Hartman-type conditions for multivalued Dirichlet problem in abstract spaces

Jan  Andres; Luisa Malaguti; Martina  Pavlačková

doi:10.3934/proc.2015.0038

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2015, 2015(special): 38-55. doi: 10.3934/proc.2015.0038

Hartman-type conditions for multivalued Dirichlet problem in abstract spaces

Jan Andres ^1, , Luisa Malaguti ^2, and Martina Pavlačková ^1,

1.	Dept. of Math. Analysis and Appl. of Mathematics, Fac. of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic, Czech Republic
2.	Dept. of Engineering Sciences and Methods, University of Modena and Reggio Emilia, Reggio Emilia, I-42122

Received August 2014 Revised January 2015 Published November 2015

Abstract
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Under a Creative Commons license

The classical Hartman's Theorem in [18] for the solvability of the vector Dirichlet problem will be generalized and extended in several directions. We will consider its multivalued versions for Marchaud and upper-Carathéodory right-hand sides with only certain amount of compactness in Banach spaces. Advanced topological methods are combined with a bound sets technique. Besides the existence, the localization of solutions can be obtained in this way.

Keywords: Dirichlet problem, topological methods, multivalued operators, bound sets technique., Hartman-type conditions, abstract spaces.

Mathematics Subject Classification: Primary: 34A60; Secondary: 34B15, 47H0.

Citation: Jan Andres, Luisa Malaguti, Martina Pavlačková. Hartman-type conditions for multivalued Dirichlet problem in abstract spaces. Conference Publications, 2015, 2015 (special) : 38-55. doi: 10.3934/proc.2015.0038

References:

[1]	P. Amster and J. Haddad, A Hartman-Nagumo type conditions for a class of contractible domains, Topol. Methods Nonlinear Anal., 41 (2013), 287-304.
[2]	J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Topological Fixed Point Theory and Its Applications, Vol. 1, Kluwer, Dordrecht, 2003.
[3]	J. Andres, L. Malaguti and M. Pavlačková, Dirichlet problem in Banach spaces: the bound sets approach, Bound. Value Probl., 25 (2013), 1-21.
[4]	J. Andres, L. Malaguti and M. Pavlačková, On second-order boundary value problems in Banach spaces: a bound sets approach, Topol. Methods Nonlinear Anal., 37 (2011), 303-341.
[5]	J. Andres, L. Malaguti and M. Pavlačková, Scorza-Dragoni approach to Dirichlet problem in Banach spaces, Bound. Value Probl., 23 (2014), 1-24.
[6]	J. Andres, L. Malaguti and V. Taddei, On boundary value problems in Banach spaces, Dynam. Systems Appl., 18 (2009), 275-302.
[7]	S. R. Bernfeld and V. Lakshmikantham, An introduction to nonlinear boundary value Problems,, Mathematics in Science and Engineering, ().
[8]	S. Cecchini, L. Malaguti and V. Taddei, Strictly localized bounding functions and Floquet boundary value problems, Electron. J. Qual. Theory Differ. Equ., 47 (2011), 1-18.
[9]	J. Chandra, V. Lakshmikantham and A. R. Mitchell, Existence of solutions of boundary value problems for nonlinear second-order systems in Banach space, Nonlinear Anal., 2 (1978), 157-168.
[10]	K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992.
[11]	L.H. Erbe and W. Krawcewicz, Nonlinear boundary value problems for differential inclusions $y^{''}\in F(t,y,y')$, Ann. Polon. Math., 54 (1991), 195-226.
[12]	L. Erbe, C. C. Tisdell and P. J. Y. Wong, On systems of boundary value problems for differential inclusions, Acta Math. Sin. (Engl. Ser.), 23 (2007), 549-556.
[13]	C. Fabry and P. Habets, The Picard boundary value problem for nonlinear second order vector differential equations, J . Differential Equations, 42 (1981), 186-198.
[14]	M. Frigon, Boundary and periodic value problems for systems of differential equations under Bernstein-Nagumo growth condition, Differential Integral Equations, 8 (1995), 1789-1804.
[15]	R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, LNM 568, Springer, Berlin, 1977.
[16]	R. E. Gaines and J. Mawhin, Ordinary differential equations with nonlinear boundary conditions, J . Differential Equations, 26 (1977), 200-222.
[17]	A. Granas, R. B. Guenther and J. W. Lee, Some general existence principles in the Carathéodory theory of nonlinear differential system, J. Math. Pures Appl., 70 (1991), 153-196.
[18]	P. Hartman, On boundary value problems for systems of ordinary, nonlinear, second order differential equations, Trans. Amer. Math. Soc., 96 (1960), 493-509.
[19]	P. Hartman, Ordinary Differential Equations, Willey, New York, 1964.
[20]	S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I. Theory. Mathematics and its Applications, 419, Kluwer, Dordrecht, 1997.
[21]	M. I. Kamenskii, V. V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces,, De Gruyter Series in Nonlinear Analysis and Applications, ().
[22]	M. Kožušníková, A bounding functions approach to multivalued Dirichlet problem, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 55 (2007), 1-19.
[23]	A. Lasota and J. A. Yorke, Existence of solutions of two-point boundary value problems for nonlinear systems, J . Differential Equations, 11 (1972), 509-518.
[24]	N. H. Loc and K. Schmitt, Bernstein-Nagumo conditions and solutions to nonlinear differential inequalities, Nonlinear Anal., 75 (2012), 4664-4671.
[25]	J. Mawhin, Boundary value problems for nonlinear second order vector differential equations, J . Differential Equations, 16 (1974), 257-269.
[26]	J. Mawhin, The Bernstein-Nagumo problem and two-point boundary value problems for ordinary differential equations In: Qualitative Theory of Differential Equations (ed. M. Farkas), Budapest, 1981, pp. 709-740.
[27]	J. Mawhin, Two point boundary value problems for nonlinear second order differential equations in Hilbert spaces, Tôkoku Math. J., 32 (1980), 225-233.
[28]	J. Mawhin, Some boundary value problems for Hartman-type perturbation of the ordinary vector $p$-Laplacian, Nonlinear Anal., 40 (2000), Ser. A: Theory Methods, 497-503.
[29]	N. S. Papageorgiou and S. Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, 19, Springer, Berlin, 2009.
[30]	M. Pavlačková, A bound sets technique for Dirichlet problem with an upper-Caratheodory right-hand side, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 49 (2010), 95-106.
[31]	M. Pavlačková, A Scorza-Dragoni approach to Dirichlet problem with an upper-Carathéodory right-hand side, Topol. Meth. Nonlin. Anal., 44 (2014), 239-247.
[32]	K. Schmitt, Randwertaufgaben für gewöhnliche Differentialgleichungen,, Proc. Steiermark. Math. Symposium, ().
[33]	K. Schmitt and P. Volkmann, Boundary value problems for second order differential equations in convex subsets of a Banach space, Trans. Amer. Math. Soc., 218 (1976), 397-405.
[34]	K. Schmitt and R.C. Thompson, Boundary value problems for infinite systems of second-order differential equations, J . Differential Equations,18 (1975), 277-295.

show all references

References:

[1]	P. Amster and J. Haddad, A Hartman-Nagumo type conditions for a class of contractible domains, Topol. Methods Nonlinear Anal., 41 (2013), 287-304.
[2]	J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Topological Fixed Point Theory and Its Applications, Vol. 1, Kluwer, Dordrecht, 2003.
[3]	J. Andres, L. Malaguti and M. Pavlačková, Dirichlet problem in Banach spaces: the bound sets approach, Bound. Value Probl., 25 (2013), 1-21.
[4]	J. Andres, L. Malaguti and M. Pavlačková, On second-order boundary value problems in Banach spaces: a bound sets approach, Topol. Methods Nonlinear Anal., 37 (2011), 303-341.
[5]	J. Andres, L. Malaguti and M. Pavlačková, Scorza-Dragoni approach to Dirichlet problem in Banach spaces, Bound. Value Probl., 23 (2014), 1-24.
[6]	J. Andres, L. Malaguti and V. Taddei, On boundary value problems in Banach spaces, Dynam. Systems Appl., 18 (2009), 275-302.
[7]	S. R. Bernfeld and V. Lakshmikantham, An introduction to nonlinear boundary value Problems,, Mathematics in Science and Engineering, ().
[8]	S. Cecchini, L. Malaguti and V. Taddei, Strictly localized bounding functions and Floquet boundary value problems, Electron. J. Qual. Theory Differ. Equ., 47 (2011), 1-18.
[9]	J. Chandra, V. Lakshmikantham and A. R. Mitchell, Existence of solutions of boundary value problems for nonlinear second-order systems in Banach space, Nonlinear Anal., 2 (1978), 157-168.
[10]	K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992.
[11]	L.H. Erbe and W. Krawcewicz, Nonlinear boundary value problems for differential inclusions $y^{''}\in F(t,y,y')$, Ann. Polon. Math., 54 (1991), 195-226.
[12]	L. Erbe, C. C. Tisdell and P. J. Y. Wong, On systems of boundary value problems for differential inclusions, Acta Math. Sin. (Engl. Ser.), 23 (2007), 549-556.
[13]	C. Fabry and P. Habets, The Picard boundary value problem for nonlinear second order vector differential equations, J . Differential Equations, 42 (1981), 186-198.
[14]	M. Frigon, Boundary and periodic value problems for systems of differential equations under Bernstein-Nagumo growth condition, Differential Integral Equations, 8 (1995), 1789-1804.
[15]	R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, LNM 568, Springer, Berlin, 1977.
[16]	R. E. Gaines and J. Mawhin, Ordinary differential equations with nonlinear boundary conditions, J . Differential Equations, 26 (1977), 200-222.
[17]	A. Granas, R. B. Guenther and J. W. Lee, Some general existence principles in the Carathéodory theory of nonlinear differential system, J. Math. Pures Appl., 70 (1991), 153-196.
[18]	P. Hartman, On boundary value problems for systems of ordinary, nonlinear, second order differential equations, Trans. Amer. Math. Soc., 96 (1960), 493-509.
[19]	P. Hartman, Ordinary Differential Equations, Willey, New York, 1964.
[20]	S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I. Theory. Mathematics and its Applications, 419, Kluwer, Dordrecht, 1997.
[21]	M. I. Kamenskii, V. V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces,, De Gruyter Series in Nonlinear Analysis and Applications, ().
[22]	M. Kožušníková, A bounding functions approach to multivalued Dirichlet problem, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 55 (2007), 1-19.
[23]	A. Lasota and J. A. Yorke, Existence of solutions of two-point boundary value problems for nonlinear systems, J . Differential Equations, 11 (1972), 509-518.
[24]	N. H. Loc and K. Schmitt, Bernstein-Nagumo conditions and solutions to nonlinear differential inequalities, Nonlinear Anal., 75 (2012), 4664-4671.
[25]	J. Mawhin, Boundary value problems for nonlinear second order vector differential equations, J . Differential Equations, 16 (1974), 257-269.
[26]	J. Mawhin, The Bernstein-Nagumo problem and two-point boundary value problems for ordinary differential equations In: Qualitative Theory of Differential Equations (ed. M. Farkas), Budapest, 1981, pp. 709-740.
[27]	J. Mawhin, Two point boundary value problems for nonlinear second order differential equations in Hilbert spaces, Tôkoku Math. J., 32 (1980), 225-233.
[28]	J. Mawhin, Some boundary value problems for Hartman-type perturbation of the ordinary vector $p$-Laplacian, Nonlinear Anal., 40 (2000), Ser. A: Theory Methods, 497-503.
[29]	N. S. Papageorgiou and S. Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, 19, Springer, Berlin, 2009.
[30]	M. Pavlačková, A bound sets technique for Dirichlet problem with an upper-Caratheodory right-hand side, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 49 (2010), 95-106.
[31]	M. Pavlačková, A Scorza-Dragoni approach to Dirichlet problem with an upper-Carathéodory right-hand side, Topol. Meth. Nonlin. Anal., 44 (2014), 239-247.
[32]	K. Schmitt, Randwertaufgaben für gewöhnliche Differentialgleichungen,, Proc. Steiermark. Math. Symposium, ().
[33]	K. Schmitt and P. Volkmann, Boundary value problems for second order differential equations in convex subsets of a Banach space, Trans. Amer. Math. Soc., 218 (1976), 397-405.
[34]	K. Schmitt and R.C. Thompson, Boundary value problems for infinite systems of second-order differential equations, J . Differential Equations,18 (1975), 277-295.

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