Upload
ngotu
View
225
Download
0
Embed Size (px)
Citation preview
Chapter 6 Stability of Ishikawa Iteration for Nonexpansive Type Condition
This Chapter contains some results on stability of Ishikawa iteration procedures
for pair of multi-valued maps satisfying nonexpansive type condition. The nonexpansive
type condition used is a generalization of many well known contractive conditions. Our
result generalizes many well established results on stability of ishikawa iterates available
in the literature.
Preliminaries
Let (X, d) be a complete metric space, T a selfmap of X. Let 푥 ∈ X,
푥 =f(T, 푥 ) denote an iteration procedure which yields a sequence of points
{푥 }. Suppose that {푥 } converges to a fixed point p of T. Let {푦 }⊂ 푋, 휀 =
푑 푦 , 푓(푇, 푦 ) . If lim휀 = 0, implies that 푦 =p, then the iteration procedure is
said to be T stable.
The study of iteration procedures was initiated by Urbe [144], however
Harder and Hicks [46] gave a formal definition of stability of iterative procedures.
From literature it appears that Ostrowski [91] was the first to discuss stability of
iteration procedures in metric space. Because of increasing use of computational
mathematics and revolution in computer programming convergency and stability
results for certain classes of mappings have been extensively studied by many
authors(see [14], [23], [24], [46], [47], [52], [57], [80], [81], [82], [89] and the
references there in). Harder and Hicks [46], [47] pointed out that the study of
stability is theoretically as well as numerically interesting.
Estelar
Stability of Ishikawa Iteration for nonexpansive type condition
104 | P a g e
Mann in 1953 introduced iterative schemes and employed this iterative
scheme to obtain the convergence of functions to fixed point. Rhoades [116]
established results which showed that continuous self maps of a closed and
bounded interval converges to a fixed point of the function but Mann iteration
process sometime fails to converge for Lipschitzian pseudo-contractive maps.
Ishikawa [57] introduced a new iterative process for the convergence of such
maps. In last decade an extensive work has been done by researcher around
convergence of Mann and Ishikawa iteration process for single-valued and
multivalued mappings under various contractive conditions (see [14], [23], [24],
[46], [47], [52], [57], [80], [81], [82], [89] and the references there in).
B.E. Rhoades [116] established the following result for the convergence of
Mann iteration procedure.
Theorem6.1. Let T be a self-map of a closed convex subset K of a real Banach
space (X, d). let {푥 } be a general Mann iteration of T that converges to a
point p ∈ X. If there exists the constants 훼,훽, 훾 ≥ 0,훿 < 1 such that
‖푇푥 − 푇푝‖ ≤ 훼‖푥 − 푝‖ + 훽‖푥 − 푇푥 ‖+ 훾‖푝 − 푇푥 ‖
+ 훿 max{‖푝 − 푇푝‖,‖푥 − 푝‖}
Then p is fixed point of T.
It is to be noted that, in above theorem, if T is continuous then Mann
iteration process converges to fixed point of T. But if it is not then there is no
guarantee that Mann iterative process converges and even if it converges, it will
not necessarily converge to a fixed point of T. Hu et.al [52] solved this problem
by extending above theorem to the Ishikawa iteration process.
Theorem6.2.[52]. Let K be a compact convex subset of a Hilbert space, 푇:퐾 →
퐾 a Lipschitzian pseudo-contractive map and 푥 ∈ 푋. Then the Ishikawa iteration
{푥 } defined as:
Estelar
Stability of Ishikawa Iteration for nonexpansive type condition
105 | P a g e
푥 = 훼 푇[훽 푇푥 + (1 − 훽 )푥 ] + (1− 훼 )푥 ,
where {훼 } and {훽 } are sequence of positive number that satisfy following
three conditions.
I. 0 ≤ 훼 ≤ 훽 ≤ 1, for all positive integer n.
II. lim → 훽 = 0,
III. s∑ 훼 훽 = ∞,
converges strongly to a fixed point of T.
We need the following Lemma due to Nadler [88] for our main result.
Lemma6.1.[88]. If A, B ∈ CB(X) and a ∈ A, then for 휖 > 0 there exists b ∈ B
such that
d(a, b)≤ H(a, B) + 휖.
The Ishikawa iteration scheme for two multivalued mappings is defined as
follows:
Definition6.1. Let K is a nonempty subset of X. and S,T : 퐾 → 퐶퐵(푋). Ishikawa
iteration is defined as:
푥 ∈ 퐾 푦 = (1 − 훽 )푥 + 훽 푎 ,푎 ∈ 푇푥 푥 = (1 − 훼 )푥 + 훼 푏 , 푏 ∈ 푆푦
(6.1)
Where 0 ≤ 훼 , 훽 ≤ 1 for all n.
Recently for Ishikawa iterates for multivalued mappings on a Banach
space Rhoades [111] established a generic theorem with number of corollaries.
