]>
2013
6
2
ISSN 2008-1898
91
On the stability of an affine functional equation
On the stability of an affine functional equation
en
en
In this paper, we obtain the general solution and we prove the generalized Hyers-Ulam stability for an affine
functional equation.
60
67
Liviu
Cădariu
Department of Mathematics
''Politehnica'' University of Timişoara
Romania
liviu.cadariu@mat.upt.ro; lcadariu@yahoo.com
Laura
Găvruţa
Department of Mathematics
''Politehnica'' University of Timişoara
Romania
laura.gavruta@mat.upt.ro
Paşc
Găvruţa
Department of Mathematics
''Politehnica'' University of Timişoara
Romania
pgavruta@yahoo.com
Generalized Ulam-Hyers stability
affine functional equation
direct method
fixed points
Article.1.pdf
[
[1]
T. Aoki , On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66
##[2]
J. A. Baker , The stability of certain functional equations, Proc. AMS, 112(3) (1991), 729-732
##[3]
J. Brzdęk, K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Analysis - TMA, 74 (2011), 6861-6867
##[4]
L. Cădariu, L. Găvruţa, P. Găvruţa , Weighted space method for the stability of some nonlinear equations, Appl. Anal. Discrete Math., 6 (2012), 126-139
##[5]
L. Cădariu, L. Găvruţa, P. Găvruţa, Fixed points and generalized Hyers-Ulam stability, Abstr. Appl. Anal. Article ID 712743, 2012 (2012), 1-10
##[6]
L. Cădariu, V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure and Appl. Math., Art. 4, 4(1) (2003), -
##[7]
L. Cădariu, V. Radu, On the stability of the Cauchy functional equation: a fixed points approach, Iteration theory (ECIT '02), (J. Sousa Ramos, D. Gronau, C. Mira, L. Reich, A. N. Sharkovsky - Eds.)Grazer Math. Ber., 346 (2004), 43-52
##[8]
L. Cădariu L., V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications, Article ID 749392, 2008 (2008), 1-15
##[9]
L. Cădariu L., V. Radu, A general fixed point method for the stability of Cauchy functional equation, in Functional Equations in Mathematical Analysis, Th. M. Rassias, J. Brzdek (Eds.), Series Springer Optimization and Its Applications 52, (2011)
##[10]
L. Cădariu L., V. Radu , A general fixed point method for the stability of the monomial functional equation , Carpathian J. Math., 28 (2012), 25-36
##[11]
I.-S. Chang, H.-M. Kim, On the Hyers-Ulam Stability of Quadratic Functional Equations, 3(3), (2002)
##[12]
S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publishing Company, New Jersey, London , Singapore Hong Kong (2002)
##[13]
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
##[14]
G. L. Forti, An existence and stability theorem for a class of functional equations , Stochastica, 4 (1980), 23-30
##[15]
G. L. Forti , Hyers-Ulam stability of functional equations in several variables, Aeq. Math., 50 (1995), 143-90
##[16]
G. L. Forti, Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. , 295(1) (2004), 127-133
##[17]
Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. , 14 (1991), 431-434
##[18]
L. Găvruţa, Matkowski contractions and Hyers-Ulam stability, Bul. Şt. Univ. ''Politehnica'' Timişoara, Seria Mat.-Fiz., 53(67) (2008), 32-35
##[19]
P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436
##[20]
P. Găvruţa, On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings, J. Math. Anal. Appl., 261 (2001), 543-553
##[21]
P. Găvruţa, L. Găvruţa, A new method for the generalized Hyers-Ulam-Rassias stability, Int. J. Nonlinear Anal. Appl., 1 (2010), 11-18
##[22]
D. H. Hyers, On the stability of the linear functional equation, Prod. Natl. Acad. Sci. USA, 27 (1941), 222-224
##[23]
D. H. Hyers, G. Isac G., Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel (1998)
##[24]
S.-M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Series Springer Optimization and Its Applications, Springer (2011)
##[25]
D. Miheţ , The Hyers-Ulam stability for two functional equations in a single variable, Banach J. Math. Anal. Appl., 2 (2008), 48-52
##[26]
V. Radu, The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4 (2003), 91-96
##[27]
Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300
##[28]
Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23-130
##[29]
S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York (1964)
]
Uniform exponential stability for evolution families on the half-line
Uniform exponential stability for evolution families on the half-line
en
en
In this paper we give a characterization for the uniform exponential stability of evolution families \(\{\Phi(t; t_0)\}_{t\geq t_0}\)
on \(\mathbb{R}_+\) that do not have an exponential growth, using the hypothesis that the pairs of function spaces
\((L^1(X);L^\infty(X))\) and \((L^p(X);L^q(X)), (p; q) \neq (1;\infty)\), are admissible to the evolution families.
