]>
2012
5
4
ISSN 2008-1898
54
On Banach contraction principle in a cone metric space
On Banach contraction principle in a cone metric space
en
en
The object of this paper is to establish a generalized form of Banach contraction principle for a cone metric
space which is not necessarily normal. This happens to be a generalization of all different forms of Banach
contraction Principle, which have been arrived at in L. G. Huang and X. Zhang [L. G. Huang and X.
Zhang, J. Math. Anal. Appl 332 (2007), 1468-1476] and Sh. Rezapour, R. Hamlbarani [Sh. Rezapour, R.
Hamlbarani, J. Math. Anal. Appl. 345 (2008) 719-724] and D. Ilic, V. Rakocevic [D. Ilic, V. Rakocevic,
Applied Mathematics Letters 22 (2009), 728-731]. It also results that the theorem on quasi contraction of
Ćirić [L. J. B. Ćirić, Proc. American Mathematical Society 45 (1974), 999-1006]. for a complete metric
space also holds good in a complete cone metric space. All the results presented in this paper are new.
252
258
Shobha
Jain
Quantum School of Technology
India
shobajain1@yahoo.com
Shishir
Jain
Shri Vaishnav Institute of Technology and Science
India
jainshishir11@rediffmail.com
Lal Bahadur
Jain
Retd. Principal, Govt. Arts and Commerce College )
India
lalbahdurjain11@yahoo.com
Cone metric space
common fixed points.
Article.1.pdf
[
[1]
V. Berinde, Itrative approximation of fixed points, Springer Verlag, (2007)
##[2]
L. J. B. Ćirić, A generalization of Banach contraction princple, Proc. American Mathematical Society , 45 (1974), 999-1006
##[3]
L. G. Huang, X. Zhang , Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468-1476
##[4]
D. Ilic, V. Rakocevic, Quasi-contraction on a cone metric space, Applied Mathematics Letters, 22 (2009), 728-731
##[5]
Sh. Jain, Sh. Jain, L. B. Jain , Compatibilty and weak compatibility for four self maps in a cone metric space, Bulletin of Mathematical analysis and application, 1 (2010), 1-18
##[6]
Sh. Jain, Sh. Jain, L. B. Jain, Weakly compatibile maps in a cone metric space, Rendiconti Del Seminario Matematica, 68 (2010), 115-225
##[7]
B. V. Kvedaras, A. V. Kibenko, A. I. Perov, On some boundary value problems, Litov. matem. sbornik , 5 (1965), 69-84
##[8]
E. M. Mukhamadiev, V. J. Stetsenko, Fixed point principle in generalized metric space, Izvestija AN Tadzh. SSR, fiz.-mat. i geol.-chem. nauki., 10 (1969), 8-19
##[9]
A. I. Perov , The Cauchy problem for systems of ordinary differential equations, In , Approximate methods of solving differential equations, Kiev, Naukova Dumka, 1964 (12), 115-134
##[10]
A. I. Perov, A. V. Kibenko, An approach to studying boundary value problems, Izvestija AN SSSR, ser. math. , 30 (1966), 249-264
##[11]
h. Rezapour, R. Hamlbarani , Some notes on the paper ''Cone metric spaces and fixed point theorems of contractive mappings'', J. Math. Anal. Appl., 345 (2008), 719-724
##[12]
R. Vasuki , A Fixed Point Theorem for a sequence of Maps satisfying a new contractive type contraction in Menger Space, Math Japonica , 35 (1990), 1099-1102
##[13]
P. P. Zabrejko, K-metric and K-normed linear spaces , Survey Collect. Math. , 48 (1997), 825-859
]
On coupled generalised Banach and Kannan type contractions
On coupled generalised Banach and Kannan type contractions
en
en
In this paper we have proved two theorems in which we have established the existence of coupled fixed
point results in partially ordered complete metric spaces for generalised coupled Banach and Kannan type
mappings. The generalisation has been accomplished by following the line of argument given by Geraghty
[Proc. Amer. Math. Soc., 40 (1973), 604-608] . Here the mapping are assumed to satisfy certain contractive
type inequalities. We have illustrated our result with two examples. First example is presented to show that
our result is a proper generalizations of the corresponding results of Bhaskar et al [Nonlinear Anal. TMA,
65 (7) (2006), 1379-1393].
