]>
2019
12
3
ISSN 2008-1898
56
A new extension of exponential distribution with statistical properties and applications
A new extension of exponential distribution with statistical properties and applications
en
en
A new extension of exponential distribution, named as the \(\textit{Type I half logistic exponential distribution}\) is introduced in this paper. Explicit expressions for the moments, probability weighted, quantile function, mean deviation, order statistics, and Renyi entropy are investigated. Parameter estimates of the new distribution are obtained based on maximum likelihood procedure. Two real data sets are employed to show the usefulness of the new distribution.
135
145
Abdullah M.
Almarashi
Statistics Department, Faculty of Science
King AbdulAziz University
Kingdom of Saudi Arabia
aalmarashi@kau.edu.sa
M.
Elgarhy
Vice Presidency for Graduate Studies and Scientific Research
University of Jeddah
KSA
m_elgarhy85@yahoo.com
Mamhoud M.
Elsehetry
Institute of Statistical Studies and Research (ISSR), Department of Mathematical Statistics
Cairo University
Egypt
m_elgarhy85@yahoo.com
B. M.
Golam Kibria
Department of Mathematics and Statistics
Florida International University
USA
kibriag@fiu.edu
Ali
Algarni
Statistics Department, Faculty of Science
King AbdulAziz University
Kingdom of Saudi Arabia
ahalgarni@kau.edu.sa
Exponential distribution
maximum likelihood method
moments
order statistics
type I half logistic-G distributions
Article.1.pdf
[
[1]
A. M. Almarashi, M. Elgarhy , A new muth generated family of distributions with applications , J. Nonlinear Sci. Appl., 11 (2018), 1171-1184
##[2]
A. Alzaatreh, C. Lee, F. Famoye , A new method for generating families of continuous distributions, Metron, 71 (2013), 63-79
##[3]
M. Bourguignon, R. B. Silva, G. M. Cordeiro , The Weibull–G family of probability distributions , J. Data Sci., 12 (2014), 53-68
##[4]
G. M. Cordeiro, M. Alizadeh, P. R. D. Marinho, The type I half-logistic family of distributions, J. Stat. Comput. Simul., 86 (2015), 707-728
##[5]
G. M. Cordeiro, M. Alizadeh, E. M. M. Ortega, The exponentiated half-logistic family of distributions: Properties and applications, J. Probab. Stat., 2014 (2014), 1-21
##[6]
G. M. Cordeiro, M. de Castro, A new family of generalized distributions, J. Stat. Comput. Simul., 81 (2011), 883-893
##[7]
H. A. David , Order statistics, John Wiley & Sons, New York (1981)
##[8]
M. Elgarhy, M. Haq, G. Ozel, A new exponentiated extended family of distributions with Applications, Gazi University J. Sci., 30 (2017), 101-115
##[9]
M. Elgarhy, A. S. Hassan, M. Rashed , Garhy-generated family of distributions with application, Math. Theory Model., 6 (2016), 1-15
##[10]
N. Eugene, C. Lee, F. Famoye, The beta-normal distribution and its applications, Commun. Stat. Theory Methods, 31 (2002), 497-512
##[11]
M. Haq, M. Elgarhy , The odd Frechet-G family of probability distributions, J. Stat. Appl. Prob., 7 (2018), 185-201
##[12]
A. S. Hassan, M. Elgarhy , A New family of exponentiated Weibull-generated distributions, Int. J. Math. Appl., 4 (2016), 135-148
##[13]
A. S. Hassan, M. Elgarhy, Kumaraswamy Weibull-generated family of distributions with applications , Adv. Appl. Stat., 48 (2016), 205-239
##[14]
A. S. Hassan, M. Elgarhy, M. Shakil , Type II half Logistic family of distributions with applications , Pak. J. Stat. Oper. Res., 13 (2017), 245-264
##[15]
D. Kundu, M. Z. Raqab , Estimation of R = P (Y < X) for three-parameter Weibull distribution, Stat. Prob. Lett., 79 (2009), 1839-1846
##[16]
M. M. Ristic, N. Balakrishnan , The gamma-exponentiated exponential distribution, J. Stat. Comput. Simul., 82 (2012), 1191-1206
##[17]
K. Zografos, N. Balakrishnan, On families of beta- and generalized gamma-generated distributions and associated inference, Statistical Methodology, 6 (2009), 344-362
]
The particular solutions of some types of Euler-Cauchy ODE using the differential transform method
The particular solutions of some types of Euler-Cauchy ODE using the differential transform method
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en
In this paper, we apply the differential transform method to find the
particular solutions of some types of Euler-Cauchy ordinary differential
equations. The first model is a special case of the nonhomogeneous \(n^{\rm th}\) order ordinary differential equations of Euler-Cauchy equation. The
second model under consideration in this paper is the nonhomogeneous second
order differential equation of Euler-Cauchy equation with a bulge function.