68
73
Petre
Preda
Department of Mathematics
West University of Timişoara
Romania
preda@math.uvt.ro; rmuresan@math.uvt.ro
Raluca
Mureşan
Department of Mathematics
West University of Timişoara
Romania
rmuresan@math.uvt.ro
Evolution family
admissibility
uniform exponential stability.
Article.2.pdf
[
[1]
L. Barreira, C. Valls, Admissibility for nonuniform exponential contractions , J. Diff. Eq., 249 (2010), 2889-2904
##[2]
L. Barreira, C. Valls, Regularity of center manifolds under nonuniform hyperbolicity, Discrete and Continuous Dynamical Systems, 30 (2011), 55-76
##[3]
C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Diferential Equations, Math. Surveys Monogr., vol. 70, Amer. Math. Soc., Providence, RI (1999)
##[4]
W. A. Coppel, Dichotomies in Stability Theory, Lect. Notes Math., vol. 629, Springer-Verlag, New-York (1978)
##[5]
J. L. Daleckij, M. G. Krein, Stability of Diferential Equations in Banach Space, Amer. Math. Soc., Providence, RI (1974)
##[6]
R. Datko , Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. Math. Anal., 3 (1972), 428-445
##[7]
P. Hartman, Ordinary Differential Equations, Wiley, New-York, London, Sydney (1964)
##[8]
B. M. Levitan, V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge (1982)
##[9]
J. L. Massera, J. J. Schäffer, Linear Diferential Equations and Function Spaces, Academic Press, New York (1966)
##[10]
N. van Minh, N. T. Huy , Exponential dichotomy of evolution equations and admissibility of function spaces on the half line, J. Funct. Anal., 235 (2006), 330-354
##[11]
N. van Minh, N. T. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44
##[12]
N. van Minh, F. Rägiger, R. Schnaubelt , Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line, Integr. Equ. Oper. Theory, 32 (1998), 332-353
##[13]
O. Perron, Die stabilitätsfrage bei diferentialgeighungen, Math. Z., 32 (1930), 703-728
##[14]
P. Preda, A. Pogan, C. Preda , Admissibility and exponential dichotomy of evolutionary processes on half-line, Rend. Sem. Mat. Univ. Pol. Torino, 61 (2003), 461-473
##[15]
P. Preda, A. Pogan, C. Preda, Schffer spaces anduniform exponential stability of linear skew-product semi ows, J. Diff. Eq., 2005 (212), 191-207
##[16]
P. Preda, A. Pogan, C. Preda, Schffer spaces and exponential dichotomy for evolutionary processes, J. Diff. Eq., 230 (2006), 378-391
]
Some strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative variables
Some strong sufficient conditions for cyclic homogeneous polynomial inequalities of degree four in nonnegative variables
en
en
We establish some strong sufficient conditions that the inequality \(f_4(x; y; z) \geq 0\) holds for all nonnegative
real numbers \(x; y; z\), where\( f_4(x; y; z)\) is a cyclic homogeneous polynomial of degree four. In addition, in the
case \(f_4(1; 1; 1) = 0\) and also in the case when the inequality \(f_4(x; y; z) \geq 0\) does not hold for all real numbers
\(x; y; z\), we conjecture that the proposed sufficient conditions are also necessary that\( f_4(x; y; z) \geq 0\) for all
nonnegative real numbers \(x; y; z\). Several applications are given to show the effectiveness of the proposed
methods.
74
85
Yuanzhe
Zhou
The School of Physics and Technology at Wuhan University
China
Vasile
Cirtoaje
Department of Automatic Control and Computers
University of Ploiesti
Romania
vcirtoaje@upg-ploiesti.ro
Cyclic homogeneous polynomial
strong sufficient conditions
necessary and sufficient conditions
nonnegative real variables.
