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# Some convergence results for iterative sequences of Prešić type and applications

Mohammad Saeed Khan1, Maher Berzig2 and Bessem Samet3*

Author Affiliations

1 Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, Al-Khod, Post Box 36, PCode 123, Muscat, Sultanate of Oman

2 Ecole Supérieure des Sciences et Techniques de Tunis, 5, Avenue Taha Hussein-Tunis, B.P. 56, Bab Menara-1008, Tunisie

3 Department of Mathematics, King Saud University, Riyadh, Saudi Arabia

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Advances in Difference Equations 2012, 2012:38  doi:10.1186/1687-1847-2012-38

 Received: 22 November 2011 Accepted: 29 March 2012 Published: 29 March 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this article, we study the convergence of iterative sequences of Prešić type involving new general classes of operators in the setting of metric spaces. As application, we derive some convergence results for a class of nonlinear matrix difference equations. Numerical experiments are also presented to illustrate the convergence algorithms.

Mathematics Subject Classification 2000: 54H25; 47H10; 15A24; 65H05.

##### Keywords:
iterative sequence; convergence; difference equation; fixed point; matrix

### 1 Introduction

In 1922, Banach proved the following famous fixed point theorem.

Theorem 1.1 (Banach [1]) Let (X, d) be a complete metric space and f : X X be a contractive mapping, that is, there exists δ ∈ [0, 1) such that

Then f has a unique fixed point, that is, there exists a unique x* X such that x* = fx*. Moreover, for any x0 X, the iterative sequence xn+1 = fxn converges to x*.

This theorem called the Banach contraction principle is a simple and powerful theorem with a wide range of application, including iterative methods for solving linear, nonlinear, differential, integral, and difference equations. Many generalizations and extensions of the Banach contraction principle exist in the literature. For more details, we refer the reader to [2-28].

Consider the k-th order nonlinear difference equation

(1)

with the initial values x0,..., xk-1 X, where k is a positive integer (k ≥ 1) and f : Xk X. Equation (1) can be studied by means of fixed point theory in view of the fact that x* ∈ X is a solution to (1)) if and only if x* is a fixed point of f, that is, x* = f(x*, ..., x*). One of the most important results in this direction has been obtained by Prešić in [22] by generalizing the Banach contraction principle in the following way.

Theorem 1.2 (Prešić [22]) Let (X,d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

for all x0, ..., xk X, where δ1, ..., δk are positive constants such that δ1 + ... + δk ∈ (0,1). Then f has a unique fixed point x* ∈ X, that is, there exists a unique x* ∈ X such that x* = f(x*, ..., x*). Moreover, for any initial values x0, ..., xk-1 X, the iterative sequence {xn} defined by (1) converges to x*.

It is easy to show that for k = 1, Theorem 1.2 reduces to the Banach contraction principle. So, Theorem 1.2 is a generalization of the Banach fixed point theorem.

In [13], Ćirić and Prešić generalized Theorem 1.2 as follows.

Theorem 1.3 (Ćirić and Prešić [13]) Let (X,d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

for all x0, ..., xk X, where λ ∈ (0,1) is a constant. Then f has a unique fixed point x* ∈ X, that is, there exists a unique x* ∈ X such that x* = f(x*,..., x*). Moreover, for any initial values x0, ..., xk-1 X, the iterative sequence {xn} defined by (1) converges to x*.

The applicability of the result due to Ćirić and Prešić to the study of global asymptotic stability of the equilibrium for the nonlinear difference Equation (1) is revealed, for example, in the recent article [8].

Other generalizations were obtained by Păcurar in [20,21].

Theorem 1.4 (Păcurar [20]) Let (X, d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

for all x0, ..., xk X, where a is a constant such that 0 < ak(k + 1) < 1. Then f has a unique fixed point x* ∈ X, that is, there exists a unique x* ∈ X such that x* = f(x*, ..., x*). Moreover, for any initial values x0, ..., xk-1 X, the iterative sequence {xn} defined by (1) converges to x*.

In the particular case k = 1, from Theorem 1.4, we obtain Kannan's fixed point theorem for discontinuous mappings in [15].

