and   1( )( ) .Tx n C    HenceTx S  for every x S . Let 1 2,x x S  with 1 2x x  . Since G  is nondecreasing, 1 2Tx Tx , that is, T  is an increasing mapping. Then by the Knaster-Tarski fixed point theorem, there exists a positive x S  such that Tx x . Thus  ( )x n  is a bounded nonoscilatory solution of equation (1.1), which completes the proof. Theorem 2.2. Assume that  01 ( ) ,p p n p G     is nondecreasing and  (2.1) holds, then equation (1.1) has a bounded nonoscillatory solution.   ISSN 2347-1921                                                           5233 | P a g e                                                O c t o b e r  1 2 ,  2 0 1 5   Proof: Let B  be a Banach space as defined in Theorem 2.1. We can define a partial ordering as follows: for given 1 2 1 2, ,x x B x x   means that 1 2( ) ( )x n x n for 0 0n n N . Define      3 4 0: ( ) , ,S x B C x n C n n       where 3C  and 4C  are positive constants such that    0 3 4( 1) (1 ) .p C p C     If  1 3 0( ) , ,x n C n n   then 1x S  and 1 infx S . In addition, if *S S   , then      3 4 0* : ( ) , , , .S x B x n C C n n           Let  2 0 3 4 0( ) sup : , .x n C C n n        Then 2x S   and  2 sup *x S . From the condition (2.1) there exists 1 0n n  with     1 0n n            (2.4) sufficiently large that    0 313[( 1) ]( ) , .( )s np CQ s n nG C         (2.5) For x S , we define    1/11 0 11( ) ( ) ( ( )) ,( )( ) ( )( )( ), .s nx n Q s G x s n nTx n p nTx n n n n                    For 1n n  and x S ,  by making use of (2.5), we obtain         1/4 4 4 4 41 1 1( )( ) (1 ) ,Tx n C C C p C Cp p p            and     1/3 31/0 33 331/3 0 33 0 3 331( )( ) ( ) ( )( )[( 1) ]1( )( ) ( )1[( 1) ]( )1( ),s nTx n C G C Q sp np CC G Cp n G CC p Cp nC p C Cp nC                                     ThusTx S  for every x S . Let 1 2,x x S  with 1 2x x  . Since G  is nondecreasing, 1 2Tx Tx , that is, T  is an increasing mapping. Then by the Knaster-Tarski fixed point theorem, there exists a positive x S  such that Tx x . Thus  ( )x n  is a bounded nonoscilatory solution of equation (1.1), which completes the proof.   ISSN 2347-1921                                                           5234 | P a g e                                                O c t o b e r  1 2 ,  2 0 1 5   Theorem 2.3. Assume that  0 ( ) 1,p n p G    is nondecreasing and      0( , ) ,dn n s cQ n s           (2.6) then equation (1.2) has a bounded nonoscillatory solution. Proof: Let B  be a Banach space as defined in Theorem 2.1. We can define a partial ordering as follows: for given 1 2 1 2, ,x x B x x   means that 1 2( ) ( )x n x n for 0 0n n N . Define      5 6 0: ( ) , ,S x B C x n C n n       where 5C  and 6C  are positive constants such that    5 6( 1) .C p C    If  1 5 0( ) , ,x n C n n   then 1x S  and 1 infx S . In addition, if *S S   , then      5 6 0* : ( ) , , , .S x B x n C C n n           Let  2 0 5 6 0( ) sup : , .x n C C n n        Then 2x S   and  2 sup *x S . From the condition (2.6) there exists 1 0n n  with     1 0 max{ , }n n d          sufficiently large that    616[(1 ) ]( , ) , .( )ds n i cp CQ s i n nG C           For x S , we define    1/11 0 1( ) ( ) ( , ) ( ( )) ,( )( )( )( ), .ds n i cp n x n Q s i G x s i n nTx nTx n n n n               The remaining part of the proof is similar to that of Theorem 2.1, and hence the details are omitted. Theorem 2.4. Assume that  01 ( ) ,p p n p G     is nondecreasing and (2.6) holds, then equation (1.2) has a bounded nonoscillatory solution. Proof: Let B  be a Banach space as defined in Theorem 2.1. We can define a partial ordering as follows: for given 1 2 1 2, ,x x B x x   means that 1 2( ) ( )x n x n for 0 0n n N . Define      7 8 0: ( ) , ,S x B C x n C n n       where 7C  and 8C  are positive constants such that    0 7 8( 1) ( 1) .p C p C     If  1 7 0( ) , ,x n C n n   then 1x S  and 1 infx S . In addition, if *S S   , then      7 8 0* : ( ) , , , .S x B x n C C n n           Let  2 0 7 8 0( ) sup : , .x n C C n n        Then 2x S   and  2 sup *x S . From the condition (2.6) there exists 1 0n n  with    ISSN 2347-1921                                                           5235 | P a g e                                                O c t o b e r  1 2 ,  2 0 1 5      1 0n n d            sufficiently large that    0 717[( 1) ]( , ) , .