Theorem6.3. Let X be a Banach space, K a closed convex subset of X. S and T are
multivalued mappings from K to CB(X). Suppose that the Ishikawa iteration
scheme (6.1), with {훼 } satisfying:
Estelar
Stability of Ishikawa Iteration for nonexpansive type condition
106 | P a g e
(i) 0 ≤ 훼 , 훽 ≤ 1 for all n,
(ii) lim 푖푛푓훼 = 푑 > 0and {푎 },{푏 }, satisfying
‖푎 − 푏 ‖ ≤ 퐻(푇푥 ,푆푦 ) + 휖 with lim 휖 = 0 (6.2)
Converges to a point p. If there exists non-negative number 훼,훽, 훾, 훿 with 훽 ≤
1 such that for sufficiently large n, S and T satisfying
퐻(푇푥 푆푦 ) ≤ 훼‖푥 − 푏 ‖ + 훽‖푥 − 푎 ‖ (6.3)
and
퐻(푆푝,푇푥 ) ≤ 훼‖푥 − 푝‖ + 훾푑(푥 ,푇푥 ) + 훿푑(푝,푇푥 )
+훽 max {푑(푝, 푆푝),푑(푥 ,푆푝)} (6.4)
then p is a fixed point of S. If also
퐻(푆푝,푇푝) ≤ 훽[푑(푝,푇푝) + 푑(푝, 푠푝)]. (6.5)
then p is a common fixed point of S and T.
Recently Singh and Dimri [129] extended and generalized the
results of Hu et.al [52]and Rhoades [111], [116] and established a common fixed
point theorem for Ishikawa iterates for a pair of multivalued mappings in Banach
space satisfying nonexpansive type condition. Now we generalize the result of
Singh and Dimri [133] for more general nonexpansive type condition. Our result
serves as a generalization of results due to Hu et.al [52] Rhoades [111], [116] and
Singh and Dimri [129], as well as several well known results.
Main results
Theorem6.4. Let K be a nonempty, closed, convex subset of Banach space X and
T, S: → 퐶퐵(푋) satisfying
Estelar
Stability of Ishikawa Iteration for nonexpansive type condition
107 | P a g e
H(Tx, Ty) ≤ a ‖푥 − 푦‖ +b{d(x, Tx)+ d(y, Sy)}
+ c [d(x, Sy) + d(y, Tx)], (6.6)
Where a >0, 0 ≤ b<1,0 ≤ c<1 are such that
a+ b+ c = 1.
for all x, y ∈ K. If there exists 푥 ∈ 퐾 such that the {푥 }satisfying (6.1), (6.2)with
(i) 0 ≤ 훼 , 훽 ≤ 1 for all n,
(ii) lim 푖푛푓훼 = 푑 > 0 and {푎 }, {푏 }, satisfying
‖푎 − 푏 ‖ ≤ 퐻(푇푥 ,푆푦 ) + 휖
(iii)훽 =0
Converges to a point p, then p is a common fixed point of S and T.
Proof. Since in Theorem6.3 Rhaodes [111] showed that the conditions (6.3),(6.4)
and (6.5) are enough for the convergence of Ishikawa iteration. So for the proof of
our theorem it is sufficient to show that S and T satisfy the above mention
conditions,
From (6.6) we have
퐻(푇푥 ,푆푦 ) ≤ a ‖푥 − 푦 ‖ +b{d(푥 , T푥 )+ d(푦 , S푦 )}
+ c [d(푥 , S푦 ) + d(푦 , T푥 )], (6.7)
From (6.1) we have
Estelar
Stability of Ishikawa Iteration for nonexpansive type condition
108 | P a g e
⎩⎪⎪⎪⎨
⎪⎪⎪⎧‖푥 − 푦 ‖ = 훽 ‖푥 − 푎 ‖ 푑(푥 ,푇푥 ) ≤ ‖푥 − 푎 ‖ 푑(푦 ,푆푦 ) ≤ ‖푦 − 푏 ‖ ≤ ‖푦 − 푥 ‖+ ‖푥 − 푏 ‖
≤ 훽 ‖푥 − 푎 ‖ + ‖푥 − 푏 ‖,푑(푥 ,푆푦 ) ≤ ‖푥 − 푏 ‖ 푑(푦 ,푇푥 ) ≤ ‖푦 − 푎 ‖ = ‖푦 − 푥 ‖ + ‖푥 − 푎 ‖
≤ (1 + 훽 )‖푥 − 푎 ‖
(6.8)
Now using (6.8) we get
d(푥 , T푥 )+ d(푦 , S푦 ) ≤ ‖푥 − 푎 ‖ +훽 ‖푥 − 푎 ‖ + ‖푥 − 푏 ‖
≤ (1+ 훽 ) ‖푥 − 푎 ‖+ ‖푥 − 푏 ‖ (6.9)
Also
d(푥 , S푦 ) + d(푦 , T푥 ) ≤ ‖푥 − 푏 ‖ + (1 + 훽 )‖푥 − 푎 ‖ (6.10)
using (6.10), (6.9) and (6.8) in (6.7) we get
퐻(푇푥 ,푆푦 ) ≤ a‖푥 − 푎 ‖ + b {(1+ 훽 ) ‖푥 − 푎 ‖+ ‖푥 − 푏 ‖}
c{‖푥 − 푏 ‖ + (1 + 훽 )‖푥 − 푎 ‖ }
