Volume 2014, Issue 1 470181

Research Article

Open Access

On Positive Solutions and Mann Iterative Schemes of a Third Order Difference Equation

Zeqing Liu,

Zeqing Liu

orcid.org/0000-0001-5919-8822

Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China lnnu.edu.cn

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Heng Wu,

Heng Wu

Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China lnnu.edu.cn

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Shin Min Kang,

Shin Min Kang

orcid.org/0000-0002-5294-4447

Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea gnu.ac.kr

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Young Chel Kwun,

Corresponding Author

Young Chel Kwun

[email protected]

orcid.org/0000-0002-5028-0640

Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea donga.ac.kr

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Zeqing Liu,

Zeqing Liu

orcid.org/0000-0001-5919-8822

Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China lnnu.edu.cn

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Heng Wu,

Heng Wu

Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China lnnu.edu.cn

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Shin Min Kang,

Shin Min Kang

orcid.org/0000-0002-5294-4447

Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea gnu.ac.kr

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Young Chel Kwun,

Corresponding Author

Young Chel Kwun

[email protected]

orcid.org/0000-0002-5028-0640

Department of Mathematics, Dong-A University, Pusan 614-714, Republic of Korea donga.ac.kr

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First published: 28 January 2014

https://doi.org/10.1155/2014/470181

Citations: 2

Academic Editor: Zhi-Bo Huang

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Abstract

The existence of uncountably many positive solutions and convergence of the Mann iterative schemes for a third order nonlinear neutral delay difference equation are proved. Six examples are given to illustrate the results presented in this paper.

1. Introduction and Preliminaries

Recently, many researchers studied the oscillation, nonoscillation, and existence of solutions for linear and nonlinear second and third order difference equations and systems see, for example, [1–23] and the references cited therein. By means of the Reccati transformation techniques, Saker [18] discussed the third order difference equation

(1)

and presented some sufficient conditions which ensure that all solutions are to be oscillatory or tend to zero. Utilizing the Schauder fixed point theorem, Yan and Liu [22] proved the existence of a bounded nonoscillatory solution for the third order difference equation

(2)

Agarwal [2] established the oscillatory and asymptotic properties for the third order nonlinear difference equation

(3)

Andruch-Sobiło and Migda [4] studied the third order linear difference equation of neutral type

(4)

and obtained sufficient conditions which ensure that all solutions of the equation are oscillatory. Grace and Hamedani [6] discussed the difference equation

(5)

and gave some new criteria for the oscillation of all solutions and all bounded solutions.

Our goal is to discuss solvability and convergence of the Mann iterative schemes for the following third order nonlinear neutral delay difference equation:

(6)

where τ, k, n₀ ∈ ℕ,

, h,

, and

(7)

By employing the Banach fixed point theorem and some new techniques, we establish the existence of uncountably many positive solutions of (6), conceive a few Mann iterative schemes for approximating these positive solutions, and prove their convergence and the error estimates. Six nontrivial examples are included.

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

(8)

and

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

(9)

It is easy to see 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.

Lemma 1. Let {p_t} _t∈ℕ be a nonnegative sequence and τ ∈ ℕ.

(i)
If then.
(ii)
If then.

Proof. Note that

(10)

that is,

(11)

As in the proof of (10), we infer that

(12)

which implies that

(13)

This completes the proof.

2. Uncountably Many Positive Solutions and Mann Iterative Schemes

In this section, using the Banach fixed point theorem and Mann iterative schemes, we establish the existence of uncountably many positive solutions of (6), prove convergence of the Mann iterative schemes relative to these positive solutions, and compute the error estimates between the Mann iterative schemes and the positive solutions.

Theorem 2. Assume that there exist two constants M and N with M > N > 0 and four nonnegative sequences, , and satisfying

(14)

(15)

(16)

(17)

(18)

Then one has the following.

(a)
For any L ∈ (N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that, for each, the Mann iterative sequence generated by the scheme
(19)

converges to a positive solution

of (6) with lim⁡_n→∞z_n = +∞ and has the following error estimate:

(20)

where

is an arbitrary sequence in [0,1] such that

(21)

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

Proof. Firstly, we show that (a) holds. Put L ∈ (N, M). It follows from (16)~(18) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(22)

(23)

(24)

Define a mapping

(25)

for each

. In light of (14), (15), (22), (23), and (25), we obtain that for each

(26)

which yield that

(27)

which implies that S_L is a contraction in A(N, M). The Banach fixed point theorem and (27) ensure that S_L has a unique fixed point

; that is,

(28)

which mean that

(29)

which yields that

(30)

which gives that

(31)

which together with (24) implies that

is a positive solution of (6) in A(N, M). Note that

(32)

which guarantees that lim⁡_n→∞z_n = +∞. It follows from (19), (22), (24), (25), and (27) that for any m ∈ ℕ₀ and n ≥ T

(33)

which implies that

(34)

That is, (20) holds. Thus Lemma 1, (20), and (21) guarantee that lim⁡_m→∞x_m = z.