259
270
B. S.
Choudhury
Department of Mathematics
Bengal Engineering and Science University
India
binayak12@yahoo.co.in
Amaresh
Kundu
Department of Mathematics
Siliguri Institute of Technology
India
kunduamaresh@yahoo.com
Partially ordered set
Contractive-type mapping
Mixed monotone property
Coupled fixed point.
Article.2.pdf
[
[1]
R. P. Agarwal, M. A. El-Gebeily, D. O'Regan, Generalized contractions in partially ordered metric spaces , Appl. Anal., 87(1) (2008), 109-116
##[2]
A. Amini-Harandi, H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal. TMA, 72(5) (2010), 2238-2242
##[3]
H. Aydi , Some coupled fixed point results on partial metric spaces, Int. J. Math. Math. Sci., Article ID 647091, doi:10.1155/2011/647091. , 2001 (2011), 1-11
##[4]
B. S. Choudhury, P. Maity, Coupled fixed point results in generalized metric spaces, Math. Comput. Modelling, 54 (2011), 73-79
##[5]
B. S. Choudhury, K. Das, Fixed points of generalized Kannan type mappings in generalized Menger spaces , Commun. Korean Math. Soc., 24 (2009), 529-537
##[6]
B. S. Choudhury, N. Metiya, A. Kundu , Coupled coincidence point theorems in ordered metric Spaces, Ann. Univ. Ferrara, 57 (2011), 1-16
##[7]
B. S. Choudhury, A. Kundu , A coupled coincidence point result in partially ordered metric spaces for compatible mappings, Nonlinear Anal. TMA, 73 (2010), 2524-2531
##[8]
B. S. Choudhury, K. Das, S. K. Bhandari, A fixed point theorem for Kannan type mappings in 2-menger space using a control function, Bull. Math. Anal. Appl., 3 (2011), 141-148
##[9]
B. S. Choudhury, A. Kundu, A Kannan-like contraction in partially ordered spaces, Demonstratio Mathematica, (to appear), -
##[10]
J. Caballero, J. Harjani, K. Sadarangani, Contractive-like mapping principles in ordered metric spaces and application to ordinary differential equations, Fixed Point Theory and Appl., Article ID 916064, doi:10.1155/2010/916064. , (2010), 1-14
##[11]
Lj. B. Ćirić, V. Lakshmikantham , Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces, Stoch. Anal. Appl., 27 (6) (2009), 1246-1259
##[12]
Lj. B. Ćirić, D. Mihet, R. Saadati, Monotone generalized contractions in partially ordered probabilistic metric spaces, Topol. Appl., 156 (2009), 2838-2844
##[13]
L. Ćirić, N. Cakić, M. Rajović, J. S. Ume, Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory and Appl., Article ID 131294, 11 pages (2008)
##[14]
E. H. Connell, Properties of fixed point spaces, Proc. Amer. Math. Soc., 10 (1959), 974-979
##[15]
M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604-608
##[16]
T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. TMA, 65 (7) (2006), 1379-1393
##[17]
X. Hu, Common Coupled fixed point theorems for contractive mappings in fuzzy metric spaces, Fixed Point Theory and Appl., Article ID 363716, doi:10.1155/2011/363716., (2011), 1-14
##[18]
J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces, Nonlinear Anal., 74 (2011), 768-774
##[19]
L. Janos, On mappings contractive in the sense of Kannan, Proc. Amer. Math. Soc., 61(1) (1976), 171-175
##[20]
R. Kannan, Some results on fixed points , Bull. Cal. Math. Soc., 60 (1968), 71-76
##[21]
R. Kannan, Some results on fixed points-II, Amer. Math. Monthly, 76 (1969), 405-408
##[22]