Theorem 1.5 (Păcurar [21]) Let (X, d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

for all x0, ..., xk X, where δ1, ..., δk are positive constants such that and

(2)

with L ≥ 0. Then f has a unique fixed point x* X, that is, there exists a unique x* X such that x* = f(x*, ..., x*). Moreover, for any initial values x0,..., xk-1 X, the iterative sequence {xn} defined by (1) converges to x*.

In the particular case k = 1, the contractive condition (2) reduces to strict almost contraction (see [4-7]).

Note that these approaches are motivated by the currently increasing interest in the study of nonlinear difference equations which appear in many interesting examples from system theory, economics, inventory analysis, probability models for learning, approximate solutions of ordinary and partial differential equations just to mention a few [29-31]. We refer the reader to [32-34] for a detailed study of the theory of difference equations.

For other studies in this direction, we refer the reader to [23,25,35,36].

In this article, we study the convergence of the iterative sequence (1) for more general classes of operators. Presented theorems extend and generalize many existing results in the literature including Theorems 1.1, 1.2, 1.4, and 1.5. We present also an application to a class of nonlinear difference matrix equations and we validate our results with numerical experiments.

### 2 Main results

In order to prove our main results we shall need the following lemmas.

Lemma 2.1 Let k be a positive integer and α1, α2, ..., αk ≥ 0 such that . If { Δn} is a sequence of positive numbers satisfying

then there exist L ≥ 0 and τ ∈ (0,1) such that Δ n n for all n ≥ 1.

Lemma 2.2 Let {an}, {bn} be two sequences of positive real numbers and q ∈ (0,1) such that an+1 qan + bn, n ≥ 0 and bn → 0 as n → ∞. Then an → 0 as n → ∞.

Let Θ be the set of functions θ : [0, ∞)4 → [0, ∞) satisfying the following conditions:

(i) θ is continuous,

(ii) for all t1, t2, t3, t4 ∈ [0, ∞),

Example 2.1 The following functions belong to Θ:

(1) θ(t1, t2, t3, t4) = L min{t1, t2, t3, t4}, L > 0 t1, t2, t3, t4 ≥ 0.

(2) θ(t1, t2, t3, t4) = L ln(1 + t1t2t3t4), L > 0 t1,t2,t3,t4 ≥ 0.

(3) θ(t1, t2, t3, t4) = L ln(1 + t1) ln(1 + t2) ln(1 + t3) ln(1 + t4), L > 0 t1,t2,t3,t4 ≥ 0.

(4) θ(t1, t2, t3, t4) = Lt1t2t3t4, L > 0 t1, t2, t3, t4 ≥ 0.

(5) , L > 0 t1, t2, t3, t4 ≥ 0.

Our first result is the following.

Theorem 2.1 Let (X,d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

(3)

for all x0,..., xk X, where δ1,..., δk + 1 are positive constants such that 2A + δ ∈ (0,1) with and . Then f has a unique fixed point x* ∈ X, that is, there exists a unique x* ∈ X such that x* = f(x*,..., x*). Moreover, for any z0 X, the iterative sequence {zn} defined by

converges to x*.

Proof. Define the mapping F : X X by

Using (3), for all x, y X, we have

Thus, we have

(4)

where

Now, let z0 be an arbitrary element of X. Define the sequence {zn} by

Using (4), we have

On the other hand, from the property (ii) of the function θ, we have

Then we get

for all n = 1, 2,.... This implies that

for all n = 1, 2,.... Since we have 2A + δ ∈ (0,1), then {zn} is a Cauchy sequence in (X, d). Now, since (X, d) is complete, there exists x* X such that zn x* as n → ∞. We shall prove that x* is a fixed point of F, that is, x* = Fx*. Using (4), we have

where

Thus we have

Letting n → ∞ in the above inequality, and using the properties (i) and (ii) of θ, we obtain

which implies (since 1 - A > 0) that x* = Fx* = f(x*, ..., x*).

Now, we shall prove that x* is the unique fixed point of F. Suppose that y* ∈ X is another fixed point of F, that is, y* = Fy* = f(y*,..., y*). Using (4), we have

On the other hand, we have

Then we get

which implies (since δ < 1) that x* = y*.

