# American Institute of Mathematical Sciences

July  2017, 22(5): 1987-1998. doi: 10.3934/dcdsb.2017117

## The averaging of fuzzy hyperbolic differential inclusions

 1 Department of Applied Mathematics, Odessa State Academy Civil Engineering and Architecture, 4, Didrihsona street, 65029 Odessa, Ukraine 2 Department of Mathematics, Odessa National Polytechnic University, 1, Shevchenko avenue, 65044 Odessa, Ukraine

* Corresponding author: Andrej V. Plotnikov

Received  January 2016 Revised  February 2016 Published  March 2017

In this paper the substantiation of a possibility of application of partial averaging method for hyperbolic differential inclusions with the fuzzy right-hand side with the small parameters is considered.

Citation: Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117
##### References:
 [1] S. Abbasbandy, T. A. Viranloo, O. Lopez-Pouso and J. J. Nieto, Numerical methods for fuzzy differential inclusions, Computers & Mathematics with Applications, 48 (2004), 1633-1641. doi: 10.1016/j.camwa.2004.03.009. Google Scholar [2] R. P. Agarwal, D. O'Regan and V. Lakshmikantham, A stacking theorem approach for fuzzy differential equations, Nonlinear Analysis, 55 (2003), 299-312. doi: 10.1016/S0362-546X(03)00241-4. Google Scholar [3] R. P. Agarwal, D. O'Regan and V. Lakshmikantham, Maximal solutions and existence theory for fuzzy differential and integral equations, Journal of Applied Analysis, 11 (2005), 171-186. doi: 10.1515/JAA.2005.171. Google Scholar [4] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations Second edition. Grundlehren der Mathematischen Wissenschaften, 250 Springer-Verlag, New York, 1988.Google Scholar [5] J.-P. Aubin, Fuzzy differential inclusions, Problems of control and information theory, 19 (1990), 55-67. Google Scholar [6] V. A. Baĭdosov, Differential inclusions with fuzzy right-hand side, Soviet Math. Dokl., 40 (1990), 567-569. Google Scholar [7] V. A. Baĭdosov, Fuzzy differential inclusions, J. of Appl. Math. and Mechan., 54 (1990), 8-13. doi: 10.1016/0021-8928(90)90080-T. Google Scholar [8] M. Benchohra, A note on an hyperbolic differential inclusion in Banach spaces, Bull. Belg. Math. Soc. Simon Stevin, 9 (2002), 101-107. Google Scholar [9] M. Benchohra and S. K. Ntouyas, An existence theorem for an hyperbolic differential inclusion in banach spaces, Discuss. Math. Differ. Incl. Control Optim., 22 (2002), 5-16. doi: 10.7151/dmdico.1029. Google Scholar [10] M. Benchohra, L. Gorniewicz, S. K. Ntouyas and A. Ouahab, Impulsive hyperbolic differential inclusions with variable times, Topol. Methods Nonlinear Anal., 22 (2003), 319-329. doi: 10.12775/TMNA.2003.042. Google Scholar [11] M. Benchohra, J. Henderson and S. K. Ntouyas, ImpulsiveDifferential Equations and Inclusions Contemporary Mathematics and Its Applications, 2 Hindawi Publishing Corporation, New York, 2006.Google Scholar [12] A. M. Bertone, R. M. Jafelice, L. C. de Barros and R. C. Bassanezi, On fuzzy solutions for partial differential equations, Fuzzy Set and Systems, 219 (2013), 68-80. doi: 10.1016/j.fss.2012.12.002. Google Scholar [13] N. N. Bogoliubov and Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations International Monographs on Advanced Mathematics and Physics Hindustan Publishing Corp. , Delhi, Gordon and Breach Science Publishers, New York, 1961.Google Scholar [14] N. N. Bogoliubov, Yu. A. Mitropolsky and A. M. Samoĭlenko, Methods of Accelerated Convergence in Nonlinear Mechanics Hindustan Publishing Corp. , Delhi; Springer-Verlag, Berlin-New York, 1976.Google Scholar [15] A. Cernea, Some second-order necessary conditions for nonconvex hyperbolic differential inclusion problems, J. Math. Anal. Appl., 253 (2001), 616-639. doi: 10.1006/jmaa.2000.7170. Google Scholar [16] A. Cernea, On the set of solutions of some nonconvex nonclosed hyperbolic differential inclusions, Czechoslovak Math. J., 52 (2002), 215-224. doi: 10.1023/A:1021787808233. Google Scholar [17] F. S. De Blasi and J. Myjak, On the structure of the set of solutions of the Darboux problem of hyperbolic equations, Proc. Edinburgh Math. Soc.