January  2018, 23(1): 295-329. doi: 10.3934/dcdsb.2018021

The Krasnosel'skii formula for parabolic differential inclusions with state constraints

1. 

Institute of Mathematics, Technical University of Łódź, Poland

2. 

Faculty of Mathematics and Computer Sciences, Nicolaus Copernicus University, Poland

* The first author was partially supported by the Polish National Science Center under grant 2013/09/B/ST1/01963

Received  December 2016 Published  January 2018

We consider a constrained semilinear evolution inclusion of parabolic type involving an $m$-dissipative linear operator and a source term of multivalued type in a Banach space and topological properties of the solution map. We establish the $R_δ$-description of the set of solutions surviving in the constraining area and show a relation between the fixed point index of the Krasnosel'skii-Poincaré operator of translation along trajectories associated with the problem and the appropriately defined constrained degree of the right-hand side in the equation. This provides topological tools appropriate to obtain results on the existence of periodic solutions to studied differential problems.

Citation: Wojciech Kryszewski, Dorota Gabor, Jakub Siemianowski. The Krasnosel'skii formula for parabolic differential inclusions with state constraints. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 295-329. doi: 10.3934/dcdsb.2018021
References:
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S. AizicoviciN. Papageorgiu and V. Staicu, Periodic solutions of nonlinear evolution inclusions in Banach spaces, J. Nonlinear Convex Anal., 7 (2006), 163-177. 

[2]

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapova, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness And Condensing Operators, Basel–Berlin–Boston, Birkhäuser Verlag, 1992. doi: 10.1007/978-3-0348-5727-7.

[3]

Ch. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, A Hitchhiker's Guide 3rd edition, Springer-Verlag, Berlin 2006.

[4]

W. Arendt and A. F. M. ter Elst, From forms to semigroups, in Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, (eds. W. Arendt et al. ), Birkhäuser/Springer Basel AG, Basel, 221 (2012), 47–69. doi: 10.1007/978-3-0348-0297-0_4.

[5]

J. -P. Aubin, A. M. Bayen and P. Sait-Pierre, Viability Theory: New Directions 2nd edition, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16684-6.

[6]

J. -P. Aubin and I. Ekeland, Applied Nonlinear Analysis John Wiley & Sons, Inc., New York, 1984.

[7]

J. -P. Aubin and H. Frankowska, Set-valued Analysis Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.

[8]

D. Averna, Lusin type theorems for multifunctions, Scorza Dragoni's property and Carathéodory selections, Boll. Un. Mat. Ital. A(7), 8 (1994), 193-202. 

[9]

R. Bader, On the semilinear multi-valued flow under constraints and the periodic problem, Comment. Math. Univ. Carolin., 41 (2000), 719-734. 

[10]

R. Bader and W. Kryszewski, On the solution sets of differential inclusions and the periodic problem in Banach spaces, Nonlinear Anal., 54 (2003), 707-754.  doi: 10.1016/S0362-546X(03)00098-1.

[11]

Cz. Bessaga and A. Pe lczyński, Selected Topics in Infinite-Dimensional Topology, PWN— Polish Scientific Publishers, Warsaw, 1975.

[12]

D. Bothe, Flow invariance for perturbed nonlinear evolution equations, Abstr. Appl. Anal., 1 (1996), 417-433.  doi: 10.1155/S1085337596000231.

[13]

D. Bothe, Periodic solutions of a nonlinear evolution problem from heterogeneous catalysis, Differential Integral Equations, 14 (2001), 641-670. 

[14]

D. Bothe, Nonlinear Evolutions in Banach Spaces – Existence and Qualitive Theory with Applications to Reaction-Diffusion Systems, Habilitationsschrift, Techische Universität Darmstadt, 1999, http://www.mma.tu-darmstadt.de/media/mma/bilderdateien_5/publication_mma/habilschrift.pdf.

[15]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011.

[16]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[17]

O. Cârjă, M. Necula and I. I. Vrabie, Viability, Invariance and Applications Elsevier Science B. V., Amsterdam, 2007.

[18] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, The Clarendon Press, Oxford University Press, New York, 1998. 
[19]

D.-H. ChenR.-N. Wang and Y. Zhou, Nonlinear evolution inclusions: Topological characterizations of solution sets and applications, J. Funct. Anal., 265 (2013), 2039-2073.  doi: 10.1016/j.jfa.2013.05.033.

[20]

Y. Chen, Anti-periodic solutions for semilinear evolution equations, J. Math. Anal. Appl., 315 (2006), 337-348.  doi: 10.1016/j.jmaa.2005.08.001.

