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Existence of nontrivial solutions for equations of $p(x)$Laplace type without Ambrosetti and Rabinowitz condition
On the properties of solutions set for measure driven differential inclusions
1.  Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61614 Poznań, Poland 
2.  Faculty of Electrical Engineering and Computer Science, Stefan cel Mare University, Universitatii 13, 720229 Suceava, Romania 
References:
[1] 
J.P. Aubin and H. Frankowska, "SetValued Analysis", Birkhäuser, Boston, 1990. Google Scholar 
[2] 
P. Billingsley, Weak convergence of measures: Applications in probability, in "DCBMSNSF Regional Conference Series in Applied Mathematics", 1971. Google Scholar 
[3] 
A.M. Bruckner, J.B. Bruckner and B.S. Thomson, "Real Analysis", PrenticeHall, 1997. Google Scholar 
[4] 
C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions", in Lecture Notes in Math. 580, Springer, Berlin, 1977. Google Scholar 
[5] 
M. Cichoń and B. Satco, Measure differential inclusions  between continuous and discrete, Adv. Diff. Equations 2014, 2014:56, 18 pp. Google Scholar 
[6] 
G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equations 4 (1991), 739765. Google Scholar 
[7] 
M. Federson, J.G. Mesquita and A. Slavik, Measure functional differential equations and functional dynamic equations on time scales, J. Diff. Equations 252 (2012), 38163847. Google Scholar 
[8] 
D. Fraňková, Regulated functions, Math. Bohem. 116 (1991), 2059. Google Scholar 
[9] 
Fremlin, D.H., Measure Theory. Vol. 2, Torres Fremlin, Colchester (2003). Google Scholar 
[10] 
Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on a parameter, Funct. Differ. Equ. 16 (2009), 299313. Google Scholar 
[11] 
R. Lucchetti, G. Salinetti and R. JB. Wets, Uniform convergence of probability measures: topological criteria, Jour. Multivariate Anal. 51 (1994), 252264. Google Scholar 
[12] 
J. Lygeros, M. Quincampoix and T. Rze.zuchowski, Impulse differential inclusions driven by discrete measures, in "Hybrid Systems: Computation and Control", Lecture Notes in Computer Science 4416 (2007), 385398. Google Scholar 
[13] 
B. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Syst. Estimation and Control 4 (1994), 121. Google Scholar 
[14] 
B. Miller and E.Y. Rubinovitch, "Impulsive Control in Continuous and DiscreteContinuous Systems", Kluwer Academic Publishers, Dordrecht, 2003. Google Scholar 
[15] 
G. A. Monteiro and M. Tvrdý, Generalized linear differential equations in a Banach space: Continuous dependence on a parameter, Discrete Contin. Dyn. Syst. 33 (2013), 283303. Google Scholar 
[16] 
W.R. Pestman, Measurability of linear operators in the Skorokhod topology, Bull. Belg. Math. Soc. 2 (1995), 381388. Google Scholar 
[17] 
R.R. Rao, Relations between weak and uniform convergence of measures with applications The Annals of Mathematical Statistics 33 (1962), 659680. Google Scholar 
[18] 
S. Saks, "Theory of the Integral", Monografie Matematyczne, Warszawa, 1937. Google Scholar 
[19] 
Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations. Boundary Problems and Adjoints", Dordrecht, Praha, 1979. Google Scholar 
[20] 
A.N. Sesekin and S.T. Zavalishchin, "Dynamic Impulse Systems", Dordrecht, Kluwer Academic, 1997. Google Scholar 
[21] 
G.N. Silva and R.B. Vinter, Measure driven differential inclusions, J. Math. Anal. Appl. 202 (1996), 727746. Google Scholar 
[22] 
A. Slavik, Wellposedness results for abstract generalized differential equations and measure functional differential equations, Journal of Differential Equations 259 (2015), 666707. Google Scholar 
