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July  2020, 13(7): 1899-1919. doi: 10.3934/dcdss.2020149

Evolution fractional differential problems with impulses and nonlocal conditions

Irene Benedetti 1,, , Valeri Obukhovskii 2, and  Valentina Taddei 3,

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, I-06123 Perugia, Italy
2. 

Faculty of Physics and Mathematics, Voronezh State Pedagogical University, 394043 Voronezh, Russia
3. 

Dipartimento di Scienze e Metodi dell'Ingegneria, Università di Modena e Reggio Emilia, I-42122 Reggio Emilia, Italy

* Corresponding author: Irene Benedetti

Dedicated to Professor Patrizia Pucci on the occasion of her 65th birthday

Received  September 2018 Revised  October 2018 Published  November 2019

We obtain existence results for mild solutions of a fractional differential inclusion subjected to impulses and nonlocal initial conditions. By means of a technique based on the weak topology in connection with the Glicksberg-Ky Fan Fixed Point Theorem we are able to avoid any hypotheses of compactness on the semigroup and on the nonlinear term and at the same time we do not need to assume hypotheses of monotonicity or Lipschitz regularity neither on the nonlinear term, nor on the impulse functions, nor on the nonlocal condition. An application to a fractional diffusion process complete the discussion of the studied problem. 200 words.

Keywords: Caputo fractional derivative, impulse functions, nonlocal initial conditions, evolution inclusions in abstract spaces.
Mathematics Subject Classification: Primary: 34A08, 34A37, 34B10; Secondary: 34G25.
Citation: Irene Benedetti, Valeri Obukhovskii, Valentina Taddei. Evolution fractional differential problems with impulses and nonlocal conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1899-1919. doi: 10.3934/dcdss.2020149
References:
[1]

K. Aissani and M. Benchohra, Impulsive fractional differential inclusions with infinite delay, Electronic Journal of Differential Equations, 2013 (2013), 13 pp.

[2]

K. Balachandran, S. Kiruthika and J. J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun Nonlinear Sci Numer Simulat, 16 (2011), 1970-1977.  doi: 10.1016/j.cnsns.2010.08.005.

[3]

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

[4]

I. Benedetti, V. Obukovskii and V. Taddei, On noncompact fractional order differential inclusions with generalized boundary condition and impulses in a Banach space, Journal of Function Spaces, (2015), Art. ID 651359, 10 pp. doi: 10.1155/2015/651359.

[5]

I. Benedetti, V. Obukovskii and V. Taddei, On generalized boundary value problems for a class of fractional differential inclusions, Fractional Calculus and Applied Analysis, 20 (2017), 1424-1446.  doi: 10.1515/fca-2017-0075.

[6]

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

[7]

A. Chadha and D. N. Pandey, Existence of a mild solution for impulsive neutral fractional differential equations with nonlocal conditions, Differential Equations and Applications, 7 (2015), 151-168.  doi: 10.7153/dea-07-09.

[8]

A. Chauhan and J. Dabas, Existence of mild solutions for impulsive fractional-order semilinear evolution equations with nonlocal conditions, Electronic Journal of Differential Equations, 2011 (2011), 10 pp.

[9]

N. Dunford and J. T. Schwartz, Linear Operators. Part I. General theory, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988.

[10]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.

[11]

I. Ekeland and R. Teman, Convex Anaysis and Variational Problems, Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., Inc., New York, 1976.

[12]

H. Ergören and A. Kiliçman, Non-local boundary value problems for impulsive fractional integro-differential equations in Banach spaces, Bound. Value Probl., 2012 (2012), 15 pp. doi: 10.1186/1687-2770-2012-145.

[13]

K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.

[14]

F.-D. Ge, H.-C. Zhou and C.-H. Kou, Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique, Applied Mathematics and Computation, 275 (2016), 107-120.  doi: 10.1016/j.amc.2015.11.056.

[15]

I. L. Glicksberg, A further generalization of the Kakutani fixed theorem with application to Nash equilibrium points, Proc. Amer. Math. Soc., 3 (1952), 170-174.  doi: 10.2307/2032478.

[16]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications, 7. Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110870893.