Theorem 2.2 Let (X, d) be a complete metric space, k a positive integer and f : Xk X. Suppose that

(5)

for all x0, ..., xk X, where δ1, ..., δk are positive constants such that δ ∈ (0,1) with , and B ≥ 0. Then

(a) there exists a unique x* ∈ X such that x* = f(x*,..., x*);

(b) the sequence {xn} defined by

(6)

converges to x* for any x0, ..., xk-1 X.

Proof. Applying Theorem 2.1 with δk + 1 = 0, and remarking that ∈ Θ, we obtain immediately (a). Now, we shall prove (b). Let x0,..., xk-1 X and xn = f(xn-k,..., xn-1), n k. Then by (5), the property (ii) of θ and since x* = Fx* = f(x*,..., x*), we have

Since k is a fixed positive integer, then we may denote

Then we get

Similarly we get that

Denoting

we get

Continuing this process, for n k, we obtain

Denoting

the above inequality becomes

(7)

Now, we shall prove that the sequence {En} given by

converges to 0 as n → ∞.

For n k, from (5), we have

As d(xn, f(xn-k,..., xn-1) = 0, the above inequality leads to

According to Lemma 2.1, this implies the existence of τ ∈ (0,1) and L ≥ 0 such that

Now, En is a finite sum of sequences converging to 0, so it is convergent to 0.

Finally, using (7) and applying Lemma 2.2 with an = d(xn, x*) and bn = En + 1-k, we get that d(xn, x*) → 0 as n → ∞, that is, the iterative sequence {xn} converges to the unique fixed point of f

Remark 2.1 In the particular case θ(t1, t2, t3, t4) = min{t1, t2, t3, t4}, from Theorem 2.2 we obtain Păcurar's result (see Theorem 1.5).

Now, we shall prove the following result.

Theorem 2.3 Let (X, d) be a complete metric space, k a positive integer and f : Xk X.

Suppose that

(8)

for all x0, ..., xk X, where a is a positive constant such that A ∈ (0, 1/2) with . Then

(a) there exists a unique x* ∈ X such that x* = f(x*, ..., x*);

(b) the sequence {xn} defined by

(9)

converges to x* for any x0, ..., xk-1 X, with a rate estimated by

(10)

where L ≥ 0, τ ∈ (0, 1) and M = τ1-k + 2τ2-k + ⋯ + k.

Proof. (a) follows immediately from Theorem 2.1 with δ = 0 and δk+1 = a. Now, we shall prove (b). Let x0, ..., xk-1 X and xn = f(xn-k, ..., xn-1), n k. Then by (8), the property (ii) of θ and since x* = Fx* = f(x*,..., x*), we have

(11)

Using (4), for all i = 0,1,..., k - 1, we get

This implies that

(12)

Now, combining (12) with (11), we obtain

Similarly, one can show that

(13)

This implies that

Define the sequence { Δp} by

We get that

Since , we can apply Lemma 2.1 to deduce that there exist L ≥ 0 and τ ∈ (0,1) such that

(14)

This implies that Δp → 0 as p → ∞, that is, xp x* as p → ∞. Finally, (10) follows from (14) and (13).

Remark 2.2 Many results can be derived from our Theorems 2.1, 2.2 and 2.3 with respect to particular choices of θ (see Example 2.1).

Remark 2.3 Clearly, Theorem 1.4 of Păcurar is a particular case of our Theorem 2.3.

### 3 Application: convergence of the recursive matrix sequence

In the last few years there has been a constantly increasing interest in developing the theory and numerical approaches for Hermitian positive definite (HPD) solutions to different classes of nonlinear matrix equations (see [37-41]). In this section, basing on Theorem 1.3 of Ćirić and Prešić, we shall study the nonlinear matrix difference equation

(15)

where Q is an N × N positive definite matrix, A and B are arbitrary N × N matrices, α and β are real numbers. Here, A* denotes the conjugate transpose of the matrix A.