≤[ a+b+c] ‖푥 − 푎 ‖ +[ b+c] ‖푥 − 푏 ‖
and since a+b+c=1 implies b+c <1, we can say that (6.3) is satisfied. Again
using (6.6) we get
퐻(푇푥 ,푆푝) ≤ a ‖푥 − 푝‖ +b{d(푥 , T푥 )+ d(푝, S푝)}
+ c [d(푥 , S푝) + d(푝, T푥 )],
≤ a ‖푥 − 푝‖ +b{‖푥 − 푎 ‖ +d(푝, S푝)}
+ c [d(푥 , S푝) + d(푝, 푎 )] (6.11)
Since (6.3) is satisfied, therefore using (6.2), we get
Estelar
Stability of Ishikawa Iteration for nonexpansive type condition
109 | P a g e
‖푥 − 푎 ‖ ≤ ‖푥 − 푏 ‖ + ‖푏 − 푎 ‖
≤ ‖푥 − 푏 ‖ + 퐻(푇푥 ,푆푦 ) + 휖
≤ ‖푥 − 푏 ‖ + ‖푥 − 푏 ‖
+(푏 + 푐)‖푥 − 푎 ‖ + 휖
Since lim‖푥 − 푏 ‖= 0, we obtain
lim sup‖푥 − 푎 ‖ ≤ (푏 + 푐)lim 푠푢푝‖푥 − 푎 ‖
and since 0≤ (푏 + 푐) ≤ 1, which implies,
lim ‖푥 − 푎 ‖ = 0. (6.12)
Also
‖푝 − 푎 ‖ ≤ ‖푝 − 푥 ‖ + ‖푥 − 푎 ‖ (6.13)
Using (6.11), (6.12) and (6.13) we get
퐻(푇푥 ,푆푝) ≤ a ‖푥 − 푝‖+b d(푝, S푝) + c[d(푥 , S푝)+ ‖푝 − 푥 ‖ + ‖푥 − 푎 ‖]
≤ (푎 + 푐)‖푥 − 푝‖ + b d(푝, S푝) + c d(푥 , S푝). (6.14)
The conditions (푏 + 푐) > 0 and 0 ≤ b<1,0 ≤ c<1 implies that condition (6.4)
is a special case of (6.14). Hence (6.14) implies (6.4). Since (6.3) and (6.4) is
satisfied hence by Theorem 6.1 p is a fixed point of S. Also we can see that
condition (6.5) is also satisfied hence p is a common fixed point of S, T.
Estelar
Stability of Ishikawa Iteration for nonexpansive type condition
110 | P a g e
Remark1. It is remarkable that the nonexpansive type condition used in Theorem
6.3 contains the condition used in Singh and Dimri [129], and many
other contractive conditions.
Remark2. The result established by Singh and Dimri [129] and Rhoades
[111][116] are special cases of our result and can be obtained as
corollaries.
*****
Estelar
Bibliography
1. Aamri, M. and Moutawakil, D.: Common fixed points under contractive
conditions in symmetric spaces, Appl. Math. E-notes, 3(2003), 156-162.
2. Aamri, M. and Moutawakil, D.: Some new common fixed point
theorems under srrict contractive conditions, J. Math. Anal. Appl., 270
(2002), 181-188.
3. Aamri, M., Bassou, A. and Moutawakil, D.E.: Common fixed points for
weakly compatible maps in symmetric spaces with applications to
probabilistic spaces, Appl. Math. E-notes, 5(2005), 171-175.
4. Alghamdi, M.A., Radenovic, S., Shahzad, N.: On some generalizations
of commuting mappings, Abstr. Appl. Anal., 2012 (2012).
5. Aliouche, A. and Merghadi, F.: A common fixed point theorem via a
generalized contractive condition, Annal. Math. info., 36(2009), 3-14.
6. Altun, I and Durmaz, G.: Some fixed point theorems on ordered cone
metric spaces, Rend. Cir. Mat. Palero, 58 (2009), 319-325.
7. Altun, I., Damjanovic, B. and Doric, D.: Fixed point and common fixed
point theorems on ordered cone metric spaces, Appl. Math. Lett., 23
(2010), 310-316.
8. Amini-Harandi, A. and Emami, H.: A fixed point theorem for
contraction type maps in partially ordered metric spaces and application to
ordinary differential equations, Nonlinear Anal., 72 (2010), 2238-2242.
