]>
2014
7
5
ISSN 2008-1898
79
Lefschetz type theorems for a class of noncompact mappings
Lefschetz type theorems for a class of noncompact mappings
en
en
In this paper we present new fixed point results for general compact absorbing type contractions in new
extension spaces.
288
295
Donal
ORegan
School of Mathematics, Statistics and Applied Mathematics
National University of Ireland
Ireland
donal.oregan@nuigalway.ie
Extension spaces
fixed point theory
compact absorbing contractions.
Article.1.pdf
[
[1]
R. P. Agarwal, D.O'Regan, A Lefschetz fixed point theorem for admissible maps in Fréchet spaces, Dynamic Systems and Applications, 16 (2007), 1-12
##[2]
R. P. Agarwal, D.O'Regan, Fixed point theory for compact absorbing contractive admissible type maps , Applicable Analysis, 87 (2008), 497-508
##[3]
J. Andres, L. Gorniewicz, Fixed point theorems on admissible multiretracts applicable to dynamical systems, Fixed Point Theory, 12 (2011), 255-264
##[4]
H. Ben-El-Mechaiekh, The coincidence problem for compositions of set valued maps, Bull. Austral. Math. Soc., 41 (1990), 421-434
##[5]
H. Ben-El-Mechaiekh, Spaces and maps approximation and fixed points, Jour. Computational and Appl. Mathematics, 113 (2000), 283-308
##[6]
H. Ben-El-Mechaiekh, P. Deguire, General fixed point theorems for non-convex set valued maps, C.R. Acad. Sci. Paris, 312 (1991), 433-438
##[7]
R. Engelking, General Topology, Heldermann Verlag, Berlin (1989)
##[8]
G. Fournier, L. Gorniewicz, The Lefschetz fixed point theorem for multi-valued maps of non-metrizable spaces , Fundamenta Mathematicae, 92 (1976), 213-222
##[9]
L. Gorniewicz, Topological fixed point theory of multivalued mappings, Kluwer Acad. Publishers, Dordrecht (1999)
##[10]
L. Gorniewicz, A. Granas, Some general theorems in coincidence theory, J. Math. Pures et Appl., 60 (1981), 361-373
##[11]
L. Gorniewicz, M. Slosarski , Fixed points of mappings in Klee admissible spaces, Control and Cybernetics, 36 (2007), 825-832
##[12]
A. Granas, J. Dugundji, Fixed point theory, Springer , New York (2003)
##[13]
J. L. Kelley, General Topology, D. Van Nostrand Reinhold Co., New York (1955)
##[14]
D. O'Regan , Fixed point theory on extension type spaces and essential maps on topological spaces, Fixed Point Theory and Applications, 2004 (2004), 13-20
##[15]
D. O'Regan, Fixed point theory for compact absorbing contractions in extension type spaces, CUBO, 12 (2010), 199-215
##[16]
D. O'Regan, Fixed point theory in generalized approximate neighborhood extension spaces, Fixed Point Theory, 12 (2011), 155-164
##[17]
D. O'Regan , Lefschetz fixed point theorems in generalized neighborhood extension spaces with respect to a map, Rend. Circ. Mat. Palermo, 59 (2010), 319-330
##[18]
D. O'Regan , Fixed point theory for extension type maps in topological spaces, Applicable Analysis, 88 (2009), 301-308
##[19]
D. O'Regan, Periodic points for compact absorbing contractions in extension type spaces, Commun. Appl. Anal., 14 (2010), 1-11
]
Additive \(\rho\)--functional inequalities
Additive \(\rho\)--functional inequalities
en
en
In this paper, we solve the additive \(\rho\)-functional inequalities
\[\|f(x + y) - f(x) - f(y)\| \leq \| \rho( 2f (\frac{ x + y}{ 2}) - f(x) - f(y) ) \|, \qquad (1)\] ;
\[\|2f (\frac{ x + y}{ 2}) - f(x) - f(y)\| \leq \| \rho(f(x + y) - f(x) - f(y) ) \|, \qquad (2)\] ;
where \(\rho\) is a fixed non-Archimedean number with \(|\rho|<1\) or \(\rho\) is a fixed complex number with \(|\rho|<1\).