This study showed that this method is powerful and efficient in finding the
particular solution for Euler-Cauchy ODE and capable of reducing the size of
calculations comparing with other methods.
146
151
Meshari
Alesemi
Mathematics Department, Faculty of Science
Jazan University
Kingdom of Saudi Arabia
malesem@jazanu.edu.sa
M. A.
El-Moneam
Mathematics Department, Faculty of Science
Jazan University
Kingdom of Saudi Arabia
maliahmedibrahim@jazanu.edu.sa
Bader S.
Bader
Mathematics Department, Faculty of Science
Jazan University
Kingdom of Saudi Arabia
bsaad1757@gmail.com
E. S.
Aly
Mathematics Department, Faculty of Science
Jazan University
Kingdom of Saudi Arabia
elkhateeb@jazanu.edu.sa
Differential equations
differential transform method
Euler-Cauchy equations
Article.2.pdf
[
[1]
M. S. Abdualrab, A formula for solving a special case of Euler-Cauchy ODE., Int. Math. Forum, 4 (2009), 1997-2000
##[2]
J. Ali , One dimensional differential transform method for some higher order boundary value problems in finite domain, Int. J. Contemp. Math. Sci., 7 (2012), 263-272
##[3]
M. T. Alquran, Applying differential transform method to nonlinear partial differential equations: a modified approach, Appl. Appl. Math., 7 (2012), 155-163
##[4]
A. Aslanov, Determination of convergence intervals of the series solutions of EmdenFowler equations using polytropes and isothermal spheres, Phy. Lett. A, 372 (2008), 3555-3561
##[5]
K. Batiha, B. Batiha, A new algorithm for solving linear ordinary differential equations, World Appl. Sci. J., 15 (2011), 1774-1779
##[6]
C. Bervillier, Status of the differential transformation method, Appl. Math. Comput., 218 (2012), 10158-10170
##[7]
J. Biazar, M. Eslami, Differential transform method for quadratic Riccati differential equation, Int. J. Nonlinear Sci., 9 (2010), 444-447
##[8]
S.-H. Chang, I.-L. Chang, A new algorithm for calculating one dimensional differential transform of nonlinear functions, Appl. Math. Comput., 195 (2008), 799-805
##[9]
E. A. Elmabrouk, F. Abdewahid, Useful Formulas for One-dimensional Differential Transform, Britsh J. Appl. Sci. Tech., 18 (2016), 1-8
##[10]
V. S. Ert ürk, Application of differential transformation method to linear sixth-order boundary value problems, Appl. Math. Sci. (Ruse), 1 (2007), 51-58
##[11]
V. S. Ertürk, Approximate Solutions of a Class of Nonlinear Differential Equations by Using Differential Transformation Method, Int. J. Pure Appl. Math., 30 (2006), 403-407
##[12]
G. G. Ev Pukhov, Differential transforms and circuit theory, Circuit Theory Appl., 10 (2008), 265-276
##[13]
B. Ghil, H. Kim, The Solution of Euler-Cauchy Equation Using Laplace Transform, Int. J. math. Anal., 9 (2015), 2611-2618
##[14]
P. Haarsa, S. Pothat , The Reduction of Order on Cauchy-Euler Equation with a Bulge Function, Appl. Math. Sci., 9 (2015), 1139-1143
##[15]
I. H. A. H. Hassan, V. S. Ertürk, Solution of differential types of the linear and nonlinear higher-order boundary value problems by differential transformation method, Eur. J. Pure Appl. Math., 2 (2009), 426-447
##[16]
K. Parand, Z. Roozbahani, F. Bayat Babolghani, Solving nonlinear Lane-Emden type equations with unsupervised combined artificial neural networks, Int. J. Industrial Mathematics, 5 (2013), 1-12
##[17]
M. A. Soliman, Y. Al-Zeghayer, Aproximate analytical solution for the isothermal Lane Emden equation in a spherical geometry, Revist Mexicanade Astronmiay Atrofisca, 15 (2015), 173-180
##[18]
A.-M. Wazwas, A new algorithm for solving differential equations of LaneEmden type, Applied Math. Comput., 118 (2001), 287-310
##[19]
A.-M. Wazwas, The modified decomposition method for analytic treatment of differential equations, Appl. Math. Comput., 173 (2006), 165-176
##[20]
A. Yildrim, T. Özis, Solutions of singular IVPs of LaneEmden type by the variational iteration method, Nonlinear Anal., 70 (2009), 2480-2484
##[21]
E. M. E. Zayed, M. A. El-Moneam, Some oscillation criteria for second order nonlinear functional ordinary differential equations, Acta Math. Sci. Ser. B (Engl. Ed.), 27 (2007), 602-610
##[22]
