Existence results of delay and fractional differential equations via fuzzy weakly contraction mapping principle

• Tabassum, Rehana ^[1] ; Azam, Akbar ^[1] ; Mohammed, Shehu Shagari ^[2]
  1. [1] COMSATS Institute of Information Technology

     COMSATS Institute of Information Technology

     Pakistán

     
  2. [2] Ahmadu Bello University

     Ahmadu Bello University

     Chikaji, Nigeria

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 20, Nº. 2, 2019, págs. 449-469
• Idioma: inglés
• DOI: 10.4995/agt.2019.11683
• Enlaces
  • Texto completo
• Resumen

  • The purpose of this article is to extend the results derived through former articles with respect to the notion of weak contraction into intuitionistic fuzzy weak contraction in the context of (T,N,∝) -cut set of an intuitionistic fuzzy set. We intend to prove common fixed point theorem for a pair of intuitionistic fuzzy mappings satisfying weakly contractive condition in a complete metric space which generalizes many results existing in the literature. Moreover, concrete results on existence of the solution of a delay differential equation and a system of Riemann-Liouville Cauchy type problems have been derived. In addition, we also present illustrative examples to substantiate the usability of our main result.

• Referencias bibliográficas
  • H. M. Abu-Donia, Common fixed points theorems for fuzzy mappings in metric space under $varphi $-contraction condition, Chaos Solitons &...
  • A. Z. Al-Abedeen, Existence theorem on differential equation of generalized order, Al-Rafidain J. Sci. Mosul University, Iraq, 1 (1976), 95-104.
  • Y. I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in: New Results in Operator Theory and Its Applications,...
  • H. L. Arora and J. G. Alshamani, Stability of differential equations of noninteger order through fixed point in the large, Indian J. Pure...
  • K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and Systems 20, no. 1 (1986), 87-96. https://doi.org/10.1016/S0165-0114(86)80034-3
  • A. Azam, M. Arshad and P. Vetro, On a pair of fuzzy $varphi$-contractive mappings, Mathematical and Computer Modelling 52, no. 1 (2010), 207-214....
  • A. Azam and M. Rashid, A fuzzy coincidence theorem with applications in a function space, Journal of Intelligent and Fuzzy Systems 27, no....
  • A. Azam, R. Tabassum and M. Rashid, Coincidence and fixed point theorems of intuitionistic fuzzy mappings with applications, Journal of Mathematical...
  • A. Azam and R. Tabassum, Existence of common coincidence point of intuitionistic fuzzy maps, Journal of Intelligent and Fuzzy Systems 35 (2018),...
  • J. S. Bae, Fixed point theorems for weakly contractive multivalued maps, Journal of Mathematical Analysis and Applications 284, no. 2 (2003),...
  • I. Beg and M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point...
  • M. A. Al-Bassam, Some existence theorems on differential equations of generalized order, J. Reine Angew. Math. 218, no. 1 (1965), 70-78. https://doi.org/10.1515/crll.1965.218.70
  • V. Berinde, Approximating fixed points of weak contractions, Fixed Point Theory 4 (2003), 131-142.
  • S. M. Ciupe, B. L. de Bivort, D. M. Bortz and P. W. Nelson, Estimates of kinetic parameters from HIV patient data during primary infection...
  • K. Cooke, Y. Kuang and B. Li, Analyses of an antiviral immune response model with time delays, Canad. Appl. Math. Quart. 6, no. 4 (1998),...
  • K. L. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math....
  • P. Z. Daffer and H. Kaneko, Fixed points of generalized contractive multi-valued mappings, Journal of Mathematical Analysis and Applications...
  • S. K. De, R. Biswas and A. R. Roy, An application of intuitionistic fuzzy sets in medical diagnosis, Fuzzy Sets and Systems 117, no. 2 (2001),...
  • D. Delbosco and L. Rodino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Appl. 204, no. 2 (1996), 609-625....
  • S. Heilpern, Fuzzy mappings and fixed point theorems, Journal of Mathematical Analysis and Applications 83, no. 2 (1981), 566-569. https://doi.org/10.1016/0022-247X(81)90141-4
  • Z. Jia, L. Amselang and P. Gros, Content-based image retrieval from a large image database, Pattern Recognition 11, no. 5 (2008), 1479-1495....
  • A. Kharal, Homeopathic drug selection using intuitionistic fuzzy sets, Homeopathy 98, no. 1 (2009), 35-39. https://doi.org/10.1016/j.homp.2008.10.003
  • A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited,...
  • S. Konjik, L. Oparnica and D. Zorica, Waves in viscoelastic media described by a linear fractional model, Integral Transforms Spec. Funct....
  • A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Nauka, Moscow, 1968.
  • D. F. Li, Multiattribute decision making models and methods using intuitionistic fuzzy sets, J. Comput. Syst. Sci. 70 (2005), 73-85. https://doi.org/10.1016/j.jcss.2004.06.002
  • D. Martinetti, V. Janis and S. Montes, Cuts of intuitionistic fuzzy sets respecting fuzzy connectives, Information Sciences 232 (2013), 267-275....
  • S. B. Nadler Jr, Multi-valued contraction mappings, Pacific Journal of Mathematics 30, no. 2 (1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475
  • P. W. Nelson, J. D. Murray and A. S. Perelson, A model of HIV-1 pathogenesis that includesan intracellular delay. Math. Biosci. 163 (2000),...
  • B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis 4, no. 47 (2001), 2683-2693. https://doi.org/10.1016/S0362-546X(01)00388-1
  • A. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives : Theory and Applcations, Gordon and Breach Science Publishers,...
  • A. M. A. El-Sayed and A. G. Ibrahim, Multivalued fractional differential equations, Appl. Math. Comp. 68, no. 1 (1995), 15-25. https://doi.org/10.1016/0096-3003(94)00080-N
  • A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods, results and problems, I. Appl. Anal. 78, no. 1-2 (2001),...
  • P. Turchin and A. D. Taylor, Complex dynamics in ecological time series, Ecology 73 (1992), 289-305. https://doi.org/10.2307/1938740
  • D. Valerjo, D. Machadoa and J. T. Kryakova, Some pioneers of the applications of fractional calculus, Fract. Calc. Appl. Anal. 17 (2014),...
  • B. Vielle and G. Chauvet, Delay equation analysis of human respiratory stability, Math. Biosci. 152, no. 2 (1998), 105-122. https://doi.org/10.1016/S0025-5564(98)10028-7
  • M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth, J. Math. Biol. 47, no. 3 (2003), 270-294. https://doi.org/10.1007/s00285-003-0211-0
  • Y. H. Shen, F. X. Wang and W. Chen, A note on intuitionistic fuzzy mappings, Iranian Journal of Fuzzy Systems 9, no. 5 (2012), 63-76.
  • L. A. Zadeh, Fuzzy sets, Information and Control 8, no. 3 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X