Estelar
Bibliography
112 | P a g e
9. Azam, A. and Arshad, M.: Fixed points of a sequence of locally
contractive multivalued maps, Comput. Math. Appl., 57 (2009), 96-100.
10. Babu, G.V.R., Alemayehu, G.M.: Common fixed point theorems for
occasionally weakly compatible maps satisfying property (E. A) using an
inequality involving quadratic terms, Appl. Math. Lett., 24(2011), 975–
981.
11. Balasubrmaniam, P., Murlishankar, S.M. and Pant, R.P.: Common
Fixed Points of Four Mappings in a Fuzzy Metric Space, J. Fuzzy Math.,
10-2 (2002), 379-384.
12. Barros, L. C., Bassanezi, R. C., Tonelli, P. A.: Fuzzy modeling in
population dynamics. Ecol Model., 128(2000), 27–33.
13. Beg, I. and Butt, A. R.: Common fixed point for generalized set valued
contractions satisfying an implicit relation in partially ordered metric
spaces, Math.Commun., 15 (2010), 65-75.
14. Berinde, V.: Iterative approximation of fixed points, Fditura Efemeride,
Baia Mare, 2002.
15. Bhatt A., Chandra H. and Sahu D.R.: Common fixed point theorems for
occasionally weakly compatible mappings under relaxed conditions,
Nonlinear Anal., 73 (2010), 176– 182. 16. Bonsall, F. F.: Lectures on some fixed point theorems of functional
analysis, Tata Institute, Bombay, 1962.
17. Bouhadjera, H. and Godet-Thobie, C.: Common fixed point theorems
for pair of subcompatible maps, 17 June 2009, arXiv: 0906.3159v1 [math.
FA].
Estelar
Bibliography
113 | P a g e
18. Boyce, W. B.: Commuting functions with common fixed point, Trans.
Amer. Math. Soc., 137 (1969), 77-92.
19. Boyd, D. W. and Wong, J. S.: On nonlinear contractions, Proc. Amer.
Math. Soc., 20 (1969), 458-464.
20. Browder, F. E.: Fixed point theorems for non-compact mappings in
Hilbert space, Proc. Natl. Acad. Sci. USA, 53 (1965), 1272–1276.
21. Caristi, J.: Fixed point theorems for mappings satisfying inwardness
conditions, Trans. Amer. Math. Soc., 215(1976), 241–251.
22. Chandra, M., Mishra, S.N., Singh, S.L., and Rhoades, B.E.:
Coincidence and fixed points of nonexapansive type multi-valued and
single-valued maps, Indian J. pure Appl. Math., 26(5)(1995), 393-401.
23. Chidume, C.E. and Osilike, M.O.: Fixed point iterations for quasi-
contractive maps in uniformly smooth Banach spaces, Bull. Kor. Math.
Soc., 30(1993), 201-212.
24. Chidume, C.E.: Approximation of fixed points of quasi-contractive
mappings in ܮ spaces, Indian J. Pure appl. Math., 22(1991), 273-286.
25. Cho, S.H.: On common fixed points in fuzzy metric space, Int. Math.
Forum, 1(10) 2006, 471-479.
26. Cho, Y.J., Pathak, H.K., Kang, S.M. and Jung, J.S.: Common Fixed
Points of Compatible Maps of Type () on Fuzzy Metric Spaces, Fuzzy
Sets and Systems, 93 (1998), 99-111.
Estelar
Bibliography
114 | P a g e
27. Chugh, R. and Kumar, S.: Common Fixed Point Theorem in Fuzzy
Metric Spaces, Bull. Calcutta Math. Soc., 94(1)(2002), 17-22.
28. Chugh, Renu and Kumar, Sanjay: Minimal commutativity and common
fixed points, J. Indian Math. Soc., 70(1-4)(2003), 169-177.
29. Chugh, Renu and Savita: Common fixed points of four R-weakly
commuting mappings, J. Indian Math. Soc., 70(1-4) (2003), 185-189.
30. Ciric, Lj. B.: A generalization of Banach contraction principle, Proc.
Amer. Math. Soc., 45(1974), 267-273.
31. Ciric, Lj. B.: Fixed Point for generalized multivalued contraction, Math.
Vesnik, 9(1972), 265-272.
32. Ciric, Lj. B.: On some nonexpansive type mappings and fixed points,
Indian J. Pure Appl. Math., 24(1993), 145-149.
33. Ciric´, Lj., Bessem, S. and Calogero, V.: Common fixed point theorems
for families of occasionally weakly compatible mappings, Math. Comput.
Modell., 53(2011), 631–636.
34. Collatz, L.: Functional analysis and numerical mathematics, Academic
Press, New York, 1966.
35. Connell, E. H.: Properties of fixed point spaces, Proc. Amer. Math. Soc.,
10 (1959), 974–979
36. Doric, D.,Kadelburg, Z.,Radenovic, S.: A note on occasionally weakly
compatible mappings and common fixed points, Fixed Point Theory
(2013).
