Applied Mathematics
Vol.07 No.14(2016), Article ID:70159,20 pages
10.4236/am.2016.714139
On the Asymptotic Behavior of Second Order Quasilinear Difference Equations
Vadivel Sadhasivam1, Pon Sundar2, Annamalai Santhi1
1PG and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, Namakkal, India
2Om Muruga College of Arts and Science, Salem, India
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 15 June 2016; accepted 26 August 2016; published 29 August 2016
ABSTRACT
In this paper, we investigate the asymptotic behavior of the following quasilinear difference equations
(E)
where
,
. We classified the solutions into six types by means of their asymptotic behavior. We establish the necessary and/or sufficient conditions for such equations to possess a solution of each of these six types.
Keywords:
Asymptotic Behavior, Positive Solutions, Homogeneous, Quasilinear Difference Equations
1. Introduction
Recently, the asymptotic properties of the solutions of second order differential equations [1] [2] difference equations of the type (E) and/or related equations have been investigated by many authors, for example see, [3] - [19] and the references cited there in. Following this trend, we investigate the existence of these six types of solutions of the Equation (E) showing the necessary and/or sufficient conditions can be obtained for the existence of those solutions. For the general backward on difference equations, the reader is referred to the monographs [20] - [24] .
In 1996, PJY Wang and R.P. Agarwal [25] considered the quasilinear equation
(1)
and obtained oscillation criteria for the Equation (1).
In 1996, E. Thandapani, M.M.S. Manuel and R.P. Agarwal [26] have studied the quasi-linear difference equation
(2)
In 2000, Pon Sundaram and E. Thandapani [27] considered the following quasi-linear functional difference equation
(3)
and they have established necessary and sufficient conditions for the solutions of Equation (3) to have various types of nonoscillatory solutions. Further they have established some new oscillation conditions for the oscillation of solutions of Equation (3).
In 1997, E. Thandapani and R. Arul [28] studied, the following quasi-linear equation
(4)
They established necessary and sufficient conditions for the solutions of (4) to have various type of nono- scillatory solutions.
In 2004, E. Thandapani et al. [29] studied the equation
(5)
and established conditions for the existence of non-oscillatory solutions.
S.S. Cheng and W.T. Patula [30] studied the difference equation
(6)
where and proved an existence theorem for Equation (6).
In 2002, M. Mizukanmi et al. [1] discussed the asymptotic behavior of the following equation
(7)
Discrete models are more suitable for understanding the problems in Economics, genetics, population dynamics etc. In the qualitative theory of difference equations asymptotic behavior of solutions plays a vital role. Motivated by this, we consider the discrete analogue of (7) of the form
(8)
where,
and
is the forward difference operator defined by
We assume the following conditions on Equation (8)
1) and
are positive constants
2) is a real sequence such that
for all
.
For simplicity, we often employ the notation
interms of which Equation (8) can be expressed in
By a solution of Equation (8), we mean a real sequence, together with
exists and satisfies Equation (8) for all
.
We here call Equation (8) super-homogeneous or sub-homogeneous according as α < β or α > β If α = β Equation (8) is often called half-linear. Our attention is mainly paid to the super-homogeneous and sub-homo- geneous cases, and the half-linear is almost excluded from our consideration.
2. The Classification of All Solutions of Equation (8)
To classify all solutions of Equation (8), we need the following lemma.
Lemma 1. Let be a local solutions of Equation (8) near
and
, be its right
maximal interval of existence. Then we have either near w or
near w. That is
does
not charge strictly its sign infinitely many times as.
The classification of all (local) solutions of Equation (8) are given on the basis of Lemma 1. Since the proof is easy, we leave it to the reader.
Proposition 1. Each local solution of Equation (8) falls into exactly one of the following six types.
1) Singular solution of the first kind: type there exist a
such that
2) Decaying solution: type (D), can be continued to ¥, and satisfies
for all large n, and
3) Asymptotically constant solution: type (AC) can be continued to ¥, and satisfies
for all large n and
4) Asymptotically linear solution: type (AL) can be continued to ¥ and satisfies
for all large n and
5) Asymptotically super-linear solution: type (AS) can be continued to ¥ and satisfies
for all large n and
6) Singular solution of second kind: type
has the finite escape time; that is, there exists a
such that
3. Main Results for the Super-Homogeneous Equations
Before we list our main results for the case. Throughout this section we assume that
Theorem 2. Equation (8) has no solution of type.