Using the direct method, we prove the Hyers-Ulam stability of the additive \(\rho\)-functional inequalities (1)
and (2) in non-Archimedean Banach spaces and in complex Banach spaces, and prove the Hyers-Ulam
stability of additive \(\rho\)-functional equations associated with the additive \(\rho\)-functional inequalities (1) and (2)
in non-Archimedean Banach spaces and in complex Banach spaces.
296
310
Choonkil
Park
Research Institute for Natural Sciences
Hanyang University
Republic of Korea
baak@hanyang.ac.kr
Hyers-Ulam stability
additive \(\rho\)-functional equation
additive \(\rho\)-functional inequality
non-Archimedean normed space
Banach space.
Article.2.pdf
[
[1]
T. Aoki , On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66
##[2]
A. Bahyrycz, M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar., 142 (2014), 353-365
##[3]
M. Balcerowski, On the functional equations related to a problem of Z. Boros and Z. Daróczy, Acta Math. Hungar., 138 (2013), 329-340
##[4]
Z. Daróczy, Gy. Maksa , A functional equation involving comparable weighted quasi-arithmetic means, Acta Math. Hungar., 138 (2013), 147-155
##[5]
W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math., 71 (2006), 149-161
##[6]
W. Fechner, On some functional inequalities related to the logarithmic mean, Acta Math. Hungar., 128 (2010), 31-45
##[7]
P. Găvruţa , A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-43
##[8]
A. Gilányi , Eine zur Parallelogrammgleichung äquivalente Ungleichung, Aequationes Math., 62 (2001), 303-309
##[9]
A. Gilányi , On a problem by K. Nikodem, Math. Inequal. Appl., 5 (2002), 707-710
##[10]
D. H. Hyers , On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27 (1941), 222-224
##[11]
M. S. Moslehian, Gh. Sadeghi, A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.- TMA, 69 (2008), 3405-3408
##[12]
C. Park, Y. Cho, M. Han , Functional inequalities associated with Jordan-von Neumann-type additive functional equations , J. Inequal. Appl., Article ID 41820 , 2007 (2007), 1-13
##[13]
W. Prager, J. Schwaiger, A system of two inhomogeneous linear functional equations, Acta Math. Hungar., 140 (2013), 377-406
##[14]
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300
##[15]
J. Rätz , On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math., 66 (2003), 191-200
##[16]
S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. , New York (1960)
]
Some new generalizations of Ostrowski type inequalities on time scales involving combination of \(\triangle\)-integral means
Some new generalizations of Ostrowski type inequalities on time scales involving combination of \(\triangle\)-integral means
en
en
In this paper we obtain some new generalizations of Ostrowski type inequalities on time scales involving
combination of \(\triangle\)-integral means, i.e., a new Ostrowski type inequality on time scales involving combination
of \(\triangle\)-integral means, two Ostrowski type inequalities for two functions on time scales, and some new
perturbed Ostrowski type inequalities on time scales. We also give some other interesting inequalities as
special cases.
311
324
Yong
Jiang
College of Mathematics and Statistics
Nanjing University of Information Science and Technology
China
jiang@nuist.edu.cn
Hüseyin
Rüzgar
Department of Mathematics, Faculty of Science and Arts
University of Nigde
Turkey
091908002@nigde.edu.tr
Wenjun
Liu
College of Mathematics and Statistics
Nanjing University of Information Science and Technology
China
wjliu@nuist.edu.cn
Adnan
Tuna
Department of Mathematics, Faculty of Science and Arts
University of Nigde
Turkey
atuna@nigde.edu.tr
Ostrowski inequality
perturbed Ostrowski inequality
\(\triangle\)-integral means
time scales.