[21]

A. Ćwiszewski, Topological degree methods for perturbations of operators generating compact $C_0$ semigroups, J. Differential Equations, 220 (2006), 434-477.  doi: 10.1016/j.jde.2005.04.007.

[22]

A. Ćwiszewski, Positive periodic solutions of parabolic evolutions problems: a translation along trajectories approach, Cent. Eur. J. Math., 9 (2011), 244-268.  doi: 10.2478/s11533-011-0010-6.

[23]

A. Ćwiszewski and P. Kokocki, Krasnosel'skii type formula and translation along trajectories method for evolution equations, Discrete Contin. Dyn. Syst., 22 (2008), 605-628.  doi: 10.3934/dcds.2008.22.605.

[24]

A. Ćwiszewski and W. Kryszewski, Constrained topological degree and positive solutions of fully nonlinear boundary value problems, J. Differential Equations, 247 (2009), 2235-2269.  doi: 10.1016/j.jde.2009.06.025.

[25]

N. Dancer, Degree theory on convex sets and applications to bifurcation, in Calculus of Variations and Partial Differential Equations (Pisa, 1996), (eds. G. Buttazzo. A. Marino and M. K. V. Murthy), Springer, Berlin, (2000), 185-225. doi: 10.1007/978-3-642-57186-2_8.

[26]

J. DiestelW. M. Ruess and W. Schachermayer, On weak compactness in $L^1(μ, X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453.  doi: 10.2307/2160321.

[27]

A. Djebali, L. Górniewicz and A. Ouahab, Solution Sets for Differential Equations and Inclusions Walter de Gruyter & Co., Berlin, 2013. doi: 10.1515/9783110293562.

[28]

K. -J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations Springer-Verlag, New York, 2000.

[29]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[30]

D. Gabor and W. Kryszewski, A global bifurcation index for set-valued perturbations of Fredholm operator, Nonlinear Anal., 73 (2010), 2714-2736.  doi: 10.1016/j.na.2010.06.055.

[31]

L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings Springer, Dordrecht, 2006.

[32]

L. Górniewicz, Topological structure of solution sets: Current results, Arch. Math. (Brno), 36 (2000), 343-382. 

[33]

A. Granas and J. Dugundji, Fixed Point Theory Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.

[34]

N. Hirano and N. Shioji, Invariant sets for nonlinear evolution equations, {C}auchy problems and periodic problems, Abstr. Appl. Anal., 3 (2004), 183-203.  doi: 10.1155/S1085337504311073.

[35]

S. T. Hu and N. S. Papageorgiou, Handbook of multivalued analysis. Vol. I Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.

[36]

S. T. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4615-4665-8_17.

[37]

D. M. Hyman, On decreasing sequences of compact absolute retracts, Fund. Math., 64 (1969), 91-97.  doi: 10.4064/fm-64-1-91-97.

[38]

M. JuniewiczH. T. Nguyen and J. Ziemińska, Carathéodory CM-selectors for oppositely semicontinuous multifunctions of two variables, Bull. Pol. Acad. Sci. Math., 50 (2002), 47-57. 

[39]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110870893.

[40]

A. Kanigowski and W. Kryszewski, Perron-Frobenius and Krein-Rutman theorem for tangentially positive operators, Centr. Eur. J. Math., 10 (2012), 2240-2263.  doi: 10.2478/s11533-012-0118-3.

[41]

M. A. Krasnosel'skiǐ and P. P. Zabreǐko, Geometrical Methods of Nonlinear Analysis Springer-Verlag, Berlin, 1984.

[42]

W. Kryszewski and J. Siemianowski, The Bolzano mean-value theorem and partial differential equations, J. Math. Anal. Appl., 457 (2018), 1452-1477.  doi: 10.1016/j.jmaa.2017.01.040.

[43]

A. Kucia, Scorza Dragoni type theorems, Fund. Math., 138 (1991), 197-203.  doi: 10.4064/fm-138-3-197-203.

[44]

R. C. Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc., 83 (1977), 495-552.  doi: 10.1090/S0002-9904-1977-14321-8.

[45]

V. Obukhovski, P. P. Zecca, N. Van Loi and S. Kornev, Method of Guiding Functions in Problems of Nonlinear Analysis Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-37070-0.

[46]

N. H. Pavel, Invariant sets for a class of semi-linear equations of evolution, Nonlinear Anal., 1 (1977), 187-196.  doi: 10.1016/0362-546X(77)90009-8.

[47]

N. H. Pavel, Differential Equations, Flow Invariance and Applications, Research Notes in Mathematics, 113. Boston–London–Melbourne: Pitman Advanced Publishing Program, 1984.