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) and T_c ≥ n₀ + τ + β and a mappingsatisfying (22)~(27), where θ, L, and T are replaced by θ_c, L_c, and T_c, respectively, and the mappinghas a fixed point, which is a positive solution of (6) in A(N, M) with. It follows that

(35)

which together with (14) and (20) means that for n ≥ max⁡⁡{T₁, T₂}

(36)

which yields that

(37)

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

Theorem 3. Assume that there exist two constants M and N with M > N > 0 and four nonnegative sequences, , , andsatisfying (14), (15), and

(38)

(39)

(40)

Then one has the following.

(a)
For any L ∈ (N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that, for each, the Mann iterative sequencegenerated by the scheme
(41)
converges to a positive solutionof (6) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), whereis an arbitrary sequence in [0,1] satisfying (21).
(b)
Equation (6) possesses uncountably many positive solutions in A(N, M).

Proof. Let L ∈ (N, M). It follows from (38)~(40) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(42)

(43)

(44)

Define a mapping

(45)

for each

. Using (14), (15), (42), (43), and (45), we get that for each

and n ≥ T

(46)

which imply (27). The rest of the proof is similar to the proof of Theorem 2 and is omitted. This completes the proof.

Theorem 4. Assume that there exist three constants b, M, and N with (1 − b)M > N > 0 and four nonnegative sequences, , andsatisfying (14), (15), (38), (39) and

(47)

Then one has the following.

(a)
For any L ∈ (bM + N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that, for any, the Mann iterative sequence generated by the scheme
(48)
converges to a positive solutionof (6) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), whereis an arbitrary sequence in [0,1] satisfying (21).
(b)
Equation (6) possesses uncountably many positive solutions in A(N, M).

Proof. Put L ∈ (bM + N, M). It follows from (38), (39), and (47) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(49)

Define a mapping

(50)

for each

. In view of (14), (15), and (49) and (50), we obtain that for each

and n ≥ T

(51)

which imply (27). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 5. Assume that there exist constants b, M, and N with (1 + b)M > N > 0 and four nonnegative sequences, , , and satisfying (14), (15), (38), (39), and

(52)

Then one has the following.

(a)
For any L ∈ (N, (1 + b)M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that, for any, the Mann iterative sequencegenerated by (48) converges to a positive solutionof (6) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), whereis an arbitrary sequence in [0,1] satisfying (21).
(b)
Equation (6) possesses uncountably many positive solutions in A(N, M).

Proof. Put L ∈ (N, (1 + b)M). It follows from (38), (39), and (52) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(53)

(54)

(55)

Define a mapping

by (50). By virtue of (15), (50), (53), and (55), we infer that for all

and n ≥ T

(56)

That is, (27) holds. The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 6. Assume that there exist constants b, M, and N with (1 − 1/b)M > N > 0 and four nonnegative sequences, , , and satisfying (14), (15), (38), (39), and

(57)

Then one has the following.

(a)
For any L ∈ ((1/b)M + N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that, for any , the Mann iterative sequence generated by the scheme
(58)
converges to a positive solutionof (6) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), whereis an arbitrary sequence in [0,1] satisfying (21).
(b)
Equation (6) possesses uncountably many positive solutions in A(N, M).

Proof. Put L ∈ ((1/b)M + N, M). It follows from (38), (39), and (57) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(59)

(60)

(61)

Define a mapping

(62)

for each

. In view of (14), (15), and (59)~(62), we obtain that for each

and n ≥ T

(63)

which imply (27). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 7. Assume that there exist constants b, M, and N with (1 + 1/b)M > N > 0 and four nonnegative sequences, , , and satisfying (14), (15), (38), (39) and

(64)

Then one has the following.

(a)
For any L ∈ (−(1 + 1/b)M, −N), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that, for any, the Mann iterative sequencegenerated by the scheme
(65)
converges to a positive solution of (6) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), where is an arbitrary sequence in [0,1] satisfying (21).
(b)
Equation (6) possesses uncountably many positive solutions in A(N, M).

