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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

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.  Google Scholar

[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. Google Scholar

[3]

J. Andres, L. Malaguti and M. Pavlačková, Dirichlet problem in Banach spaces: the bound sets approach, Bound. Value Probl., 25 (2013), 1-21. Google Scholar

[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.  Google Scholar

[5]

J. Andres, L. Malaguti and M. Pavlačková, Scorza-Dragoni approach to Dirichlet problem in Banach spaces, Bound. Value Probl., 23 (2014), 1-24. Google Scholar

[6]

J. Andres, L. Malaguti and V. Taddei, On boundary value problems in Banach spaces, Dynam. Systems Appl., 18 (2009), 275-302.  Google Scholar

[7]

S. R. Bernfeld and V. Lakshmikantham, An introduction to nonlinear boundary value Problems,, Mathematics in Science and Engineering, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, LNM 568, Springer, Berlin, 1977.  Google Scholar

[16]

R. E. Gaines and J. Mawhin, Ordinary differential equations with nonlinear boundary conditions, J . Differential Equations, 26 (1977), 200-222.  Google Scholar

[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.  Google Scholar

[18]

P. Hartman, On boundary value problems for systems of ordinary, nonlinear, second order differential equations, Trans. Amer. Math. Soc., 96 (1960), 493-509. Google Scholar

[19]

P. Hartman, Ordinary Differential Equations, Willey, New York, 1964. Google Scholar

[20]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I. Theory. Mathematics and its Applications, 419, Kluwer, Dordrecht, 1997. Google Scholar

[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, ().   Google Scholar

[22]

M. Kožušníková, A bounding functions approach to multivalued Dirichlet problem, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 55 (2007), 1-19.  Google Scholar

[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. Google Scholar

[24]

N. H. Loc and K. Schmitt, Bernstein-Nagumo conditions and solutions to nonlinear differential inequalities, Nonlinear Anal., 75 (2012), 4664-4671.  Google Scholar

[25]

J. Mawhin, Boundary value problems for nonlinear second order vector differential equations, J . Differential Equations, 16 (1974), 257-269. Google Scholar

[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.  Google Scholar

[27]

J. Mawhin, Two point boundary value problems for nonlinear second order differential equations in Hilbert spaces, Tôkoku Math. J., 32 (1980), 225-233. Google Scholar

[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. Google Scholar

[29]

N. S. Papageorgiou and S. Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, 19, Springer, Berlin, 2009. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[32]

K. Schmitt, Randwertaufgaben für gewöhnliche Differentialgleichungen,, Proc. Steiermark. Math. Symposium, ().   Google Scholar

[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. Google Scholar

[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. Google Scholar

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.  Google Scholar

[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. Google Scholar

[3]

J. Andres, L. Malaguti and M. Pavlačková, Dirichlet problem in Banach spaces: the bound sets approach, Bound. Value Probl., 25 (2013), 1-21. Google Scholar

[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.  Google Scholar

[5]

J. Andres, L. Malaguti and M. Pavlačková, Scorza-Dragoni approach to Dirichlet problem in Banach spaces, Bound. Value Probl., 23 (2014), 1-24. Google Scholar

[6]

J. Andres, L. Malaguti and V. Taddei, On boundary value problems in Banach spaces, Dynam. Systems Appl., 18 (2009), 275-302.  Google Scholar

[7]

S. R. Bernfeld and V. Lakshmikantham, An introduction to nonlinear boundary value Problems,, Mathematics in Science and Engineering, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Deimling, Multivalued Differential Equations, De Gruyter, Berlin, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, LNM 568, Springer, Berlin, 1977.  Google Scholar

[16]

R. E. Gaines and J. Mawhin, Ordinary differential equations with nonlinear boundary conditions, J . Differential Equations, 26 (1977), 200-222.  Google Scholar

[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.  Google Scholar

[18]

P. Hartman, On boundary value problems for systems of ordinary, nonlinear, second order differential equations, Trans. Amer. Math. Soc., 96 (1960), 493-509. Google Scholar

[19]

P. Hartman, Ordinary Differential Equations, Willey, New York, 1964. Google Scholar

[20]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I. Theory. Mathematics and its Applications, 419, Kluwer, Dordrecht, 1997. Google Scholar

[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, ().   Google Scholar

[22]

M. Kožušníková, A bounding functions approach to multivalued Dirichlet problem, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 55 (2007), 1-19.  Google Scholar

[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. Google Scholar

[24]

N. H. Loc and K. Schmitt, Bernstein-Nagumo conditions and solutions to nonlinear differential inequalities, Nonlinear Anal., 75 (2012), 4664-4671.  Google Scholar

[25]

J. Mawhin, Boundary value problems for nonlinear second order vector differential equations, J . Differential Equations, 16 (1974), 257-269. Google Scholar

[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.  Google Scholar

[27]

J. Mawhin, Two point boundary value problems for nonlinear second order differential equations in Hilbert spaces, Tôkoku Math. J., 32 (1980), 225-233. Google Scholar

[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. Google Scholar

[29]

N. S. Papageorgiou and S. Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, 19, Springer, Berlin, 2009. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[32]

K. Schmitt, Randwertaufgaben für gewöhnliche Differentialgleichungen,, Proc. Steiermark. Math. Symposium, ().   Google Scholar

[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. Google Scholar

[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. Google Scholar

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