37. Dugundji, J. and Granas, A.: Fixed Point Theory, vol. 1, PWNPolish
Scientific Publishers, Warszawa 1982.
Estelar
Bibliography
115 | P a g e
38. Feng, Y. and Liu, S.: Fixed Points theorems for multi-valued contractive
mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl.,
317(2006), 103-112.
39. Gairola, Ajay, Singh, Jay and Joshi, M.C.: Coincidence and Fixed Point
for Weakly Reciprocally Continuous Single-Valued and Multi-Valued
Maps, (in press) Demonstratio mathematica.
40. George, A. and Veeramani, P.: On Some Results in Fuzzy Metric
Spaces, Fuzzy Sets and Systems, 64(1994), 395-399.
41. George, A. and Veeramani, P.: On some results of analysis for the fuzzy
metric space, Fuzzy sets and systems, 90(1997), 365-368.
42. Giles, R. A.: A computer program for fuzzy reasoning, Fuzzy Sets Syst.
4(2000), 221–234.
43. Gopal, D. and Imdad, M.: Some new common fixed point theorems in
fuzzy metric spaces, Ann Univ Ferrara (2011), doi: 10.1007/s11565-011-
0126-4.
44. Gopal, D., Pant, R. P. and Ranadive, A. S.: Common fixed Points of
absorbing maps, Bulletin of the Marathwada Math. Soc., 9(1) (2008), 43-
48.
45. Grabiec, M., Fixed Points in Fuzzy Metric Space, Fuzzy Sets and
Systems, 27(1988), 385-389.
46. Harder, A. M., and Hicks, T. L.: A stable iteration procedure for
nonexpansive mappings, Math. Japonica, 33(1988), 693-706.
Estelar
Bibliography
116 | P a g e
47. Harder, A. M., and Hicks, T. L.: A stable iteration procedure for
nonexpansive mappings, Math. Japonica, 33(5)(1988), 687-692.
48. Harder, A.M.: Fixed point theory and stability results for fixed point
iterative procedures, Ph.D. thesis, University of Missouri-Rolla, Missouri,
1987.
49. Harjani, J. and Sadarangani, K.: Generalized contractions in partially
ordered metric spaces and applications to ordinary differential equations,
Nonlinear Anal., 72 (2010), 1188-1197.
50. Hicks, T.L. and Rhoades, B.E.: Fixed point in symmetric space with
application to probabilistic space, Nonlinear analysis, 36 (1999), 331-344.
51. Hicks, T.L: Fixed Point theorem for multivalued mappings, Indian J. Pure
Appl. Math., 29(2)(1998), 133-137.
52. Hu, T., Chi, Huang, j. and Rhoades, B.E.: A general principle for
Ishikawa iteration for multivalued mappings, Indian J. Pure Appl. Math.,
28(8)(1997), 1091-1098.
53. Hussain, N., Khamsi, M. A. and Latif, A.: Common fixed points for
JHoperators and occasionally weakly biased pairs under relaxed
conditions, Nonlinear Anal., 74 (2011), 2133-2140.
54. Imdad, M. and Ali, Javid: Reciprocal continuity and common fixed
points of nonself mappings, Taiwanese Journal of Mathematics, 13(5)
(2009), 1457-1473.
55. Imdad, M., Ali, Javid and Khan, Ladlay: Coincidence fixed points in
symmetric spaces under strict contractions, J. Math. Anal. Appl., 320
(2006), 352 – 360.
Estelar
Bibliography
117 | P a g e
56. Imdad, M., Ali, Javid and Tanveer, M.: Remarks on some recent
metrical common fixed point theorems, Appl. Math. Lett., (2011), doi:
10.1016/j.aml.2011.01.045.
57. Ishikawa, S.I.: Fixed point by new iteration method, Proc. Am. Math.
Soc., 143(1974), 147-150.
58. Itoh, S. and Takahashi, W.: Single-valued mappings, multivalued
mappings and fixed point theorems, J. Math. Anal. Appl., 59(1977), 514-
521.
59. Joshi, M. C. and Bose, R. K.: Some topics on nonlinear functional
analysis, Wiley East, Ltd, New Delhi, (1985).
60. Joshi, M.C., Joshi, L.K. and Dimri, R.C.: Fixed Point theorems for
multivalued mappings in symmetric spaces, Demonstratio mathematica,
XL(3)(2007), 733-738.
61. Jungck, G., and Rhoades, B. E.: Fixed point theorems for occasionally
weakly compatible mappings, Fixed Point Theory, 7(2)(2006), 287-296.
62. Jungck, G., and Rhoades, B. E.: Fixed points for set-valued functions
without continuity, Indian J. Pure Appl. Math., 29(3)(1998), 227-238.
63. Jungck, G., Commuting mappings and fixed points, Amer. Math.
Monthly ,83(4)(1976), 261-263.