E. M. E. Zayed, S. R. Grace, H. El-Metwally, M. A. El-Moneam, The oscillatory behavior of second order nonlinear functional differential equations, Arab. J. Sci. Eng. Sect. A Sci., 31 (2006), 23-30
##[23]
J. K. Zhou, Differential transformation and its applications for electrical circuits, Huarjung University Press, wuuhahn (1986)
]
A new class of distributions based on the zero truncated Poisson distribution with properties and applications
A new class of distributions based on the zero truncated Poisson distribution with properties and applications
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en
We study a new family of distributions defined by the minimum of the Poisson
random number of independent and identically distributed random variables
having the Topp Leone-G distribution. Some mathematical properties of the
new family are derived. Maximum likelihood estimation of the model
parameters is investigated. Two special models of the new family are
discussed. We perform three applications to real data sets to show the
potentiality of the proposed family. In order to test the validity of the
new family, a modified Chi-squared goodness-of-fit test based on
Nikulin-Rao-Robson statistics is proposed theoretically.
152
164
T. H. M.
Abouelmagd
Management Information System Department
Taibah University
Saudi Arabia
tabouelmagd@taibahu.edu.sa
Mohammed S.
Hamed
Management Information System Department
Taibah University
Saudi Arabia
moswilem@gmail.com
Laba
Handique
Department of Statistics
Dibrugarh University
India
handiquelaba@gmail.com
Hafida
Goual
Laboratory of Probability and Statistics
University of Badji Mokhtar
Algeria
goual.hafida@gmail.com
M. Masoom
Ali
Department of Mathematical Sciences
Ball State University
USA
mali@bsu.edu
Haitham M.
Yousof
Department of Statistics, Mathematics and Insurance
Benha University
Egypt
haitham.yousof@fcom.bu.edu.eg
Mustafa C.
Korkmaz
Department of Measurement and Evaluation
ArtvinCoruh University
Turkey
mcagatay@artvin.edu.tr
Topp Leone-G family
order statistics
maximum likelihood estimation
quantile function
generating function
moments
Article.3.pdf
[
[1]
M. Alizadeh, H. M. Yousof, A. Z. Afify, G. M. Cordeiro, M. Mansoor, The complementary generalized transmuted Poisson-G family of distributions, Austrian J. Stat., 47 (2018), 51-71
##[2]
M. Alizadeh, H. M. Yousof, M. Rasekhi, E. Altun, The odd log-logistic Poisson-G family of distributions, J. Math. Exten., 12 (2018)
##[3]
G. R. Aryal, H. M. Yousof, The exponentiated generalized-G Poisson family of distributions, Stoch. Qual. Control, 32 (2017), 7-23
##[4]
T. Bjerkedal , Acquisition of resistance in Guinea pigs infected with different doses of virulent tubercle bacilli , Amer. J. Hygiene, 72 (1960), 130-148
##[5]
A. J. Gross, V. Clark, Survival Distributions: Reliability Applications in the Biometrical Sciences, John Wiley, New York (1975)
##[6]
D. Kundu, M. Z. Raqab, Estimation of R = P (Y < X ) for three parameter Weibull distribution, Statist. Probab. Lett., 79 (2009), 1839-1846
##[7]
M. S. Nikulin, Chi-square test for continuous distribution with shift and scale parameters, Theory Probab. Appl., 19 (1973), 559-568
##[8]
K. C. Rao, D. S. Robson, A chi-square statistic for goodness-of-fit for tests with in the exponential family, Comm. Statist., 3 (1974), 1139-1153
##[9]
Y. Sangsanit, W. Bodhisuwan, The Topp-Leone generator of distributions: properties and inferences, Songklanakarin J. Sci. Technol., 38 (2016), 537-548
##[10]
D. S. Shibu, M. R. Irshad, Extended new generalized Lindley distribution, Statistica, 76 (2016), 41-56
##[11]
V. Voinov, M. Nikulin, N. Balakrishnan, Chi-Squared Goodness of Fit Tests with Applications, Academic Press, London (2013)
]
The odd Frechet inverse Weibull distribution with application
The odd Frechet inverse Weibull distribution with application
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en
A new three parameters distribution called the odd Frechet inverse Weibull (OFIW) distribution is introduced. The reliability analysis of the new model is discussed. Several of its mathematical properties are studied. The maximum likelihood (ML) estimation are derived for OFIW parameters. The importance and flexibility of the OFIW is assessed using one real data set.
