Common Fixed Point Theorems for the Pair of Mappings in Hilbert Space

Durdana Lateef and A. Bhattacharya

Department of Mathematics, Jadavpur University, Kolkata- 700 032 (India).

 

ABSTRACT:

In this paper common fixed point theorem for the pair of mapping satisfying different contractive condition in Hilbert space has been proved.

KEYWORDS: Fixed point; Hilbert space; contraction mapping; Banach space

Copy the following to cite this article: Lateef D, Bhattacharya A. Common Fixed Point Theorems for the Pair of Mappings in Hilbert Space . Orient. J. Comp. Sci. and Technol;3(2)

Copy the following to cite this URL: Lateef D, Bhattacharya A. Common Fixed Point Theorems for the Pair of Mappings in Hilbert Space. Orient. J. Comp. Sci. and Technol;3(2). Available from: http://www.computerscijournal.org/?p=2342

Introduction

Most of fixed point theorems for mappings in metric spaces satisfying different contraction conditions may be extended to the abstract spaces, like Hilbert, Banach and locally convex spaces etc with some modifications. Some such interesting classes of contraction by Ciric¹,Dotson² proved fixed point theorems for non-expansive mappings on star shaped subsets of Banach spaces (i.e. || Tx-Ty|| ≤ ||x-y|| for x,y ∈ C). Then T has a fixed point in C. Pandhare and Waghmode³ have proved class of pairs of generalized contraction type mapping in Hilbert space on the line of Ciric¹ and proved some common fixed point theorems and some such interesting classes of contraction introduced by Kannan⁴. Sayyed and Badshah⁵ proved a class of pair of generalized contraction type mapping in Hilbert space. The result of this theorem is inspired by the results due to Dubey⁶, Naimpally and Singh⁷.

Definition

Let X be a Banach space and C be a non-empty subset of X. Let T₁, T₂ : C → C be two mappings. The iteration scheme called I-scheme is defined as follows :x0 ∈ C, …(1)
y2n = β2nT1x2n + (1- β2n)x2n, n ≥ 0
x2n+1 = (1-α2n)x2n + α2nT2y2n, n≥ 0 …(2)
y2n+1 = β2n+1T1x2n+1 + (1- β2n+1)x2n+1, n ≥ 0
x2n+2 = (1-α2n+1)x2n+1 + α2n+1 T2y2n+1, n≥0 …(3)

In the Ishikawa scheme, {α2n}, {β2n} satisfy

0 ≤ α2n ≤ β2n ≤ 1, for all n lim
n→∞ . β2n = 0 and Σα2nβ2n = ∞.

In this paper we shall make the assumption that

(i) 0 ≤ α2n ≤ β2n ≤ 1, for all n,
(ii) lim _n→∞ α2n = α2n > 0, and
(iii) lim _n→∞ β2n = β2n < 1.

We know that Banach space is Hilbert if and only if its norm satisfies the parallelogram law i.e. every x,y ∈ X (Hilbert space).

||x + y||2 + ||x – y||2 = 2||x||2 + 2||y||2 …(4)
which implies, ||x + y||2 ≤ 2||x||2 + 2||y||2 …(5)

We often use this inequality throughout the result.

Further, we prove the result concerning the existence of common fixed point of pairs of mappings satisfying the contraction condition of the type

||Tx-Ty||2 ≤ h Max { ||x – y||2, ||x – Tx||2,
||y – Ty||2,1/4(||x-Ty||2 +||y – Tx||2)}   …(6)

Theorem

Let X be a Hilbert space and C be a closed, convex subset of X. Let T₁ and T₂ be two sets of mapping satisfying

||T1x–T2y||2 ≤ hMax{||x – y||2, ||x – T1x||2,
||y–T2y||2,1/4(||x–T2y||2+||y–T1x||2)} …(7)

where h is real number satisfying 0 ≤ h < 1. If there exists a point x₀ such that the I-scheme for T₁ and T₂ defined by (2) and (3) converges to a point p, then p is common fixed point of T₁ and T₂.

Proof

It follows from (2) that x_2n+1 – x_2n = α_2n (T₂y_2n– x_2n). Since x_2n → p, ||x_2n+1 – x_2n|| → 0. Since {α_2n} is bounded away from zero, ||T₂y_2n – x_2n|| → 0. It also → 0. Since T₁ and T₂ satisfiesfollowsthat||p–T₂y_n|| (7), we have

||T1x2n–T2y2n||2≤hMax{||x2n-y2n||2,||x2n–T1x2n||2, ||y2n – T2y2n||2, 1/4 (||x2n–T2y2n||2+||y2n–T1x2n||2)}      …(8)