Article.3.pdf
[
[1]
R. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl., 4 (2001), 535-557
##[2]
F. Ahmad, P. Cerone, S. S. Dragomir, N. A. Mir, On some bounds of Ostrowski and Čebyšev type, J. Math. Inequal., 4 (2010), 53-65
##[3]
F. M. Atici, D. C. Biles, A. Lebedinsky, An application of time scales to economics, Math. Comput. Modelling, 43 (2006), 718-726
##[4]
M. Bohner, M. Fan, J. M. Zhang, Periodicity of scalar dynamic equations and applications to population models, J. Math. Anal. Appl., 330 (2007), 1-9
##[5]
M. Bohner, A. Peterson , Dynamic equations on time scales, Birkhäuser Boston, Boston, MA (2001)
##[6]
M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Birkhäuser Boston, Boston, MA (2003)
##[7]
M. Bohner, T. Matthews, The Grüss inequality on time scales, Commun. Math. Anal., (electronic)., 3 (2007), 1-8
##[8]
M. Bohner, T. Matthews, Ostrowski inequalities on time scales, JIPAM. J. Inequal. Pure Appl. Math., 9 (2008), 1-8
##[9]
M. Bohner, T. Mathews, A. Tuna , Diamond-alpha Grüss type inequalities on time scales, Int. J. Dyn. Syst. Differ. Equ., 3 (2011), 234-247
##[10]
M. Bohner, T. Mathews, A. Tuna, Weighted Ostrowski-Grüss inequalities on time scales, Afr. Diaspora J. Math., 12 (2011), 89-99
##[11]
P. Cerone, A new Ostrowski type inequality involving integral means over end intervals, Tamkang J. Math., 33 (2002), 109-118
##[12]
C. Dinu, Ostrowski type inequalities on time scales, An. Univ. Craiova Ser. Mat. Inform., 34 (2007), 43-58
##[13]
S. S. Dragomir, P. Cerone, J. Roumeliotis, S. A. Wang, A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 42(90) (1999), 301-314
##[14]
S. S. Dragomir, A generalization of the Ostrowski integral inequality for mappings whose derivatives belong to \(L_p[a; b]\) and applications in numerical integration, J. Math. Anal. Appl., 255 (2001), 605-626
##[15]
S. S. Dragomir, Refinements of the generalised trapezoid and Ostrowski inequalities for functions of bounded variation, Arch. Math. (Basel), 91 (2008), 450-460
##[16]
S. S. Dragomir, Ostrowski's type inequalities for some classes of continuous functions of selfadjoint operators in Hilbert spaces, Comput. Math. Appl., 62 (2011), 4439-4448
##[17]
S. S. Dragomir, Refinements of the Ostrowski inequality in terms of the cumulative variation and applications, Analysis (Berlin), 34 (2014), 223-240
##[18]
S. Hilger, Ein Mabkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, PhD thesis, Univarsi. Würzburg (1988)
##[19]
S. Hussain , Generalization of Ostrowski and Čebyšev type inequalities involving many functions, Aequationes Math., 85 (2013), 409-419
##[20]
S. Hussain, M. A. Latif, M. Alomari , Generalized double-integral Ostrowski type inequalities on time scales, Appl. Math. Lett., 24 (2011), 1461-1467
##[21]
V. Kac, P. Cheung, Quantum Calculus, Springer-Verlag, New York (2002)
##[22]
W. N. Li, Some delay integral inequalities on time scales , Comput. Math. Appl., 59 (2010), 1929-1936
##[23]
W. N. Li, W. H. Sheng, Some Gronwall type inequalities on time scales, J. Math. Inequal., 4 (2010), 67-76
##[24]
V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic systems on measure chains, Kluwer Acad. Publ., Dordrecht (1996)
##[25]
G. Lapenta , Particle simulations of space weather, J. Comput. Phys., 231 (2012), 795-821
##[26]