165
172
Aisha
Fayomi
Statistics Department, Faculty of Science
King AbdulAziz University
Kingdom of Saudi Arabia
afayomi@kau.edu.sa
Odd Frechet family
inverse Weibull distribution
order statistics
maximum likelihood
Article.4.pdf
[
[1]
S. Abbas, S. A. Taqi, F. Mustafa, M. Murtaza, M. Q. Shahbaz, Topp-Leone inverse Weibull distribution: theory and application, Eur. J. Pure Appl. Math., 10 (2017), 1005-1022
##[2]
S. J. Almalki, J. Yuan, A new modified Weibull distribution, Reliab. Eng. Sys. Safety, 111 (2013), 164-170
##[3]
F. R. S. de Gusmao, E. M. M. Ortega, G. M. Cordeiro, The generalized inverse Weibull distribution, Statist. Papers, 52 (2011), 591-619
##[4]
I. Elbatal, Y. M. El Gebaly, E. A. Amin, The beta generalized inverse Weibull geometric distribution and its applications, Pak. J. Stat. Oper. Res., 13 (2017), 75-90
##[5]
A. J. Gross, V. A. Clark, Survival distributions: Reliability applications in the biomedical sciences , John Wiley & Sons, New York (1975)
##[6]
S. Hanook, M. Q. Shahbaz, M. Mohsin, B. M. G. Kibria, A Note On beta inverse Weibull distribution, Comm. Statist. Theory Methods, 42 (2013), 320-335
##[7]
M. Haq, M. Elgarhy, The odd Fréchet-G family of probability distributions, J. Statist. Appl. Prob., 7 (2018), 185-201
##[8]
R. Jiang, D. N. P. Murthy, P. Ji, Models involving two inverse Weibull distributions, Reliab. Eng. Sys. Safety, 73 (2001), 73-81
##[9]
R. Jiang, M. J. Zuo, H. X. Li, Weibull and inverse Weibull mixture models allowing negative weights, Reliab. Eng. Sys. Safety, 66 (1999), 227-234
##[10]
A. Z. Keller, A. R. R. Kamath, Alternative reliability models for mechanical systems, Third Int. Conf. Reliab. Maintainabil. (Toulse, France), 1982 (1982), 411-415
##[11]
M. S. Khan, R. King , Modified inverse Weibull distribution , J. Statist. Appl. Prob., 1 (2012), 115-132
##[12]
M. S. Khan, R. King, I. Hudson, Characterizations of the transmuted Inverse Weibull distribution, Anziam J., 55 (2014), 197-217
##[13]
M. S. Khan, G. R. Pasha, A. H. Pasha, Theoretical analysis of inverse Weibull distribution, WSEAS Tran. Math., 7 (2008), 30-38
##[14]
D. V. Lindley, Fiducial distributions and Bayes’ theorem, J. Roy. Statist. Soc. Ser. B, 20 (1958), 102-107
##[15]
W. Nelson, Applied Life Data Analysis New York, John Wiley & Sons, New York (1982)
##[16]
M. Q. Shahbaz, S. Shahbaz, N. S. Butt, The Kumaraswamy inverse Weibull Distribution, Pak. J. Stat. Oper. Res., 3 (2012), 479-489
##[17]
R. Shanker, Aradhana distribution and its Applications, Int. J. Statist. Appl., 6 (2016), 23-34
##[18]
R. Shanker, Sujatha distribution and its Applications, Statist. Transition-New Series, 17 (2016), 1-20
##[19]
R. Shanker, K. K. Shukla, H. Fesshaye, A generalization of Sujatha distribution and its applications with real lifetime data, J. Institute Sci. Tech., 22 (2017), 66-83
]
Poisson Burr X Weibull distribution
Poisson Burr X Weibull distribution
en
en
The main goal of this paper is to introduce a continuous distributions based
on the zero truncated Poisson which accommodates increasing, bathtub,
decreasing, J-shaped, constant and unimodal shapes of monotone failure
rates. A comprehensive account of some of its mathematical properties are
provided. The new probability density function can be expressed as a linear
combination of exponentiated Weibull densities. The method of the maximum
likelihood is used to estimate the model parameters. Empirically, we proved
the importance and flexibility of the new distribution in modeling two data
sets.