Theorem 3. Equation (8) has a solution of type (D) if and only if
(9)
Theorem 4. Equation (8) has a solution of type (AC) if and only if
(10)
Theorem 5. Equation (8) has a solution of type (AL) if and only if
(11)
Theorem 6. Equation (8) has a solution of type (AS) if (11) holds.
Theorem 7. Equation (8) does not have solutions of type (AS) if there are constants and
satisfying
(12)
and
(13)
Remark 1. The set of all pairs satisfying inequalities (13) is not empty. In fact, the pair
belongs to it.
Theorem 8. Equation (8) has a solutions of type.
Remark 2. Theorem 7 has the same conclusion that these are not solutions of type (AS). However, Theorem 7 is still valid for the case that p is nonnegative. For example, it is formed by this extended version of Theorem 7 that the equation
does not have solutions of type (AS).
Example 1 Let, consider the Equation (8) with
(14)
For this equation, we have the following results:
1) Equation (14) has a solution of type (D) if and only if (Theorem 3).
2) Equation (14) has a solution of type (AC) if and only if (Theorem 4).
3) Equation (14) has a solution of type (AL) if and only if (Theorem 5).
4) Equation (14) has a solution of type (AS) if and only if (Theorem 6).
4. Main Results for the Sub-Homogeneous Equation
Below we list our main results for the case. Throughout this section we assume that
.
Theorem 9. Equation (8) has a solutions of type.
Theorem 10. Equation (8) has a solution of type (D) if
(15)
Theorem 11. Equation (8) does not have solutions of type (D) if
(16)
Theorem 12. Equation (8) does not have solutions of type (D) if there are constants and
satisfying
(17)
and
(18)
Remark 3. The set of all pairs satisfying inequalities (18) is not empty. In fact, the pair
belongs to it.
Theorem 13. Equation (8) has a solution of type (AC) if and only if (15) holds.
Theorem 14. Equation (8) has a solution of type (AL) if and only if
Theorem 15. Equation (8) has a solution of type (AS) if and only if
(19)
Theorem 16. Equation (8) has no solutions of type.
Example 2. Let and consider the Equation (14) again.
We have the following results:
1) Equation (14) has a solution of type (D) if and only if (Theorem 10 and 11).
2) Equation (14) has a solution of type (AC) if and only if (Theorem 14).
3) Equation (14) has a solution of type (AL) if and only if (Theorem 15).
4) Equation (14) has a solution of type (AS) if and only if (Theorem 16).
5. Auxillary Lemma
In this section, we collect axillary lemmas, which are mainly concerned with local solution of Equation (8). A comparison lemma of the following type is useful, and will be used in many places.
Lemma 2. Suppose that are such that
for
. Let
and
be solutions of the equations
respectively. If and
, then
and
for a < n ≤ b.
Proof. We have
(20)
(21)
By the hypotheses we have in some right neighborhood of a. If
for some point in a < n ≤ b, we can find a c such that a < c ≤ b satisfying
for a < n < c and
. But, this yields a contradiction, because
Hence we see that for
. Returning to (20), we find that
for
. The proof is complete.
The uniqueness of local solutions with non-zero initial data can be easily proved. That is, for given,
and
, Equation (8) has a unique local solution
satisfying
,
provided that
. The uniqueness of the trivial solution can be concluded for the case
.
Lemma 3. Let α ≤ β and. If
is a local solution of Equation (1) satisfying
then
for
.
Proof. Assume the contrary. We may suppose that for
. Then, we can find
such that
satisfying
and
for
. Summing (8), we obtain
We therefore have
(22)
(23)
Put. We see that
for
and w is nondecreas- ing. From (22) and (23), we can get
Let. Then from this observation we see that
where
Consequently, we have
(24)
If α = β, from (24), we have,
. This is a contradiction because
. If α < β, from
(24) we have,
. This is also a contraction because
.
The proof is complete.
Lemma 4. Let. Then all local solutions of Equation (8) can be continued to ¥ and
, that is, all solutions of Equation (8) exist on the whole interval
.
Proof. Let be a local solution of Equation (8) is a neighborhood of
. Suppose the contrary that the right maximal interval of existence of
is of the form
,
. Then, it is easily seen that
. Summing (8) twice, we have
where and
. Accordingly,
Put. Then,
Put moreover. Then, as in the proof of Lemma 3, we have
(25)
where. Since
, there is a
such that
such that
for
. Therefore it follows from (25) that
(26)
Let. Then, using discrete Gronwall’s inequality, we see that
, which is a contradiction.
Next let. Then (26) implies that
Since, we have
. This is a contradiction too. Hence
can be continued to ¥. The continuability to the left end point
is verified in a similar way. The proof is complete.