Article.3.pdf
[
[1]
T. Ando, , Some Homogeneous Cyclic Inequalities of Three Variable of Degree Three and Four, , The Australian Journal of Mathematical Analysis and Applications, vol. 7, issue 2, art. 12, [ONLINE:http://ajmaa.org/ cgi-bin/paper.pl?string=v7n2/V7I2P11.tex] (2011)
##[2]
V. Cirtoaje, Algebraic Inequalities-Old and New Methods, GIL Publishing House, (2006)
##[3]
V. Cirtoaje, On the Cyclic Homogeneous Polynomial Inequalities of Degree Four, Journal of Inequalities in Pure and Applied Mathematics, vol. 10, issue 3, art. 67, [ONLINE:http://www.emis.de/journals/JIPAM/ article1123.html] (2009)
##[4]
V. Cirtoaje, Y. Zhou, Necessary and Sufficient Conditions for Cyclic Homogeneous Polynomial Inequalities of Degree Four in Real Variables, The Australian Journal of Mathematical Analysis and Applications, vol. 9, issue 1, art. 15, [ONLINEhttp://ajmaa.org/cgi-bin/paper.pl?string=v9n1/V9I1P15.tex] (2012)
##[5]
V. Cirtoaje, Y. Zhou, Some Strong Sufficient Conditions for Cyclic Homogeneous Polynomial Inequalities of Degree Four in Real Variables, Journal of Nonlinear Analysis and Applications, vol. 2012, art. 151, [ON- LINE:http://www.ispacs.com/jnaa/?p=jnaa_articles] (2012)
##[6]
P. K. Hung, Secrets in Inequalities, vol. 2, GIL Publishing House (2008)
##[7]
, Art of Problem Solving, [ONLINE:http://www.artofproblemsolving.com/Forum/viewtopic. php?t=448153,498608], (2011)
]
Coupled coincidence point theorems for nonlinear contractions under \((F,g)\)-invariant set in cone metric spaces
Coupled coincidence point theorems for nonlinear contractions under \((F,g)\)-invariant set in cone metric spaces
en
en
We extend the recent results of coupled coincidence point theorems of Shatanawi et. al. (2012) by weakening
the concept of mixed g-monotone property. We also give an example of a nonlinear contraction mapping,
which is not applied to the existence of coupled coincidence point by the results of Shatanawi et. al. but
can be applied to our results. The main results extend and unify the results of Shatanawi et. al. and many
results of the coupled fixed point theorems of Sintunavarat et. al.
86
96
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
Coincidence point
Cone metric space
C-distance
Fixed point
(F
g)-invariant set.
Article.4.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]
T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. , 65 (2006), 1379-1393
##[3]
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
##[4]
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
##[5]
L. G. Huang, X. Zhang, Cone meric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468-1476
##[6]
Sh. Jain, Sh. Jain, L. B. Jain, On Banach contraction principle in a cone metric space, J.Nonliear Sci. Appl., 5 (2012), 252-258
##[7]
O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon. , 44 (1996), 381-391
##[8]
E. Karapinar, Couple fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. , 59 (2010), 3656-3668
##[9]
V. Lakshmikantham, L. Cirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. , 70 (2009), 4341-4349
##[10]
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
##[11]
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
##[12]
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
##[13]
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
##[14]
B. Samet, C. Vetro, Coupled fixed point, F-invariant set and fixed point of N-order, Ann. Funct. Anal., 1 (2010), 46-56
##[15]
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, doi:10.1155/2012/312078. , (2012), 1-15
##[16]
W. Sintunavarat, Y. J. Cho, P. Kumam, Coupled fixed point theorems for weak contraction mappings under F-invariant set, Abstr. Appl. Anal., doi:10.1155/2012/324874. , 15 pages ()
##[17]
D. Turkoglu, M. Abuloha, Cone metric spaces and fixed point theorems in diametrically contractive mappings, Acta Math. Sin., 26 (2010), 489-496
]
An abstract point of view on iterative approximation schemes of fixed points for multivalued operators
An abstract point of view on iterative approximation schemes of fixed points for multivalued operators
en
en
In this paper we will present an abstract point of view on iterative approximation schemes of fixed points for
multivalued operators. More precisely, we suppose that the algorithms are convergent and we will study the
impact of this hypothesis in the theory of operatorial inclusiosns: data dependence, stability and Gronwall
type lemmas. Some open problems are also presented.
97
107
Adrian
Petruşel
Department of Mathematics
Babeş-Bolyai University
Romania
petrusel@math.ubbcluj.ro
Ioan A.
Rus
Department of Mathematics
Babeş-Bolyai University
Romania
iarus@math.ubbcluj.ro
multivalued operator
fixed point
strict fixed point
iterative scheme
multivalued Picard operator
multivalued weakly Picard operator.
