Abstract
In this paper, a new class of fuzzy differential inclusions with resolvent operators in Banach spaces using \((H(\cdot ,\cdot ),\eta )\)-monotone operators is introduced and studied. A continuous selection theorem and fixed point theory are used to establish the existence of solutions. Finally, as applications, we consider special cases of fuzzy differential inclusions with general A-monotone operators. Some examples are given to illustrate our results.
Similar content being viewed by others
References
Agarwal RP, Benchohra M, Nieto JJ, Ouahab A (2010a) Some results for integral inclusions of Volterra type in Banach spaces. Adv Differ Equ 798067:1–37
Agarwal RP, Lakshmikantham V, Nieto JJ (2010b) On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal 72:2859–2862
Ahmadian A, Salahshour S, Baleanu D, Amirkhani H, Yunus R (2015) 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. J Comput Phys 294:562–584
Allahviranloo T, Armand A, Gouyandeh Z (2014) Fuzzy fractional differential equations under generalized fuzzy Caputo derivative. J Intell Fuzzy Syst 26:1481–1490
Arshad S (2013) On existence and uniqueness of solution of fuzzy fractional differential equations. Iran J Fuzzy Syst 10:137–151
Aubin JP, Cellina A (1984) Differential inclusions. Springer, Berlin
Aubin JP, Ekeland I (1984) Applied nonlinear analysis. Wiley, New York
Balooee J, Cho YJ (2013) Algorithms for solutions of extended general mixed variational inequalities and fixed points. Optim Lett 7:1929–1955
Bohnenblust HF, Karlin S (1950) On a theorem of Ville. In: Contribution to the theory of games. Princeton University Press, Princeton
Bragdi M, Debbouche A, Baleanu D (2013) Existence of solutions for fractional differential inclusions with separated boundary conditions in Banach space. Adv Math Phys. https://doi.org/10.1155/2013/426061
Chang SS (1991) Variational inequality and complementarity problem theory with applications. Shanghai Scientific and Technological Literature, Shanghai
Chang SS, Zhu YG (1989) On variational inequalities for fuzzy mappings. Fuzzy Sets Syst 32:359–367
Chang YK, Nieto JJ (2009) Some new existence results for fractional differential inclusions with boundary conditions. Math Comput Model 49:605–609
Cui YS, Lan HY, Chen YC (2008) On implicit fuzzy proximal dynamical systems involving general \(A\)-monotone operators in Banach spaces. In: The 5th international conference on fuzzy systems and knowledge discovery, Jinan
Deimling K (1992) Multivalued differential equations. Walter de Gruyter, Berlin
Edsberg L (2015) Introduction to computation and modeling for differential equations. Wiley, New York
Fang YP, Huang NJ (2003) \(H\)-monotone operator and resolvent operator technique for variational inclusions. Appl Math Comput 145:795–803
Fang YP, Huang NJ, Thompson HB (2005) A new system of variational inclusions with \((H,\eta )\)-monotone operators in Hilbert spaces. Comput Math Appl 49:365–374
Friesz TL, Bernstein DH, Mehta NJ, Tobin RL, Ganjlizadeh S (1994) Day-to-day dynamic network disequilibria and idealized traveler information systems. Oper Res 42:1120–1136
Guo DJ (1985) Nonlinear functional analysis. Shandong Sciences and Technology Press, Jinan
Hu SC, Papageorgiou NS (1997) Handbook of multivalued analysis, vol I. Theory. Kluwer Academic Publishers, Dordrecht
Huang NJ, Fang YP (2003) A new class of generalized variational inclusions involving maximal \(\eta \)-monotone mappings. Publ Math Debr 62:83–98
Hüllermeier E (1997) An approach to modelling and simulation of uncertain dynamical systems. Int J Uncertain Fuzziness Knowl-Based Syst 5:117–137
Hung NV, Tam VM, Köbis E, Yao JC (2019a) Existence of solutions and algorithm for generalized vector quasi-complementarity problems with application to traffic network problems. J Nonlinear Convex Anal 20:1751–177
Hung NV, Tam VM, Yao JC (2019b) Existence and convergence theorems for split general random variational inclusions with random fuzzy mappings. Linear Nonlinear Anal 5:51–65
Hung NV, Tam VM, Tuan N, O’Regan D (2019c) Regularized gap functions and error bounds for generalized mixed weak vector quasivariational inequality problems in fuzzy environments. Fuzzy Sets Syst. https://doi.org/10.1016/j.fss.2019.09.015
