Fuzzy fixed point results and applications to ordinary fuzzy differential equations in complex valued metric spaces

Yıl 2019, Cilt: 48 Sayı: 6, 1712 - 1728, 08.12.2019

Muhammad Sarwar , Humaira - Tongxing Li

https://doi.org/10.15672/HJMS.2018.633

Cited By: 2

Öz

The main purpose of this article is to discuss the existence of the common solution of second-order nonlinear boundary value problems

$$\mathfrak{x}''(\jmath)=\Bbbk(\jmath,\mathfrak{x}(\jmath),\mathfrak{x}'(\jmath)),\quad\text{if}\:\jmath\in[0,\Lambda],\quad\Lambda>0,$$

$$\mathfrak{x}(\jmath_1)=\mathfrak{x}_1,\quad\mathfrak{x}(\jmath_2)=\mathfrak{x}_2,\quad\jmath_1,\jmath_2\in[0,\Lambda]$$

where $\Bbbk:[0,\Lambda]\times\mathfrak{S}(\mathcal{S})\times\mathfrak{S}(\mathcal{S})\rightarrow\mathfrak{S}(\mathcal{S})$ is a continuous function and $\mathfrak{S}(\mathcal{S})$ is a family of fuzzy sets.

  In this regard we obtain common fixed point results for two pairs of fuzzy mappings satisfying rational contractive condition in the setting of complex valued metric spaces. Our results improve those reported in the existing literature.

Anahtar Kelimeler

Complex valued metric space, Fuzzy mapping, Fixed point, Common fixed point, Cauchy sequence, Contractive condition