The following lemma establishes more than is stated in Theorem 8. Accordingly the proof of Theorem 8 will be omitted.
Lemma 5. Let and
and
be given. Then there exists an
such that the right maximal interval of existence of each solution
of Equation (1) satisfying
and
is a finite interval
,
, and
.
Proof. Let be fixed, and put
. There is an
satisfying
We first claim that the solution of Equation (8) with the initial condition,
does not exist on
; that is
blow up at some
. To see this suppose the contrary that
exists at least
. By the definition of m, we have
Summing the inequality form N to yields
and hence
Finally, summing the above inequality both sides from N to, we obtain
which is a contradiction to the choice of M. Hence must blow up at some
,
.
If and
, then Lemma 2 implies that
on the common interval of existence of y and z and therefore
blows up at some point before
. The proof is complete.
6. Nonnegative Nonincreasing Solutions
The main objective of this section is to prove the following theorem.
Theorem 17. For each, the problem
has exactly one solution such that
is defined for
and satisfies
(27)
Furthermore, if is a solution for
of Equation (1) satisfying
and
then
Remark 4.
1) In the case, employing Lemma 3, we can strengthen (27) to the property that
(28)
2) In the case, all local solutions of Equation (8) can be continued to the whole interval
Hence in this case property (6.2) always holds for all solutions
with
and
[resp
].
The property of nonnegative nonincreasing solutions described in Theorem 17 will play important roles through the paper. This section is entirely derided to proving Theorem 17. To this end we prepare several lemmas.
Lemma 6. Let and t be a bounded function on
. Then, the two point boundary value problem
(29)
has a solution.
Proof. Let be a constant such that
We first claim that with each, we can associate a unique constant
satisfying
(30)
Further this satisfies
(31)
To see this let be fixed, and consider the function
If, then
. If
, then
. Since
I is a strictly increasing continuous function, there is a unique constant satisfying
, namely (30). Then (31) is clearly satisfied.
By (31), we see that there is a constant satisfying
for all
. Choose
so large that
Consider the Banach space BN of all real sequences with the supernum norm
.
Now we define the set and the mapping
by
and
respectively. Then the boundary value problem (29) is equivalent to finding a fixed element of. We show that F has a fixed element in Y (via) the Schavder fixed point theorem
Hence F maps Y into itself.
Next, to see the continuity of F, assume that be a sequence converging to
uniformly in
. We must prove that
converges to
uniformly in
. As a first step, we show that
. Assume that this is not the case. Then because of the boundedness of
,
there is a subsequence satisfying
for some finite value
. Noting the relation
We have
This contradicts the uniqueness of the number. Hence
. Then we find similarly that
uniformly on
.
It will be easily seen that the sets
are uniformly bounded on. Then
is compact.
From the above observations we see that F has a fixed element in Y. Then this fixed element is a solution of boundary value problem (29) is easily proved. The proof is now complete.
Lemma 7. Let and
. Then the two point boundary value problem
(32)
has a solution such that
and
for
.
Proof. Define the bounded function f on by
By Lemma 6, the boundary value problem
has a solution y.
We show that y satisfies for
. If this is not the case, we can find an interval
such that
on
and
. The definition of f implies that y
satisfies the equation on
. Hence
is a linear function on
.
Obviously that this is a contradiction. We see therefore that on
.
Since and
on
, by the definition of t, we find that
on. Hence
, which implies that y is a desired solution of problem (32). The proof is complete.
Proof of Theorem 17. The uniqueness of satisfying the properties mentored here is easily established as in the proof Lemma 2. Therefore we prove only the existence of such a
.
By Lemma 7, for each, we have a solution
of the boundary value problem
satisfying and
for
let us extend each
over the interval
by defining
for
. Below we show that
contains a subsequence converging to a desired solution of (8).
As a first step, we prove that
(33)
In fact, if this is not case, then for some i. Since
. Lemma 2 implies that
for
. Putting
, we have
a con-
tradiction. Accordingly (33) holds, and so exists, since
on
for any,
is uniformly bounded on each compact subinterval of
. Noting that
is nondecreasing and nonpositive on
, we have
Hence is equicontinuous on each compact subinterval of
. From these consideration we
find that there is a subsequence and a function
satisfying
uni-
formly on each compact subinterval of. Finally we shall show that
is a desired solution of Equation (8). Let
be fixed arbitrarily. For all sufficiently large
’s we have
letting, we obtain
Taking difference in this above equality, we are that solves Equation (8) on
. That
satisfies (27) is evident. The proof of Theorem 17 is complete.
7. Proofs of Main Results for the Super-Homogeneous Equations
Throughout this section, we assume that.
