June  2017, 37(6): 2977-2998. doi: 10.3934/dcds.2017128

An approximation solvability method for nonlocal semilinear differential problems in Banach spaces

1. 

Department of Mathematics and Computer Sciences, University of Perugia, Italy

2. 

Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh city, Viet Nam

3. 

Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh city, Viet Nam

4. 

Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, Italy

* Corresponding author: Nguyen Van Loi

Received  September 2016 Revised  January 2017 Published  February 2017

A new approximation solvability method is developed for the study of semilinear differential equations with nonlocal conditions without the compactness of the semigroup and of the nonlinearity. The method is based on the Yosida approximations of the generator of C0-semigroup, the continuation principle, and the weak topology. It is shown how the abstract result can be applied to study the reaction-diffusion models.

Citation: Irene Benedetti, Nguyen Van Loi, Valentina Taddei. An approximation solvability method for nonlocal semilinear differential problems in Banach spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2977-2998. doi: 10.3934/dcds.2017128
References:
[1]

F. Achleitner and C. Kuehn, On bounded positive stationary solutions for a nonlocal Fisher-KPP equation, Nonlin. Anal.: TMA, 112 (2015), 15-29.  doi: 10.1016/j.na.2014.09.004.

[2]

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

[3] J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer Academic Publishers, Dordrecht, 2003.  doi: 10.1007/978-94-017-0407-6.
[4]

R. B. Banks, Growth and Diffusion Phenomena: Mathematical Frameworks and Applications Texts in Applied Mathematics 14, Springer-Verlag, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-662-03052-3.

[5] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976. 
[6]

I. Benedetti, N. V. Loi, L. Malaguti and V. Obukhovskii, An approximation solvability method for nonlocal differential problems in Hilbert spaces Commun. Contemp. Math. (2016), 1650002, 34pp. doi: 10.1142/S0219199716500024.

[7]

I. BenedettiN. V. LoiL. Malaguti and V. Taddei, Nonlocal diffusion second order partial differential equations, J. Diff. Equ., 262 (2017), 1499-1523.  doi: 10.1016/j.jde.2016.10.019.

[8]

I. BenedettiL. Malaguti and V. Taddei, Nonlocal semilinear evolution equations without strong compactness: Theory and applications, Bound. Val. Probl., 2013 (2013), 18pp.  doi: 10.1186/1687-2770-2013-60.

[9]

I. BenedettiL. Malaguti and V. Taddei, Two-points b.v.p. for multivalued equations with weakly regular r.h.s., Nonlin. Anal.: TMA., 74 (2011), 3657-3670.  doi: 10.1016/j.na.2011.02.046.

[10]

I. BenedettiL. Malaguti and V. Taddei, Semilinear differential inclusions via weak topologies, J. Math. Anal. Appl., 368 (2010), 90-102.  doi: 10.1016/j.jmaa.2010.03.002.

[11]

H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.  doi: 10.1088/0951-7715/22/12/002.

[12]

S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly-valued functions, Ann. of Math. (2), 39 (1938), 913-944.  doi: 10.2307/1968472.

[13]

F. E. Browder and D. G. de Figueiredo, J-monotone nonlinear operators in Banach spaces, Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math., 28 (1966), 412-420. 

[14]

F. E. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Funct. Anal., 3 (1969), 217-245.  doi: 10.1016/0022-1236(69)90041-X.

[15]

L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.  doi: 10.1016/0022-247X(91)90164-U.

[16] R. Ekland, Convex Analysis and Variational Problems, North Holland, Amsterdam, 1979.  doi: 10.1137/1.9781611971088.
[17]

M. Furi and A. M. Pera, A continuation method on locally convex spaces and applications to ordinary differential equations on noncompact intervals, Ann. Polon. Math., 47 (1987), 331-346. 

[18]

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

[19]

I. Korobenko and E. Braverman, A logistic model with a carrying capacity driven diffusion, Canad. Appl. Math. Quarterly, 17 (2009), 85-104. 

[20]

N. Van Loi, Method of guiding functions for differential inclusions in a hilbert space, Differen. Urav. , 46 (2010), 1433{1443 (in Russian); [English Translation in: Diff. Equat. , 46 (2010), 1438{1447. doi: 10.1134/S0012266110100071.

[21]

Y. Mir and F. Dubeau, Linear and logistic models with time dependent coefficients, Elect. J. Diff. Equ., 18 (2016), 17pp-104. 

[22]

N. Papageorgiou, Existence of solutions for boundary value problems of semilinear evolution inclusions, Indian J. Pure Appl. Math, 23 (1992), 477-488. 

[23]

W. V. Petryshyn, Using degree theory for densely defined A-proper maps in the solvability of semilinear equations with unbounded and noninvertible linear part, Nonlin. Anal.: TMA., 4 (1980), 259-281.  doi: 10.1016/0362-546X(80)90053-X.

