Advances in Analysis

Download PDF (400.7 KB) PP. 232 - 246 Pub. Date: October 20, 2017

DOI: 10.22606/aan.2017.24002

• Yupin Wang^*
  School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P R China
• Shurong Sun
  
  School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P R China
• Zhenlai Han
  
  School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, P R China

In this paper, we investigate the existence and uniqueness of solution to boundary value problems for a class of nonlinear fuzzy factional differential equations involving the fuzzy gH- fractional Caputo derivative. By means of the Schauder fixed point theorem in semi-linear spaces and integral inequality technique, some qualitative results of solutions are obtained. An example is provided from which new results are found.

Fuzzy fractional differential equations, generalized Hukuhara differentiability, boundary value problems, Schauder fixed point theorem, generalized Gronwall inequality.

[1] A. Khastan, J. Nieto, “Periodic boundary value problems for first-order linear differential equations with uncertainty under generalized differentiability,” Information Sciences, vol. 222, pp. 544–558, 2013.

[2] H. Wang, “Monotone iterative method for boundary value problems of fuzzy differential equations,” Journal of Intelligent and Fuzzy Systems, vol. 30, pp. 831–843, 2016.

[3] T. Allahviranloo, Z. Gouyandeh, A. Armand, A. Hasanoglu, “On fuzzy solutions for heat equation based on generalized Hukuhara differentiability,” Fuzzy Sets and Systems, vol. 265, pp. 1–23, 2015.

[4] M. Guo, X. Xue, R. Li, “Impulsive functional differential inclusions and fuzzy population models,” Fuzzy Sets and Systems, vol. 138, pp. 601–615, 2003.

[5] R. Jafelice, L. Barros, R. Bassanez, F. Gomide, “Fuzzy modeling in symptomatic HIV virus infected population,” Bulletin of Mathematical Biology, vol. 66, pp. 1597–1620, 2004.

[6] A. Ahmadian, S. Salahshour, D. Baleanu, H. Amirkhani, R. Yunus, “Tau method for the numerical solution of a fuzzy fractional kinetic model and its application to the oil palm frond as a promising source of xylose,” Journal of Computational Physics, vol. 294, pp. 562–584, 2015.

[7] S. Paul, S. Mondal, P. Bhattacharya, “Discussion on fuzzy quota harvesting model in fuzzy environment: fuzzy differential equation approach,” Modeling Earth Systems and Environment, vol. 2, pp. 1–15, 2016.

[8] R. Agarwal, V. Lakshmikantham, J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,” Nonlinear Analysis, vol. 72, pp. 2859–2862, 2010.

[9] R. Agarwal, S. Arshad, D. O’Regan, V. Lupulescu, “A Schauder fixed point theorem in semilinear spaces and applications,” Fixed Point Theory and Applications, vol. 306, pp. 1–13, 2013.

[10] A. Khastan, J. Nieto, R. Rodríguez-López, “Schauder fixed-point theorem in semilinear spaces and its application to fractional differential equations with uncertainty,” Fixed Point Theory and Applications, vol. 21, pp. 1–14, 2014.

[11] T. Allahviranloo, A. Armand, Z. Gouyandeh, “Fuzzy fractional differential equations under generalized fuzzy Caputo derivative,” Journal of Intelligent and Fuzzy Systems, vol. 26, pp. 1481–1490, 2014.

[12] V. Lupulescu, “Fractional calculus for interval-valued functions,” Fuzzy Sets and Systems, vol. 265, pp. 63–85, 2015.

[13] V. Hgo, “Fuzzy fractional functional integral and differential equations,” Fuzzy Sets and Systems, vol. 280, pp. 58–90, 2015.

[14] N. Hoa, “Fuzzy fractional functional differential equations under Caputo gH-differentiability,” Communications in Nonlinear Science and Numerical Simulation, vol. 22, pp. 1134–1157, 2015.

[15] S. Salahshour, A. Ahmadian, N. Senu, D. Baleanu, P. Agarwal, “On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem,” Entropy, vol. 17, pp. 885–902, 2015.

[16] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific, Singapore, 1994.

[17] V. Lakshmikantham, R. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis, London, 2003.

[18] V. Lakshmikantham, B. Gnana, D. Vasundhara, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers, 2006.

[19] L. Gomes, L. Barros, B. Bede, Fuzzy Differential Equations in Various Approaches, Springer International Publishing, 2015.

[20] G. Anastassiou, S. Gal, “On a fuzzy trigonometric approximation theorem of Weierstrass-type,” The Journal of Fuzzy Mathematics, vol. 3, pp. 701–708, 2001.

[21] M. Puri, D. Ralescu, “Differentials of fuzzy functions,” Journal of Mathematical Analysis and Applications, vol. 91, pp. 552–558, 1983.

[22] H. Román-Flores, M. Rojas-Medar, “Embedding of level-continuous fuzzy sets on Banach spaces,” Information Sciences, vol. 114, pp. 227–247, 2002.

[23] O. Kaleva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, pp. 301–317, 1987.

[24] R. Agarwal, S. Arshad, D. O’Regan, V. Lupulescu, “Fuzzy fractional integral equations under compactness type condition,” Fractional Calculus and Applied Analysis, vol. 15, pp. 572–590, 2012.

[25] L. Stefanini, “A generalization of Hukuhara difference and division for interval and fuzzy arithmetic,” Fuzzy Sets and Systems, vol. 161, pp. 1564–1584, 2010.

[26] B. Bede, L. Stefanini, “Generalized differentiability of fuzzy-valued functions,” Fuzzy Sets and Systems, vol. 230, pp. 119–141, 2013.

[27] H. Ye, J. Gao, Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, pp. 1075–1081, 2007.