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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.   Google Scholar

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.   Google Scholar

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

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