W. J. Liu, Q.-A. Ngô, An Ostrowski-Grüss type inequality on time scales, Comput. Math. Appl., 58 (2009), 1207-1210
##[27]
W. J. Liu, Q.-A. Ngô, A generalization of Ostrowski inequality on time scales for k points, Appl. Math. Comput., 203 (2008), 754-760
##[28]
W. J. Liu, Q.-A. Ngô, W. B. Chen, A perturbed Ostrowski-type inequality on time scales for k points for functions whose second derivatives are bounded, J. Inequal. Appl., Art. ID 597241, 2008 (2008), 1-12
##[29]
W. J. Liu, Q.-A. Ngô, W. B. Chen , A new generalization of Ostrowski type inequality on time scales, An. Ştiinţ. Univ. ''Ovidius'' Constanţa Ser. Mat., 17 (2009), 101-114
##[30]
W. J. Liu, Q.-A. Ngô, W. B. Chen, Ostrowski type inequalities on time scales for double integrals, Acta Appl. Math., 110 (2010), 477-497
##[31]
W. J. Liu, A. Tuna, Weighted Ostrowski, Trapezoid and Grüss type inequalities on time scales, J. Math. Inequal., 6 (2012), 381-399
##[32]
W. J. Liu, A. Tuna, Y. Jiang, New weighted Ostrowski and Ostrowski-Grüss Type inequalities on time scales, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.), 60 (2014), 57-76
##[33]
W. J. Liu, A. Tuna, Y. Jiang, On weighted Ostrowski type, Trapezoid type, Grüss type and Ostrowski-Grüss like inequalities on time scales, Appl. Anal., 93 (2014), 551-571
##[34]
Q.-A. Ngo, W. J. Liu, A sharp Grüss type inequality on time scales and application to the sharp Ostrowski- Grüss inequality, Commun. Math. Anal., 6 (2009), 33-41
##[35]
A. Ostrowski , Über die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert , Comment. Math. Helv., 10 (1937), 226-227
##[36]
M. Z. Sarikaya, N. Aktan, H. Yildirim, On weighted Čebyšev-Grüss type inequalities on time scales, J. Math. Inequal., 2 (2008), 185-195
##[37]
M. Z. Sarikaya, New weighted Ostrowski and Čebyšev type inequalities on time scales, Comput. Math. Appl., 60 (2010), 1510-1514
##[38]
C. Soria-Hoyo, F. Pontiga, A. Castellanos, A PIC based procedure for the integration of multiple time scale problems in gas discharge physics, J. Comput. Phys., 228 (2009), 1017-1029
##[39]
K.-L. Tseng, S. R. Hwang, S. S. Dragomir, Generalizations of weighted Ostrowski type inequalities for mappings of bounded variation and their applications, Comput. Math. Appl., 55 (2008), 1785-1793
##[40]
A. Tuna, D. Daghan, Generalization of Ostrowski and Ostrowski-Grüss type inequalities on time scales, Comput. Math. Appl., 60 (2010), 803-811
##[41]
A. Tuna, Y. Jiang, W. J. Liu, Weighted Ostrowski, Ostrowski-Grüss and Ostrowski- Čebyhev Type Inequalities on Time Scales, Publ. Math. Debrecen, 81 (2012), 81-102
]
Visco-resolvent algorithms for monotone operators and nonexpansive mappings
Visco-resolvent algorithms for monotone operators and nonexpansive mappings
en
en
Two new type of visco-resolvent algorithms for finding a zero of the sum of two monotone operators and a
fixed point of a nonexpansive mapping in a Hilbert space are investigated. The algorithms consist of the
zeros and the fixed points of the considered problems in which one operator is replaced with its resolvent
and a viscosity term is added. Strong convergence of the algorithms are shown. As special cases, we can
approach to the minimum norm common element of the zero of the sum of two monotone operators and the
fixed point of a nonexpansive mapping without using the metric projection. Some applications are included.
