# American Institute of Mathematical Sciences

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

## Hartman-type conditions for multivalued Dirichlet problem in abstract spaces

 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.
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.   Google Scholar [2] J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems,, Topological Fixed Point Theory and Its Applications, (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.   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.   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.   Google Scholar [6] J. Andres, L. Malaguti and V. Taddei, On boundary value problems in Banach spaces,, Dynam. Systems Appl., 18 (2009), 275.   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.   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.   Google Scholar [10] K. Deimling, Multivalued Differential Equations,, De Gruyter, (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.   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.   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.   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.   Google Scholar [15] R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations,, LNM 568, (1977).   Google Scholar [16] R. E. Gaines and J. Mawhin, Ordinary differential equations with nonlinear boundary conditions,, J . Differential Equations, 26 (1977), 200.   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.   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.   Google Scholar [19] P. Hartman, Ordinary Differential Equations,, Willey, (1964).   Google Scholar [20] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis,, Vol. I. Theory. Mathematics and its Applications, (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.   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.   Google Scholar [24] N. H. Loc and K. Schmitt, Bernstein-Nagumo conditions and solutions to nonlinear differential inequalities,, Nonlinear Anal., 75 (2012), 4664.   Google Scholar [25] J. Mawhin, Boundary value problems for nonlinear second order vector differential equations,, J . Differential Equations, 16 (1974), 257.   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), (1981), 709.   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.   Google Scholar [28] J. Mawhin, Some boundary value problems for Hartman-type perturbation of the ordinary vector $p$-Laplacian,, Nonlinear Anal., 40 (2000), 497.   Google Scholar [29] N. S. Papageorgiou and S. Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, 19,, Springer, (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.   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.   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.   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.   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.   Google Scholar [2] J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems,, Topological Fixed Point Theory and Its Applications, (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.   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.   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.   Google Scholar [6] J. Andres, L. Malaguti and V. Taddei, On boundary value problems in Banach spaces,, Dynam. Systems Appl., 18 (2009), 275.   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.   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.   Google Scholar [10] K. Deimling, Multivalued Differential Equations,, De Gruyter, (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.   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.   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.   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.   Google Scholar [15] R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations,, LNM 568, (1977).   Google Scholar [16] R. E. Gaines and J. Mawhin, Ordinary differential equations with nonlinear boundary conditions,, J . Differential Equations, 26 (1977), 200.   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.   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.   Google Scholar [19] P. Hartman, Ordinary Differential Equations,, Willey, (1964).   Google Scholar [20] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis,, Vol. I. Theory. Mathematics and its Applications, (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.   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.   Google Scholar [24] N. H. Loc and K. Schmitt, Bernstein-Nagumo conditions and solutions to nonlinear differential inequalities,, Nonlinear Anal., 75 (2012), 4664.   Google Scholar [25] J. Mawhin, Boundary value problems for nonlinear second order vector differential equations,, J . Differential Equations, 16 (1974), 257.   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), (1981), 709.   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.   Google Scholar [28] J. Mawhin, Some boundary value problems for Hartman-type perturbation of the ordinary vector $p$-Laplacian,, Nonlinear Anal., 40 (2000), 497.   Google Scholar [29] N. S. Papageorgiou and S. Th. Kyritsi-Yiallourou, Handbook of Applied Analysis, Advances in Mechanics and Mathematics, 19,, Springer, (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.   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.   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.   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.   Google Scholar
 [1] Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets. Conference Publications, 2007, 2007 (Special) : 181-190. doi: 10.3934/proc.2007.2007.181 [2] Mustapha Cheggag, Angelo Favini, Rabah Labbas, Stéphane Maingot, Ahmed Medeghri. Complete abstract differential equations of elliptic type with general Robin boundary conditions, in UMD spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 523-538. doi: 10.3934/dcdss.2011.4.523 [3] Isabeau Birindelli, Francoise Demengel. The dirichlet problem for singluar fully nonlinear operators. Conference Publications, 2007, 2007 (Special) : 110-121. doi: 10.3934/proc.2007.2007.110 [4] Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295 [5] Poongodi Rathinasamy, Murugesu Rangasamy, Nirmalkumar Rajendran. Exact controllability results for a class of abstract nonlocal Cauchy problem with impulsive conditions. Evolution Equations & Control Theory, 2017, 6 (4) : 599-613. doi: 10.3934/eect.2017030 [6] Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105 [7] Peter Monk, Virginia Selgas. Sampling type methods for an inverse waveguide problem. Inverse Problems & Imaging, 2012, 6 (4) : 709-747. doi: 10.3934/ipi.2012.6.709 [8] Steve Hofmann, Dorina Mitrea, Marius Mitrea, Andrew J. Morris. Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets. Electronic Research Announcements, 2014, 21: 8-18. doi: 10.3934/era.2014.21.8 [9] Svetlana Pastukhova, Valeria Chiadò Piat. Homogenization of multivalued monotone operators with variable growth exponent. Networks & Heterogeneous Media, 2020, 15 (2) : 281-305. doi: 10.3934/nhm.2020013 [10] Bo Guan, Heming Jiao. The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 701-714. doi: 10.3934/dcds.2016.36.701 [11] Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569 [12] Elisa Sovrano. Ambrosetti-Prodi type result to a Neumann problem via a topological approach. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 345-355. doi: 10.3934/dcdss.2018019 [13] Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337 [14] Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061 [15] Hermann Brunner, Jingtang Ma. Abstract cascading multigrid preconditioners in Besov spaces. Communications on Pure & Applied Analysis, 2006, 5 (2) : 349-365. doi: 10.3934/cpaa.2006.5.349 [16] Zhan-Dong Mei, Jigen Peng, Yang Zhang. On general fractional abstract Cauchy problem. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2753-2772. doi: 10.3934/cpaa.2013.12.2753 [17] Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417 [18] Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5353-5378. doi: 10.3934/dcds.2015.35.5353 [19] Luisa Arlotti. Explicit transport semigroup associated to abstract boundary conditions. Conference Publications, 2011, 2011 (Special) : 102-111. doi: 10.3934/proc.2011.2011.102 [20] Francesca Papalini. Strongly nonlinear multivalued systems involving singular $\Phi$-Laplacian operators. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1025-1040. doi: 10.3934/cpaa.2010.9.1025

Impact Factor: