This paper is concerned with solvability of the second-order nonlinear neutral delay difference equation . Utilizing the Banach fixed point theorem and some new techniques, we show the existence of uncountably many unbounded positive solutions for the difference equation, suggest several Mann-type iterative schemes with errors, and discuss the error estimates between the unbounded positive solutions and the sequences generated by the Mann iterative schemes. Four nontrivial examples are given to illustrate the results presented in this paper.

1. Introduction and Preliminaries

Recently, the oscillation, nonoscillation, asymptotic behavior, and existence of solutions of different classes of linear and nonlinear second-order difference equations have been studied by many authors; see, for example, [1–26] and the references cited therein. Using the Banach fixed point theorem, Jinfa [5] discussed the existence of a bounded nonoscillatory solution for the second-order neutral delay difference equation with positive and negative coefficients:

(1)

under the condition p ≠ −1. Luo and Bainov [13] and M. Migda and J. Migda [16] considered the asymptotic behaviors of nonoscillatory solutions for the second-order neutral difference equation with maxima:

(2)

and the second-order neutral difference equation:

(3)

respectively. Meng and Yan [15] studied the existence of bounded nonoscillatory solutions for the second-order nonlinear nonautonomous neutral delay difference equation:

(4)

Applying the cone compression and expansion theorem in Fréchet spaces, Tian and Ge [21] established the existence of multiple positive solutions of the second-order discrete equation on the half-line:

(5)

with certain boundary value conditions. But to the best of our knowledge, results on multiplicity of unbounded solutions for neutral delay difference equations are very scarce in the literature. Nothing has been done with the existence of uncountably many unbounded positive solutions for (1)~(5) and any other second-order neutral delay difference equations:

Inspired and motivated by the results in [1–26], in this paper we introduce and study the second-order nonlinear neutral delay difference equation:

(6)

where τ, k, n₀ ∈ ℕ,

, and

(7)

By means of the Banach fixed point theorem and some new techniques, we establish sufficient conditions for the existence of uncountably many unbounded positive solutions of (6), suggest a few Mann iterative schemes with errors for approximating these unbounded positive solutions, and prove their convergence and the error estimates. The results obtained in this paper extend the result in [5]. Four nontrivial examples are interested in the text to illustrate the importance of our results.

Throughout this paper, we assume that Δ is the forward difference operator defined by Δx_n = x_n+1 − x_n, ℝ = (−∞, +∞), ℝ⁺ = [0, +∞), ℤ, ℕ₀, and ℕ denote the sets of all integers, nonnegative integers, and positive integers, respectively,

(8)

represents the Banach space of all real sequences on ℕ_β with norm

(9)

It is clear that A(N, M) is a closed and convex subset of

. By a solution of (6), we mean a sequence

with a positive integer T ≥ n₀ + τ + β such that (6) holds for all n ≥ T.

The following lemmas play important roles in this paper.

Lemma 1 (see [27].)Let ,, and be four nonnegative sequences satisfying the inequality

(10)

where

, lim _n→∞β_n = 0, and

. Then lim _n→∞α_n = 0.

Lemma 2 (see [11].)Let τ, n₀ ∈ ℕ and be a nonnegative sequence. Then

(a)
;
(b)
.

2. Existence of Uncountably Many Unbounded Positive Solutions

Using the Banach fixed point theorem and the Mann iterative schemes with errors, we next discuss the existence of uncountably many unbounded positive solutions of (6), prove that the Mann iterative schemes with errors converge to these unbounded positive solutions, and compute the error estimates between the Mann iterative schemes with errors and the unbounded positive solutions.

Theorem 3. Assume that there exist two constants M and N with M > N > 0 and four nonnegative sequences , and satisfying

(11)

(12)

(13)

(14)

(15)

Then

(a) for any L ∈ (N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that for each , the Mann iterative sequence with errors generated by the scheme

(16)

converges to an unbounded positive solution x ∈ A(N, M) of (6) and has the following error estimate:

(17)

where

is an arbitrary sequence in A(N, M) and

and

are any sequences in [0,1] such that

(18)

(b)
equation (6) possesses uncountably many unbounded positive solutions in A(N, M).