325
344
Peize
Li
Department of Mathematics
Tianjin Polytechnic University
China
tjlipeize@163.com
Shin Min
Kang
Department of Mathematics and the RINS
Gyeongsang National University
Korea
smkang@gnu.ac.kr
Li-Jun
Zhu
School of Mathematics and Information Science
Beifang University of Nationalities
China
zhulijun1995@sohu.co
Monotone operator
nonexpansive mapping
zero point
fixed point
resolvent.
Article.4.pdf
[
[1]
K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, On a strongly nonexpansive sequence in Hilbert spaces, J. Nonlinear Convex Anal., 8 (2007), 471-489
##[2]
H. H. Bauschke, J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review, 38 (1996), 367-426
##[3]
H. H. Bauschke, P. L. Combettes, A Dykstra-like algorithm for two monotone operators, Pacific J. Optim., 4 (2008), 383-391
##[4]
H. H. Bauschke, P. L. Combettes, S. Reich , The asymptotic behavior of the composition of two resolvents , Nonlinear Anal., 60 (2005), 283-301
##[5]
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145
##[6]
Y. Censor, S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications, Oxford University Press, New York, USA (1997)
##[7]
P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504
##[8]
P. L. Combettes, S. A. Hirstoaga , Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136
##[9]
P. L. Combettes, S. A. Hirstoaga , Approximating curves for nonexpansive and monotone operators, J. Convex Anal., 13 (2006), 633-646
##[10]
P. L. Combettes, S. A. Hirstoaga, Visco-penalization of the sum of two monotone operators, Nonlinear Anal., 69 (2008), 579-591
##[11]
F. Ding, T. Chen , Iterative least squares solutions of coupled Sylvester matrix equations, Systems Control Lett., 54 (2005), 95-107
##[12]
J. Eckstein, D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318
##[13]
Y. P. Fang, N. J. Huang, H-Monotone operator resolvent operator technique for quasi-variational inclusions, Appl. Math. Comput., 145 (2003), 795-803
##[14]
K. Geobel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press (1990)
##[15]
L. J. Lin, Variational inclusions problems with applications to Ekeland's variational principle, fixed point and optimization problems , J. Global Optim., 39 (2007), 509-527
##[16]
P.-L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979
##[17]
X. Liu, Y. Cui, Common minimal-norm fixed point of a finite family of nonexpansive mappings, Nonlinear Anal., 73 (2010), 76-83
##[18]
X. Lu, H. K. Xu, X. Yin , Hybrid methods for a class of monotone variational inequalities, Nonlinear Anal., 71 (2009), 1032-1041
##[19]
A. Moudafi, On the regularization of the sum of two maximal monotone operators, Nonlinear Anal., 42 (2009), 1203-1208
##[20]
G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72 (1979), 383-390
##[21]
J. W. Peng, Y. Wang, D. S. Shyu, J. C. Yao, Common solutions of an iterative scheme for variational inclusions, equilibrium problems and fixed point problems, J. Inequal. Appl., Article ID 720371, 2008 (2008), 1-15
##[22]
S. M. Robinson, Generalized equation and their solutions, part I, basic theory, Math Program. Study, 10 (1979), 128-141
##[23]
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216
##[24]
R. T. Rockafellar , Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898
##[25]
A. Sabharwal, L. C. Potter, Convexly constrained linear inverse problems: iterative least-squares and regularization, IEEE Trans. Signal Process, 46 (1998), 2345-2352
##[26]
T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), 103-123
##[27]
S. Takahashi, W. Takahashi, M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147 (2010), 27-41
##[28]
W. Takahashi, M. Toyoda , Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428
##[29]
H. K. Xu, Iterative algorithms for nonlinear operators , J. London Math. Soc., 2 (2002), 1-17
##[30]
Y. Yao, R. Chen, H. K. Xu , Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 3447-3456
##[31]
Y. Yao, Y. C. Liou , Composite algorithms for minimization over the solutions of equilibrium problems and fixed point problems , Abstr. Appl. Anal., Article ID 763506, 2010 (2010), 1-19
##[32]
S. S. Zhang, H. W. Lee Joseph, C. K. Chan, Algorithms of common solutions for quasi variational inclusion and fixed point problems , Appl. Math. Mech. (English Ed.), 29 (2008), 571-581
]
Coupled coincidence point theorems for mappings without mixed monotone property under c-distance in cone metric spaces
Coupled coincidence point theorems for mappings without mixed monotone property under c-distance in cone metric spaces
en
en
Fixed point theory in the field of partially ordered metric spaces has been an area of attraction since
the appearance of Ran and Reurings theorem and Nieto and Rodríguez-López theorem. One of the most
significant hypotheses of these theorems was the mixed monotone property which has been avoided and
replaced by the notion of invariant set in recent years and many statements have been proved using the
concept of invariant set. In this paper we show that the invariant condition guides us to prove many similar
results to which we were exposed to using the mixed monotone property. We present some examples in
support of applicability of our results.