64. Jungck, G.: Common fixed points for non-continuous non-self maps on
nonmetric spaces, Far East Journal of Mathematical Sciences, 4(1996),
199- 215.
Estelar
Bibliography
118 | P a g e
65. Jungck, G.: Compatible mappings and common fixed points, Internat. J.
Math. Sci., 9(1986), 771-779.
66. Kaleva, O.: The completion of fuzzy metric spaces, J. Math. Anal. Appl.,
109(1985), 194-198.
67. Kamran, T.: Coincidence and fixed points for hybrid strict contractions,
J. Math. Anal. Appl., 299 (2004), 235-283.
68. Kamran, T.: Fixed points of asymptotically regular noncompatible maps,
Demonstrato Math., XXXVIII (2) (2005), 485-494.
69. Kaneko, H., Sessa, S.: Fixed point for compatible multivalued and single-
valued mappings, Internat. J. Math. Math. Sci., 12(2)(1989), 257-262.
70. Kaneko, H.: A common fixed point of weakly commuting multivalued
mappings, Math. Japon., 33(5)(1988), 741-744.
71. Kannan, R.: Some results on fixed points - II, Amer. Math. Month., 76
(1969), 405-408.
72. Kannan, R.: Some results on fixed points, Bull. Cal. Math. Soc., 60
(1968), 71-76.
73. Kirk, W. and Sims, B.: Handbook of Metric Fixed Point Theorey,
Kluwer Academic Publishers, Dordrecht (2001).
74. Klim, D. and Wardowski, D.: Fixed point theorems for set-valued
contractions in complete metric spaces, J. Math. Anal. Appl.,
334(1)(2007), 132–139.
Estelar
Bibliography
119 | P a g e
75. Kramosil, O. and Michalek, J.: Fuzzy Metric and Statistical Metric
Spaces, Kybernetika, 11(1975), 326-334.
76. Kumar S. and Pant, B. D.: A common fixed point theorem in
probabilistic metric space using implicit relation, Filomat, 22(2008), 43-
52.
77. Kumar, S. and Chugh, R.: Common fixed point theorems using minimal
commutativity and reciprocal continuity conditions in metric space,
Scientiae Mathematicae Japonicae, 56 (2)(2002), 269-275.
78. Liu, Z., Sun Wei, Kang Shin Min, Ume, Jeong Sheok: On Fixed Point
theorem for multivalued Contraction, Fixed Point theory and applications,
2010.
79. Lopez, J.R. and Romaguera, S.: The Hausdorff fuzzy metric on compact
sets, fuzzy Sets and Systems, 147(2004), 273-283.
80. Mann, W.R.: Mean value methods in iteration, Proc. Amer., Math., Soc.,
4(1953), 506-510.
81. Mathpal, P.C.: Existence of Coincidence and fixed points of mappings
and stability of iterative procedure, PhD thesis, G.B. P.U. Ag. &Tech.
(2010).
82. Matkowski, J. and Singh, S.L.: Round-off stability of functional
iteration on product spaces, Indian J. Math., 39(3)(1997), 275-286.
83. Meir, A. and Keeler, E.: A theorem on contraction mappings, J. Math.
Anal. Appl., 28 (1969), 326 – 329.
Estelar
Bibliography
120 | P a g e
84. Mihet, D.: A note on a paper of Hicks and Rhoades, Nonlinear Analysis,
65(7) (2006), 1411-1413.
85. Moutawakil, D.: A fixed point theorem for multi-valued mappings in
symmetric spaces, Appl. Math. E-Notes, 4(2004), 26-32.
86. Muralisankar, S. and Kalpana, G.: Common fixed point theorem in
intuitionistic fuzzy metric space using general contractive condition of
integral type, Int. J. Cont. Math. Sci., 4(11)( 2009), 505 – 518.
87. Murthy, P. P.: Important tools and possible applications of metric fixed
point theory, Nonlinear Analysis, 47 (2001), 3479-3490.
88. Nadler, S.B. Jr.: Multi-valued Contraction mappings, Pacific J. Math.
30(1969), 475-488.
89. Ortega, J.M. and Rheinboldt, W.C.: Iterative solution of nonlinear
equations in several variables, Academic Press, New York, 1970.
90. Oslike, M.O.: Stability results for fixed point iterative procedures, J. Nig.
Math. Soc., 14/15(1995/1996), 17-29.
91. Ostrowski, A.M.: The round –off stability of iterations, Z. Angew. Math.
Mech., 47(1967), 177-181.
92. Pant, R. P. and Bisht, R. K.: Common fixed point theorems under a new
continuity condition, Ann Univ Ferrara, DOI: 10.1007/S 11565-011-0141-
5.
93. Pant, R. P. and Pant, V.: Some fixed point theorem in fuzzy metric
space, J. Fuzzy Math., 16(3)(2008), 599-611.