Proof. First of all we show that (a) holds. Set L ∈ (N, M). It follows from (13), (14), and (15) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(19)

(20)

(21)

Define a mapping

(22)

for each

. In view of (11), (12), (19), (20), and (22), we deduce that for each

and for all n ≥ T

(23)

which yield that

(24)

which means that S_L is a contraction in A(N, M). It follows from the Banach fixed point theorem that S_L has a unique fixed point

, that is,

(25)

which imply that

(26)

which yields that

(27)

which together with (21) gives that

is an unbounded positive solution of (6) in A(N, M). It follows from (16), (19), (21), (22), and (24) that for any m ∈ ℕ₀ and n ≥ T

(28)

which implies that

(29)

That is, (17) holds. Thus, Lemma 1 and (17) and (18) guarantee that lim _m→∞x_m = x.

Next we show that (b) holds. Let L₁, L₂ ∈ (N, M) and L₁ ≠ L₂. As in the proof of (a), we deduce similarly that for each c ∈ {1,2} there exist constants θ_c ∈ (0,1), T_c ≥ n₀ + τ + β, and a mapping satisfying (19)~(24), where θ, L, and T are replaced by θ_c, L_c and T_c, respectively, and the mapping has a fixed point , which is an unbounded positive solution of (6) in A(N, M), that is,

(30)

which together with (11) and (17) implies that for n ≥ max {T₁, T₂}

(31)

which yields that

(32)

that is, z¹ ≠ z². This completes the proof.

Theorem 4. Assume that there exist two constants M and N with M > N > 0 and four nonnegative sequences , , , and satisfying (11), (12),

(33)

(34)

(35)

Then

(a) for any L ∈ (N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that for each , the Mann iterative sequence with errors generated by the scheme

(36)

converges to an unbounded positive solution x ∈ A(N, M) of (6) and has the error estimate (17), where

is an arbitrary sequence in A(N, M), and

and

are any sequences in [0,1] satisfying (18);

(b) equation (6) possesses uncountably many unbounded positive solutions in A(N, M).

Proof. Let L ∈ (N, M). It follows from (33)~(35) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(37)

(38)

(39)

Define a mapping

(40)

for each

. Using (11), (12), (37), (38), and (40), we get that for each

and n ≥ T

(41)

which imply (24). Consequently, (24) means that S_L is a contraction in A(N, M) and has a unique fixed point

, which is also an unbounded positive solution of (6) in A(N, M). The rest of the proof is similar to the proof of Theorem 3 and is omitted. This completes the proof.

Theorem 5. Assume that there exist three constants a, M, and N with (1 − a)M > N > 0 and four nonnegative sequences , , , and satisfying (11), (12), (33), (34) and

(42)

Then

(a) for any L ∈ (aM + N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that for any , the Mann iterative sequence with errors generated by the scheme

(43)

converges to an unbounded positive solution x ∈ A(N, M) of (6) and has the error estimate (17), where

is an arbitrary sequence in A(N, M) and

and

are any sequences in [0,1] satisfying (18);

(b) equation (6) possesses uncountably many unbounded positive solutions in A(N, M).

Proof. Put L ∈ (aM + N, M). It follows from (33), (34), and (42) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(44)

Define a mapping

(45)

for each

. In view of (11), (12), and (44) and (45), we obtain that for each

and n ≥ T,

(46)

which give (24), in turn, which implies that S_L is a contraction in A(N, M) and possesses a unique fixed point

, which is an unbounded positive solution of (6) in A(N, M). The rest of the proof is similar to that of Theorem 3 and is omitted. This completes the proof.

Theorem 6. Assume that there exist constants a, M, and N with (1 + a)M > N > 0 and four nonnegative sequences , , , and satisfying (11), (12), (33), (34), and

(47)

Then

(a) for any L ∈ (N, (1 + a)M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that for any and the Mann iterative sequence with errors generated by (43) converges to an unbounded positive solution x ∈ A(N, M) of (6) and has the error estimate (17), where is an arbitrary sequence in A(N, M), and are any sequences in [0,1] satisfying (18);

(b) equation (6) possesses uncountably many unbounded positive solutions in A(N, M).

Proof. Put L ∈ (N, (1 + a)M). It follows from (33), (34), and (47) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(48)

(49)

(50)

Define a mapping

by (45). By virtue of (12), (45), (48), and (50), we easily verify that

(51)

which yield that S_L(A(N, M))⊆A(N, M). The rest of the proof is similar to that of Theorem 5 and is omitted. This completes the proof.

Remark 7. Theorems 3~6 extend and improve Theorem 1 in [5].

3. Examples

In this section we suggest four examples to explain the results presented in Section 2. Note that Theorem 1 in [5] is useless for all these examples.