(Series 2), 29 (1986), 7-14. doi: 10.1017/S0013091500017351. Google Scholar [18] F. S. De Blasi and J. Myjak, On the set of solutions of a differential inclusion, Bull. Inst. Math. Acad. Sinica, 14 (1986), 271-275. Google Scholar [19] P. Diamond and P. E. Kloeden, Metric Spaces of Fuzzy Sets. Theory and Applications World Scientific Publishing Co. , Inc. , River Edge, NJ, 1994.Google Scholar [20] S. Djebali, L. Górniewicz and A. Ouahab, Solution Sets for Differential Equations and Inclusions De Gruyter Series in Nonlinear Analysis and Applications, 18 Walter de Gruyter & Co. , Berlin, 2013.Google Scholar [21] D. Dutta Majumder and K. K. Majumdar, Complexity analysis, uncertainty management and fuzzy dynamical systems: a cybernetic approach, Kybernetes, 33 (2004), 1143-1184. doi: 10.1108/03684920410534489. Google Scholar [22] P. Fatou, Sur le mouvement d'un systéme soumis á des forces á courte période, Bull. Soc. Math. France, 56 (1928), 98-139. Google Scholar [23] J. R. Graef, J. Henderson and A. Ouahab, Impulsive Differential Inclusions. A Fixed Point Approach De Gruyter Series in Nonlinear Analysis and Applications, 20 De Gruyter, Berlin, 2013.Google Scholar [24] M. Guo, X. Xue and R. Li, Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets and Systems, 138 (2003), 601-615. doi: 10.1016/S0165-0114(02)00522-5. Google Scholar [25] J. K. Hale, Theory of Functional Differential Equations Second edition. Applied Mathematical Sciences, 3 Springer-Verlag, New York-Heidelberg, 1977.Google Scholar [26] E. Hullermeier, An approach to modelling and simulation of uncertain dynamical system, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 5 (1997), 117-137. doi: 10.1142/S0218488597000117. Google Scholar [27] R. M. Jafelice, C. G. Almeida, J. F. Meyer and H. L. Vasconcelos, Fuzzy parameter in a partial differential equation model for population dispersal of leaf-cutting ants, Nonlinear Anal. Real World Appl., 12 (2011), 3397-3412. doi: 10.1016/j.nonrwa.2011.06.003. Google Scholar [28] S. Klymchuk, A. Plotnikov and N. Skripnik, Overview of V.A. Plotnikov's research on averaging of differential inclusions, Phys. D, 241 (2012), 1932-1947. doi: 10.1016/j.physd.2011.05.004. Google Scholar [29] N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1947.Google Scholar [30] V. Lakshmikantham, T. Granna Bhaskar and J. Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces Cambridge Scientific Publishers, Cambridge, UK, 2006.Google Scholar [31] V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions Taylor & Francis, London, UK, 2003.Google Scholar [32] V. L. Lažar, Ulam-Hyers stability for partial differential inclusions, Electron. J. Qual. Theory Differ. Equ., 21 (2012), 1-19. doi: 10.14232/ejqtde.2012.1.21. Google Scholar [33] P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems Applied Mathematical Sciences, 72 Springer-Verlag, New York, 1988.Google Scholar [34] A. Lomtatidze and J. Šremr, Caratheodory solutions to a hyperbolic differential inequality with a non-positive coefficient