[17]

L. V. Kantorovich and G. P. Akilov, Functional Analysis, Second edition, Pergamon Press, Oxford-Elmsford, N.Y., 1982.

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[19]

A. N. Kochubeĭ, Diffusion of fractional order, Differential Equations, 26 (1990), 485-492. 

[20]

A. N. Kochubeĭ, The Cauchy problem for evolution equations of fractional order, Differential Equations, 25 (1989), 967-974. 

[21]

S. Q. Liang and R. Mei, Existence of mild solutions for fractional impulsive neutral evolution equations with nonlocal conditions, Advances in Difference Equations, 2014 (2014), 16 pp. doi: 10.1186/1687-1847-2014-101.

[22]

F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solutions and Fractals, 7 (1996), 1461-1477.  doi: 10.1016/0960-0779(95)00125-5.

[23]

K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.

[24]

J. Mu, Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions, Bound. Value Probl., 2012 (2012), 12 pp. doi: 10.1186/1687-2770-2012-71.

[25]

R. R. Nigmatullin, Fractional integral and its physical interpretation, Theoretical and Mathematical Physics, 90 Issue 3 (1992), 242–251.

[26]

I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering 198, Academic Press, Inc., San Diego, CA 1999.

[27]

L. Schwartz, Cours d'Analyse I, Second Edition, Hermann, Paris 1981.

[28]

X.-B. Shua, Y. Z. Lai and Y. M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Analysis, 74 (2011), 2003-2011.  doi: 10.1016/j.na.2010.11.007.

[29]

N. K. Tomar and J. Dabas, Controllability of impulsive fractional order semilinear evolution equations with nonlocal conditions, Journal of Nonlinear Evolution Equations and Applications, 2012 (2012), 57-67. 

[30]

M. Väth, Ideal Spaces, Lect. Notes Math., no. 1664, Springer, Berlin, Heidelberg, 1997.

[31]

I. I. Vrabie, $C_0$-Semigroups and Applications, North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam, 2003.

[32]

J. R. Wang, M. Fečkan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynamics of PDE, 8 (2011), 345-361.  doi: 10.4310/DPDE.2011.v8.n4.a3.

[33]

J. R. Wang and A. G. Ibrahim, Existence and controllability results for nonlocal fractional impulsive differential inclusions in Banach spaces, Journal of Function Spaces and Applications, 2013 (2013), Art. ID 518306, 16 pp. doi: 10.1155/2013/518306.

[34]

L. Z. Zhang and Y. Liang, Monotone iterative technique for impulsive fractional evolution equations with noncompact semigroup, Advances in Difference Equations, 2015 (2015), 15 pp. doi: 10.1186/s13662-015-0665-6.

[35]

Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier/Academic Press, London, 2016. doi: doi.

show all references

References:
[1]

K. Aissani and M. Benchohra, Impulsive fractional differential inclusions with infinite delay, Electronic Journal of Differential Equations, 2013 (2013), 13 pp.

[2]

K. Balachandran, S. Kiruthika and J. J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun Nonlinear Sci Numer Simulat, 16 (2011), 1970-1977.  doi: 10.1016/j.cnsns.2010.08.005.

[3]

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

[4]

I. Benedetti, V. Obukovskii and V. Taddei, On noncompact fractional order differential inclusions with generalized boundary condition and impulses in a Banach space, Journal of Function Spaces, (2015), Art. ID 651359, 10 pp. doi: 10.1155/2015/651359.

[5]

I. Benedetti, V. Obukovskii and V. Taddei, On generalized boundary value problems for a class of fractional differential inclusions, Fractional Calculus and Applied Analysis, 20 (2017), 1424-1446.  doi: 10.1515/fca-2017-0075.

[6]

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

[7]

A. Chadha and D. N. Pandey, Existence of a mild solution for impulsive neutral fractional differential equations with nonlocal conditions, Differential Equations and Applications, 7 (2015), 151-168.  doi: 10.7153/dea-07-09.

[8]

A. Chauhan and J. Dabas, Existence of mild solutions for impulsive fractional-order semilinear evolution equations with nonlocal conditions, Electronic Journal of Differential Equations, 2011 (2011), 10 pp.