Example 8. Consider the second-order nonlinear neutral delay difference equation:

(52)

where τ ∈ ℕ is fixed. Let n₀ = 3, k = 1, β = min {3 − τ, 1}, M and N two positive constants with M > N and

(53)

It is easy to see that (11), (12), and (15) are satisfied. Note that

(54)

(55)

which together with Lemma 2 yield that (13) and (14) hold. It follows from Theorem 3 that (52) possesses uncountably many unbounded positive solutions in A(N, M). On the other hand, for any L ∈ (N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that for each

, the Mann iterative sequence with errors

generated by (16) converges to an unbounded positive solution x ∈ A(N, M) of (52) and has the error estimate (17), where

is an arbitrary sequence in A(N, M) and

and

are any sequences in [0,1] satisfying (18).

Example 9. Consider the second-order nonlinear neutral delay difference equation:

(56)

where τ ∈ ℕ is fixed. Let n₀ = 5, k = 2, β = 5 − τ, M and N two positive constants with M > N and

(57)

It is clear that (11), (12), and (35) are fulfilled. Note that

(58)

which yields that

(59)

Thus, Theorem 4 guarantees that (56) possesses uncountably unbounded positive solutions in A(N, M). On the other hand, for any L ∈ (N, M), there exist θ ∈ (0,1) and T ≥ τ + n₀ + β such that the Mann iterative sequence with error

generated by (36) converges to an unbounded positive solution x ∈ A(N, M) of (56) and has the error estimate (17), where

is an arbitrary sequence in A(N, M) and

and

are any sequences in [0,1] satisfying (18).

Example 10. Consider the second-order nonlinear neutral delay difference equation:

(60)

where τ ∈ ℕ is fixed. Let n₀ = 7, k = 2, a = 3/4, β = min {7 − τ, 5}, M and N two positive constants with M > 4N and

(61)

It is not difficult to verify that (11), (12), and (42) are fulfilled. Note that

(62)

That is, (33) and (34) are satisfied. Consequently Theorem 5 implies that (60) possesses uncountably many unbounded positive solutions in A(N, M). On the other hand, for any L ∈ ((3/4)M + N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that the Mann iterative sequence with error

generated by (43) converges to an unbounded positive solution x ∈ A(N, M) of (60) and has the error estimate (17), where

is an arbitrary sequence in A(N, M) and

and

are any sequences in [0,1] satisfying (18).

Example 11. Consider the second-order nonlinear neutral delay difference equation:

(63)

where τ ∈ ℕ is fixed. Let n₀ = 11, k = 1, a = −4/5, β = min {11 − τ, 7}, M and N two positive constants with M > 5N and

(64)

Obviously, (11), (12), and (50) are satisfied. Note that

(65)

which gives that

(66)

That is, (33) and (34) hold. Thus, Theorem 6 shows that (63) possesses uncountably many unbounded positive solutions in A(N, M). On the other hand, for any L ∈ (N, M/5), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that the Mann iterative sequence with error

generated by (43) converges to an unbounded positive solution x ∈ A(N, M) of (63) and has the error estimate (17), where

is an arbitrary sequence in A(N, M) and

and

are any sequences in [0,1] satisfying (18).

Acknowledgments

This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2002165).