Now, ||y2n– x2n||2 =||β2nT1x2n +(1- β2n) x2n– x2n||2 =||β2nT1x2n+x2n -β2nx2n– x2n||2 = ||β2n(T1x2n – x2n) ||2 = β2n 2 || (T1x2n – T2y2n) + (T2y2n – x2n) ||2 ≤2β2n 2||T1x2n–T2y2n||2+2β2n 2||(T2y2n–x2n)||2 ≤ 2||T1x2n–T2y2n||2+2||(T2y2n–x2n)||2       …(9)

||y2n – T2y2n||2 = ||β2nT1x2n + (1- β2n) x2n – T2y2n||2 =||β2nT1x2n+(1-β2n)x2n-T2y2n+β2nT2y2n–β2nT2y2n||2 =||β2n(T1x2n – T2y2n) + (1- β2n) (x2n – T2y2n) ||2 ≤ 2β2n 2 ||T1x2n – T2y2n||2 + 2(1- β2n) || x2n – T2y2n||2 ≤ 2||T1x2n – T2y2n||2 + 2||x2n – T2y2n||2      …(10)

||y2n – T1x2n||2 = ||β2nT1x2n + (1- β2n) x2n – T1x2n||2 = || (1- β2n)(x2n – T1x2n) ||2 = (1- β2n)2 ||x2n – T1x2n||2 = (1-β2n)2 ||(x2n – T2y2n) + (T2y2n – T1x2n)||2 ≤2(1-β2n)2||x2n–T2y2n||2+2(1-β2n)2||T2y2n–T1x2n||2 ≤ 2||x2n–T2y2n||2 + 2 ||T2y2n – T1x2n||2      …(11)

from (8), (9), (10) and (11) can be written as :

||T1x2n–T2y2n||2≤hmax {2||T1x2n – T2y2n||2 + 2 ||T2y2n – x2n||2), 2||x2n – T2y2n||2 + 2 ||T2y2n – T1x2n||2, 2||T1x2n – T2y2n||2 + 2||x2n – T2y2n||2), 1/4(3||x2n–T2y2n||2+2||T2y2n– T1x2n||2,)} ≤ h (2||T1x2n – T2y2n||2 + 2||T2y2n – x2n||2) ≤ 2h/1-2h ||x2n – T2y2n||2 Taking limit as n → ∞, we get ||T1x2n – T2y2n|| → 0. It follows that ||x2n – T1x2n||2 ≤ 2||x2n – T2y2n||2 + 2||T2y2n – T1x2n||2 → 0. And ||p – T1x2n||2 ≤ 2||p – x2n||2 + 2||x2n – T1y2n||2 → 0 as n → ∞.

If x2n, p satisfies (7), we have ||T1x2n–T2p||2≤ h max {||x2n – p||2, ||x2n – T1x2n||2, ||p – T2p||2, 1/4(||x2n – T2p||2 + ||p – T1x2n||2)} ≤ h max {||x2n – p||2, ||x2n – T1x2n||2, ||p – x2n + x2n – T2p||2, 1/4(||x2n–T1x2n+T1x2n–T2p||2+||p – T1x2n||2)}

Using inequality (5),we have||T1x2n–T2p||2≤h max {||x2n – p||2, ||x2n – T1x2n||2, 2||x2n – p||2, + 4||x2n–T1x2n||2 + 4||T1x2n – T2p||2, 1/4(2||x2n – T1x2n||2 + 2||T1x2n – T2p||2 + ||p – T1x2n||2)}

Taking limit as n → ∞, we get ||T1x2n – T2p|| → 0.

Finally, ||p – T2p||2 = ||p – T1x2n + T1x2n – T2p||2 ≤ 2||p – T1x2n||2 + 2||T1x2n – T2p||2 → 0, as n → ∞.

Showing that p = T2 p. Similarly, we can prove that p = T1 p. Thus p is a common fixed point of T1 and T2. This completes the proof.

References

1. Ciric, L.B.,”Generalized contraction and fixed point theorms” Publ. Inst. Math. (Beograd) (N.S.), 12: 19-26 (1971)
2. Dotson, J.W.G., “Fixed point theorems for non-expesive mappings on star shaped subsets of Banach spaces”J.Lon.Math. Soc.4(1972),403-410.
3. Pandhare, D.M. and Waghmod, B.B., “Generalized contraction and fixed point theorems in Hilbert spaces” Acta Ciencia Indica XXIIM: 145-150 (1997).
4. Kannan, R., “Fixed point theorems in reflexive Banach spaces” proc. Amer. Math. Soc. 18:111-118 (1973).
5. Sayyed, F. and Badshah, V.H., “Generalized contraction and common fixed point theorem in Hilbert space” J. Indian Acad. Math. 23: 2 (2001).
6. Dubey, B.N., “Generalization of fixed point theorems of Naimpally and sing” Acta Ciencia Indica.XVII, M. 3: 509-514 (1991).
7. Naimpally, S.A. and Singh, K.L., “Extensions of some fixed point theorems of Rhoades” J. Math. Anal. Appl.96: 437-446 (1983).

---

This work is licensed under a Creative Commons Attribution 4.0 International License.