ORIGINAL_ARTICLE
$r$-fuzzy regular semi open sets in smooth topological spaces
In this paper, we introduce and study the concept of $r$-fuzzy regular semi open (closed) sets in smooth topological spaces. By using $r$-fuzzy regular semi open (closed) sets, we define a new fuzzy closure operator namely $r$-fuzzy regular semi interior (closure) operator. Also, we introduce fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. Moreover, we investigate the relationship among fuzzy regular semi continuous and fuzzy regular semi irresolute mappings. Finally, we have given some counter examples to show that these types of mappings are not equivalent.
http://scma.maragheh.ac.ir/article_22080_6b6d5d86c4a101f32db67cdc439daa70.pdf
2017-04-01T11:23:20
2018-01-19T11:23:20
1
17
$r$-fuzzy regular semi open (closed) sets
$r$-fuzzy regular semi interior (closure) operator
Fuzzy regular semi continuous (irresolute) maps
Appachi
Vadivel
avmaths@gmail.com
true
1
Department of Mathematics, Annamalai University, Annamalai Nagar-608002, Tamil Nadu, India.
Department of Mathematics, Annamalai University, Annamalai Nagar-608002, Tamil Nadu, India.
Department of Mathematics, Annamalai University, Annamalai Nagar-608002, Tamil Nadu, India.
LEAD_AUTHOR
Elangovan
Elavarasan
maths.aras@gmail.com
true
2
Department of Mathematics, Annamalai University, Annamalai Nagar-608002, Tamil Nadu, India.
Department of Mathematics, Annamalai University, Annamalai Nagar-608002, Tamil Nadu, India.
Department of Mathematics, Annamalai University, Annamalai Nagar-608002, Tamil Nadu, India.
AUTHOR
[1] R. Badard, Smooth axiomatics, First IFSA Congress Palma de Mallorca, 1986.
1
[2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968) 182-190.
2
[3] K.C. Chattopadhyay and S.K. Samanta, Fuzzy topology, Fuzzy Sets and Systems, 54 (1993) 207-212.
3
[4] K.C. Chattopadhyay, R.N. Hazra, and S.K. Samanta, Gradation of openness, Fuzzy Sets and Systems, 49 (2) (1992) 237-242.
4
[5] R.N. Hazra, S.K. Samanta, and K.C. Chattopadhyay, Fuzzy topology redefined, Fuzzy Sets and Systems, 45 (1992) 79-82.
5
[6] E.E. Kerre and M.A. Fath Alla, On fuzzy regular semi open sets and $alpha S^{*}$-closed fuzzy topological spaces, The Journal of Fuzzy Mathematics, 11 (1) (2003) 225-235.
6
[7] S.J. Lee and E.P. Lee, Fuzzy $r$-regular open sets and fuzzy almost $r$-continuous maps, Bull. Korean Math. Soc., 39 (3) (2002) 441-453.
7
[8] A.S. Mashhour, M.H. Ghanim, and M.A. Fath Alla, $alpha$-separation axioms and $alpha$-compactness in fuzzy topological spaces, Rocky Moutain Journal of Mathematics, 16 (3) (1986) 591-600.
8
[9] A.A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systems, 48 (1992) 371-375.
9
[10] A.A. Ramadan, S.E. Abbas, and Y.C. Kim, Fuzzy irresolute mappings in smooth fuzzy topological spaces, The Journal of Fuzzy Mathematics, 9 (4) (2001) 865-877.
10
[11] A.P. Šostak, On a fuzzy topological structure, Rend. Circ. Matem. Palermo Ser II, 11 (1986), 89-103.
11
[12] A. Vadivel and E. Elavarasan, Applications of $r$-generalized regular fuzzy closed sets, Annals of Fuzzy Mathematics and Informatics, (Accepted).
12
[13] A.M. Zahran, Fuzzy regular semi open sets and $alpha S$-closed spaces, The Journal of Fuzzy Mathematics, 2 (4) (1997), 579-586.
13
ORIGINAL_ARTICLE
Dynamic equivalence relation on the fuzzy measure algebras
The main goal of the present paper is to extend classical results from the measure theory and dynamical systems to the fuzzy subset setting. In this paper, the notion of dynamic equivalence relation is introduced and then it is proved that this relation is an equivalence relation. Also, a new metric on the collection of all equivalence classes is introduced and it is proved that this metric is complete.
http://scma.maragheh.ac.ir/article_22015_386a0c0212ed48b855025307bda3aa1e.pdf
2017-04-01T11:23:20
2018-01-19T11:23:20
19
28
Fuzzy measure algebra
Dynamical system
Equivalence relation
Uniformly continuous
Roya
Ghasemkhani
roya.ghasemkhani@gmail.com
true
1
Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran.
Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran.
Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran.
AUTHOR
Uosef
Mohammadi
u.mohamadi@ujiroft.ac.ir
true
2
Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran.
Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran.
Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran.
LEAD_AUTHOR
[1] G.A. Afrozi, S. Shakeri, and S.H. Rasouli, On the fuzzy metric spaces, The Journal of Mathematics and Computer Science Vol .2 No.3 (2011) 475-482.
1
[2] J.R. Brown, Ergodic theory and Topological Dynamics, Academic Pres, New York, 1976.
2
[3] D. Dumitrescu, C. Haloiu, and A. Dumetrescu, Generators of fuzzy dynamical systems, Fuzzy Sets and Systems, 113 (2000) 447-452.
3
[4] M. Ebrahimi, U. Mohamadi, m-Generators of fuzzy dynamical systems, Cankaya University Journal of Science and Engineering, 9 (2012), 167-182.
4
[5] P.R. Halmos, Measure Theory, Springer-Verlag New York, 1950.
5
[6] P.E. Kloeden, Fuzzy dynamical systems, Fuzzy Sets and Systems, 7 (1982) 275-296.
6
[7] U. Mohamadi, Weighted information function of dynamical systems, Journal of mathematics and computer science, 10 (2014) 72-77.
7
[8] P. Walters, An Introduction to Ergodic Theory, Springer Verlag, 1982.
8
[9] L.A. Zadeh, Fuzzy Sets, Inform. and control, 8 (1965) 338-352.
9
ORIGINAL_ARTICLE
Fuzzy weakly $e$-closed functions
In this paper, we introduce and characterize fuzzy wea-kly $e$-closed functions in fuzzy topological spaces and the relationship between these mappings and some properties of them are investigated.
http://scma.maragheh.ac.ir/article_23649_83ca0155b8a1e0d756e5be1c689630fe.pdf
2017-04-01T11:23:20
2018-01-19T11:23:20
29
37
Fuzzy topology
Fuzzy $e$-closed functions
Fuzzy weakly $e$-closed functions
Fuzzy contra $e$-open functions
Veerappan
Chandrasekar
vckkc3895@gmail.com
true
1
Department of Mathematics, Kandaswami Kandar's College, P-velur, Tamil Nadu-638 182, India.
Department of Mathematics, Kandaswami Kandar's College, P-velur, Tamil Nadu-638 182, India.
Department of Mathematics, Kandaswami Kandar's College, P-velur, Tamil Nadu-638 182, India.
AUTHOR
Somasundaram
Parimala
pspmaths@gmail.com
true
2
Research Scholar (Part Time), Department of Mathematics, Kandaswami Kandar's College, P-velur, Tamil Nadu-638 182, India.
Research Scholar (Part Time), Department of Mathematics, Kandaswami Kandar's College, P-velur, Tamil Nadu-638 182, India.
Research Scholar (Part Time), Department of Mathematics, Kandaswami Kandar's College, P-velur, Tamil Nadu-638 182, India.
LEAD_AUTHOR
[1] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190.
1
[2] V. Chandrasekar and S. Parimala, Fuzzy $e$-regular spaces and strongly $e$-irresolute mappings, accepted in Sahand Communications in Mathematical Analysis.
2
[3] K.C. Chattopadhyay, R.N. Hazra, and S.K. Samanta, Gradation of openness, Fuzzy Sets and Systems, 49 (2) (1992), 237-242.
3
[4] K.C. Chattopadhyay and S.K. Samanta, Fuzzy topology, Fuzzy Sets and Systems, 54 (1993), 207-212.
4
[5] U. Hohle, Upper semicontinuous fuzzy sets and applications, J. Math. Anall. Appl., 78 (1980), 659-673.
5
[6] U. Hohle and A.P. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, The Hand-books of fuzzy sets series, 3, Kluwer academic publishers, Dordrecht (Chapter 3), (1999).
6
[7] U. Hohle and A.P. Šostak, A general theory of fuzzy topological spaces, Fuzzy Sets and Systems, 73 (1995), 131-149.
7
[8] Y.C. Kim, $delta$-closure operators in fuzzy bitopological spaces, Far East J. Math. Sci., 2 (5) (2000), 791-808.
8
[9] Y.C. Kim, $r$-fuzzy $alpha$-open and $r$-fuzzy preopen sets in fuzzy bitopological spaces, Far East J. Math. Sci. Spec. (III), (2000), 315-334.