and delayed arguments, Bound. Value Probl., 2014 (2014), 1-13. doi: 10.1186/1687-2770-2014-52. Google Scholar [35] H. V. Long, N. T. K. Son, N. T. M. Ha and L. H. Son, The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations, Fuzzy Optimization and Decision Making, 13 (2014), 435-462. doi: 10.1007/s10700-014-9186-0. Google Scholar [36] S. Nedelcu and A. Cernea, On the existence of solutions for nonconvex impulsive hyperbolic differential inclusions, Ann. Univ. Buchar. Math. Ser., 1 (2010), 275-284. Google Scholar [37] C. V. Negoiţă and D. A. Ralescu, Application of Fuzzy Sets to Systems Analysis Interdisciplinary Systems Research, 11 Birkhäuser Verlag, Basel-Stuttgart, 1975.Google Scholar [38] A. I. Panasyuk and V. I. Panasyuk, An equation generated by a differential inclusion, Mat. Zametki, 27 (1980), 429-437. doi: 10.1007/BF01140170. Google Scholar [39] N. A. Perestyuk, V. A. Plotnikov, A. M. Samoĭlenko and N. V. Skripnik, Differential Equations with Impulse Effects: Multivalued Right-Hand Sides with Discontinuities de Gruyter Studies in Mathematics, 40 Walter de Gruyter & Co. , Berlin, 2011.Google Scholar [40] A. V. Plotnikov, T. A. Komleva and L. I. Plotnikova, The partial averaging of differential inclusions with fuzzy right-hand side, J. Adv. Res. Dyn. Control Syst., 2 (2010), 26-34. Google Scholar [41] A. V. Plotnikov and T. A. Komleva, The partial averaging of fuzzy differential inclusions on finite interval International Journal of Differential Equations 2014 (2014), Article ID 307941, 5 pages.Google Scholar [42] A. V. Plotnikov, A procedure of complete averaging for fuzzy differential inclusions on a finite segment, Ukrainian Math. J, 67 (2015), 421-430. doi: 10.1007/s11253-015-1090-4. Google Scholar [43] V. A. Plotnikov, A. V. Plotnikov and A. N. Vityuk, Differential Equations with a Multivalued Right-Hand Side. Asymptotic Methods AstroPrint, Odessa, Ukraine, 1999.Google Scholar [44] M. L. Puri and D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986), 409-422. doi: 10.1016/0022-247X(86)90093-4. Google Scholar [45] A. M. Samoĭlenko and M. O. Perestyuk, Impulsive Differential Equations World Scientific Publishing Co. , Inc. , River Edge, NJ, 1995.Google Scholar [46] J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems Applied Mathematical Sciences, 59 Springer-Verlag, New York, 1985.Google Scholar [47] J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems 2nd edition, Appl. Math. Sci. , 59 Springer-Verlag, New York, 2007.Google Scholar [48] J. Šremr, Absolutely continuous functions of two variables in the sense of Caratheodory, Electron. J. Diff. Equ., 2010 (2010), 1-11. Google Scholar [49] V. Staicu, On a non-convex hyperbolic differential inclusion, Proc. Edinburgh Math. Soc., 35 (1992), 375-382. doi: 10.1017/S0013091500005666. Google Scholar [50] G. Teodoru, A characterization of the solutions of the Darboux problem for the equation ${\partial ^2}z/\partial x\partial y \in F(x,y,z)$, An. Stiint. Univ. Al. I. Cuza Iasi Mat., 33 (1987), 33-38. Google Scholar [51] A. N. Vityuk, Properties of solutions of hyperbolic differential equations with many-valued right-hand sides, Mat. Fiz. Nelin. Mekh., 15 (1991), 59-62. Google Scholar [52] A. N. Vityuk, Equation of the integral funnel of a partial differential inclusion, Dokl. Ukr. Akad. Nauk, Ser. A, 9 (1992), 19-20. Google Scholar [53] A. N. Vityuk, On solutions of hyperbolic differential inclusions with nonconvex right-hand side, Ukrainian Math. J., 47 (1995), 617-621. doi: 10.1007/BF01056048. Google Scholar [54] A. N. Vityuk, On an R-solution generated by a differential inclusion of hyperbolic type, Differential Equations, 30 (1994), 1578-1586. Google Scholar [55] A. N. Vityuk, Continuous dependence of the R-solution generated by a differential hyperbolic inclusion on parameters, Ukrainian Math. J., 47 (1995), 1625-1631. doi: 10.1007/BF01060163. Google Scholar