[24] L. Schwartz, Cours d'Analyse, 2, Hermann, Paris, 1981. 
[25]

I. Singer, Bases in Banach Spaces I Springer Verlag, Berlin, Heildelberg, New York, 1970.

[26]

I. I. Vrabie, C0-Semigroups and Applications North-Holland Mathematics Studies 191, North-Holland Publishing Co. , Amsterdam, 2003.

show all references

References:
[1]

F. Achleitner and C. Kuehn, On bounded positive stationary solutions for a nonlocal Fisher-KPP equation, Nonlin. Anal.: TMA, 112 (2015), 15-29.  doi: 10.1016/j.na.2014.09.004.

[2]

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

[3] J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer Academic Publishers, Dordrecht, 2003.  doi: 10.1007/978-94-017-0407-6.
[4]

R. B. Banks, Growth and Diffusion Phenomena: Mathematical Frameworks and Applications Texts in Applied Mathematics 14, Springer-Verlag, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-662-03052-3.

[5] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976. 
[6]

I. Benedetti, N. V. Loi, L. Malaguti and V. Obukhovskii, An approximation solvability method for nonlocal differential problems in Hilbert spaces Commun. Contemp. Math. (2016), 1650002, 34pp. doi: 10.1142/S0219199716500024.

[7]

I. BenedettiN. V. LoiL. Malaguti and V. Taddei, Nonlocal diffusion second order partial differential equations, J. Diff. Equ., 262 (2017), 1499-1523.  doi: 10.1016/j.jde.2016.10.019.

[8]

I. BenedettiL. Malaguti and V. Taddei, Nonlocal semilinear evolution equations without strong compactness: Theory and applications, Bound. Val. Probl., 2013 (2013), 18pp.  doi: 10.1186/1687-2770-2013-60.

[9]

I. BenedettiL. Malaguti and V. Taddei, Two-points b.v.p. for multivalued equations with weakly regular r.h.s., Nonlin. Anal.: TMA., 74 (2011), 3657-3670.  doi: 10.1016/j.na.2011.02.046.

[10]

I. BenedettiL. Malaguti and V. Taddei, Semilinear differential inclusions via weak topologies, J. Math. Anal. Appl., 368 (2010), 90-102.  doi: 10.1016/j.jmaa.2010.03.002.

[11]

H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.  doi: 10.1088/0951-7715/22/12/002.

[12]

S. Bochner and A. E. Taylor, Linear functionals on certain spaces of abstractly-valued functions, Ann. of Math. (2), 39 (1938), 913-944.  doi: 10.2307/1968472.

[13]

F. E. Browder and D. G. de Figueiredo, J-monotone nonlinear operators in Banach spaces, Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math., 28 (1966), 412-420. 

[14]

F. E. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Funct. Anal., 3 (1969), 217-245.  doi: 10.1016/0022-1236(69)90041-X.

[15]

L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.  doi: 10.1016/0022-247X(91)90164-U.

[16] R. Ekland, Convex Analysis and Variational Problems, North Holland, Amsterdam, 1979.  doi: 10.1137/1.9781611971088.
[17]

M. Furi and A. M. Pera, A continuation method on locally convex spaces and applications to ordinary differential equations on noncompact intervals, Ann. Polon. Math., 47 (1987), 331-346. 

[18]

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

[19]

I. Korobenko and E. Braverman, A logistic model with a carrying capacity driven diffusion, Canad. Appl. Math. Quarterly, 17 (2009), 85-104. 

[20]

N. Van Loi, Method of guiding functions for differential inclusions in a hilbert space, Differen. Urav. , 46 (2010), 1433{1443 (in Russian); [English Translation in: Diff. Equat. , 46 (2010), 1438{1447. doi: 10.1134/S0012266110100071.

[21]

Y. Mir and F. Dubeau, Linear and logistic models with time dependent coefficients, Elect. J. Diff. Equ., 18 (2016), 17pp-104. 

[22]

N. Papageorgiou, Existence of solutions for boundary value problems of semilinear evolution inclusions, Indian J. Pure Appl. Math, 23 (1992), 477-488. 

[23]

W. V. Petryshyn, Using degree theory for densely defined A-proper maps in the solvability of semilinear equations with unbounded and noninvertible linear part, Nonlin. Anal.: TMA., 4 (1980), 259-281.  doi: 10.1016/0362-546X(80)90053-X.

[24] L. Schwartz, Cours d'Analyse, 2, Hermann, Paris, 1981. 
[25]

I. Singer, Bases in Banach Spaces I Springer Verlag, Berlin, Heildelberg, New York, 1970.

[26]

I. I. Vrabie, C0-Semigroups and Applications North-Holland Mathematics Studies 191, North-Holland Publishing Co. , Amsterdam, 2003.

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