M. Kikkaw, T. Suzuki , Some similarity between contractions and Kannan mappings, Fixed Point Theory and Appl., Article ID 649749. , 2008 (2008), 1-8
##[23]
E. Karapinar, Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl., 59 (2010), 3656-3668
##[24]
V. Lakshmikantham, L. Ciric , Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. TMA, 70 (2009), 4341-4349
##[25]
J. J. Nieto, R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22(3) (2005), 223-239
##[26]
J. J. Nieto, R. R. Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta. Math. Sin., (Engl. Ser.), 23(12) (2007), 2205-2212
##[27]
A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132(5) (2004), 1435-1443
##[28]
D. A. Ruiz, A. J. Melado, A continuation method for weakly Kannan maps, Fixed Point Theory and Appl., Article ID 321594, doi:10.1155/2010/321594. , 2010 (2010), 1-12
##[29]
B. Samet, H. Yazidi , Coupled fixed point theorems in partially ordered ''-chainable Metric spaces, TJMCS, 1 (3) (2010), 142-151
##[30]
N. Shioji, T. Suzuki and W. Takahashi, Contractive mappings, Kannan mappings and metric completeness, Proc. Amer. Math. Soc., 126 (1998), 3117-3124
##[31]
Y. Enjouji, M. Nakanishi, T. Suzuki, A generalization of Kannan's fixed point theorem, Fixed Point Theory and Appl., Article ID 192872, doi:10.1155/2009/192872. , 2009 (2009), 1-10
##[32]
P. V. Subrahmanyam, Completeness and fixed points, Monatsh. Math., 80 (1975), 325-330
]
Existence of solutions for fractional impulsive neutral functional infinite delay integrodifferential equations with nonlocal conditions
Existence of solutions for fractional impulsive neutral functional infinite delay integrodifferential equations with nonlocal conditions
en
en
This paper is mainly concerned with the existence of solutions for fractional impulsive neutral functional
integrodifferential equations with nonlocal initial conditions and infinite delay. The results are obtained by
the fixed point theorem.
271
280
A.
Anguraj
Department of Mathematics
P. S. G. College of Arts and Science
India
angurajpsg@yahoo.com
M. Latha
Maheswari
Department of Mathematics with CA
P. S. G. College of Arts and Science
India
lathamahespsg@gmail.com
Existence of solution
Fractional
Integrodifferential equations
Impulsive conditions
Nonlocal conditions
Fixed point theorem.
Article.3.pdf
[
[1]
R. P. Agarwal, M. Benchohra, B. A. Slimani, Existence results for differential equations with fractional order and impulses, Memoirs on Differential Equations and Mathematical physics, 44 (2008), 1-21
##[2]
A. Anguraj, P. Karthikeyan, G. M. N'Guérékata, Nonlocal cauchy problem for some fractional abstract integro- differential equations in Banach spaces, Communications in Mathematical Analysis, 6 (1) (2009), 31-35
##[3]
A. Anguraj, K. Karthikeyan, Y. K. Chang , Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators, Nonlinear Analysis: Theory Methods and Applications, 71 (2009), 4377-4386
##[4]
K. Balachandran, J. J. Trujillo , The non local cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces, Nonlinear Analysis: Theory Methods and Applications, 72 (12) (2010), 4587-4593
##[5]
K. Balachandran, S. Kiruthika, J. J. Trujillo , Existence results for fractional impulsive integrodifferential equations in Banach spaces , Commun Nonlinear Sci Numer Simulat, 16 (4) (2011), 1970-1977