Article.5.pdf
[
[1]
J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer Acad. Publ., Dordrecht, 2003. , , , ()
##[2]
J.-P. Aubin and A. Cellina, Differential Inclusions, Springer Verlag, Berlin, 1984. , , , ()
##[3]
V. Berinde, Iterative Approximations of Fixed Points, Springer Verlag, Berlin, 2007., , , ()
##[4]
[4] L.M. Blumenthal, Theory and Applications of Distance Geometry, Oxford University Press, 1953. , , , ()
##[5]
S. Carl and S. Heikkilä, Fixed Point Theory in Ordered Sets and Applications, Springer, Berlin, 2011., , , ()
##[6]
[6] S.S. Chang and K.-K. Tan, Iteration processes for approximating fixed points of operators by monotone type, Bull. Austral. Math. Soc., 57(1998), 433-445. , , , (), -
##[7]
C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, Springer Verlag, Berlin, 2009. [, , , ()
##[8]
8] H. Covitz and S.B. Nadler jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math., 8(1970), 5-11. , , , (), -
##[9]
R. Espínola, A. Petruşel, Existence and data dependence of fixed points for multivalued operators on gauge spaces, J. Math. Anal. Appl., 309(2005), 420-432., , , (), -
##[10]
[10] Y. Feng and S. Liu, Fixed point theorems for multivalued increasing operators in partial ordered spaces, Soochow J. Math., 30(2004), No.4, 461-469. , , , (), -
##[11]
M. Fréchet, Les espaces abstraits, Gauthier-Villars, Paris, 1928., , , ()
##[12]
[12] V. Glăvan and V. Guţu, Shadowing and stability in set-valued dynamics (Preprint). , , , ()
##[13]
L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Kluwer Acad. Publ., Dordrecht, 1999., , , ()
##[14]
[14] S. Gudder and F. Schroeck, Generalized convexity, SIAM J. Math. Anal. 11(1980), 984-1001. , , , (), -
##[15]
S. Hu, N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I and II, Kluwer Acad. Publ., Dordrecht, 1997 and 1999. , , , ()
##[16]
[16] J.R. Jachymski, Fixed point theorems in metric and uniform spaces via the Knaster-Tarski principle, Nonlinear Anal., 32(1998) No.2, 225-233. , , , (), -
##[17]
M.A. Khamsi and W.A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, Wiley-Interscience, New York, 2001. [, , , ()
##[18]
18] W.A. Kirk and B. Sims (Editors), Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht, 2001. , , , ()
##[19]
S.B. Nadler jr., Multivalued contraction mappings, Pacific J. Math., 30(1969), 475-488. , , , (), -
##[20]
[20] K. Neammanee, A. Kaewkhao, On multivalued weak contraction mappings, J. Math. Research, 3(2011), No. 2, 151-156. , , , (), -
##[21]
K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer Acad. Publ., Dordrecht, 2000. , , , ()
##[22]
[22] B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comput. Math. Appl., 54(2007), No. 6, 872-877. , , , (), -
##[23]
A. Petruşel, Starshaped and fixed points, Seminar on Fixed Point Theory, Babes-Bolyai Univ., 1987, 19-24. , , , (), -
##[24]
[24] A. Petruşel, Operatorial Inclusions, House of the Book of Science, Cluj-Napoca, 2002. , , , ()
##[25]
A. Petruşel, Multivalued weakly Picard operators and applications, Sci. Math. Jpn., 59(2004), 169-202., , , (), -
##[26]
[26] A. Petruşel and I.A. Rus, Multivalued Picard and weakly Picard operators, Proc. 6th International Conference on Fixed Point Theory and Applications, Valencia, Spain, July 19-26, 2003 (E. Llorens Fuster, J. Garcia Falset, B. Sims-Eds.), Yokohama Publ., 2004, 207-226. , , , (), -
##[27]
A. Petruşel and I.A. Rus, The theory of a metric fixed point theorem for multivalued operators, Proc. 9th International Conference on Fixed Point Theory and its Applications, Changhua, Taiwan, July 16-22, 2009, (L.J. Lin, A. Petruşel, H.K. Xu-Eds.), Yokohama Publ. 2010, 161-175., , , (), -
##[28]
[28] A. Petruşel and G. Petruşel, Multivalued Picard operators, J. Nonlinear Convex Anal., 13(2012), No. 1, 157-171. , , , (), -
##[29]
A. Petruşel, I.A. Rus and J.-C. Yao, Well-posedness in the generalized sense of the fixed point problems, Taiwanese J. Math., 11(2007), No.3, 903-914. , , , (), -
##[30]