Proof. Put L ∈ (−(1 + 1/b)M, −N). It follows from (38), (39), and (64) that there exist θ ∈ (0,1) and T ≥ n₀ + τ + β satisfying

(66)

(67)

(68)

Define a mapping

(69)

for each

. Making use of (15), (66), (68), and (69), we conclude that

(70)

which yield (27). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

3. Examples

In this section, we suggest six examples to explain the results presented in Section 2.

Example 1. Consider the third order nonlinear neutral delay difference equation

(71)

where τ ∈ ℕ is fixed. Let n₀ = 4, k = 1, and β = min⁡{4 − τ, 1}, and let M and N be two positive constants with M > N and

(72)

It is easy to see that (14), (15), and (18) are satisfied. Note that

(73)

which together with Lemma 1 yield that (16) and (17) hold. It follows from Theorem 2 that (71) possesses uncountably many 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

generated by (19) converges to a positive solution

of (71) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), where

is an arbitrary sequence in [0,1] satisfying (21).

Example 2. Consider the third order nonlinear neutral delay difference equation

(74)

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

(75)

It is clear that (14), (15), and (40) are fulfilled. Note that

(76)

which means that

(77)

Observe that

(78)

which yields that

(79)

Thus Theorem 3 guarantees that (74) possesses uncountably 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

generated by (41) converges to a positive solution

of (74) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), where

is an arbitrary sequence in [0,1] satisfying (21).

Example 3. Consider the third order nonlinear neutral delay difference equation

(80)

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

(81)

It is not difficult to verify that (14), (15), and (47) are fulfilled. Note that

(82)

which implies that

(83)

Observe that

(84)

which means that

(85)

That is, (38) and (39) hold. Consequently Theorem 4 implies that (80) possesses uncountably many 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

generated by (41) converges to a positive solution

of (80) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), where

is an arbitrary sequence in [0,1] satisfying (21).

Example 4. Consider the third order nonlinear neutral delay difference equation

(86)

where τ ∈ ℕ is fixed. Let n₀ = 9, k = 1, b = −5/6, and β = 9 − τ, and let M and N be two positive constants with M > 6N and

(87)

Obviously, (14), (15), and (52) are satisfied. Note that

(88)

which implies that

(89)

Notice that

(90)

which gives that

(91)

That is, (38) and (39) hold. Thus Theorem 5 shows that (86) possesses uncountably many positive solutions in A(N, M). On the other hand, for any L ∈ (N, (1/6)M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that the Mann iterative sequence

generated by (48) converges to a positive solution

of (86) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), where

is an arbitrary sequence in [0,1] satisfying (21).

Example 5. Consider the third order nonlinear neutral delay difference equation

(92)

where τ ∈ ℕ is fixed. Let n₀ = 3, k = 1, b = π/2, and β = min⁡⁡{3 − τ, 1}, and let M and N be two positive constants with (1 − 2/π)M > N and

(93)

Clearly, (14), (15), and (61) are satisfied. Note that

(94)

which means that

(95)

which implies that

(96)

That is, (38) and (39) hold. Consequently Theorem 6 implies that (92) possesses uncountably many positive solutions in A(N, M). On the other hand, for any L ∈ ((2/π)M + N, M), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that the Mann iterative sequence

generated by (58) converges to a positive solution

of (92) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), where

is an arbitrary sequence in [0,1] satisfying (21).

Example 6. Consider the third order nonlinear neutral delay difference equation

(97)

where τ ∈ ℕ is fixed. Let n₀ = 6, k = 1, b = −2, and β = min⁡{6 − τ, 3}, and let M and N be two positive constants with (1/2)M > N and

(98)

Obviously, (14), (15), and (64) are satisfied. Note that

(99)

which means that

(100)

It is clear that

(101)

which implies that

(102)

That is, (38) and (39) hold. Consequently Theorem 7 implies that (97) possesses uncountably many positive solutions in A(N, M). On the other hand, for any L ∈ (−M/2, −N), there exist θ ∈ (0,1) and T ≥ n₀ + τ + β such that the Mann iterative sequence

generated by (65) converges to a positive solution

of (97) with lim⁡_n→∞z_n = +∞ and has the error estimate (20), where

is an arbitrary sequence in [0,1] satisfying (21).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380).

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On Positive Solutions and Mann Iterative Schemes of a Third Order Difference Equation

Abstract

1. Introduction and Preliminaries

2. Uncountably Many Positive Solutions and Mann Iterative Schemes

3. Examples

Conflict of Interests

Acknowledgment

References

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On Positive Solutions and Mann Iterative Schemes of a Third Order Difference Equation

Abstract

1. Introduction and Preliminaries

2. Uncountably Many Positive Solutions and Mann Iterative Schemes

3. Examples

Conflict of Interests

Acknowledgment

References

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References

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