We first review the Thompson metric on the open convex cone P(N) (N ≥ 2), the set of all N × N Hermitian positive definite matrices. We endow P(N) with the Thompson metric defined by

where M(A/B) = inf{λ > 0 : A ≤ λB} = λ+(B-1/2AB-1/2), the maximal eigenvalue of B-1/2AB-1/2. Here, X Y means that Y - X is positive semi-definite and X < Y means that Y - X is positive definite. Thompson [42] has proved that P(n) is a complete metric space with respect to the Thompson metric d and d(A, B) = |log( A-1/2BA-1/2)|, where |⋅| stands for the spectral norm. The Thompson metric exists on any open normal convex cones of real Banach spaces; in particular, the open convex cone of positive definite operators of a Hilbert space. It is invariant under the matrix inversion and congruence transformations, that is,

(16)

for any nonsingular matrix M. The other useful result is the nonpositive curvature property of the Thompson metric, that is,

(17)

By the invariant properties of the metric, we then have

(18)

for any X, Y P(N) and nonsingular matrix M.

Lemma 3.1 [40]For all A, B, C, D P(N), we have

In particular,

#### 3.1 A convergence result

We shall prove the following convergence result.

Theorem 3.1 Suppose that λ = max{|α|, |β|} ∈ (0,1). Then

(i) Equation (15) has a unique equilibrium point in P(N), that is, there exists a unique U P(N) such that

(ii) for any X0, X1 > 0, the iterative sequence {Xn} defined by (15) converges to U.

Proof. Define the mapping f : P(N) × P(N) → P(N) by

Using Lemma 3.1 and properties (16)-(18), for all X, Y, Z P(N), we have

Thus we proved that

for all X, Y, Z P(N). Since λ ∈ (0, 1), (i) and (ii) follow immediately from Theorem 1.3.

### 3.2 Numerical experiments

All programs are written in MATLAB version 7.1.

We consider the iterative sequence {Xn} defined by

(19)

where

and

It is clear that from our Theorem 3.1, Eq.(19) has a unique equilibrium point U P(3). We denote by Rm (m ≥ 1) the residual error at the iteration m, that is,

where |⋅| is the spectral norm.

After 40 iterations, we obtain

with residual error

The convergence history of the algorithm (19) is given by Figure 1.

Figure 1. Convergence history for Equation (19).

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

### References

1. Banach, S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund Math. 3, 133–181 (1922)

2. Agarwal, RP, Meehan, M, O'Regan, D: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)

3. Azam, A, Arshad, M, Beg, I: Fixed points of fuzzy contractive and fuzzy locally contractive maps. Chaos Solitons Fract. 42(5), 2836–2841 (2009). Publisher Full Text

4. Babu, GVR, Sandhya, ML, Kameswari, MVR: A note on a fixed point theorem of Berinde on weak contractions. Carpathian J Math. 24, 8–12 (2008)

5. Berinde, V: On the approximation of fixed points of weak contractive mappings. Carpathian J Math. 19(1), 7–22 (2003)

6. Berinde, V: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal Forum. 9, 43–53 (2004)

7. Berinde, V: Iterative Approximation of Fixed Points. Lecture Notes in Mathematics, Springer, Berlin (2007)

8. Chen, YZ: A Prešić type contractive condition and its applications. Nonlinear Anal. 71, 2012–2017 (2009). Publisher Full Text

9. Ćirić, LB: Generalized contractions and fixed point theorems. Publ L'Inst Math. 12(26), 19–26 (1971). PubMed Abstract

10. Ćirić, LB: On contraction type mappings. Math Balkanica. 1, 52–57 (1971)

11. Ćirić, LB: On a family of contractive maps and fixed points. Publ L'Inst Math. 17, 45–51 (1974). PubMed Abstract

12. Ćirić, LB: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos Solitons Fract. 42(1), 146–154 (2009). Publisher Full Text

13. Ćirić, LB, Prešić, SB: On Presic type generalisation of Banach contraction mapping principle. Acta Math Univ Com. LXXVI(2), 143–147 (2007)

14. Imdad, M, Ali, J, Tanveer, M: Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces. Chaos Solitons Fract. 42(5), 3121–3129 (2009). Publisher Full Text