173
183
T. H. M.
Abouelmagd
Management Information System Department
Department of Statistics, Mathematics and Insurance
Taibah University
Benha University
Saudi Arabia
Egypt
tabouelmagd@taibahu.edu.sa
Mohammed S.
Hamed
Management Information System Department
Department of Statistics, Mathematics and Insurance
Taibah University
Benha University
Saudi Arabia
Egypt
moswilem@gmail.com
Haitham M.
Yousof
Department of Statistics, Mathematics and Insurance
Benha University
Egypt
haitham.yousof@fcom.bu.edu.eg
Truncated Poisson
moments
maximum likelihood estimation
function
generating function
Article.5.pdf
[
[1]
T. H. M. Abouelmagd, A new flexible distribution based on the zero truncated Poisson distribution: mathematical properties and applications to lifetime data, Biostatistics and Biometrics, 1–7. (2018)
##[2]
V. G. Cancho, F. Louzada-Neto, G. D. C. Barriga, The Poisson-exponential lifetime distribution, Comput. Statist. Data Anal., 55 (2011), 677-686
##[3]
M. Chahkandi, M. Ganjali, On some lifetime distributions with decreasing failure rate, Comput. Statist. Data Anal., 53 (2009), 4433-4440
##[4]
M. Ç. Korkmaz, M. Erişoğlu , The Burr XII-Geometric Distribution, J. Selçuk Univer. Nat. Appl. Sci., 3 (2014), 75-87
##[5]
M. Ç. Korkmaz, C. Kuş, H. Erol, The mixed Weibull-negative binomial distribution, Selçuk J. Appl. Math., Special Issue, (2011), 21-33
##[6]
M. Ç. Korkmaz, H. M. Yousof, G. G. Hamedani, M. M. Ali, Marshall-Olkin Generalized G Poisson family of distributions, Pakistan J. Statist., 34 (2018), 251-267
##[7]
C. Kuş, A new lifetime distribution, Comput. Statist. Data Anal., 51 (2007), 4497-4509
##[8]
M. W. A. Ramos, P. R. D. Marinho, G. M. Cordeiro, R. V. da Silva, G. G. Hamedani , The Kumaraswamy-G Poisson family of distributions, J. Stat. Theory Appl., 14 (2015), 222-239
##[9]
H. M. Yousof, A. Z. Afify, G. G. Hamedani, G. Aryal , The Burr X generator of distributions for lifetime data, J. Stat. Theory Appl., 16 (2017), 288-305
]
Ekeland's variational principle in complete quasi-G-metric spaces
Ekeland's variational principle in complete quasi-G-metric spaces
en
en
In this paper, by concept of \(\Gamma\)-function which is define on q-G-m (quasi-\(G\)-metric) space, we
establish a generalized Ekeland's variational principle in the setting of lower semicontinuous
from above. As application we prove generalized flower petal theorem in q-G-m.
184
191
E.