show all references

##### References:
 [1] S. Abbasbandy, T. A. Viranloo, O. Lopez-Pouso and J. J. Nieto, Numerical methods for fuzzy differential inclusions, Computers & Mathematics with Applications, 48 (2004), 1633-1641. doi: 10.1016/j.camwa.2004.03.009. Google Scholar [2] R. P. Agarwal, D. O'Regan and V. Lakshmikantham, A stacking theorem approach for fuzzy differential equations, Nonlinear Analysis, 55 (2003), 299-312. doi: 10.1016/S0362-546X(03)00241-4. Google Scholar [3] R. P. Agarwal, D. O'Regan and V. Lakshmikantham, Maximal solutions and existence theory for fuzzy differential and integral equations, Journal of Applied Analysis, 11 (2005), 171-186. doi: 10.1515/JAA.2005.171. Google Scholar [4] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations Second edition. Grundlehren der Mathematischen Wissenschaften, 250 Springer-Verlag, New York, 1988.Google Scholar [5] J.-P. Aubin, Fuzzy differential inclusions, Problems of control and information theory, 19 (1990), 55-67. Google Scholar [6] V. A. Baĭdosov, Differential inclusions with fuzzy right-hand side, Soviet Math. Dokl., 40 (1990), 567-569. Google Scholar [7] V. A. Baĭdosov, Fuzzy differential inclusions, J. of Appl. Math. and Mechan., 54 (1990), 8-13. doi: 10.1016/0021-8928(90)90080-T. Google Scholar [8] M. Benchohra, A note on an hyperbolic differential inclusion in Banach spaces, Bull. Belg. Math. Soc. Simon Stevin, 9 (2002), 101-107. Google Scholar [9] M. Benchohra and S. K. Ntouyas, An existence theorem for an hyperbolic differential inclusion in banach spaces, Discuss. Math. Differ. Incl. Control Optim., 22 (2002), 5-16. doi: 10.7151/dmdico.1029. Google Scholar [10] M. Benchohra, L. Gorniewicz, S. K. Ntouyas and A. Ouahab, Impulsive hyperbolic differential inclusions with variable times, Topol. Methods Nonlinear Anal., 22 (2003), 319-329. doi: 10.12775/TMNA.2003.042. Google Scholar [11] M. Benchohra, J. Henderson and S. K. Ntouyas, ImpulsiveDifferential Equations and Inclusions Contemporary Mathematics and Its Applications, 2 Hindawi Publishing Corporation, New York, 2006.Google Scholar [12] A. M. Bertone, R. M. Jafelice, L. C. de Barros and R. C. Bassanezi, On fuzzy solutions for partial differential equations, Fuzzy Set and Systems, 219 (2013), 68-80. doi: 10.1016/j.fss.2012.12.002. Google Scholar [13] N. N. Bogoliubov and Yu. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations International Monographs on Advanced Mathematics and Physics Hindustan Publishing Corp. , Delhi, Gordon and Breach Science Publishers, New York, 1961.Google Scholar [14] N. N. Bogoliubov, Yu. A. Mitropolsky and A. M. Samoĭlenko, Methods of Accelerated Convergence in Nonlinear Mechanics Hindustan Publishing Corp. , Delhi; Springer-Verlag, Berlin-New York, 1976.Google Scholar [15] A. Cernea, Some second-order necessary conditions for nonconvex hyperbolic differential inclusion problems, J. Math. Anal. Appl., 253 (2001), 616-639. doi: 10.1006/jmaa.2000.7170. Google Scholar [16] A. Cernea, On the set of solutions of some nonconvex nonclosed hyperbolic differential inclusions, Czechoslovak Math. J., 52 (2002), 215-224. doi: 10.1023/A:1021787808233. Google Scholar [17] F. S. De Blasi and J. Myjak, On the structure of the set of solutions of the Darboux problem of hyperbolic equations, Proc. Edinburgh Math. Soc.