15. Kannan, R: Some results on fixed points. Bull Calcutta Math Soc. 60, 71–76 (1968)

16. Khan, MS, Swaleh, M, Sessa, S: Fixed point theorems by altering distances between the points. Bull Aust Math Soc. 30, 1–9 (1984). Publisher Full Text

17. Meir, A, Keeler, E: A theorem on contraction mappings. J Math Anal Appl. 28, 326–329 (1969). Publisher Full Text

18. Miheţ, D: Fixed point theorems in probabilistic metric spaces. Chaos Solitons Fract. 41(2), 1014–1019 (2009). Publisher Full Text

19. Nadler, SB Jr.: Multi-valued contraction mappings. Pac J Math. 30, 475–488 (1969)

20. Păcurar, M: Approximating common fixed points of Presić-Kannan type operators by a multi-step iterative method. An Şt Univ Ovidius Constanţa. 17(1), 153–168 (2009). PubMed Abstract

21. Păcurar, M: Fixed points of almost Persić operators by a k-step iterative method. An Ştiinţ Univ Al I. Cuza Iaşi. Mat (N.S.) Tomul. LVII, (2011). PubMed Abstract

22. Prešić, SB: Sur une classe d'inéquations aux differences finies et sur la convergence de certaines suites. Pub de l'Inst Math Belgrade. 5(19), 75–78 (1965)

23. Rao, KPR, Kishore, GNV, Mustaq Ali, Md: A generalization of the Banach contraction principle of Presic type for three maps. Math Sci. 3(3), 273–280 (2009)

24. Rhoades, BE: A comparison of various definitions of contractive mappings. Trans Am Math Soc. 226, 257–290 (1977)

25. Rus, IA: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)

26. Smart, DR: Fixed Point Theorems. Cambridge University Press, Cambridge (1974)

27. Suzuki, T: A generalized Banach contraction principle which characterizes metric completeness. Proc Am Math Soc. 136, 1861–1869 (2008)

28. Vetro, F: On approximating curves associated with nonexpansive mappings. Carpathian J Math. 27(1), 142–147 (2011)

29. Godunov, SK, Ryabenku, VS: Difference Schemes. North-Holland, Amsterdam (1987)

30. Goldberg, S: Introduction to Difference Equations. Wiley, New York (1958)

31. Johnson, RM: Theory and Applications of Linear Differential and Difference Equations. Wiley, New York (1984)

32. Agarwal, RP: Difference Equations and Inequalities. Marcel Dekker, New York (1992)

33. Kelley, WG, Peterson, A: Diference Equations. Academic press, New York (1990)

34. Lakshmikautham, V, Frigiante, D: Theory of Difference Equations. Academic press, New York (1990)

35. Rus, IA: An iterative method for the solution of the equation x = f(x, ..., x). Rev Anal Numer Theory Approx. 10(1), 95–100 (1981)

36. Rus, IA: An Abstract Point of View in the Nonlinar Difference Equations. pp. 272–276. Editura Carpatica, Cluj-Napoca (1999)

37. Dehgham, M, Hajarian, M: An efficient algorithm for solving general coupled matrix equations and its application. Math Comput Model. 51, 1118–1134 (2010). Publisher Full Text

38. Duan, X, Liao, A: On Hermitian positive definite solution of the matrix equation . J Comput Appl Math. 229, 27–36 (2009). Publisher Full Text

39. Liao, A, Yao, G, Duan, X: Thompson metric method for solving a class of nonlinear matrix equation. Appl Math Comput. 216, 1831–1836 (2010). Publisher Full Text

40. Lim, Y: Solving the nonlinear matrix equation via a contraction principle. Linear Algebra Appl. 430, 1380–1383 (2009). Publisher Full Text

41. Zhoua, B, Duana, G, Li, Z: Gradient based iterative algorithm for solving coupled matrix equations. Syst Control Lett. 58, 327–333 (2009). Publisher Full Text

42. Thompson, AC: On certain contraction mappings in a partially ordered vector space. Proc Am Math soc. 14, 438–443 (1963)