Proof of Theorem 2. The theorem is an immediate consequence of the uniqueness of the trivial solution (Lemma 3).
Proof of Theorem 4. Necessity Part: Let be a positive solution of Equation (8) for
of type
(AC). It is easy to see that and
as
. Hence summing (8) twice, we have
from which we find that
This is equivalent to (10).
Sufficiency Part: Let (10) hold. Fix an and choose
so that
We introduce the Banach space of all bounded, real sequences
with norm
.
Define the set and the mapping
by
We below show via the Schauder-Tychonoff fixed point theorem that F has at least one fixed element in Y. Firstly, let. Then
Thus, and hence
. Secondly, to see the continuity of F, let
be a sequence in Y covering to
uniformly on each compact subinterval of
since
is bounded for
and
The Lebesgue dominated convergence theorem implies that uniformly on each compact subinterval of
since for
,
The set is uniformly bounded on
. This implies that
is compact.
From there observations we find that F has a proved element y in Y such that. That this y is a solution of Equation (1) of type (AC) is easily proved. The proof is complete.
Proof of Theorem 3. Sufficiency Part: Let be a solution of Equation (8) satisfying
for
. The existence of such a solution is ensured by Theorem 17. Obviously,
is either of type (D) or type (AC). Theorem 4 shows that under assumption (9), Equation (8) does not posses solutions of type (AC). Hence
must be of type (D).
Necessity Part: Let be a positive solution of Equation (8) for
of type (D). Clearly
satisfies
To verify (9), suppose the contrary that (9) fails to hold. Then, nothing that is decreasing for
, we have
Accordingly,
The left hand side tends to ¥ as because of
, where as the right hand side tends to 0 as
. This contradiction verifies (9). The proof is complete.
Proof of Theorem 5. Necessity Part: Let be a positive solution of Equation (8) near ¥ of type (AL). There is a constant
and
satisfying
(34)
Summation of Equation (8) from to
yields
Since, this in equality implies that
(35)
Combining (35) with (34), we find that (11) holds.
Sufficiency Part: We fix arbitrarily, and choose
large enough so that
Let be the Banach space as in the proof Theorem 4. Define the set
as follows
The mapping defined by
As in the proof of the sufficiency part of Theorem 4, we can show that F has a fixed element by the Schavder-Tyehonoff fixed point Theorem
Taking D twice for this formula we see that is a positive solution of Equation (8) for
.
L’Hospital’s rule shows that. Thus
is a solution of Equation (8) of type (AL). The proof
is complete.
Lemma 8. Let. If (11) holds, then there is a positive solution of Equation (8) for
of type (AL) satisfying
.
Proof. By Theorem 5, there is an (AL)-type positive solution of Equation (8) defined in some neigh-
borhood of. Let
be a positive solution of Equation (8) for
satisfying and
,
for
. Take a
such that
and
for
. By Lemma 2 if
is sufficiently elver to
, then the solution
of Equation (8) with
and
exists at least on
and satisfies
Then Lemma 2 again implies that as long as
exists. Since
and
exists for
, this means that
exists for
and satisfies
,
. Then we have
Noting that is the unique solution of (8) satisfying
and passing through the point
we have
. Therefore
is of type (AL). The proof is complete.
Proof of Theorem 6. For, we denote by
, the unique solution of Equation (8) with in initial condition
and
. The maximal interval of existence of
may be finite or infinite.
Define the set by
We know by Lemma 8 that and by Lemma 5 that
for all sufficiently large
. Hence
exists. For
there are three possibilities:
1)
2) and
is of type (AS)
3) and
is of type
.
To prove the theorem, we below show that case (b) occurs. For simplicity, we write for
below.
Suppose that the case (a) occurs. Then and
. By condition
(11) we can find a satisfying
Choose close enough to
so that
exists at least on
and
. Then, for such a
,
can be extended to
, and satisfies
. In fact, if this is not the case, there is
satisfying
for
and
. It follows therefore that
for
. Summing the Equation (8) (with
) for
yields
This contradiction implies that exists for
and satisfies
. These observations show that
, which contradicts the definition of
. Hence case (a) does not occur.
Next, suppose that case (c) occurs. Let be the point such that
. By Le- mma 5, there is an
such that solution
of Equation (8) satisfying
,
must
blow up at some finite. For sufficiently small
, we have
. Then if
is sufficiently close to
, then
can be continued at least to
, and satisfies
,
. Then, even through
can be continued to N,
blows up at some finite point by the definition of M. This fact shows that such a
does not belong to S, contradicting the definition of
, again. Consequently case (b) occurs, and hence the proof of Theorem 6 is complete.
