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.

[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.

[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.

[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.

[5]

J.-P. Aubin, Fuzzy differential inclusions, Problems of control and information theory, 19 (1990), 55-67.

[6]

V. A. Baĭdosov, Differential inclusions with fuzzy right-hand side, Soviet Math. Dokl., 40 (1990), 567-569.

[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.

[8]

M. Benchohra, A note on an hyperbolic differential inclusion in Banach spaces, Bull. Belg. Math. Soc. Simon Stevin, 9 (2002), 101-107.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

P. Diamond and P. E. Kloeden, Metric Spaces of Fuzzy Sets. Theory and Applications World Scientific Publishing Co. , Inc. , River Edge, NJ, 1994.

[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.

[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.

[22]

P. Fatou, Sur le mouvement d'un systéme soumis á des forces á courte période, Bull. Soc. Math. France, 56 (1928), 98-139.

[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.

[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.

[25]

J. K. Hale, Theory of Functional Differential Equations Second edition. Applied Mathematical Sciences, 3 Springer-Verlag, New York-Heidelberg, 1977.

[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.

[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.

[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.

[29]

N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1947.

[30]

V. Lakshmikantham, T. Granna Bhaskar and J. Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces Cambridge Scientific Publishers, Cambridge, UK, 2006.

[31]

V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions Taylor & Francis, London, UK, 2003.

[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.

[33]

P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems Applied Mathematical Sciences, 72 Springer-Verlag, New York, 1988.

[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.

[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.

[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.

[37]

C. V. Negoiţă and D. A. Ralescu, Application of Fuzzy Sets to Systems Analysis Interdisciplinary Systems Research, 11 Birkhäuser Verlag, Basel-Stuttgart, 1975.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[45]

A. M. Samoĭlenko and M. O. Perestyuk, Impulsive Differential Equations World Scientific Publishing Co. , Inc. , River Edge, NJ, 1995.

[46]

J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems Applied Mathematical Sciences, 59 Springer-Verlag, New York, 1985.

[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.

[48]

J. Šremr, Absolutely continuous functions of two variables in the sense of Caratheodory, Electron. J. Diff. Equ., 2010 (2010), 1-11.

[49]

V. Staicu, On a non-convex hyperbolic differential inclusion, Proc. Edinburgh Math. Soc., 35 (1992), 375-382. doi: 10.1017/S0013091500005666.

[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.

[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.

[52]

A. N. Vityuk, Equation of the integral funnel of a partial differential inclusion, Dokl. Ukr. Akad. Nauk, Ser. A, 9 (1992), 19-20.

[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.

[54]

A. N. Vityuk, On an R-solution generated by a differential inclusion of hyperbolic type, Differential Equations, 30 (1994), 1578-1586.

[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.

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.

[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.

[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.

[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.

[5]

J.-P. Aubin, Fuzzy differential inclusions, Problems of control and information theory, 19 (1990), 55-67.

[6]

V. A. Baĭdosov, Differential inclusions with fuzzy right-hand side, Soviet Math. Dokl., 40 (1990), 567-569.

[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.

[8]

M. Benchohra, A note on an hyperbolic differential inclusion in Banach spaces, Bull. Belg. Math. Soc. Simon Stevin, 9 (2002), 101-107.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

P. Diamond and P. E. Kloeden, Metric Spaces of Fuzzy Sets. Theory and Applications World Scientific Publishing Co. , Inc. , River Edge, NJ, 1994.

[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.

[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.

[22]

P. Fatou, Sur le mouvement d'un systéme soumis á des forces á courte période, Bull. Soc. Math. France, 56 (1928), 98-139.

[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.

[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.

[25]

J. K. Hale, Theory of Functional Differential Equations Second edition. Applied Mathematical Sciences, 3 Springer-Verlag, New York-Heidelberg, 1977.

[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.

[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.

[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.

[29]

N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics Princeton University Press, Princeton, 1947.

[30]

V. Lakshmikantham, T. Granna Bhaskar and J. Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces Cambridge Scientific Publishers, Cambridge, UK, 2006.

[31]

V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions Taylor & Francis, London, UK, 2003.

[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.

[33]

P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems Applied Mathematical Sciences, 72 Springer-Verlag, New York, 1988.

[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.

[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.

[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.

[37]

C. V. Negoiţă and D. A. Ralescu, Application of Fuzzy Sets to Systems Analysis Interdisciplinary Systems Research, 11 Birkhäuser Verlag, Basel-Stuttgart, 1975.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[45]

A. M. Samoĭlenko and M. O. Perestyuk, Impulsive Differential Equations World Scientific Publishing Co. , Inc. , River Edge, NJ, 1995.

[46]

J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems Applied Mathematical Sciences, 59 Springer-Verlag, New York, 1985.

[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.

[48]

J. Šremr, Absolutely continuous functions of two variables in the sense of Caratheodory, Electron. J. Diff. Equ., 2010 (2010), 1-11.

[49]

V. Staicu, On a non-convex hyperbolic differential inclusion, Proc. Edinburgh Math. Soc., 35 (1992), 375-382. doi: 10.1017/S0013091500005666.

[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.

[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.

[52]

A. N. Vityuk, Equation of the integral funnel of a partial differential inclusion, Dokl. Ukr. Akad. Nauk, Ser. A, 9 (1992), 19-20.

[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.

[54]

A. N. Vityuk, On an R-solution generated by a differential inclusion of hyperbolic type, Differential Equations, 30 (1994), 1578-1586.

[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.

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