##[6]
M. Benchohra, D. Seba, Impulsive fractional differential equations in Banach spaces, Electronic Journal of Qualitative Theory of Differential Equations Special Edition I, 8 (2009), 1-14
##[7]
M. Benchohra, B. A. Slimani, Existence and Uniqueness of solutions to impulsive fractional differential equations, Electronic Journal of Differential Equations, 10 (2009), 1-11
##[8]
B. Bonilla, M. Rivero, L. Rodriguez-Germa, J. J. Trujillo, Fractional differential equations as alternative models to nonlinear differential equations, Applied Mathematics and Computation, 187 (2007), 79-88
##[9]
Y. K. Chang, Controllability of impulsive functional differential systems with infinite delay in Banach spaces, Chaos, Solitons and Fractals, 33 (2007), 1601-1609
##[10]
Gisele M. Mophou, Existence and uniqueness of solutions to impulsive fractional differential equations, Nonlinear Analysis, 72 (2010), 1604-1615
##[11]
JH. He, Approximate analytical solution for seepage ow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167 (1998), 57-68
##[12]
E. Hernandez, D. O'Regan, K. Balachandran, On recent developments in the theory of abstract differential equations with fractional derivatives, Nonlinear Analysis: Theory, Methods and Applications, 73(10) (2010), 3462-3471
##[13]
R. Hilfer, Applications of Fractional Calculus in physics, World Scientific, Singapore (2000)
##[14]
V. Lakshmikantham, Theory of fractional differential equations, Nonlinear Analysis, Theory methods and Applications, 60 (10) (2008), 3337-3343
##[15]
G. M. Mophou, G. M. N'Guérékata, Existence of mild solution for some fractional differential equations with nonlocal conditions, Semigroup Forum, 79 (2) (2009), 322-335
##[16]
Mouffak Benchohra, Samira Hamani , The method of upper and lower solutions and impulsive fractional differential inclusions, Nonlinear Analysis:Hybrid Systems, 3 (2009), 433-440
##[17]
B. N. Sadovskii , On a fixed point principle, Functional Analysis and its Applications, 1 (2) (1967), 74-76
##[18]
Xianmin Zhang, Xiyue Huang, Zuohua Liu , The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay , Nonlinear Analysis:Hybrid Systems, 4 (2010), 775-781
]
Existence of unbounded positive solutions for BVPs of singular fractional differential equations
Existence of unbounded positive solutions for BVPs of singular fractional differential equations
en
en
In this article, we establish the existence of multiple unbounded positive solutions to the boundary value
problem of the nonlinear singular fractional differential equation
\[
\begin{cases}
D^\alpha_{ 0^+}u(t) + f(t; u(t)) = 0; t \in (0; 1); 1 < \alpha < 2,\\
[I^{2-\alpha}_{ 0^+} u(t)]'|_{t=0} = 0\\
u(1) = 0.
\end{cases}
\]
Our analysis relies on the well known fixed point theorems in the cones in Banach spaces. Here \(f\) is singular
at \(t = 0\) and \(t = 1\).
281
293
Yuji
Liu
Department of Mathematics
Guangdong University of Business Studies
P. R. China
liuyuji888@sohu.com
Haiping
Shi
Basic Courses Department
Guangdong Construction Vocational Technology Institute
P. R. China
haipingshi@sohu.com
Singular fractional differential equation
boundary value problem
unbounded positive solution
Fixed Point Theorem.