345
358
Rakesh
Batra
Department of Mathematics, Hans Raj College
University of Delhi
India
rakeshbatra.30@gmail.com
Sachin
Vashistha
Department of Mathematics, Hindu College
University of Delhi
India
vashistha_sachin@rediffmail.com
Rajesh
Kumar
Department of Mathematics,Hindu College
University of Delhi
India
rajeshhctm@rediffmail.com
fixed point
coincidence point
cone metric space
c-distance
(F
g)-invariant set.
Article.5.pdf
[
[1]
R. Batra, S. Vashistha, Coupled coincidence point theorems for nonlinear contractions under c-distance in cone metric spaces, Ann. Funct. Anal., 4 (2013), 138-148
##[2]
R. Batra, S. Vashistha , Coupled coincidence point theorems for nonlinear contractions under (F; g)-invariant set in cone metric spaces, J. Nonlinear Sci. Appl., 6 (2013), 86-96
##[3]
R. Batra, S. Vashistha , Some coupled coincidence point results under c-distance in cone metric spaces, Eng. Math. Lett., 2 (2013), 90-114
##[4]
T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications , Nonlinear Anal., 65 (2006), 1379-1393
##[5]
Y. J. Cho, R. Saadati, S. Wang, Common fixed point theorems on generalized distance in ordered cone metric spaces, Comput. Math. Appl. , 61 (2011), 1254-1260
##[6]
Y. J. Cho, Z. Kadelburg, R. Saadati, W. Shatanawi , Coupled fixed point theorems under weak contractions , Discrete Dyn. Nat. Soc. Article ID 184534, (2012), 1-9
##[7]
L. G. Huang, X. Zhang , Cone meric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. , 332 (2007), 1468-1476
##[8]
Sh. Jain, Sh. Jain, L. B. Jain , On Banach contraction principle in a cone metric space , J.Nonliear Sci. Appl., 5 (2012), 252-258
##[9]
O. Kada, T. Suzuki, W. Takahashi , Nonconvex minimization theorems and fixed point theorems in complete metric spaces , Math. Japon., 44 (1996), 381-391
##[10]
E. Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl., 59 (2010), 3656-3668
##[11]
V. Lakshmikantham, L. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., 70 (2009), 4341-4349
##[12]
H. K. Nashine, B. Samet, C. Vetro , Coupled coincidence points for compatible mappings satisfying mixed monotone property, J. Nonlinear Sci. Appl., 5 (2012), 104-114
##[13]
J. J. Nieto, R. Rodríguez-López , Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser. ), 23 (2007), 2205-2212
##[14]
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 (2004), 1435-1443
##[15]
K. P. R. Rao, S. Hima Bindu, Md. Mustaq Ali , Coupled fixed point theorems in d-complete topological spaces, J. Nonlinear Sci. Appl., 5 (2012), 186-194
##[16]
B. Samet, C. Vetro , Coupled fixed point, F-invariant set and fixed point of N-order , Ann. Funct. Anal., 1 (2010), 46-56
##[17]
W. Shatanawi, E. Karapinar, H. Aydi , Coupled coincidence points in partially ordered cone metric spaces with a c-distance, J. Appl. Math, Article ID 312078, (2012), 1-15
##[18]
W. Sintunavarat, Y. J. Cho, P. Kumam, Coupled fixed point theorems for weak contraction mappings under F-invariant set, Abstr. Appl. Anal. , (), 1-15
##[19]