[23] 
M. Tvrdý, "Differential and Integral Equations in the Space of Regulated Functions", Habil. Thesis, Praha, 2001. Google Scholar 
show all references
References:
[1] 
J.P. Aubin and H. Frankowska, "SetValued Analysis", Birkhäuser, Boston, 1990. Google Scholar 
[2] 
P. Billingsley, Weak convergence of measures: Applications in probability, in "DCBMSNSF Regional Conference Series in Applied Mathematics", 1971. Google Scholar 
[3] 
A.M. Bruckner, J.B. Bruckner and B.S. Thomson, "Real Analysis", PrenticeHall, 1997. Google Scholar 
[4] 
C. Castaing and M. Valadier, "Convex Analysis and Measurable Multifunctions", in Lecture Notes in Math. 580, Springer, Berlin, 1977. Google Scholar 
[5] 
M. Cichoń and B. Satco, Measure differential inclusions  between continuous and discrete, Adv. Diff. Equations 2014, 2014:56, 18 pp. Google Scholar 
[6] 
G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equations 4 (1991), 739765. Google Scholar 
[7] 
M. Federson, J.G. Mesquita and A. Slavik, Measure functional differential equations and functional dynamic equations on time scales, J. Diff. Equations 252 (2012), 38163847. Google Scholar 
[8] 
D. Fraňková, Regulated functions, Math. Bohem. 116 (1991), 2059. Google Scholar 
[9] 
Fremlin, D.H., Measure Theory. Vol. 2, Torres Fremlin, Colchester (2003). Google Scholar 
[10] 
Z. Halas and M. Tvrdý, Continuous dependence of solutions of generalized linear differential equations on a parameter, Funct. Differ. Equ. 16 (2009), 299313. Google Scholar 
[11] 
R. Lucchetti, G. Salinetti and R. JB. Wets, Uniform convergence of probability measures: topological criteria, Jour. Multivariate Anal. 51 (1994), 252264. Google Scholar 
[12] 
J. Lygeros, M. Quincampoix and T. Rze.zuchowski, Impulse differential inclusions driven by discrete measures, in "Hybrid Systems: Computation and Control", Lecture Notes in Computer Science 4416 (2007), 385398. Google Scholar 
[13] 
B. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Syst. Estimation and Control 4 (1994), 121. Google Scholar 
[14] 
B. Miller and E.Y. Rubinovitch, "Impulsive Control in Continuous and DiscreteContinuous Systems", Kluwer Academic Publishers, Dordrecht, 2003. Google Scholar 
[15] 
G. A. Monteiro and M. Tvrdý, Generalized linear differential equations in a Banach space: Continuous dependence on a parameter, Discrete Contin. Dyn. Syst. 33 (2013), 283303. Google Scholar 
[16] 
W.R. Pestman, Measurability of linear operators in the Skorokhod topology, Bull. Belg. Math. Soc. 2 (1995), 381388. Google Scholar 
[17] 
R.R. Rao, Relations between weak and uniform convergence of measures with applications The Annals of Mathematical Statistics 33 (1962), 659680. Google Scholar 
[18] 
S. Saks, "Theory of the Integral", Monografie Matematyczne, Warszawa, 1937. Google Scholar 
[19] 
Š. Schwabik, M. Tvrdý and O. Vejvoda, "Differential and Integral Equations. Boundary Problems and Adjoints", Dordrecht, Praha, 1979. Google Scholar 
[20] 
A.N. Sesekin and S.T. Zavalishchin, "Dynamic Impulse Systems", Dordrecht, Kluwer Academic, 1997. Google Scholar 
[21] 
G.N. Silva and R.B. Vinter, Measure driven differential inclusions, J. Math. Anal. Appl. 202 (1996), 727746. Google Scholar 
[22] 
A. Slavik, Wellposedness results for abstract generalized differential equations and measure functional differential equations, Journal of Differential Equations 259 (2015), 666707. Google Scholar 
[23] 
M. Tvrdý, "Differential and Integral Equations in the Space of Regulated Functions", Habil. Thesis, Praha, 2001. Google Scholar 
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