References

1 Abu-Risha M. H., Oscillation of second-order linear difference equations, Applied Mathematics Letters. (2000) 13, no. 1, 129–135, https://doi.org/10.1016/S0893-9659(99)00156-1, MR1750979, ZBL0947.39005.
10.1016/S0893-9659(99)00156-1
Web of Science® Google Scholar
2 Agarwal R. P., Difference Equations and Inequalities, 2000, 228, 2nd edition, Marcel Dekker, New York, NY, USA, Monographs and Textbooks in Pure and Applied Mathematics, MR1740241.
10.1201/9781420027020
Google Scholar
3 Agarwal R. P., Manuel M. M. S., and Thandapani E., Oscillatory and nonoscillatory behavior of second-order neutral delay difference equations. II, Applied Mathematics Letters. (1997) 10, no. 2, 103–109, https://doi.org/10.1016/S0893-9659(97)00019-0, MR1436505, ZBL0905.34065.
10.1016/S0893-9659(97)00019-0
Web of Science® Google Scholar
4 Cheng S. S., Li H. J., and Patula W. T., Bounded and zero convergent solutions of second-order difference equations, Journal of Mathematical Analysis and Applications. (1989) 141, no. 2, 463–483, https://doi.org/10.1016/0022-247X(89)90191-1, MR1009057, ZBL0698.39002.
10.1016/0022-247X(89)90191-1
Web of Science® Google Scholar
5 Jinfa C., Existence of a nonoscillatory solution of a second-order linear neutral difference equation, Applied Mathematics Letters. (2007) 20, no. 8, 892–899, https://doi.org/10.1016/j.aml.2006.06.021, MR2323128, ZBL1144.39004.
10.1016/j.aml.2006.06.021
Web of Science® Google Scholar
6 Li W. T. and Cheng S. S., Oscillation criteria for a nonlinear difference equation, Computers & Mathematics with Applications. (1998) 36, no. 8, 87–94, https://doi.org/10.1016/S0898-1221(98)00185-0, MR1653847, ZBL0932.39006.
10.1016/S0898-1221(98)00185-0
Web of Science® Google Scholar
7 Li W.-T. and Cheng S.-S., Remarks on two recent oscillation theorems for second-order linear difference equations, Applied Mathematics Letters. (2003) 16, no. 2, 161–163, https://doi.org/10.1016/S0893-9659(03)80026-5, MR1962310, ZBL1052.39009.
10.1016/S0893-9659(03)80026-5
Web of Science® Google Scholar
8 Li W.-T. and Saker S. H., Oscillation of second-order sublinear neutral delay difference equations, Applied Mathematics and Computation. (2003) 146, no. 2-3, 543–551, https://doi.org/10.1016/S0096-3003(02)00604-5, MR2008571, ZBL1052.39010.
10.1016/S0096-3003(02)00604-5
Web of Science® Google Scholar
9 Li X. and Zhu D., New results for the asymptotic behavior of a nonlinear second-order difference equation, Applied Mathematics Letters. (2003) 16, no. 5, 627–633, https://doi.org/10.1016/S0893-9659(03)00057-0, MR1986025, ZBL1049.39007.
10.1016/S0893-9659(03)00057-0
Web of Science® Google Scholar
10 Liang H. and Weng P., Existence and multiple solutions for a second-order difference boundary value problem via critical point theory, Journal of Mathematical Analysis and Applications. (2007) 326, no. 1, 511–520, https://doi.org/10.1016/j.jmaa.2006.03.017, MR2277799, ZBL1112.39008.
10.1016/j.jmaa.2006.03.017
Web of Science® Google Scholar
11 Liu Z., Kang S. M., and Ume J. S., Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations, Applied Mathematics and Computation. (2009) 213, no. 2, 554–576, https://doi.org/10.1016/j.amc.2009.03.050, MR2536681, ZBL1182.39002.
10.1016/j.amc.2009.03.050
Web of Science® Google Scholar
12 Liu Z., Xu Y., and Kang S. M., Global solvability for a second order nonlinear neutral delay difference equation, Computers & Mathematics with Applications. (2009) 57, no. 4, 587–595, https://doi.org/10.1016/j.camwa.2008.09.050, MR2489080, ZBL1165.39307.
10.1016/j.camwa.2008.09.050
Web of Science® Google Scholar
13 Luo J. W. and Bainov D. D., Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima, Journal of Computational and Applied Mathematics. (2001) 131, no. 1-2, 333–341, https://doi.org/10.1016/S0377-0427(00)00264-8, MR1835720, ZBL0984.39006.
10.1016/S0377-0427(00)00264-8
Web of Science® Google Scholar
14 Ma M., Tang H., and Luo W., Periodic solutions for nonlinear second-order difference equations, Applied Mathematics and Computation. (2007) 184, no. 2, 685–694, https://doi.org/10.1016/j.amc.2006.06.069, MR2294933, ZBL1123.39005.
10.1016/j.amc.2006.06.069
Web of Science® Google Scholar
15 Meng Q. and Yan J., Bounded oscillation for second-order nonlinear delay differential equation in a critical state, Journal of Computational and Applied Mathematics. (2006) 197, no. 1, 204–211, https://doi.org/10.1016/j.cam.2005.10.033, MR2256061, ZBL1102.34325.