[30] G. Petruşel and A. Petruşel, Existence and data dependence of the strict fixed points for multivalued \(\delta\)-contractions on graphic, Pure Math. Appl., 17(2006), No. 3-4, 413-418. , , , (), -
##[31]
S.Yu. Pilyugin, Shadowing in Dynamical Systems, Springer Verlag, Berlin, 1999., , , ()
##[32]
[32] T. Puttasantiphat, Mann and Ishikawa iteration schemes for multivalued mappings in CAT(0) spaces, Appl. Math. Sci., 4(2010), No.61, 3005-3018. , , , (), -
##[33]
S. Reich, Fixed point of contractive functions, Boll. Un. Mat. Ital., 5(1972), 26-42., , , (), -
##[34]
[34] S. Reich and A.J. Zaslavski, Convergence of inexact iterative schemes for nonexpansive set-valued mappings, Fixed Point Theory Appl., 2010(2010), Article ID 518243, 10 p. , , , (), -
##[35]
I.A. Rus, Picard operators and applications, Sci. Math. Jpn., 58(2003), 191-219. , , , (), -
##[36]
[36] I.A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevance, Fixed Point Theory, 9(2008), 541-559. , , , (), -
##[37]
I.A. Rus, Strict fixed point theory, Fixed Point Theory, 4(2003), 177-183., , , (), -
##[38]
[38] I.A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001. , , , ()
##[39]
I.A. Rus, Fixed Point Structure Theory, Cluj University Press, Cluj-Napoca, 2006., , , ()
##[40]
[40] I.A. Rus, An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations, Fixed Point Theory, 13(2012), No.1, 179-192. , , , (), -
##[41]
I.A. Rus, A. Petruşel and A. Sîntămărian, Data dependence of the fixed point set of some multivalued weakly Picard operators, Nonlinear Anal., 52(2003), no. 8, 1947-1959., , , (), -
##[42]
[42] I.A. Rus, A. Petruşel and G. Petruşel, Fixed Point Theory, Cluj University Press, 2008. , , , ()
##[43]
N. Shahzad and H. Zegeye, On Mann and Ishikawa schemes for multivalued maps in Banach spaces, Nonlinear Anal., 71(2009), 838-844. , , , (), -
##[44]
[44] A. Sîntămărian, Metrical strict fixed point theorems for multivalued mappings, Sem. on Fixed Point Theory, 1997, 27-30. , , , (), -
##[45]
S.L. Singh, C. Bhatnagar and A.M. Hashim, Round-off stability of Picard iterative procedure for multivalued operators, Nonlinear Anal. Forum, 10(2005), No. 1, 13-19. , , , (), -
##[46]
[46] R.E. Smithson, Fixed point of order preserving multifunction, Proc. Amer. Math. Soc., 28(1971), 304-310. , , , (), -
##[47]
Y. Song and Y.J. Cho, Some notes on Ishikawa iteration for multi-valued mappings, Bull. Korean Math. Soc., 48(2011), No. 3, 575-584., , , (), -
##[48]
[48] Y. Song and H. Wang, Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Anal. Theory Methods Appl., 70(2009), No. 4-A, 1547-1556. , , , (), -
##[49]
W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, 2000. , , , ()
##[50]
[50] W. Takahashi, A convexity in metric spaces and nonexpansive mapping I, Kodai Math. Sem. Rep., 22(1970), 142-149. , , , (), -
##[51]
E. Tarafdar and G.X.-Z. Yuan, Set-valued contraction mapping principle, Applied Math. Letters, 8(1995), 79-81., , , (), -
##[52]
[52] G.X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, 1999., , , ()
]
Existence and Ulam-Hyers stability results for coincidence problems
Existence and Ulam-Hyers stability results for coincidence problems
en
en
Let \(X, Y\) be two nonempty sets and \(s, t : X \rightarrow Y\) be two single-valued operators.
By definition, a solution of the coincidence problem for s and \(t\) is a pair \((x^*; y^*) \in X \times Y\) such that
\[s(x^*) = t(x^*) = y^*.\]
It is well-known that a coincidence problem is, under appropriate conditions, equivalent to a fixed point
problem for a single-valued operator generated by s and t. Using this approach, we will present some
existence, uniqueness and Ulam - Hyers stability theorems for the coincidence problem mentioned above.
Some examples illustrating the main results of the paper are also given.
108
116
Oana
Mleşniţe
Department of Mathematics
Babeş-Bolyai University Cluj-Napoca
Romania
oana.mlesnite@math.ubbcluj.ro
metric space
coincidence problem
singlevalued contraction
vector-valued metric
fixed point
Ulam-Hyers stability.