( )ds n i cp CQ s i n nG C             For x S , we define    1/11 0 11( ) ( , ) ( ( )) ,( )( ) ( )( )( ), .ds n i cx n Q s i G x s i n nTx n p nTx n n n n                      The remaining part of the proof is similar to that of Theorem 2.2, and hence the details are omitted. Theorem 2.5. Assume that  0 ( , ) 1,bs ap n s p G    is nondecreasing and (2.6) holds, then equation (1.3) has a bounded nonoscillatory solution. Proof: Let B  be a Banach space as defined in Theorem 2.1. We can define a partial ordering as follows: for given 1 2 1 2, ,x x B x x   means that 1 2( ) ( )x n x n for 0 0n n N . Define      9 10 0: ( ) , ,S x B C x n C n n       where 9C  and 10C  are positive constants such that    9 10(1 ) .C p C    If  1 9 0( ) , ,x n C n n   then 1x S  and 1 infx S . In addition, if *S S   , then      9 10 0* : ( ) , , , .S x B x n C C n n           Let  2 0 9 10 0( ) sup : , .x n C C n n        Then 2x S   and  2 sup *x S . From the condition (2.6) there exists 1 0n n  with     1 0 max{ , }n n b d          sufficiently large that    10110[(1 ) ]( , ) , .( )ds n i cp CQ s i n nG C           For x S , we define    1/11 0 1( , ) ( ) ( , ) ( ( )) ,( )( )( )( ), .b ds a s n i cp n s x n s Q s i G x s i n nTx nTx n n n n                The remaining part of the proof is similar to that of Theorem 2.1, and hence the details are omitted.  3. Examples In this section, we present some examples to illustrate the main results. Example 3.1. Consider the difference equation    ISSN 2347-1921                                                           5236 | P a g e                                                O c t o b e r  1 2 ,  2 0 1 5     32 81 7( ) ( 1) ( 2) 0, 0.4 2 nx n x n x n n                  (3.1) Here 2 81 7( ) , ( ) , 34 2 np n Q n    ,  and 1, 2   . By taking ( )G x x , we see that  1( )nQ n  . Further it is easy to verify that all other conditions of Theorem 2.1 are satisfied. Therefore the equation (3.1) has a bounded nonoscillatory solution. In fact,   12nx n     is one such solution of equation (3.1). Example 3.2. Consider the difference equation    3 211 1( ) ( 3) ( ) 0, 0.2 sx n x n x n s nn s                   (3.2) Here 1 1( ) , ( , ) , 32p n Q n sn s  ,  and 1, 2c d  . By taking ( )G x x , we see that all other conditions of Theorem 2.3 are satisfied and hence every solution of equation (3.2) has a bounded nonoscillatory.  Example 3.3. Consider the difference equation    2 321 21 1( ) ( ) ( ) 0, 0.2( 1) ( )s sx n x n s x n s nn s n s                 (3.3) Here 21 1( ) , ( , ) , 1, 1, 2,2( 1) ( )p n Q n s a bn s n s      ,  and 2, 3c d  . By taking ( )G x x , we see that all other conditions of Theorem 2.5 are satisfied and hence every solution of equation (3.3) has a bounded nonoscillatory.   References [1] R.P. Agarwal, Difference Equations and Inequalities, Second Edition, Marcel Dekker, New York, 2000. [2] R.P. Agarwal, M.Bohner, S.R.Grace and D.O’Regan, Discrete Oscillation Theory, Hindawi Publ. Corp., New York, 2005.  [3] R.P. Agarwal,  S.R.Grace and D.O’Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic, 2000.  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[10] S.Lourdu Marian, M.Paul Loganathan and A.George Maria Selvam, Existence of nonoscillatory solutions of first order nonlinear neutral delay difference equations, American J. of Math. Sci. and Appl., 1(1)(2013), 21-26. [11] O.Ocalan and O.Duman, Oscillation analysis of neutral difference equations with delays, Chaos Solitons and Fractals, 39(2009), 261-270. [12] N.Parhi, Behavior of solutions of delay difference equations, Indian J. Pure Appl. Math., 33(1)(2002), 31-43.   ISSN 2347-1921                                                           5237 | P a g e                                                O c t o b e r  1 2 ,  2 0 1 5   [13] E.Thandapani, R.Arul and P.S.Raja, Oscillation of first order neutral delay difference equations, Appl. Math. E-Notes, 3(2003), 88-94. [14] E.Thandapani, R.Arul and P.S.Raja, The asymptotic behavior of nonoscillatory solutions of nonlinear neutral type difference equations, Math. Comput. Model., 39(13)(2004), 1457-1465. [15] E.Thandapani, J.R.Graef and P.W.Spikes, On existence of positive solutions and oscillations of  neutral difference equations of odd order, J.Diff. Equ. Appl., 2(1996), 175-183. [16] G.Wei, Asymptotic behavior results for nonlinear neutral delay difference equations, Appl. Math. Comput., 217(2011), 7184-7190.  

Existence of Nonoscillatory Solutions of First Order Nonlinear Neutral Dierence Equations

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Existence of Nonoscillatory Solutions of First Order Nonlinear Neutral Dierence Equations

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