9
[10] Y.C. Kim and S. E. Abbas, Several types of fuzzy regular spaces, Indian J. Pure and Appl. Math., 35 (4) (2004), 481-500.
10
[11] Y.C. Kim and Biljana Krsteska, Fuzzy $P$-regular spaces, The Journal of Fuzzy Mathematics, 14 (3) (2006), 701-722.
11
[12] Y.C. Kim, A.A. Ramadan, and S.E. Abbas, $r$-fuzzy strongly preopen sets in fuzzy topological spaces, Math. Vesnik, 55 (2003), 1-13.
12
[13] Y.C. Kim and J.W. Park, $r$-fuzzy $delta$-closure and $r$-fuzzy $theta$-closure sets, J. Korea Fuzzy Logic and Intelligent systems, 10(6) (2000), 557-563.
13
[14] T. Kubiak, On fuzzy topologies, Ph.D. Thesis, A. Mickiewicz, Poznan, (1985).
14
[15] A.A. Ramadan, Smooth topological spaces, Fuzzy Sets and Systems, 48 (1992), 371-375.
15
[16] D. Sobana, V. Chandrasekar, and A. Vadivel, Fuzzy $e$-continuity in Sostak's fuzzy topological spaces, (Submitted).
16
[17] A.P. Šostak, Basic structures of fuzzy topology, J. Math. Sci., 78 (6) (1996), 662-701.
17
[18] A.P. Šostak, Two decades of fuzzy topology: Basic ideas, Notion and results, Russian Math. Surveys, 44 (6) (1989), 125-186.
18
[19] A.P. Šostak, On a fuzzy topological structure, Rend. Circ. Matem. Palermo Ser II, 11 (1986), 89-103.
19
ORIGINAL_ARTICLE
Fixed point results in cone metric spaces endowed with a graph
In this paper, we prove the existence of fixed point for Chatterjea type mappings under $c$-distance in cone metric spaces endowed with a graph. The main results extend, generalized and unified some fixed point theorems on $c$-distance in metric and cone metric spaces.
http://scma.maragheh.ac.ir/article_23163_0b45f04bdebc0d3a7cfa6e52d1e52803.pdf
2017-04-01T11:23:20
2018-01-19T11:23:20
39
47
Normal cone
Cone metric space
$c$-distance
Fixed point
Kamal
Fallahi
fallahi1361@gmail.com
true
1
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.
LEAD_AUTHOR
Ghasem
Soleimani Rad
gh.soleimani2008@gmail.com
true
2
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.
Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran.
AUTHOR
[1] F. Bojor, Fixed point theorems for Reich type contractions on metric spaces with a graph, Nonlinear Anal. (TMA)., 75 (1) (2012) 1359-1373.
1
[2] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, New York, 2008.
2
[3] Y.J. Cho, R. Saadati, and S.H. Wang, Common fixed point theorems on generalized distance in ordered cone metric spaces, Comput. Math. Appl., 61 (2011) 1254-1260.
3
[4] K. Fallahi and A. Aghanianas, On quasi-contractions in metric spaces with a graph, Hacettepe J. Math and Statistics., 45 (4) (2016) 1033-1047.
4
[5] K. Fallahi and A. Aghanianas, Chatterjea contractions in metric spaces, Int. J. Nonlinear Anal. Appl., In press.
5
[6] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (4) (2008) 1359-1373.
6
[7] L.G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007) 1467-1475.
7
[8] S. Jankovic, Z. Kadelburg, and S. Radenovic, On cone metric spaces, a survey, Nonlinear Analysis., 74 (2011) 2591-2601.
8
[9] O. Kada, T. Suzuki, and W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., 44 (1996) 381-391.
9
[10] A. Nicolae, D. O'Regan, and A. Petrusel, Fixed point theorems for singlevalued and multivalued generalized contractions in metric spaces endowed with a graph, Georgian Math. J., 18 (2011) 307-327.
10
[11] H Rahimi, G Soleimani Rad, and P Kumam, A generalized distance in a cone metric space and new common fixed point results, U.P.B. Sci. Bull., (Series A). 77 (2) (2015) 195-206.
11
[12] H. Rahimi, G. Soleimani Rad, and P. Kumam, Generalized distance and new fixed point results, Asian-European Journal of Mathematics., 9 (2) (2016) Article ID: 1650044.
12
[13] S. Wang and B. Guo, Distance in cone metric spaces and common fixed point theorems, Appl. Math. Lett., 24 (2011) 1735-1739.