Proof of Theorem 7. The proof is done by contradiction. Let be a solution of Equation (1) of type (AS). We suppose that
exists for
and satisfies
(36)
Put
. Then
Now, we employ the Young inequality of the form
(37)
in the last inequality. It follows therefore that
where is a constant. We rewrite is inequality as
Noting (7.3) and condition (13), we obtain
where is a constant. Dividing both sides by
and summing from n to
¥, we have
because. Consequently, we have
Letting, we get a contradiction to assumption (12). This completes the proof.
As was mentioned in Section 5, the proof of Theorem 8 is omitted. In fact, a more general result is proved in Lemma 5.
8. Proofs of Main Results for the Sub-Homogeneous Equations
Throughout this section, we assume that.
Proof of Theorem 9. Let be fixed so that
and put
Then there are constants and
satisfying
and
Consider the Banach space of all real sequences
with sup norm
. Define
the subset Y of by
and
We show that the hypothesis of the Schavder fixed point theorem is satisfied for Y and F. Let. Then, obviously
for
. Moreover
Hence. The continuity of F and the boundedness of the sets FY and
can be easily established. Accordingly there is a
satisfying
. By taking difference twice, we find that
is a solution of Equation (1) for
and that
for
and
. Now, we put
It is easy to see that is a solution of equation (8) for
and is of type
. The proof is complete.
Theorems 14 and 15 can be proved easily as in the proofs of Theorems 4 and 5 respectively. We therefore omit the proofs.
Proof of Theorem 10. By our assumption we can find a positive solution of Equation (8) sat-
isfying Since
, we see by Lemma 4 that each
exists for
. We show that the
sequence has the limit function
, and it gives rise to a positive solution of Equation (8) of type (D).
We first claim that
(38)
If this is not true, then for some
and
. This means however that there are two nonnegative nonincreasing solutions of Equation (8) passing through the point
. This contradiction to
Theorem 17. We therefore have (38) and so exists observe that
satisfies
Letting, we obtain via the dominated convergence theorem
We see that is a nonnegative solution of Equation (8) satisfying
. It remains to prove that
for n ≥ n0 Fix N > n0 arbitrarily. The proof of Theorem 2 implies that there is a solution
for
and
for
. We claim that
(39)
In fact, if this fails to hold, then
By this means, as before, that there are two nonnegative nonincreasing solution fo Equation (8) passing through the point. This contradiction shows that (39) holds. Hence by letting
in (39) we have
for
. Since
is arbitrary, we see that
for
. The proof is complete.
Proof of Theorem 11. The proof is done by contradiction. Let be a positive solution of equation (8) for
of type (D). Using (16). We obtain from Equation (8)
(40)
where is a positive constant. We fix a
arbitrary and consider inequality (42) only on the interval
for a moment. A summation of (42) from n to 2N, given
From which, we have
(41)
We can find a constant satisfying
Therefore (43) implies that
from which we have
Letting, we have a contradiction. The proof is complete.
Proof of Theorem 12. The proof is done by contradiction. Let be a solution of Equation (8) of type (D). We notice first that
(42)
In fact, since, we can compute as follows
Therefore (42) holds.
We may suppose that for some and
(43)
But. Then
proceeding as in the proof of Theorem 8, we obtain
where is a constant. We obtain from (43) and assumption (18)
where is a constant. Dividing both sides by
and summing from n to ¥, we have
that is,
Letting, we get a contradiction to assumption (17) by (42). The proof is complete.
Proof of Theorem 16. Sufficiency Part: By Theorem 17 and (2) of Remark 6.2, there is a positive solution of equation (8) satisfying
. This
is either of type (AL) or of type (AS). But by Theorem 15, we see that
must be of type (AS).
Necessity Part: Let be a positive solution of Equation (8) for
of type (AS). To prove (19), we
suppose the contrary that. As in the proof of Lemma 5.3, we have
where and
let
. It follows that
where is a constant. put
. We then have
Since is of type (AS),
is unbounded for
and so is
. Accordingly, there is a
, satisfying
for
. Thus
Since, this implies the boundedness of w, which is a contraction. Hence, we must have (19). The proof is complete.
Theorem 16 is clear because of all solutions of equation (8) with exist for
[see Lemma 5].
Cite this paper
Vadivel Sadhasivam,Pon Sundar,Annamalai Santhi, (2016) On the Asymptotic Behavior of Second Order Quasilinear Difference Equations7403287. Applied Mathematics,07,1612-1631. doi: 10.4236/am.2016.714139
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