Kaynakça

• [1] T. Allahviranloo, S. Salahshour, M. Homayoun-nejad, and D. Baleanu, General solutions of fully fuzzy linear systems, Abstr. Appl. Anal. 2013, 1-9, 2013.
• [2] J. Ahmad, C. Klin-Eam, and A. Azam, Common fixed points for multivalued mappings in complex valued metric spaces with applications, Abstr. Appl. Anal. 2013, 1-12, 2013.
• [3] A. Azam, Fuzzy fixed points of fuzzy mappings via a rational inequality, Hacet. J. Math. Stat. 40, 421-431, 2011.
• [4] A. Azam and I. Beg, Common fuzzy fixed points for fuzzy mappings, Fixed Point Theory Appl. 2013, 1-11, 2013.
• [5] A. Azam, B. Fisher, and M. Khan, Common fixed point theorems in complex valued metric spaces, Numer. Funct. Anal. Optim. 32, 243-253, 2011.
• [6] A. Azam, N. Mehmood, M. Rashid, and S. Radenović, Fuzzy fixed point theorems in ordered cone metric spaces, Filomat 29, 887-896, 2015.
• [7] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3, 133-181, 1922.
• [8] P. Balasubramaniam and S. Muralisankar, Existence and uniqueness of fuzzy solution for the nonlinear fuzzy integrodifferential equations, Appl. Math. Lett. 14, 455-462, 2001.
• [9] Lj. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc. 45, 267-273, 1974.
• [10] B.K. Dass and S. Gupta, An extension of Banach contraction principle through rational expression, Indian J. Pure Appl. Math. 6, 1455-1458, 1975.
• [11] T. Došenović, D. Rakić, B. Carić, and S. Radenović, Multivalued generalizations of fixed point results in fuzzy metric spaces, Nonlinear Anal. Model. Control 21, 211-222, 2016.
• [12] V.D. Estruch and A. Vidal, A note on fixed fuzzy points for fuzzy mappings, Rend. Istit. Mat. Univ. Trieste 32, 39-45, 2001.
• [13] X. Hao and L. Liu, Mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces, Math. Methods Appl. Sci. 40, 4832-4841, 2017.
• [14] X. Hao, M. Zuo, and L. Liu, Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities, Appl. Math. Lett. 82, 24-31, 2018.
• [15] S. Heilpern, Fuzzy mappings and fixed point theorem, J. Math. Anal. Appl. 83, 566- 569, 1981.
• [16] A. Jafarian, R. Jafari, A.K. Golmankhaneh, and D. Baleanu, Solving fully fuzzy polynomials using feed-back neural networks, Int. J. Comput. Math. 92, 742-755, 2015.
• [17] V. Joshi, N. Singh, and D. Singh, ϕ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papers, Cogent Math. 2016, 1-14, 2016.
• [18] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24, 301-317, 1987.
• [19] E. Karapınar and R.P. Agarwal, Further fixed point results on G-metric spaces, Fixed Point Theory Appl. 2013, 1-14, 2013.
• [20] C. Klin-eam and C. Suanoom, Some common fixed-point theorems for generalizedcontractive- type mappings on complex-valued metric spaces, Abstr. Appl. Anal. 2013, 1-6, 2013.
• [21] M.A. Kutbi, J. Ahmad, A. Azam, and A.S. Al-Rawashdeh, Generalized common fixed point results via greatest lower bound property, J. Appl. Math. 2014, 1-11, 2014.
• [22] V. Lakshmikantham and R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, Ltd., London, 2003.
• [23] X. Liu, L. Liu, and Y. Wu, Existence of positive solutions for a singular nonlinear fractional differential equation with integral boundary conditions involving fractional derivatives, Bound. Value Probl. 2018, 1-21, 2018.
• [24] D. Min, L. Liu, and Y. Wu, Uniqueness of positive solutions for the singular fractional differential equations involving integral boundary value conditions, Bound. Value Probl. 2018, 1-18, 2018.
• [25] H.K. Nashine, C. Vetro, W. Kumam, and P. Kumam, Fixed point theorems for fuzzy mappings and applications to ordinary fuzzy differential equations, Adv. Difference Equ. 2014, 1-14, 2014.
• [26] J.J. Nieto, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets and Systems 102, 259-262, 1999.
• [27] M.L. Puri and D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114, 409-422, 1986.
• [28] S. Radenović and B.E. Rhoades, Fixed point theorem for two non-self mappings in cone metric spaces, Comput. Math. Appl. 57, 1701-1707, 2009.
• [29] S. Radenović, P. Salimi, C. Vetro, and T. Došenović, Edelstein–Suzuki-type results for self-mappings in various abstract spaces with application to functional equations, Acta Math. Sci. Ser. B Engl. Ed. 36, 94-110, 2016.
• [30] S. Salahshour, A. Ahmadian, F. Ismail, and D. Baleanu, A novel weak fuzzy solution for fuzzy linear system, Entropy 18, 1-8, 2018.
• [31] W. Shatanawi, V.Ć. Rajić, S. Radenović, and A. Al-Rawashdeh, Mizoguchi– Takahashi-type theorem in tvs-cone metric spaces, Fixed Point Theory Appl. 2012, 1-7, 2012.
• [32] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems 24, 319-330, 1987.
• [33] W. Sintunavarat, Y.J. Cho, and P. Kumam, Urysohn integral equations approach by common fixed points in complex-valued metric spaces, Adv. Difference Equ. 2013, 1-14, 2013.
• [34] W. Sintunavarat and P. Kumam, Generalized common fixed point theorems in complex-valued metric spaces and applications, J. Inequal. Appl. 2012, 1-12, 2012.
• [35] W. Sintunavarat, M.B. Zada, and M. Sarwar, Common solution of Urysohn integral equations with the help of common fixed point results in complex valued metric spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111, 531-545, 2017.
• [36] S. Song, L. Guo, and C. Feng, Global existence of solutions to fuzzy differential equations, Fuzzy Sets and Systems 115, 371-376, 2000.
• [37] D. Turkoglu and B.E. Rhoades, A fixed fuzzy point for fuzzy mapping in complete metric spaces, Math. Commun. 10, 115-121, 2005.
• [38] Y. Yang and F. Meng, Existence of positive solution for impulsive boundary value problem with p-Laplacian in Banach spaces, Math. Methods Appl. Sci. 36, 650-658, 2013.