13
ORIGINAL_ARTICLE
Approximation of fixed points for a continuous representation of nonexpansive mappings in Hilbert spaces
This paper introduces an implicit scheme for a continuous representation of nonexpansive mappings on a closed convex subset of a Hilbert space with respect to a sequence of invariant means defined on an appropriate space of bounded, continuous real valued functions of the semigroup. The main result is to prove the strong convergence of the proposed implicit scheme to the unique solution of the variational inequality on the solution of systems of equilibrium problems and the common fixed points of a sequence of nonexpansive mappings and a continuous representation of nonexpansive mappings.
http://scma.maragheh.ac.ir/article_22988_6164b5942bd9d9914849f5c337fac6fa.pdf
2017-04-01T11:23:20
2018-01-19T11:23:20
49
68
Continuous representation
Fixed point
Equilibrium problem
Nonexpansive mapping
Variational inequality
Ebrahim
Soori
sori.e@lu.ac.ir
true
1
Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran.
Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran.
Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran.
LEAD_AUTHOR
[1] V. Colao, G.L. Acedo, and G. Marino, An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Anal., 71 (2009) 2708-2715.
1
[2] P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (1) (2005) 117-136.
2
[3] K. Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge Stud. Adv. Math., Cambridge, 1990.
3
[4] N. Hirano, K. Kido, and W. Takahashi, Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear Anal., 12 (1988), 1269-1281.
4
[5] N. Hussain, M.L. Bami, and E. Soori, An implicit method for finding a common fixed point of a representation of nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., (2014), DOI: 10.1186/1687-1812-2014-238.
5
[6] K. Kido and W. Takahashi, Mean ergodic theorems for semigroups of linear continuous in Banach spaces, J. Math. Anal. Appl., 103 (1984), 387-394.
6
[7] G. Marino and H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (1) (2006) 43-52.
7
[8] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (1) (2000) 46-55.
8
[9] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967) 595-597.
9
[10] S. Plubtieng and R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 336 (2007) 455-469.
10
[11] X. Qin, M. Shang, and Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Modelling, 48 (2008) 1033-1046.
11
[12] S. Saeidi, Comments on relaxed (γ, r)-cocoercive mappings, Int. J. Nonlinear Anal. Appl., 1 (2010) 54-57.
12
[13] K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5 (2001)387-404.
13
[14] E. Soori, Strong convergence for variational inequalities and equilibrium problems and representations, Int. J. Industrial Math. 5(4) (2013) 341-354.
14
[15] W. Takahashi, Nonlinear Functional Analysis: Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, 2000.
15
[16] S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., (2006) doi: 10.1016/j.jmaa.2006.08.036.
16
[17] W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces, Nonlinear Anal., (2008), doi: 10.1016/j.na.2008.01.005.
17
[18] R.U. Verma, General convergence analysis for two-step projection methods and application to variational problems, Appl. Math. Lett., 18 (11)(2005) 12861292.
18
ORIGINAL_ARTICLE
The analytical solutions for Volterra integro-differential equations within Local fractional operators by Yang-Laplace transform
In this paper, we apply the local fractional Laplace transform method (or Yang-Laplace transform) on Volterra integro-differential equations of the second kind within the local fractional integral operators to obtain the analytical approximate solutions. The iteration procedure is based on local fractional derivative operators. This approach provides us with a convenient way to find a solution with less computation as compared with local fractional variational iteration method. Some illustrative examples are discussed. The results show that the methodology is very efficient and a simple tool for solving integral equations.
http://scma.maragheh.ac.ir/article_23686_f3ebd1266c52fc8a3318ef0f1567e9cc.pdf
2017-04-01T11:23:20
2018-01-19T11:23:20
69
76
Volterra integro-differential equation
Yang-Laplace transform
Local fractional derivative operators
Hassan Kamil
Jassim
hassan.kamil28@yahoo.com
true
1
Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq.
Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq.
Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq.
LEAD_AUTHOR
[1] H.K. Jassim, The Approximate Solutions of Fredholm Integral Equations on Cantor Sets within Local Fractional Operators, Sahand Communications in Mathematical Analysis, 16 (2016) 13-20.
1
[2] H.K. Jassim, C. Ünlü, S.P. Moshokoa, and C.M. Khalique, Local Fractional Laplace Variational Iteration Method for Solving Diffusion and Wave Equations on Cantor Sets within Local Fractional Operators, Mathematical Problems in Engineering, 2015, Article ID 309870 (2015) 1-9.
2
[3] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam,2006.