(Series 2), 29 (1986), 7-14. doi: 10.1017/S0013091500017351. Google Scholar [18] F. S. De Blasi and J. Myjak, On the set of solutions of a differential inclusion, Bull. Inst. Math. Acad. Sinica, 14 (1986), 271-275. Google Scholar [19] P. Diamond and P. E. Kloeden, Metric Spaces of Fuzzy Sets. Theory and Applications World Scientific Publishing Co. , Inc. , River Edge, NJ, 1994.Google Scholar [20] S. Djebali, L. Górniewicz and A. Ouahab, Solution Sets for Differential Equations and Inclusions De Gruyter Series in Nonlinear Analysis and Applications, 18 Walter de Gruyter & Co. , Berlin, 2013.Google Scholar [21] D. Dutta Majumder and K. K. Majumdar, Complexity analysis, uncertainty management and fuzzy dynamical systems: a cybernetic approach, Kybernetes, 33 (2004), 1143-1184. doi: 10.1108/03684920410534489. Google Scholar [22] P. Fatou, Sur le mouvement d'un systéme soumis á des forces á courte période, Bull. Soc. Math. France, 56 (1928), 98-139. Google Scholar [23] J. R. Graef, J. Henderson and A. Ouahab, Impulsive Differential Inclusions. A Fixed Point Approach De Gruyter Series in Nonlinear Analysis and Applications, 20 De Gruyter, Berlin, 2013.Google Scholar [24] M. Guo, X. Xue and R. Li, Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets and Systems, 138 (2003), 601-615. doi: 10.1016/S0165-0114(02)00522-5. Google Scholar [25] J. K. Hale, Theory of Functional Differential Equations Second edition. Applied Mathematical Sciences, 3 Springer-Verlag, New York-Heidelberg, 1977.Google Scholar [26] E. Hullermeier, An approach to modelling and simulation of uncertain dynamical system, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 5 (1997), 117-137. doi: 10.1142/S0218488597000117. Google Scholar [27] R. M. Jafelice, C. G. Almeida, J. F. Meyer and H. L. Vasconcelos, Fuzzy parameter in a partial differential equation model for population dispersal of leaf-cutting ants, Nonlinear Anal. Real World Appl., 12 (2011), 3397-3412. doi: 10.1016/j.nonrwa.2011.06.003. Google Scholar [28] S. Klymchuk, A. Plotnikov and N. Skripnik, Overview of V.A. Plotnikov's research on averaging of differential inclusions, Phys. D, 241 (2012), 1932-1947. doi: 10.1016/j.physd.2011.05.004. Google Scholar [29] N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1947.Google Scholar [30] V. Lakshmikantham, T. Granna Bhaskar and J. Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces Cambridge Scientific Publishers, Cambridge, UK, 2006.Google Scholar [31] V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions Taylor & Francis, London, UK, 2003.Google Scholar [32] V. L. Lažar, Ulam-Hyers stability for partial differential inclusions, Electron. J. Qual. Theory Differ. Equ., 21 (2012), 1-19. doi: 10.14232/ejqtde.2012.1.21. Google Scholar [33] P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems Applied Mathematical Sciences, 72 Springer-Verlag, New York, 1988.Google Scholar [34] A. Lomtatidze and J. Šremr, Caratheodory solutions to a hyperbolic differential inequality with a non-positive coefficient and delayed arguments, Bound. Value Probl., 2014 (2014), 1-13. doi: 10.1186/1687-2770-2014-52. Google Scholar [35] H. V. Long, N. T. K. Son, N. T. M. Ha and L. H. Son, The existence and uniqueness of fuzzy solutions for hyperbolic partial differential equations, Fuzzy Optimization and Decision Making, 13 (2014), 435-462. doi: 10.1007/s10700-014-9186-0. Google Scholar [36] S. Nedelcu and A. Cernea, On the existence of solutions for