Hung NV, Köbis E, Tam VM (2020a) Existence of solutions and iterative algorithms for weak vector quasi-equilibrium problems. J Nonlinear Convex Anal (accepted)
Hung NV, Tam VM, Tuan NH, O’Regan D (2020b) Convergence analysis of solution sets for fuzzy optimization problems. J Comput Appl Math 369:112615
Lakshmikantham V, Mohapatra RN (2003) Theory of fuzzy differential equations and inclusion. Taylor & Francis, London
Lakshmikantham V, Leela S (1981) Nonlinear differential equations in abstract spaces. Pergamon Press, New York
Lan HY, Cai LC (2009) Variational convergence of a newproximal algorithm for nonlinear general \(A\)-monotone operator equation systems in Banach spaces. Nonlinear Anal Ser A TMA 71:6194–6201
Lan H, Nieto JJ, Cui Y (2017) Global exponential stability of general \(A\)-monotone implicit fuzzy proximal dynamical systems in Banach spaces. Soft Comput 21:3113
Lasota A, Opial Z (1965) An application of the Kakutani? Ky Fan theorem in the theory of ordinary differential equations. Bull Acad Pol Sci Sér Sci Math Astron Phys 13:781–786
Lou J, He XF, He Z (2008) Iterative methods for solving a system of variational inclusions involving \(H\)-\(\eta \)-monotone operators in Banach spaces. Comput Math Appl 55:1532–1541
Luo XP, Huang NJ (2010) \((H,\phi )\)-\(\eta \)-monotone operators in Banach spaces with an application to variational inclusions. Appl Math Comput 216:1131–1139
Min C, Huang NJ, Zhang LH (2014) Existence of local and global solutions of fuzzy delay differential inclusions. Adv Differ Equ 2014:1–14
Min C, Huang NJ, Liu ZB, Zhang LH (2015a) Existence of solution for implicit fuzzy differential inclusions. Appl Math Mech (English Ed) 36:401–416
Min C, Liu ZB, Zhang LH, Huang NJ (2015b) On a system of fuzzy differential inclusions. Filomat 29:1231–1244
Noor MA (2002) Implicit resolvent dynamical systems for quasi variational inclusions. J Math Anal Appl 269:216–226
Sun JH, Zhang LW, Xiao XT (2008) An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces. Nonlinear Anal 69:3344–3357
Verma RU (2006a) Generalized nonlinear variational inclusion problems involving \(A\)-monotone mappings. Appl Math Lett 19:960–963
Verma RU (2006b) Sensitivity analysis for generalized strongly monotone variational inclusions based on the \((A,\eta )\)-resolvent operator technique. Appl Math Lett 19:1409–1413
Wu ZB, Min C, Huang NJ (2018) On a system of fuzzy fractional differential inclusions with projection operators. Fuzzy Sets Syst 347:70–88
Wu ZB, Zou YZ (2014) Global fractional-order projective dynamical systems. Commun Nonlinear Sci Numer Simul 19:2811–2819
Wu ZB, Zou YZ, Huang NJ (2016) A class of global fractional-order projective dynamical systems involving set-valued perturbations. Appl Math Comput 277:23–33
Xia FQ, Huang NJ (2007) Variational inclusions with a general \(H\)-monotone operator in Banach spaces. Comput Math Appl 54:24–30
Xia YS, Wang J (2000) Global exponential stability of recurrent neural network for solving optimization and related problems. IEEE Trans Neural Netw 11:1017–1022
Xu Z, Wang Z (2010) A generalized mixed variational inclusions involving \((H(\cdot,\cdot ),\eta )\)-monotone operator in Banach spaces. J Math Res 2:47–56
Yannelis NC, Prabhakar ND (1983) Existence of maximal elements and equilibria in linear topological spaces. J Math Econ 12:233–245
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Zhang QB (2007) Generalized implicit variational-like inclusion problems involving \(G\)-\(\eta \)-monotone mappings. Appl Math Lett 20:216–221
Zhu YG, Rao L (2000) Differential inclusions for fuzzy maps. Fuzzy Sets Syst 112:257–261
Acknowledgements
The authors are grateful to the editor and the referees for their valuable comments which improved the results and presentation of this article. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant number 101.01-2017.18.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anibal Tavares de Azevedo.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Van Hung, N., Tam, V.M. & O’Regan, D. Existence of solutions for a new class of fuzzy differential inclusions with resolvent operators in Banach spaces. Comp. Appl. Math. 39, 42 (2020). https://doi.org/10.1007/s40314-020-1074-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-1074-3