[9]

N. Dunford and J. T. Schwartz, Linear Operators. Part I. General theory, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1988.

[10]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.

[11]

I. Ekeland and R. Teman, Convex Anaysis and Variational Problems, Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., Inc., New York, 1976.

[12]

H. Ergören and A. Kiliçman, Non-local boundary value problems for impulsive fractional integro-differential equations in Banach spaces, Bound. Value Probl., 2012 (2012), 15 pp. doi: 10.1186/1687-2770-2012-145.

[13]

K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 121-126.  doi: 10.1073/pnas.38.2.121.

[14]

F.-D. Ge, H.-C. Zhou and C.-H. Kou, Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique, Applied Mathematics and Computation, 275 (2016), 107-120.  doi: 10.1016/j.amc.2015.11.056.

[15]

I. L. Glicksberg, A further generalization of the Kakutani fixed theorem with application to Nash equilibrium points, Proc. Amer. Math. Soc., 3 (1952), 170-174.  doi: 10.2307/2032478.

[16]

M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, De Gruyter Series in Nonlinear Analysis and Applications, 7. Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110870893.

[17]

L. V. Kantorovich and G. P. Akilov, Functional Analysis, Second edition, Pergamon Press, Oxford-Elmsford, N.Y., 1982.

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[19]

A. N. Kochubeĭ, Diffusion of fractional order, Differential Equations, 26 (1990), 485-492. 

[20]

A. N. Kochubeĭ, The Cauchy problem for evolution equations of fractional order, Differential Equations, 25 (1989), 967-974. 

[21]

S. Q. Liang and R. Mei, Existence of mild solutions for fractional impulsive neutral evolution equations with nonlocal conditions, Advances in Difference Equations, 2014 (2014), 16 pp. doi: 10.1186/1687-1847-2014-101.

[22]

F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos Solutions and Fractals, 7 (1996), 1461-1477.  doi: 10.1016/0960-0779(95)00125-5.

[23]

K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.

[24]

J. Mu, Extremal mild solutions for impulsive fractional evolution equations with nonlocal initial conditions, Bound. Value Probl., 2012 (2012), 12 pp. doi: 10.1186/1687-2770-2012-71.

[25]

R. R. Nigmatullin, Fractional integral and its physical interpretation, Theoretical and Mathematical Physics, 90 Issue 3 (1992), 242–251.

[26]

I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering 198, Academic Press, Inc., San Diego, CA 1999.

[27]

L. Schwartz, Cours d'Analyse I, Second Edition, Hermann, Paris 1981.

[28]

X.-B. Shua, Y. Z. Lai and Y. M. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Analysis, 74 (2011), 2003-2011.  doi: 10.1016/j.na.2010.11.007.

[29]

N. K. Tomar and J. Dabas, Controllability of impulsive fractional order semilinear evolution equations with nonlocal conditions, Journal of Nonlinear Evolution Equations and Applications, 2012 (2012), 57-67. 

[30]

M. Väth, Ideal Spaces, Lect. Notes Math., no. 1664, Springer, Berlin, Heidelberg, 1997.

[31]

I. I. Vrabie, $C_0$-Semigroups and Applications, North-Holland Mathematics Studies, 191. North-Holland Publishing Co., Amsterdam, 2003.

[32]

J. R. Wang, M. Fečkan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynamics of PDE, 8 (2011), 345-361.  doi: 10.4310/DPDE.2011.v8.n4.a3.

[33]

J. R. Wang and A. G. Ibrahim, Existence and controllability results for nonlocal fractional impulsive differential inclusions in Banach spaces, Journal of Function Spaces and Applications, 2013 (2013), Art. ID 518306, 16 pp. doi: 10.1155/2013/518306.

[34]

L. Z. Zhang and Y. Liang, Monotone iterative technique for impulsive fractional evolution equations with noncompact semigroup, Advances in Difference Equations, 2015 (2015), 15 pp. doi: 10.1186/s13662-015-0665-6.

[35]

Y. Zhou, Fractional Evolution Equations and Inclusions: Analysis and Control, Elsevier/Academic Press, London, 2016. doi: doi.

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