nonconvex impulsive hyperbolic differential inclusions, Ann. Univ. Buchar. Math. Ser., 1 (2010), 275-284. Google Scholar [37] C. V. Negoiţă and D. A. Ralescu, Application of Fuzzy Sets to Systems Analysis Interdisciplinary Systems Research, 11 Birkhäuser Verlag, Basel-Stuttgart, 1975.Google Scholar [38] A. I. Panasyuk and V. I. Panasyuk, An equation generated by a differential inclusion, Mat. Zametki, 27 (1980), 429-437. doi: 10.1007/BF01140170. Google Scholar [39] N. A. Perestyuk, V. A. Plotnikov, A. M. Samoĭlenko and N. V. Skripnik, Differential Equations with Impulse Effects: Multivalued Right-Hand Sides with Discontinuities de Gruyter Studies in Mathematics, 40 Walter de Gruyter & Co. , Berlin, 2011.Google Scholar [40] A. V. Plotnikov, T. A. Komleva and L. I. Plotnikova, The partial averaging of differential inclusions with fuzzy right-hand side, J. Adv. Res. Dyn. Control Syst., 2 (2010), 26-34. Google Scholar [41] A. V. Plotnikov and T. A. Komleva, The partial averaging of fuzzy differential inclusions on finite interval International Journal of Differential Equations 2014 (2014), Article ID 307941, 5 pages.Google Scholar [42] A. V. Plotnikov, A procedure of complete averaging for fuzzy differential inclusions on a finite segment, Ukrainian Math. J, 67 (2015), 421-430. doi: 10.1007/s11253-015-1090-4. Google Scholar [43] V. A. Plotnikov, A. V. Plotnikov and A. N. Vityuk, Differential Equations with a Multivalued Right-Hand Side. Asymptotic Methods AstroPrint, Odessa, Ukraine, 1999.Google Scholar [44] M. L. Puri and D. A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl., 114 (1986), 409-422. doi: 10.1016/0022-247X(86)90093-4. Google Scholar [45] A. M. Samoĭlenko and M. O. Perestyuk, Impulsive Differential Equations World Scientific Publishing Co. , Inc. , River Edge, NJ, 1995.Google Scholar [46] J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems Applied Mathematical Sciences, 59 Springer-Verlag, New York, 1985.Google Scholar [47] J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems 2nd edition, Appl. Math. Sci. , 59 Springer-Verlag, New York, 2007.Google Scholar [48] J. Šremr, Absolutely continuous functions of two variables in the sense of Caratheodory, Electron. J. Diff. Equ., 2010 (2010), 1-11. Google Scholar [49] V. Staicu, On a non-convex hyperbolic differential inclusion, Proc. Edinburgh Math. Soc., 35 (1992), 375-382. doi: 10.1017/S0013091500005666. Google Scholar [50] G. Teodoru, A characterization of the solutions of the Darboux problem for the equation ${\partial ^2}z/\partial x\partial y \in F(x,y,z)$, An. Stiint. Univ. Al. I. Cuza Iasi Mat., 33 (1987), 33-38. Google Scholar [51] A. N. Vityuk, Properties of solutions of hyperbolic differential equations with many-valued right-hand sides, Mat. Fiz. Nelin. Mekh., 15 (1991), 59-62. Google Scholar [52] A. N. Vityuk, Equation of the integral funnel of a partial differential inclusion, Dokl. Ukr. Akad. Nauk, Ser. A, 9 (1992), 19-20. Google Scholar [53] A. N. Vityuk, On solutions of hyperbolic differential inclusions with nonconvex right-hand side, Ukrainian Math. J., 47 (1995), 617-621. doi: 10.1007/BF01056048. Google Scholar [54] A. N. Vityuk, On an R-solution generated by a differential inclusion of hyperbolic type, Differential Equations, 30 (1994), 1578-1586. Google Scholar [55] A. N. Vityuk, Continuous dependence of the R-solution generated by a differential hyperbolic inclusion on parameters, Ukrainian Math. J., 47 (1995), 1625-1631. doi: 10.1007/BF01060163. Google Scholar
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