Article.4.pdf
[
[1]
K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation , Wiley, New York (1993)
##[2]
S. G. Samko, A. A. Kilbas, O. I. Marichev , Fractional Integral and Derivative, Theory and Applications, Gordon and Breach (1993)
##[3]
Z. Bai, H. Lv , Positive solutions for boundary value problems of nonlinear fractional differential equations, J. Math. Anal. Appl. , 311 (2005), 495-505
##[4]
A. A. Kilbas, J. J. Trujillo, Differential equations of fractional order: methods, results and problems-I, Applicable Analysis, 78 (2001), 153-192
##[5]
A. Arara, M. Benchohra, N. Hamidi, J. J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Analysis TMA, 72 (2010), 580-586
##[6]
Z. Bai, On positive solutions of a nonlocal fractional boundary value problem , Nonlinear Analysis, 72 (2010), 916-924
##[7]
R. Dehghant, K. Ghanbari, Triple positive solutions for boundary value problem of a nonlinear fractional differential equation, Bulletin of the Iranian Mathematical Society, 33 (2007), 1-14
##[8]
S. Z. Rida, H. M. El-Sherbiny, A. A. M. Arafa, On the solution of the fractional nonlinear Schrodinger equation, Physics Letters A, 372 (2008), 553-558
##[9]
X. Xu, D. Jiang, C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Analysis TMA, 71 (2009), 4676-4688
##[10]
F. Zhang , Existence results of positive solutions to boundary value problem for fractional differential equation, Positivity, 13 (2008), 583-599
##[11]
R. W. Leggett, L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Mathematics Journal, 28 (1979), 673-688
]
Application of fixed point theorems to best simultaneous approximation in ordered semi-convex structure
Application of fixed point theorems to best simultaneous approximation in ordered semi-convex structure
en
en
In this paper, we establish some common fixed point results for uniformly \(C_q\)-commuting asymptotically
S-nonexpansive maps in a Banach space with semi-convex structure. We also extend the main results of
Ćirić [Lj. B. Ćirić, Publ. Inst. Math., 49 (1991), 174-178] and [Lj. B. Ćirić, Arch. Math. (BRNO),
29 (1993), 145-152] to semi-convex structure and obtain common fixed point results for Banach operator
pair. The existence of invariant best simultaneous approximation in ordered semi-convex structure is also
established.
294
306
N.
Hussain
King Abdul Aziz University
Saudi Arabia
nhusain@kau.edu.sa
H. K.
Pathak
School of Studies in Mathematics
Pt. Ravishankar Shukla University
India
hkpathak@sify.com
S.
Tiwari
Shri Shankaracharya Institute of Professional Management and Technology
India
tsatyaj@yahoo.co.in
Common fixed point
uniformly \(C_q\)-commuting
asymptotically S-nonexpansive map
Banach operator pair
best simultaneous approximation
Article.5.pdf
[
[1]
F. Akbar, A. R. Khan, Common fixed point and approximation results for noncommuting maps on locally convex spaces, Fixed Point Theory and Appl., Article ID 207503, 2009 (2009), 1-14
##[2]
M. A. Al-Thagafi , Common fixed point and best approximation, J. Approx. Theory , 85 (1996), 318-323
##[3]
I. Beg, D. R. Sahu, S. D. Diwan, Approximation of fixed points of uniformly R-subweakly commuting mappings, J. Math. Anal. Appl., 324 (2006), 1105-1114
##[4]
V. Berinde, General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces, Carpathian J. Math., 24 (2008), 10-19
##[5]
J. Chen, Z. Li , Common fixed points for Banach operator pairs in best approximation, J. Math. Anal., 336 (2007), 1466-1475
##[6]
E. W. Cheney , Application of fixed point theorems to approximation theory, Theory of Approximations, Academic Press , (1976), 1-8
##[7]
LJ. B. Ćirić, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 45 (1974), 267-273
##[8]
LJ. B. Ćirić, On a common fixed point theorem of a Gregus type, Publ. Inst. Math., 49 (1991), 174-178
##[9]
LJ. B. Ćirić, On Diviccaro, Fisher and Sessa open questions, Arch. Math. (BRNO), 29 (1993), 145-152
##[10]