Estelar
Bibliography
121 | P a g e
94. Pant, R. P., Bisht, R. K. and Arora, D.: Weak reciprocal continuity and
fixed point theorems, Ann Univ Ferrara, 57 (1)(2011), 181-190.
95. Pant, R. P.: A common fixed point theorem under a new condition, Indian
J. Pure Appl. Math., 30(2)(1999), 147-152.
96. Pant, R. P.: Common fixed point of two pairs of commuting mappings,
Indian J. Pure Appl. Math., 17(1986), 187-192.
97. Pant, R. P.: Common fixed points of Lipschitz type mapping pairs, J.
Math. Anal. Appl., 240(1999), 280-283.
98. Pant, R. P.: Common fixed points of noncommuting mappings, J. Math.
Anal. Appl., 188 (1994), 436-440.
99. Pant, R. P.: Discontinuity and fixed points, J. Math. Anal. Appl., 240
(1999), 284-289.
100. Pant, R.P., Bisht, R.K.: Occasionally Weakly Compatible Mappings and
Fixed Point, Bull. Belg. Math. Soc., 19(2012), 655-661.
101. Pant, R.P.: Non-compatible mappings and common fixed points,
Soochow J. Math., 26(2000), 29-35.
102. Pant, R.P: Common fixed points of four mappings, Bull. Cal. Math. Soc.,
90(1998), 281-286.
103. Pant, R.P: Common fixed points of non-commuting mappings, J. Math.
Anal. Appl., 188(1994), 436-440.
104. Park, S. and Rhoades, B.E.: Meir- Keeler type contractive conditions.
Math. Japon., 26(1981),13-20.
Estelar
Bibliography
122 | P a g e
105. Pathak, H. K. and Khan, M. S.: A comparison of various types of
compatible maps and common fixed points, Indian J. Pure Appl. Math.,
28(4) (1997), 477–485.
106. Pathak, H. K., Cho, Y. J. and Kang, S. M.: Remarks on R-weakly
commuting mappings and common fixed point theorems, Bull. Korean
Math. Soc., 34 (1997), 247-257
107. Popa, V.: Some fixed point theorems for weakly compatible mappings,
Radovi Matimaticki, 10 (2001), 245-252.
108. Regan, D. O. and Shahzad, N.: Coincidence points and invariant
approximation results for multi maps, Acta Mathematica Sinica English
Ser. 2369 (2007), 1600- 1610.
109. Rhaodes, B.E. and Saliga, L.: Some Fixed point iterative procedures, II,
Nonlinear Anal. Forum, 6(1) (2001), 193-217.
110. Rhaodes, B.E.: A common fixed point theorem for a sequence of fuzzy
mappings, Internat. J. Mat. & Math. Sci., 3(1995), 447-450.
111. Rhaodes, B.E.: A general principle for Ishikawa iterations, 5th IWAA
Proceeding.
112. Rhoades, B. E., Park, S. and Moon, K. B.: On generalizations of the
Meir-Keeler type contraction maps, J. Math. Anal. Appl., 146(1990), 482
– 494.
113. Rhoades, B. E.: A comparison of various definitions of contractive
mappings, Amer. Math. Soc., 226(1977), 257-290.
Estelar
Bibliography
123 | P a g e
114. Rhoades, B.E., Singh, S.L., Kulshrestha, C.: Coincidence theorem for
some multivalued mappings, Int. J. Math. Math. Sci., 7 (1984), 429-434.
115. Rhoades, B.E.: Comments on two fixed point iteration methods, J. Math.
Anal. Appl., 56(1976), 741-750.
116. Rhoades, B.E.: Fixed point iterations using infinite matrices, Trans. Am.
Math. Soc., 196(1974), 161-176.
117. Rhoades, B.E.: Fixed Points theorems and stability results for Fixed Point
iteration procedures, Ind. J. Pure Appl. Math., 21(1990), 1-9.
118. Rhoades, B.E.: Fixed Points theorems and stability results for Fixed Point
iteration procedures, Indian J. Pure Appl. Math., 24(1993), 691-703.
119. Rhoades, B.E.: Some fixed Point iteration procedures, Inter. J. Math.
Math. Sci., 14(1991), 1-16.
120. Rus, I.A.: Fixed point theorem for multivalued mappings in complete
metric spaces, Math Japon., 20(1995), 21-24.
121. Sastry, K. P. R. and Murthy, S. R. K.: Common fixed points of two
partially commuting tangential selfmaps on a metric space, J. Math. Anal.
Appl., 250 (2000), 731-738.
122. Schweizer, B. and Sklar, A.: Statistical Metric Spaces, Pac. J. Math., 10
(1960), 314-334.
123. Sessa S.: On a weak commutativity condition in fixed point
considerations, Publ. Inst. Math. (Beograd) (NS), 34 (46) (1982), 149-153.
Estelar
Bibliography
124 | P a g e
124. Shahabudin, Md: Study of applications of Banach’s fixed point theorem,
Chiiiagong Univ. Stud., Part II Sci., 11 (1987), 85-91.