3
[4] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
4
[5] A.A. Neamah, Local Fractional Variational Iteration Method for Solving Volterra Integro-Differential equations within the local fractional operators, Journal of Mathematics and Statistics, 10 (2014) 401-407.
5
[6] I. Podlubny, Fractional Differential Equations, Academic Press, New York, USA, 1999.
6
[7] X.J. Yang, A New Approach to Solving Class of Local Fractional Volterra Integral Equation, Advances in Information Technology and Management, 1 (2012) 183-186.
7
[8] X.J. Yang, Picard’s approximation method for solving a class of local fractional Volterra integral equations, Advances in Intelligent Transportation Systems, 1 (2012) 67-70.
8
[9] S.P. Yan, H. Jafari, and H.K. Jassim, Local Fractional Adomian Decomposition and Function Decomposition Methods for Solving Laplace Equation within Local Fractional Operators, Advances in Mathematical Physics, 2014, Article ID 161580 (2014) 1-7.
9
[10] Y.J. Yang, S.Q. Wang, and H.K. Jassim, Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative, Abstract and Applied Analysis, 2014, Article ID 176395 (2014) 1-7.
10
[11] X.J. Yang and Y. Zhang, A new Adomian decomposition procedure scheme for solving local fractional Volterra integral equation, Advances in Information Technology and Management, 1 (2012) 158-161.
11
ORIGINAL_ARTICLE
A generalization of Kannan and Chatterjea fixed point theorems on complete $b$-metric spaces
In this paper, we give some results on the common fixed point of self-mappings defined on complete $b$-metric spaces. Our results generalize Kannan and Chatterjea fixed point theorems on complete $b$-metric spaces. In particular, we show that two self-mappings satisfying a contraction type inequality have a unique common fixed point. We also give some examples to illustrate the given results.
http://scma.maragheh.ac.ir/article_23831_6c258f4180145f5370b887cf815cd897.pdf
2017-04-01T11:23:20
2018-01-19T11:23:20
77
86
$b$-metric space
Common fixed point
Altering distance function
Hamid
Faraji
hamid_ftmath@yahoo.com
true
1
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran.
AUTHOR
Kourosh
Nourouzi
nourouzi@kntu.ac.ir
true
2
Faculty of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran.
Faculty of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran.
Faculty of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran.
LEAD_AUTHOR
[1] H. Alsulami, E. Karapınar, and H. Piri, Fixed points of generalized $F$-Suzuki type contraction in complete $b$-metric spaces, Discrete Dyn. Nat. Soc., (2015), Art. ID 969726, 8 pp.
1
[2] H. Aydi, M.F. Bota, E. Karapinar, and S. Moradi, A common fixed point for weak $phi$-contractions on $b$-metric spaces, Fixed Point Theory, 13 (2012), no. 2, 337-346.
2
[3] I.A. Bakhtin, The contraction mapping principle in almost metric space, Functional analysis, (Russian), Ulýanovsk. Gos. Ped. Inst., Ulýanovsk, (1989) 26-37.
3
[4] M.F. Bota and E. Karapinar, A note on "Some results on multi-valued weakly Jungck mappings in b-metric space", Cent. Eur. J. Math., 11 (2013), No. 9, 1711-1712.
4
[5] M.F. Bota, E. Karapinar, and O. Mlesnite, Ulam-Hyers stability results for fixed point problems via $alpha - psi$-contractive mapping in $b$-metric space, Abstr. Appl. Anal. (2013), Art. ID 825293, 6 pp.
5
[6] S.K. Chatterjea, Fixed point theorems, C. R. Acad. Bulgare Sci., 25 (1972) 727-730.
6
[7] S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993) 5-11.
7
[8] S. Czerwik, Nonlinear Set-valued contraction mappings in $b$-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998) 263-276.
8
[9] R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968) 71-76.
9
[10] M.A. Khamsi and N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal., 73 (2010), no. 9, 3123-3129.
10
[11] M.S. Khan, M. Swaleh, and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Austral. Math. Soc., 30 (1984), no. 1, 1-9.
11
[12] W. Kirk and N. Shahzad, Fixed point theory in distance spaces, Springer, Cham, 2014.
12
[13] M.A. Kutbi, E. Karapinar, J. Ahmad, and A. Azam, Some fixed point results for multi-valued mappings in $b$-metric spaces, J. Inequal. Appl. 2014, (2014:126), 11 pp.
13
[14] C.S. Wong, Common fixed points of two mappings, Pacific J. Math., 48 (1973) 299-312.
14