Article.6.pdf
[
[1]
M. Bota, A. Petruşel , Ulam-Hyers stability for operatorial equations, Analele Univ. Al.I. Cuza Iaşi, 57 (2011), 65-74
##[2]
A. Buică , Coincidence Principles and Applications, Cluj University Press, in Romanian (2001)
##[3]
L. P. Castro, A. Ramos, Hyers-Ulam-Rassias stability for a class of Volterra integral equations, Banach J. Math. Anal., 3 (2009), 36-43
##[4]
K. Goebel, A coincidence theorem, Bull. de L'Acad. Pol. des Sciences, 16 (1968), 733-735
##[5]
S.-M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory and Applications, Article ID 57064, 2007 (2007), 1-9
##[6]
A. I. Perov, On Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ. Uravn., 2 (1964), 115-134
##[7]
P. T. Petru, A. Petruşel, J. C. Yao, Ulam-Hyers stability for operatorial equations and inclusions via nonself operators, Taiwanese J. Math., 15 (2011), 2195-2212
##[8]
I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10 (2009), 305-320
##[9]
I. A. Rus, Ulam stability of ordinary differential equations, Studia Univ. Babeş-Bolyai Math., 54 (2009), 125-133
##[10]
I. A. Rus, Gronwall lemma approach to the Ulam-Hyers-Rassias stability of an integral equation, Nonlinear Analysis and Variational Problems (P.M. Pardalos et al. (eds.)), 147 Springer Optimization and Its Applications, New York, 35 (), 147-152
]
A fixed point theory for \(S\)-contractions in generalized Kasahara spaces
A fixed point theory for \(S\)-contractions in generalized Kasahara spaces
en
en
The aim of this paper is to present a fixed point theory for \(S\)-contractions in generalized Kasahara spaces
\((X;\rightarrow; d)\), where \(d : X \times X \rightarrow s(\mathbb{R}_+)\) is not necessarily an \(s(\mathbb{R}_+)\)-metric.
117
123
Alexandru-Darius
Filip
Department of Mathematics
Babeş-Bolyai University of Cluj-Napoca
Romania
darius.filip@econ.ubbcluj.ro
Fixed point
\(S\)-contraction
generalized Kasahara space
sequence of successive approximations
\(s(\mathbb{R}_+)\)-metric
Neumann matrix
Ulam-Hyers stability.
Article.7.pdf
[
[1]
V. G. Angelov, A converse to a contraction mapping theorem in uniform spaces, Nonlinear Analysis, 12 (1988), 989-996
##[2]
S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundamenta Mathematicae, 3 (1922), 133-181
##[3]
R. Caccioppoli, Un teorema generale sull'esistenza di elementi uniti in una transformazione funzionale, Rendiconti dell'Academia Nazionale dei Lincei, 11 (1930), 794-799
##[4]
R. G. Cooke, Infinite Matrices and Sequence Spaces, London, (1950)
##[5]
A.-D. Filip , Fixed point theorems in Kasahara spaces with respect to an operator and applications, Fixed Point Theory, 12 (2011), 329-340
##[6]
M. Fréchet, Les espaces abstraits , Gauthier-Villars, Paris (1928)
##[7]
M. Frigon, Fixed point and continuation results for contractions in metric and gauge spaces, Banach Center Publ., 77 (2007), 89-114
##[8]
N. Gheorghiu, Fixed point theorems in uniform spaces , An. Ştiinţ. Al. I. Cuza Univ. Iaşi, Secţ. I Mat., 1982 (28), 17-18
##[9]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222-224
##[10]
D. H. Hyers, The stability of homomorphism and related topics, in: Global Analysis - Analysis on Manifolds (Th.M. Rassias (ed.)), Teubner, Leipzig, (1983), 140-153
##[11]
K. Iséki , An approach to fixed point theorems , Math. Sem. Notes, 3 (1975), 193-202
##[12]
S. Kasahara , On some generalizations of the Banach contraction theorem , Publ. RIMS, Kyoto Univ., 12 (1976), 427-437
##[13]
S. Kasahara, Fixed point theorems in certain L-spaces, Math. Seminar Notes, 5 (1977), 29-35
##[14]
I. A. Rus , Picard operators and applications, Sci. Math. Jpn., 58 (2003), 191-219
##[15]
I. A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances, Fixed Point Theory, 9 (2008), 541-559
##[16]
I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10 (2009), 305-320
##[17]
I. A. Rus, Kasahara spaces, Sci. Math. Jpn., 72 (2010), 101-110
##[18]
I. A. Rus, A. Petruşeland, G. Petruşel , Fixed Point Theory, Cluj University Press, Cluj-Napoca (2008)
##[19]
S. M. Ulam , A Collection of Mathematical Problems, Interscience Publ., New York (1960)
##[20]
P. P. Zabrejko, T. A. Makarevich, Generalization of the Banach-Caccioppoli principle to operators on pseudo- metric spaces, Diff. Urav., 23 (1987), 1497-1504
]
Ulam-Hyers stability for coupled fixed points of contractive type operators
Ulam-Hyers stability for coupled fixed points of contractive type operators
en
en
In this paper, we present existence, uniqueness and Ulam-Hyers stability results for the coupled fixed points
of a pair of contractive type singlevalued and respectively multivalued operators on complete metric spaces.