LJ. B. Ćirić, Contractive-type non-self mappings on metric spaces of hyperbolic type, J. Math. Anal. Appl. , 317 (2006), 28-42
##[11]
M. L. Diviccaro, B. Fisher, S. Sessa , A common fixed point theorem of Gregus type, Publ. Math. Debrecean, 34 (1987), 83-89
##[12]
M. Edelstein, R. C. O'Brien, Nonexpansive mappings, asymptotic regularity and successive approximations, J. London Math. Soc. , 17 (1978), 547-554
##[13]
K. Goeble, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174
##[14]
S. P. Gudder , A general theory of convexity, Rend. Sem. Mat. Milano, 49 (1979), 89-96
##[15]
L. Habiniak, Fixed point theorems and invarient approximations, J. Approximation Theory, 56 (1989), 241-244
##[16]
N. Hussain, Common fixed points in best approximation for Banach operator pairs with Ćirić type I-contractions, J. Math. Anal. Appl., 338 (2008), 1351-1363
##[17]
N. Hussain, D. O'Regan, R. P. Agarwal, Common fixed point and invarient approximation results on non- starshaped domain, Georgian Math. J., 12 (2005), 659-669
##[18]
N. Hussain, B. E. Rhoades, \( C_q\)-commuting maps and invariant approximations, Fixed point Theory and Appl., Article ID 24543, 2006 (2006), 1-9
##[19]
N. Hussain, B. E. Rhoades, G. Jungck, Common fixed point and invariant approximation results for Gregus type I-contractions, Numer. Funct. Anal. Optimiz., 28 (2007), 1139-1151
##[20]
N. Hussain, M. Abbas, J. K. Kim, Common fixed point and invariant approximation in Menger convex metric spaces, Bull. Korean Math. Soc., 45 (2008), 671-680
##[21]
G. Jungck, Commuting mapping and fixed points, Amer. Math. Monthly, 83 (1976), 261-263
##[22]
G. Jungck, On a fixed point theorem of Fisher and Sessa, Internat. J. Math. Math. Sci., 13 (1990), 497-500
##[23]
G. Jungck, N. Hussain, Compatible maps and invarient approximations, J. Math. Anal., 325 (2007), 1003-1012
##[24]
G. Jungck, S. Sessa, Fixed point theorems in best approximation theory, Math. Japon, 42 (1995), 249-252
##[25]
A. R. Khan, N. Hussain, A. B. Thaheem, Application of fixed point theorems to invariant approximation, Approx. Theory and Appl. , 16 (2000), 48-55
##[26]
S. H. Khan, N. Hussain, Convergence theorems for nonself asymptotically nonexpansive mappings, Comput. Math. Appl., 55 (2008), 2544-2553
##[27]
V. Klee, Convexity of chebyshev sets, Math. Ann., 142 (1961), 292-304
##[28]
G. Meinardus , Invarianz bei Linearea Approximation, Arch. Rational Mech. Anal., 14 (1963), 301-303
##[29]
P. D. Milman, On best simultaneous approximation in normed linear spaces, J. Approximation Theory, 20 (1977), 223-238
##[30]
D. O'Regan, N. Hussain, Generalized I-contractions and pointwise R-subweakly commuting maps, Acta Math. Sinica, 23 (2007), 1505-1508
##[31]
H. K. Pathak, Y. J. Cho, S. M. Kang, An application of fixed point theorems in best approximation theory, Internat. J. Math. Math. Sci., 21 (1998), 467-470
##[32]
H. K. Pathak, N. Hussain, Common fixed points for Banach operator pairs with applications, Nonlinear Anal., 69 (2008), 2788-2802
##[33]
A. Petrusel, Starshaped and fixed points, Seminar on fixed point theory (Cluj-Napoca), Stud. Univ. ''Babes- Bolyai'', (1987), 19-24
##[34]
S. A. Sahab, M. S. Khan, S. Sessa, A result in best approximation theory, J. Approx. Theory, 55 (1988), 349-351
##[35]
B. N. Sahney, S. P. Singh, On best simultaneous approximation, Approximation Theory III, Academic Press , (1980), 783-789
##[36]
S. P. Singh, Application of fixed point theorems in approximation theory, Applied Nonlinear Analysis, Academic Press , (1979), 389-394
##[37]
S. P. Singh, Application of a fixed point theorem to approximation theory, J. Approx. Theory, 25 (1979), 88-89
##[38]
A. Smoluk , Invarient approximations , Mathematyka [Polish], 17 (1981), 17-22
##[39]
P. V. Subrahmanyam, An application of a fixed point theorem to best approximations, J. Approx. Theory, 20 (1977), 165-172
##[40]
P. Vijayraju, Applications of fixed point theorem to best simultaneous approximations, Indian J. Pure Appl. Math., 24 (1993), 21-26
]