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. AbbasbandyT. A. ViranlooO. 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. AgarwalD. 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. AgarwalD. 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. BenchohraL. GorniewiczS. 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. BertoneR. M. JafeliceL. 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. GuoX. 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. JafeliceC. G. AlmeidaJ. 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. KlymchukA. 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. LongN. T. K. SonN. 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. PlotnikovT. 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. AbbasbandyT. A. ViranlooO. 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. AgarwalD. 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. AgarwalD. 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. BenchohraL. GorniewiczS. 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. BertoneR. M. JafeliceL. 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. GuoX. 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. JafeliceC. G. AlmeidaJ. 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. KlymchukA. 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. LongN. T. K. SonN. 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. PlotnikovT. 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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