The approach is based on Perov type fixed point theorem for contractions in spaces endowed with vector-
valued metrics.
124
136
Cristina
Urs
Department of Mathematics
Babeş-Bolyai University Cluj-Napoca
Romania
cristina.urs@math.ubbcluj.ro
metric space
coupled fixed point
singlevalued operator
vector-valued metric
Perov type contraction.
Article.8.pdf
[
[1]
G. Allaire, S. M. Kaber, Numerical Linear Algebra, Texts in Applied Mathematics, vol. 55, Springer, New York (2008)
##[2]
M. Bota, A. Petrusel, Ulam-Hyers stability for operatorial equations, Analls of the Alexandru Ioan Cuza University Iasi, 57 (2011), 65-74
##[3]
A. Bucur, L. Guran, A. Petrusel , Fixed points for multivalued operators on a set endowed with vector-valued metrics and applications, Fixed Point Theory, 10 (2009), 19-34
##[4]
A. D. Filip, A. Petrusel, Fixed point theorems on spaces endowed with vector-valued metrics, Fixed Point Theory and Applications, Article ID 281381, 2010 (2009), 1-15
##[5]
A. D. Filip, A. Petrusel , Fixed point theorems on spaces endowed with vector-valued metrics, Fixed Point Theory and Applications, Article ID 281381, 2010 (2009), 1-15
##[6]
T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379-1393
##[7]
D. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications , Nonlinear Anal., 11 (1987), 623-632
##[8]
D. Guo, Y. J. Cho, J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Science Publishers Inc., Hauppauge, NY (2004)
##[9]
S. Hong, Fixed points for mixed monotone multivalued operators in Banach spaces with applications, J. Math. Anal. Appl., 337 (2008), 333-342
##[10]
V. Lakshmikantham, L. Ćirić, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. , 70 (2009), 4341-4349
##[11]
A. I. Perov , On the Cauchy problem for a system of ordinary differential equations, Pviblizhen. Met. Reshen. Differ. Uvavn, 2 (1964), 115-134
##[12]
P. T. Petru, A. Petrusel, J. C. Yao, Ulam-Hyers stability for operatorial equations and inclusions via nonself operators, Taiwanese J. Math., 15 (2011), 2195-2212
##[13]
A. Petrusel, Multivalued weakly Picard operators and applications, Sci. Math. Japon, 59 (2004), 169-202
##[14]
R. Precup , The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Modell, 49 (2009), 703-708
##[15]
I. A. Rus, Principles, Applications of the Fixed Point Theory, Dacia, Cluj-Napoca (1979)
##[16]
I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10 (2009), 305-320
##[17]
M. D. Rus, The method of monotone iterations for mixed monotone operators, Ph. D. Thesis, Universitatea Babes-Bolyai, Cluj-Napoca (2010)
##[18]
R. S. Varga, Matrix Iterative Analysis, Springer Series in Computational Mathematics, Vol. 27, Springer, Berlin (2000)
]
Gravity-capillary water waves generated by multiple pressure distributions
Gravity-capillary water waves generated by multiple pressure distributions
en
en
Steady two-dimensional free-surface
flows subjected to multiple localised pressure distributions are considered. The
fluid is bounded below by a rigid bottom, and above by a free-surface, and is assumed to be
inviscid and incompressible. The
flow is assumed irrotational, and the effects of both gravity and surface
tension are taken into account. Forced solitary wave solutions are found numerically, using boundary integral equation techniques, based on Cauchy integral formula. The integrodifferential equations are solved
iteratively by Newton's method. The behaviour of the forced waves is determined by the Froude number,
the Bond number, and the coefficients of the pressure forcings. Multiple families of solutions are found to
exist for particular values of the Froude number; perturbations from a uniform stream, and perturbations
from pure solitary waves. Elevation waves are only obtained in the case of a negatively forced pressure
distribution.
137
144
Charlotte
Page
School of Computing Sciences
University of East Anglia
United Kingdom
Emilian I.