[48]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[49]

J. Prüss, Periodic solutions of semilinear evolution equations., Nonlinear Anal., 3 (1979), 601-612.  doi: 10.1016/0362-546X(79)90089-0.

[50]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations American Mathematical Society, Providence, RI, 1997.

[51]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions Longman Scientific & Technical, Harlow, 1995.

[52]

I. I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc., 109 (1990), 653-661.  doi: 10.1090/S0002-9939-1990-1015686-4.

show all references

References:
[1]

S. AizicoviciN. Papageorgiu and V. Staicu, Periodic solutions of nonlinear evolution inclusions in Banach spaces, J. Nonlinear Convex Anal., 7 (2006), 163-177. 

[2]

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapova, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness And Condensing Operators, Basel–Berlin–Boston, Birkhäuser Verlag, 1992. doi: 10.1007/978-3-0348-5727-7.

[3]

Ch. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, A Hitchhiker's Guide 3rd edition, Springer-Verlag, Berlin 2006.

[4]

W. Arendt and A. F. M. ter Elst, From forms to semigroups, in Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, (eds. W. Arendt et al. ), Birkhäuser/Springer Basel AG, Basel, 221 (2012), 47–69. doi: 10.1007/978-3-0348-0297-0_4.

[5]

J. -P. Aubin, A. M. Bayen and P. Sait-Pierre, Viability Theory: New Directions 2nd edition, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16684-6.

[6]

J. -P. Aubin and I. Ekeland, Applied Nonlinear Analysis John Wiley & Sons, Inc., New York, 1984.

[7]

J. -P. Aubin and H. Frankowska, Set-valued Analysis Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.

[8]

D. Averna, Lusin type theorems for multifunctions, Scorza Dragoni's property and Carathéodory selections, Boll. Un. Mat. Ital. A(7), 8 (1994), 193-202. 

[9]

R. Bader, On the semilinear multi-valued flow under constraints and the periodic problem, Comment. Math. Univ. Carolin., 41 (2000), 719-734. 

[10]

R. Bader and W. Kryszewski, On the solution sets of differential inclusions and the periodic problem in Banach spaces, Nonlinear Anal., 54 (2003), 707-754.  doi: 10.1016/S0362-546X(03)00098-1.

[11]

Cz. Bessaga and A. Pe lczyński, Selected Topics in Infinite-Dimensional Topology, PWN— Polish Scientific Publishers, Warsaw, 1975.

[12]

D. Bothe, Flow invariance for perturbed nonlinear evolution equations, Abstr. Appl. Anal., 1 (1996), 417-433.  doi: 10.1155/S1085337596000231.

[13]

D. Bothe, Periodic solutions of a nonlinear evolution problem from heterogeneous catalysis, Differential Integral Equations, 14 (2001), 641-670. 

[14]

D. Bothe, Nonlinear Evolutions in Banach Spaces – Existence and Qualitive Theory with Applications to Reaction-Diffusion Systems, Habilitationsschrift, Techische Universität Darmstadt, 1999, http://www.mma.tu-darmstadt.de/media/mma/bilderdateien_5/publication_mma/habilschrift.pdf.

[15]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations Springer, New York, 2011.

[16]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions Dans Les Espaces de Hilbert North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.

[17]

O. Cârjă, M. Necula and I. I. Vrabie, Viability, Invariance and Applications Elsevier Science B. V., Amsterdam, 2007.

[18] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, The Clarendon Press, Oxford University Press, New York, 1998. 
[19]

D.-H. ChenR.-N. Wang and Y. Zhou, Nonlinear evolution inclusions: Topological characterizations of solution sets and applications, J. Funct. Anal., 265 (2013), 2039-2073.  doi: 10.1016/j.jfa.2013.05.033.

[20]

Y. Chen, Anti-periodic solutions for semilinear evolution equations, J. Math. Anal. Appl., 315 (2006), 337-348.  doi: 10.1016/j.jmaa.2005.08.001.

[21]

A. Ćwiszewski, Topological degree methods for perturbations of operators generating compact $C_0$ semigroups, J. Differential Equations, 220 (2006), 434-477.  doi: 10.1016/j.jde.2005.04.007.

[22]

A. Ćwiszewski, Positive periodic solutions of parabolic evolutions problems: a translation along trajectories approach, Cent. Eur. J. Math., 9 (2011), 244-268.  doi: 10.2478/s11533-011-0010-6.

[23]

A. Ćwiszewski and P. Kokocki, Krasnosel'skii type formula and translation along trajectories method for evolution equations, Discrete Contin. Dyn. Syst., 22 (2008), 605-628.  doi: 10.3934/dcds.2008.22.605.

