-
Previous Article
The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials
- CPAA Home
- This Issue
-
Next Article
Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow
A blowup alternative result for fractional nonautonomous evolution equation of Volterra type
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China |
In this article, we consider a class of fractional non-autonomous integro-differential evolution equation of Volterra type in a Banach space $E$, where the operators in linear part (possibly unbounded) depend on time $t$. Combining the theory of fractional calculus, operator semigroups and measure of noncompactness with Sadovskii's fixed point theorem, we firstly proved the local existence of mild solutions for corresponding fractional non-autonomous integro-differential evolution equation. Based on the local existence result and a piecewise extended method, we obtained a blowup alternative result for fractional non-autonomous integro-differential evolution equation of Volterra type. Finally, as a sample of application, these results are applied to a time fractional non-autonomous partial integro-differential equation of Volterra type with homogeneous Dirichlet boundary condition. This paper is a continuation of Heard and Rakin [
References:
| [1] |
R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033. Google Scholar |
| [2] |
E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph. D thesis, Department of Mathematics, Eindhoven University of Technology, 2001. Google Scholar |
| [3] |
J. |
| [4] |
P. M. Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980. Google Scholar |
| [5] |
P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744. Google Scholar |
| [6] |
P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728. Google Scholar |
| [7] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. Google Scholar |
| [8] |
M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal., 3 (2004), 197-211. Google Scholar |
| [9] |
M. M. El-Borai, K. E. El-Nadi and E. G. El-Akabawy, On some fractional evolution equations, Comput. Math. Appl., 59 (2010), 1352-1355. Google Scholar |
| [10] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, NY, USA, 1969. Google Scholar |
| [11] |
R. Gorenflo and F. Mainardi, Fractional calculus and stable probability distributions, Arch. Mech., 50 (1998), 377-388. Google Scholar |
| [12] |
H. Gou and B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 204-214. Google Scholar |
| [13] |
M. L. Heard and S. M. Rankin, A semi-linear parabolic integro-differential equation, J. Differential Equations, 71 (1988), 201-233. Google Scholar |
| [14] |
H. P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351-1371. Google Scholar |
| [15] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-verlag, New York, 1981. Google Scholar |
| [16] |
V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, New York, 1981. Google Scholar |
| [17] |
Y. Li, Existence of solutions of initial value problems for abstract semilinear evolution equations, Acta Math. Sin., 48 (2005), 1089-1094 (in Chinese). Google Scholar |
| [18] |
M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726. Google Scholar |
| [19] |
K. Li, J. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476-510. Google Scholar |
| [20] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006. Google Scholar |
| [21] |
Z. Mei, J. Peng and Y. Zhang, An operator theoretical approach to Riemann-Liouville fractional Cauchy problem, Math. Nachr., 288 (2015), 784-797. Google Scholar |
| [22] |
Z. Ouyang, Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl., 61 (2011), 860-870. Google Scholar |
| [23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-verlag, Berlin, 1983. Google Scholar |
| [24] |
M. H. M. Rashid and A. Al-Omari, Local and global existence of mild solutions for impulsive fractional semi-linear integro-differential equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3493-3503. Google Scholar |
| [25] |
H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997. Google Scholar |
| [26] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second ed., Springer-verlag, New York, 1997. Google Scholar |
| [27] |
R. N. Wang, D. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235. Google Scholar |
| [28] |
R. N. Wang, T. J. Xiao and J. Liang, A note on the fractional Cauchy problems with nonlocal conditions, Appl. Math. Lette., 24 (2011), 1435-1442. Google Scholar |
| [29] |
J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 12 (2011), 262-272. Google Scholar |
| [30] |
J. Wang, Y. Zhou and M. Fečkan, Abstract Cauchy problem for fractional differential equations, Nonlinear Dyn., 74 (2013), 685-700. Google Scholar |
| [31] |
Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077. Google Scholar |
| [32] |
B. Zhu, L. Liu and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79. Google Scholar |
and K. Goebel, Measures of Noncompactness in Banach Spaces, In Lecture Notes in Pure and Applied Mathematics, Volume 60, Marcel Dekker, New York, 1980.show all references
References:
| [1] |
R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109 (2010), 973-1033. Google Scholar |
| [2] |
E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph. D thesis, Department of Mathematics, Eindhoven University of Technology, 2001. Google Scholar |
| [3] |
J. |
| [4] |
P. M. Carvalho-Neto and G. Planas, Mild solutions to the time fractional Navier-Stokes equations in $\mathbb{R}^N$, J. Differential Equations, 259 (2015), 2948-2980. Google Scholar |
| [5] |
P. Chen and Y. Li, Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions, Results Math., 63 (2013), 731-744. Google Scholar |
| [6] |
P. Chen and Y. Li, Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions, Z. Angew. Math. Phys., 65 (2014), 711-728. Google Scholar |
| [7] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. Google Scholar |
| [8] |
M. M. El-Borai, The fundamental solutions for fractional evolution equations of parabolic type, J. Appl. Math. Stoch. Anal., 3 (2004), 197-211. Google Scholar |
| [9] |
M. M. El-Borai, K. E. El-Nadi and E. G. El-Akabawy, On some fractional evolution equations, Comput. Math. Appl., 59 (2010), 1352-1355. Google Scholar |
| [10] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, NY, USA, 1969. Google Scholar |
| [11] |
R. Gorenflo and F. Mainardi, Fractional calculus and stable probability distributions, Arch. Mech., 50 (1998), 377-388. Google Scholar |
| [12] |
H. Gou and B. Li, Local and global existence of mild solution to impulsive fractional semilinear integro-differential equation with noncompact semigroup, Commun. Nonlinear Sci. Numer. Simul., 42 (2017), 204-214. Google Scholar |
| [13] |
M. L. Heard and S. M. Rankin, A semi-linear parabolic integro-differential equation, J. Differential Equations, 71 (1988), 201-233. Google Scholar |
| [14] |
H. P. Heinz, On the behaviour of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal., 7 (1983), 1351-1371. Google Scholar |
| [15] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-verlag, New York, 1981. Google Scholar |
| [16] |
V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, New York, 1981. Google Scholar |
| [17] |
Y. Li, Existence of solutions of initial value problems for abstract semilinear evolution equations, Acta Math. Sin., 48 (2005), 1089-1094 (in Chinese). Google Scholar |
| [18] |
M. Li, C. Chen and F. B. Li, On fractional powers of generators of fractional resolvent families, J. Funct. Anal., 259 (2010), 2702-2726. Google Scholar |
| [19] |
K. Li, J. Peng and J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476-510. Google Scholar |
| [20] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science B. V., Amsterdam, 2006. Google Scholar |
| [21] |
Z. Mei, J. Peng and Y. Zhang, An operator theoretical approach to Riemann-Liouville fractional Cauchy problem, Math. Nachr., 288 (2015), 784-797. Google Scholar |
| [22] |
Z. Ouyang, Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl., 61 (2011), 860-870. Google Scholar |
| [23] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-verlag, Berlin, 1983. Google Scholar |
| [24] |
M. H. M. Rashid and A. Al-Omari, Local and global existence of mild solutions for impulsive fractional semi-linear integro-differential equation, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3493-3503. Google Scholar |
| [25] |
H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997. Google Scholar |
| [26] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, second ed., Springer-verlag, New York, 1997. Google Scholar |
| [27] |
R. N. Wang, D. H. Chen and T. J. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, J. Differential Equations, 252 (2012), 202-235. Google Scholar |
| [28] |
R. N. Wang, T. J. Xiao and J. Liang, A note on the fractional Cauchy problems with nonlocal conditions, Appl. Math. Lette., 24 (2011), 1435-1442. Google Scholar |
| [29] |
J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl., 12 (2011), 262-272. Google Scholar |
| [30] |
J. Wang, Y. Zhou and M. Fečkan, Abstract Cauchy problem for fractional differential equations, Nonlinear Dyn., 74 (2013), 685-700. Google Scholar |
| [31] |
Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl., 59 (2010), 1063-1077. Google Scholar |
| [32] |
B. Zhu, L. Liu and Y. Wu, Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79. Google Scholar |
| [1] |
Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 |
| [2] |
Miloud Moussai. Application of the bernstein polynomials for solving the nonlinear fractional type Volterra integro-differential equation with caputo fractional derivatives. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021021 |
| [3] |
Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065 |
| [4] |
Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020100 |
| [5] |
Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261 |
| [6] |
Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021030 |
| [7] |
Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053 |
| [8] |
K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038 |
| [9] |
Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064 |
| [10] |
Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020044 |
| [11] |
Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053 |
| [12] |
Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217 |
| [13] |
Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537 |
| [14] |
Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129 |
| [15] |
Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907 |
| [16] |
Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57 |
| [17] |
Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid, Fudziah Ismail. Numerical solutions of Volterra integro-differential equations using General Linear Method. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 433-444. doi: 10.3934/naco.2019042 |
| [18] |
Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022 |
| [19] |
Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249 |
| [20] |
Mahesh G. Nerurkar. Spectral and stability questions concerning evolution of non-autonomous linear systems. Conference Publications, 2001, 2001 (Special) : 270-275. doi: 10.3934/proc.2001.2001.270 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]