Părău
School of Mathematics
University of East Anglia
United Kingdom
e.parau@uea.ac.uk
Scott
Grandison
School of Computing Sciences
University of East Anglia
United Kingdom
Nonlinear waves
gravity-capillary waves
Article.9.pdf
[
[1]
B. J. Binder, F. Dias, J.-M. Vanden-Broeck , Forced solitary waves and fronts past submerged obstacles, Chaos, 15 (2005), 1-027106
##[2]
B. J. Binder, F. Dias, J.-M. Vanden-Broeck, In uences of rapid changes in a channel bottom of free-surface flows, IMA Journal of Applied Mathematics, 73 (2008), 254-273
##[3]
F. Dias, J.-M. Vanden-Broeck , Open channel flows with submerged obstructions, Fluid Mech., 206 (1989), 155-170
##[4]
F. Dias, J.-M. Vanden-Broeck, Generalised critical free-surface flows, Journal of Engineering Mathematics, 42 (2002), 291-301
##[5]
F. Dias, J.-M. Vanden-Broeck , Trapped waves between submerged obstacles, J. Fluid Mech., 509 (2004), 93-102
##[6]
L. K. Forbes, L. W. Schwartz, Free-surface flow over a semicircular obstruction, J. Fluid Mech., 114 (1982), 299-314
##[7]
L. K. Forbes, Critical free-surface flow over a semi-circular obstruction, Journal of Engineering Mathematics, 22 (1988), 3-13
##[8]
M. Maleewong, J. Asavanant, R.Grimshaw, Free surface flow under gravity and surface tension due to an applied pressure distribtion: I Bond number greather than one-third, Theoretical and Computational Fluid Dynamics, 19 (2005), 237-252
##[9]
M. Maleewong, R. Grimshaw, J. Asavanant, Free surface flow under gravity and surface tension due to an applied pressure distribtion: II Bond number less than one-third, European Journal of Mechanics B/Fluids 24 (2005), 502-521. [10] Părău, E. and Vanden-Broeck, J.-M., Nonlinear two- and three-dimensional free surface flows due to moving pressure distributions, European Journal of Mechanics B/Fluids , 21 (2002), 643-656
##[10]
J.-M. Vanden-Broeck, F. Dias, Gravity-capillary solitary waves in water of infinite depth and related free-surface flows, J. Fluid Mech. , 240 (1992), 549-557
]
On controllability for nonconvex semilinear differential inclusions
On controllability for nonconvex semilinear differential inclusions
en
en
We consider a semilinear differential inclusion and we obtain sufficient conditions for h-local controllability
along a reference trajectory.
145
151
Aurelian
Cernea
Faculty of Mathematics and Computer Science
University of Bucharest
Romania
acernea@fmi.unibuc.ro
Differential inclusion
h-local controllability
mild solution
Article.10.pdf
[
[1]
J. P. Aubin, H. Frankowska, Set-valued Analysis, Birkhauser, Berlin (1990)
##[2]
A. Cernea, Continuous version of Filippov's theorem for a semilinear differential inclusion, Stud. Cerc. Mat., 49 (1997), 319-330
##[3]
A. Cernea, Derived cones to reachable sets of semilinear differential inclusions, Proc. 19th Int. Symp. Math. Theory Networks Systems, Budapest, Ed. A. Edelmayer, (2010), 235-238
##[4]
A. Cernea, Some qualitative properties of the solution set of an infinite horizon operational differential inclusion, Revue Roumaine Math. Pures Appl. , 43 (1998), 317-328
##[5]
A. Cernea , On the relaxation theorem for semilinear differential inclusions in Banach spaces , Pure Math. Appl., 13 (2002), 441-445
##[6]
A. Cernea, On the solution set of some classes of nonconvex nonclosed differential inclusions, Portugaliae Math., 65 (2008), 485-496
##[7]
F. H. Clarke, Optimization and Nonsmooth Analysis , Wiley Interscience, New York (1983)
##[8]
F. S. De Blasi, G. Pianigiani, Evolutions inclusions in non separable Banach spaces, Comment. Math. Univ. Carolinae, 40 (1999), 227-250
##[9]
F. S. De Blasi, G. Pianigiani, V. Staicu , Topological properties of nonconvex differential inclusions of evolution type , Nonlinear Anal. , 24 (1995), 711-720
##[10]
H. Frankowska, A priori estimates for operational differential inclusions , J. Diff. Eqs. , 84 (1990), 100-128
##[11]
E. S. Polovinkin, G. V. Smirnov, An approach to differentiation of many-valued mapping and necessary condition for optimization of solution of differential inclusions, Diff. Equations. , 22 (1986), 660-668
##[12]
V. Staicu, Continuous selections of solutions sets to evolution equations , Proc. Amer. Math. Soc. , 113 (1991), 403-413
##[13]
H. D. Tuan, On controllability and extremality in nonconvex differential inclusions, J. Optim. Theory Appl. , 85 (1995), 437-474
##[14]
J. Warga, Controllability, extremality and abnormality in nonsmooth optimal control , J. Optim. Theory Appl. , 41 (1983), 239-260
]