10.1016/j.cam.2005.10.033
Web of Science® Google Scholar
16 Migda M. and Migda J., Asymptotic properties of solutions of second-order neutral difference equations, Nonlinear Analysis, Theory, Methods and Applications. (2005) 63, no. 5–7, e789–e799, https://doi.org/10.1016/j.na.2005.02.005.
10.1016/j.na.2005.02.005
Web of Science® Google Scholar
17 Saker S. H., New oscillation criteria for second-order nonlinear neutral delay difference equations, Applied Mathematics and Computation. (2003) 142, no. 1, 99–111, https://doi.org/10.1016/S0096-3003(02)00286-2, MR1978250, ZBL1028.39003.
10.1016/S0096-3003(02)00286-2
Web of Science® Google Scholar
18 Tan M.-C. and Yang E.-H., Oscillation and nonoscillation theorems for second order nonlinear difference equations, Journal of Mathematical Analysis and Applications. (2002) 276, no. 1, 239–247, https://doi.org/10.1016/S0022-247X(02)00435-3, MR1944349, ZBL1033.39015.
10.1016/S0022-247X(02)00435-3
Web of Science® Google Scholar
19 Tang X. H., Bounded oscillation of second-order delay difference equations of unstable type, Computers & Mathematics with Applications. (2002) 44, no. 8-9, 1147–1156, https://doi.org/10.1016/S0898-1221(02)00222-5, MR1937574, ZBL1035.39009.
10.1016/S0898-1221(02)00222-5
Web of Science® Google Scholar
20 Thandapani E., Manuel M. M. S., Graef J. R., and Spikes P. W., Monotone properties of certain classes of solutions of second-order difference equations, Computers & Mathematics with Applications. (1998) 36, no. 10–12, 291–297, https://doi.org/10.1016/S0898-1221(98)80030-8, MR1666147, ZBL0933.39014.
10.1016/S0898-1221(98)80030-8
Web of Science® Google Scholar
21 Tian Y. and Ge W., Multiple positive solutions of boundary value problems for second-order discrete equations on the half-line, Journal of Difference Equations and Applications. (2006) 12, no. 2, 191–208, https://doi.org/10.1080/10236190500539329, MR2205088, ZBL1103.39016.
10.1080/10236190500539329
Web of Science® Google Scholar
22 Zhang B. G., Oscillation and asymptotic behavior of second order difference equations, Journal of Mathematical Analysis and Applications. (1993) 173, no. 1, 58–68, https://doi.org/10.1006/jmaa.1993.1052, MR1205909, ZBL0780.39006.
10.1006/jmaa.1993.1052
Web of Science® Google Scholar
23 Zhang Z., Chen J., and Zhang C., Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term, Computers & Mathematics with Applications. (2001) 41, no. 12, 1487–1494, https://doi.org/10.1016/S0898-1221(01)00113-4, MR1831811, ZBL0983.39003.
10.1016/S0898-1221(01)00113-4
Web of Science® Google Scholar
24 Zhang Z. and Li Q., Oscillation theorems for second-order advanced functional difference equations, Computers & Mathematics with Applications. (1998) 36, no. 6, 11–18, https://doi.org/10.1016/S0898-1221(98)00157-6, MR1645348, ZBL0935.39005.
10.1016/S0898-1221(98)00157-6
Web of Science® Google Scholar
25 Zhang Z. G. and Yu J. S., On existence of positive solutions for neutral differential equations, Science in China. Series A. (1992) 8, 785–790.
Google Scholar
26 Zhang Z. and Zhang J., Oscillation criteria for second-order functional difference equations with “summation small” coefficient, Computers & Mathematics with Applications. (1999) 38, no. 1, 25–31, https://doi.org/10.1016/S0898-1221(99)00165-0, MR1697339, ZBL0937.39013.
10.1016/S0898-1221(99)00165-0
Web of Science® Google Scholar
27 Liu L. S., Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, Journal of Mathematical Analysis and Applications. (1995) 194, no. 1, 114–125, https://doi.org/10.1006/jmaa.1995.1289, MR1353071, ZBL0872.47031.
10.1006/jmaa.1995.1289
Web of Science® Google Scholar

All articles

Unbounded Positive Solutions and Mann Iterative Schemes of a Second-Order Nonlinear Neutral Delay Difference Equation

Abstract

1. Introduction and Preliminaries

2. Existence of Uncountably Many Unbounded Positive Solutions

3. Examples

Acknowledgments

References

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Unbounded Positive Solutions and Mann Iterative Schemes of a Second-Order Nonlinear Neutral Delay Difference Equation

Abstract

1. Introduction and Preliminaries

2. Existence of Uncountably Many Unbounded Positive Solutions

3. Examples

Acknowledgments

References

References

Related

Information