125. Shahzad, N., Kamran, T.: Coincidence points and R-weakly commuting
maps, Arch. Math. (Brno), 37(2001), 179-183.
126. Sharma, B.K., Sahu, D.R. and Bounias, M.: Common Fixed Point
Theorems for a Mixed Family of Fuzzy and Crisp Mappings, Fuzzy Sets
and Systems, 125 (2002), 261-268.
127. Sharma, S.: Common Fixed Point Theorems in Fuzzy Metric Spaces,
Fuzzy Sets and Systems, 127(2002), 345-352.
128. Simmons, G. F.: Introduction to topology and modern analysis, Tata
McGraw-Hill.
129. Singh, Amit and Dimri, R.C.:On the convergence of Ishikawa iterates to
a common fixed point for a pair of nonexpansive mappings in Banach
space, Mathematica Moravica, 14(1)(2010), 113-119.
130. Singh, Brijendra and Chauhan, M.S.: Common Fixed Points of
Compatible Maps in Fuzzy Metric Spaces, Fuzzy Sets and Systems, 115
(2000), 471-475.
131. Singh, S. L. and Mishra, S. N.: Coincidences and fixed points of
reciprocally continuous and compatible hybrid maps, Int. J. Math. Math.
Sci., 30(10)(2002), 627-635.
132. Singh, S. L. and Tomar, Anita: Weaker forms of commuting maps and
existence of fixed points, J. Korea Soc. Math. Educ. (Pure Appl. Math.), 3
(2003), 145–161.
Estelar
Bibliography
125 | P a g e
133. Singh, S. L., Cho, Y. J. and Kumar, A.: Fixed points of Meir-Keeler
type hybrid contractions, Pan American Mathematical Journal,
16(4)(2006), , 35-54.
134. Singh, S.L. and Chadha, V.: Round- off stability of iterations for
multivalued operators, C.R. Math. Rep. Acad. Sci., XVII(5)(1995), 187-
192.
135. Singh, S.L. and Kulsrestha, C.: Coincidence Theorems, Indian J. Phy.
Natur. Sci., (1983), 5-10.
136. Singh, S.L. and Mishra, S.N.: Coincidence and fixed points of non-self
hybrid contraction, J. Math. Anal. Appl., 256(2001), 486-497.
137. Singh, S.L., Bhatnagar, C. and Mishra, S.N.: Stability of Jungck-type
iterative procedures, Int. J. math. Math. Sci., 2005(2005), 3035-3043.
138. Singh, S.L., Hematalin, A. and Prashad, B.: Fixed Point theorem to
hybrid maps in symmetric spaces, Tamsui Oxford Journal of Information
and Mathematical Sciences, 27 (4)(2011), 429-448.
139. Singh, S.L.: Applications of fixed point theorems to nonlinear integro
differential equations, Riv. Mat. Univ. Parm., 4(16)(1990), 205-212.
140. Smart, D. R.: Fixed point theorems, Cambridge Univ. Press, 1980.
141. Song, G.: Comments on A Common Fixed Point Theorem in Fuzzy
Metric Space, Fuzzy Sets and Systems, 135(2003), 409-413
Estelar
Bibliography
126 | P a g e
142. Sujuki, T.: A generalized Banach contraction principle that characterizes
metric completeness, Proceedings of the American Math. Society,
136(5)(2008), 1861-1869.
143. Thagafi, A., Shahzad, N.: Generalized I- nonexapansive self maps and
invariant approximations, Acta. Math. Sin., 24(2008), 867-876.
144. Urbe, M.: Convergence of numerical iteration in solution of equations, J.
Sci. Hiroshima Univ. Ser. A., 19(1956), 479-489.
145. Vasuki, R.: A Common Fixed Point Theorem in Fuzzy Metric Space,
Fuzzy Sets and Systems, 97(1998), 395-397.
146. Vasuki, R.: Common Fixed Points for R-weakly Commuting Maps in
Fuzzy Metric Spaces, Indian J. Pure Appl. Math., 30(1999), 419-423.
147. Weiczorek, A.: Applications of fixed point theorems in game theory and
mathematical economics (Polish)., Wiadom. Math., 28(1988), 25-34.
148. Wilson, W.A.: On Semi-metric spaces, Amer. J. Math., 53(1931), 361-
373.
149. Zadeh, L.A.: Fuzzy Sets, Inform. and Control, 8(1965), 338-353.
150. Zhang, S.S., and Luo, Q.: Set-valued Caristi fixed point theorem and
Ekeland’s variational principle, Appl. Math.Mech. 10 (2) (1989) 111–113
(in Chinese), English translation: Appl. Math. Mech. (English Ed.) 10 (2)
(1989)119–121.
151. Zhang, X.: Fixed point theorems of multivalued monotone mappings in
ordered metric spaces, Appl. Math. Lett., 23 (2010), 235-240.
Estelar