Hashemi
Department of Mathematics, College of Basic Sciences
Karaj Branch, Islamic Azad University
Iran
eshagh_hashemi@yahoo.com
M. B.
Ghaemi
Department of Mathematics
Iran University of Science and Technology
Iran
mghaemi@iust.ac.ir
\( \Gamma\)-Function, q-G-m space
generalized EVP
lower semicontinuous from above function
generalized Caristi's (common) fixed point theorem
nonconvex minimax theorem
generalized flower petal theorem
Article.6.pdf
[
[1]
M. Amemiya, W. Takahashi, Fixed point theorems for fuzzy mappings in complete metric spaces, Fuzzy Sets and system, 125 (2002), 253-260
##[2]
J.-P. Aubin, J. Siegel , Fixed points and stationary points of dissipative multivalued maps, Proc. Amer. Math. Soc., 78 (1980), 391-398
##[3]
J. S. Bae, Fixed point theorems for weakly contractive multivalued maps, J. Math. Anal. Appl., 284 (2003), 690-697
##[4]
J. S. Bae, E. W. Cho, S. H. Yeom , A generalization of the Caristi-Kirk fixed point theorem and its application to mapping theorems , J. Korean Math. Soc., 31 (1994), 29-48
##[5]
J. Caristi , Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc., 215 (1976), 241-251
##[6]
Y. Chen, Y. J. Cho, L. Yang, Note on the results with lower semi-continuity, Bull Korean Math. Soc., 39 (2002), 535-541
##[7]
P. Z. Daffer, H. Kaneko, W. Li, Variational principle and fixed points, in: Set Valued Mappings With Applications in Nonlinear Analysis, 2002 (2002), 129-136
##[8]
S. Dancs, M. Hegedüs, P. Medvegyev , A general ordering and fixed point principle in complete metric spaces, Acta. Sci. Math. (Szeged), 46 (1983), 381-388
##[9]
I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474
##[10]
I. Ekeland , On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353
##[11]
I. Ekeland, Remarques sur les problémes variationnels , C. R. Acad. Sci. Paris Sér. A–B, 275 (1972), 1057-1059
##[12]
L. Gajek, D. Zagrodny , Geometric variational principle, Dissertationes Math. (Rozprawy Mat.), 340 (1995), 55-71
##[13]
A. Hamel , Remarks to an equivalent formulation of Ekelands variational principle, Optimization, 31 (1994), 233-238
##[14]
A. Hamel, A. Löhne , A minimal point theorem in uniform spaces, in: Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday, 1, 2 (2003), 557-593
##[15]
D. H. Hyers, G. Isac, T. M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific Publishing Co., River Edge (1997)
##[16]
O. Kada, T. Suzuki, W. Takahashi , Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., 44 (1996), 381-391
##[17]
Y. Kijima, On a minimization theorem, in: Nonlinear Analysis and Convex Analysis (Japanese), 1995 (1995), 59-62
##[18]
N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., 141 (1989), 177-188
##[19]
Z. Mustafa, B. Sims, A new approach to generalized metric spaces , J. Nonlinear Convex Anal., 7 (2006), 289-392
##[20]
J.-P. Penot , The drop theorem, the petal theorem and Ekelands variational principle, Nonlinear Anal., 10 (1986), 813-822
##[21]
R. Saadati, S. M. Vaezpoura, P. Vetro , B. E. Rhoades, Fixed point theorems in generalized partially ordered G-metric spaces , Math. Comput. Modelling, 52 (2010), 797-801
##[22]
N. Shioji, T. Suzuki, W. Takahashi , Contractive mappings, Kannan mappings and metric completeness , Proc. Amer. Math. Soc., 126 (1998), 3117-3124
##[23]
T. Suzuki , Generalized distance and existence theorems in complete metric spaces, J. Math. Anal. Appl., 253 (2001), 440-458
##[24]
T. Suzuki, On DowningKirks theorem , J. Math. Anal. Appl., 286 (2003), 453-458
##[25]
T. Suzuki , Generalized Caristis fixed point theorems by Bae and others, J. Math. Anal. Appl., 302 (2005), 502-508
##[26]
T. Suzuki, W. Takahashi, Fixed point theorems and characterizations of metric completeness, Topol. Methods Nonlinear Anal., 8 (1996), 371-382
##[27]
W. Takahashi, Existence theorems generalizing fixed point theorems for multivalued mappings , in: Fixed Point Theory and Applications, 1991 (1991), 397-406
##[28]
W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000)
##[29]
D. Tataru, Viscosity solutions of HamiltonJacobi equations with unbounded nonlinear terms, J. Math. Anal. Appl., 163 (1992), 345-392
##[30]
C.-K. Zhong , On Ekelands variational principle and a minimax theorem, J. Math. Anal. Appl., 205 (1997), 239-250
]