D. Turkoglu, M. Abuloha , Cone metric spaces and fixed point theorems in diametrically contractive mappings, Acta Math. Sin. (Engl. Ser. ), 26 (2010), 489-496
]
On the Ulam stability of a quadratic set-valued functional equation
On the Ulam stability of a quadratic set-valued functional equation
en
en
In this paper, we prove the Ulam stability of the following set-valued functional equation by employing the
direct method and the fixed point method, respectively,
\[f ( x -\frac{ y + z}{ 2}) \oplus f (x +\frac{ y - z}{ 2})\oplus f(x + z) = 3f(x) \oplus \frac{1}{ 2} f(y) \oplus \frac{3 }{2 }f(z).\]
359
367
Faxing
Wang
Tongda College of Nanjing University of Posts and Telecommunications
P. R. China
wangfx@njupt.edu.cn
Yonghong
Shen
School of Mathematics and Statistics
Tianshui Normal University
P. R. China
shenyonghong2008@hotmail.com
Ulam stability
Quadratic set-valued functional equation
Hausdorff distance
fixed point.
Article.6.pdf
[
[1]
T. Aoki , On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66
##[2]
C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, in: Lec. Notes in Math., vol. 580, Springer, Berlin (1977)
##[3]
H. Y. Chu, A. Kim, S. Y. Yoo, On the stability of the generalized cubic set-valued functional equation, Appl. Math. Lett., 37 (2014), 7-14
##[4]
J. B. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305-309
##[5]
G. L. Forti , Hyers-Ulam stability of functional equations in several variables, Aequat. Math., 50 (1995), 143-190
##[6]
P. Gavruca, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436
##[7]
D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222-224
##[8]
S. Y. Jang, C. Park, Y. Cho, Hyers-Ulam stability of a generalized additive set-valued functional equation, J. Inequal. Appl., 2013 (2013), 1-101
##[9]
S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, (2011)
##[10]
H. A. Kenary, H. Rezaei, On the stability of set-valued functional equations with the fixed point alternative, Fixed Point Theory Appl., 2012 (2012), 1-81
##[11]
G. Lu, C. Park , Hyers-Ulam stability of additive set-valued functional equations , Appl. Math. Lett., 24 (2011), 1312-1316
##[12]
C. Park, D. O'Regan, R. Saadati, Stability of some set-valued functional equations, Appl. Math. Lett., 24 (2011), 1910-1914
##[13]
M. Piszczek, The properties of functional inclusions and Hyers-Ulam stability, Aequat. Math., 85 (2013), 111-118
##[14]
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300
##[15]
Th. M. Rassias , On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23-130
##[16]
H. Rådström, An embedding theorem for spaces of convex sets , Proc. Amer. Math. Soc., 3 (1952), 165-169
##[17]
P. K. Sahoo, P. Kannappan, Introduction to Functional Equations, CRC Press , Boca Raton (2011)
##[18]
Y. H. Shen, Y. Y. Lan, On the general solution of a quadratic functional equation and its Ulam stability in various abstract spaces, J. Nonlinear Sci. Appl., 7 (2014), 368-378
##[19]
S. M. Ulam, Problems in Modern Mathematics, Wiley, New York (1960)
]