[24]

A. Ćwiszewski and W. Kryszewski, Constrained topological degree and positive solutions of fully nonlinear boundary value problems, J. Differential Equations, 247 (2009), 2235-2269.  doi: 10.1016/j.jde.2009.06.025.

[25]

N. Dancer, Degree theory on convex sets and applications to bifurcation, in Calculus of Variations and Partial Differential Equations (Pisa, 1996), (eds. G. Buttazzo. A. Marino and M. K. V. Murthy), Springer, Berlin, (2000), 185-225. doi: 10.1007/978-3-642-57186-2_8.

[26]

J. DiestelW. M. Ruess and W. Schachermayer, On weak compactness in $L^1(μ, X)$, Proc. Amer. Math. Soc., 118 (1993), 447-453.  doi: 10.2307/2160321.

[27]

A. Djebali, L. Górniewicz and A. Ouahab, Solution Sets for Differential Equations and Inclusions Walter de Gruyter & Co., Berlin, 2013. doi: 10.1515/9783110293562.

[28]

K. -J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations Springer-Verlag, New York, 2000.

[29]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.

[30]

D. Gabor and W. Kryszewski, A global bifurcation index for set-valued perturbations of Fredholm operator, Nonlinear Anal., 73 (2010), 2714-2736.  doi: 10.1016/j.na.2010.06.055.

[31]

L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings Springer, Dordrecht, 2006.

[32]

L. Górniewicz, Topological structure of solution sets: Current results, Arch. Math. (Brno), 36 (2000), 343-382. 

[33]

A. Granas and J. Dugundji, Fixed Point Theory Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.

[34]

N. Hirano and N. Shioji, Invariant sets for nonlinear evolution equations, {C}auchy problems and periodic problems, Abstr. Appl. Anal., 3 (2004), 183-203.  doi: 10.1155/S1085337504311073.

[35]

S. T. Hu and N. S. Papageorgiou, Handbook of multivalued analysis. Vol. I Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.

[36]

S. T. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. II Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4615-4665-8_17.

[37]

D. M. Hyman, On decreasing sequences of compact absolute retracts, Fund. Math., 64 (1969), 91-97.  doi: 10.4064/fm-64-1-91-97.

[38]

M. JuniewiczH. T. Nguyen and J. Ziemińska, Carathéodory CM-selectors for oppositely semicontinuous multifunctions of two variables, Bull. Pol. Acad. Sci. Math., 50 (2002), 47-57. 

[39]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110870893.

[40]

A. Kanigowski and W. Kryszewski, Perron-Frobenius and Krein-Rutman theorem for tangentially positive operators, Centr. Eur. J. Math., 10 (2012), 2240-2263.  doi: 10.2478/s11533-012-0118-3.

[41]

M. A. Krasnosel'skiǐ and P. P. Zabreǐko, Geometrical Methods of Nonlinear Analysis Springer-Verlag, Berlin, 1984.

[42]

W. Kryszewski and J. Siemianowski, The Bolzano mean-value theorem and partial differential equations, J. Math. Anal. Appl., 457 (2018), 1452-1477.  doi: 10.1016/j.jmaa.2017.01.040.

[43]

A. Kucia, Scorza Dragoni type theorems, Fund. Math., 138 (1991), 197-203.  doi: 10.4064/fm-138-3-197-203.

[44]

R. C. Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc., 83 (1977), 495-552.  doi: 10.1090/S0002-9904-1977-14321-8.

[45]

V. Obukhovski, P. P. Zecca, N. Van Loi and S. Kornev, Method of Guiding Functions in Problems of Nonlinear Analysis Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-37070-0.

[46]

N. H. Pavel, Invariant sets for a class of semi-linear equations of evolution, Nonlinear Anal., 1 (1977), 187-196.  doi: 10.1016/0362-546X(77)90009-8.

[47]

N. H. Pavel, Differential Equations, Flow Invariance and Applications, Research Notes in Mathematics, 113. Boston–London–Melbourne: Pitman Advanced Publishing Program, 1984.

[48]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[49]

J. Prüss, Periodic solutions of semilinear evolution equations., Nonlinear Anal., 3 (1979), 601-612.  doi: 10.1016/0362-546X(79)90089-0.

[50]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations American Mathematical Society, Providence, RI, 1997.

[51]

I. I. Vrabie, Compactness Methods for Nonlinear Evolutions Longman Scientific & Technical, Harlow, 1995.

[52]

I. I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc., 109 (1990), 653-661.  doi: 10.1090/S0002-9939-1990-1015686-4.

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