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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

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

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

[26]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

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

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

[26]

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

[1]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101

[2]

Changfeng Gui, Zhenbu Zhang. Spike solutions to a nonlocal differential equation. Communications on Pure & Applied Analysis, 2006, 5 (1) : 85-95. doi: 10.3934/cpaa.2006.5.85

[3]

J. García-Melián, Julio D. Rossi. A logistic equation with refuge and nonlocal diffusion. Communications on Pure & Applied Analysis, 2009, 8 (6) : 2037-2053. doi: 10.3934/cpaa.2009.8.2037

[4]

Moritz Kassmann, Tadele Mengesha, James Scott. Solvability of nonlocal systems related to peridynamics. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1303-1332. doi: 10.3934/cpaa.2019063

[5]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095

[6]

Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391

[7]

Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581

[8]

Liviu I. Ignat, Ademir F. Pazoto. Large time behaviour for a nonlocal diffusion - convection equation related with gas dynamics. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3575-3589. doi: 10.3934/dcds.2014.34.3575

[9]

Bhargav Kumar Kakumani, Suman Kumar Tumuluri. Asymptotic behavior of the solution of a diffusion equation with nonlocal boundary conditions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 407-419. doi: 10.3934/dcdsb.2017019

[10]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39

[11]

Giovany M. Figueiredo, Tarcyana S. Figueiredo-Sousa, Cristian Morales-Rodrigo, Antonio Suárez. Existence of positive solutions of an elliptic equation with local and nonlocal variable diffusion coefficient. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3689-3711. doi: 10.3934/dcdsb.2018311

[12]

Luis Caffarelli, Serena Dipierro, Enrico Valdinoci. A logistic equation with nonlocal interactions. Kinetic & Related Models, 2017, 10 (1) : 141-170. doi: 10.3934/krm.2017006

[13]

Dong Li, Xiaoyi Zhang. On a nonlocal aggregation model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 301-323. doi: 10.3934/dcds.2010.27.301

[14]

Armel Ovono Andami. From local to nonlocal in a diffusion model. Conference Publications, 2011, 2011 (Special) : 54-60. doi: 10.3934/proc.2011.2011.54

[15]

Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 693-719. doi: 10.3934/nhm.2016014

[16]

Mousomi Bhakta, Debangana Mukherjee. Semilinear nonlocal elliptic equations with critical and supercritical exponents. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1741-1766. doi: 10.3934/cpaa.2017085

[17]

Marco Squassina. Preface: Recent progresses in the theory of nonlinear nonlocal problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : i-i. doi: 10.3934/dcdss.201803i

[18]

Vitali Vougalter, Vitaly Volpert. On the solvability conditions for the diffusion equation with convection terms. Communications on Pure & Applied Analysis, 2012, 11 (1) : 365-373. doi: 10.3934/cpaa.2012.11.365

[19]

C. Brändle, E. Chasseigne, Raúl Ferreira. Unbounded solutions of the nonlocal heat equation. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1663-1686. doi: 10.3934/cpaa.2011.10.1663

[20]

Irene Benedetti, Valeri Obukhovskii, Valentina Taddei. Evolution fractional differential problems with impulses and nonlocal conditions. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020149

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (17)
  • HTML views (7)
  • Cited by (0)

[Back to Top]