Existence of mild solutions for fractional non-instantaneous impulsive integro-differential equations with nonlocal conditions. (English) Zbl 1488.34405
Summary: This paper is concerned with the existence of mild solutions for a class of fractional semilinear integro-differential equations having non-instantaneous impulses. The result is obtained by using noncompact semigroup theory and fixed point theorem. The obtained result is illustrated by an example at the end.
MSC:
| 34K30 | Functional-differential equations in abstract spaces |
| 45J05 | Integro-ordinary differential equations |
| 34K45 | Functional-differential equations with impulses |
| 34K10 | Boundary value problems for functional-differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
| 34K37 | Functional-differential equations with fractional derivatives |
Keywords:
fractional differential equations; nonlocal conditions; fixed point theorem; noncompact semigroup; measure of noncompactnessReferences:
| [1] | L. Bai, J.J. Nieto, Variational approach to differential equations with not instantaneous impulses, Appl. Math. Lett. 73 (2017) 44-48. · Zbl 1382.34028 |
| [2] | D.D. Bainov, V. Lakshmikantham, P.S. Simeonov, Theory of Impulsive Differential Equations, in: Series in Modern Applied Mathematics, World Scientific, Singapore, 1989. · Zbl 0719.34002 |
| [3] | J. Banas, K. Goebel, Measure of Noncompactness in Banach Space, Marcal Dekker Inc., New York, 1980. · Zbl 0441.47056 |
| [4] | M. Benchohra, J. Henderson, S. Ntouyas, Impulsive Differential Equations and Inclusions, in: Contemp. Math. Appl., Hindawi Publ. Corp., New York, 2006. · Zbl 1130.34003 |
| [5] | D. Bothe, Multivalued perturbations of m-accretive differential inclusions, Israel J. Math. 108 (1998) 109-138. · Zbl 0922.47048 |
| [6] | L. Byszewski, Theorem about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Appl. Anal. 162 (1991) 494-505. · Zbl 0748.34040 |
| [7] | P. Chen, Y. Li, Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces, Nonlinear Anal. 74 (2011) 3578-3588. · Zbl 1220.34018 |
| [8] | P. Chen, X. Zhang, Y. Li, Existence of mild solutions to partial differential equations with non-instantaneous impulses, Electron. J. Differential Equations 241 (2016) 1-11. · Zbl 1350.34047 |
| [9] | K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl. 179 (1993) 630-637. · Zbl 0798.35076 |
| [10] | M.M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals 14 (2002) 433-440. · Zbl 1005.34051 |
| [11] | H. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector valued functions, Nonlinear Anal. 7 (1983) 1351-1371. · Zbl 0528.47046 |
| [12] | E. Hernández, D. O’Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc. 141 (2013) 1641-1649. · Zbl 1266.34101 |
| [13] | P. Kumar, D.N. Pandey, D. Bahuguna, On a new class of abstract impulsive functional differential equations of fractional order, J. Nonlinear Sci. Appl. 7 (2014) 102-114. · Zbl 1312.34117 |
| [14] | J. Liang, J.H. Liu, T.J. Xiao, Nonlocal impulsive problems for integrodifferential equations, Math. Comput. Modelling 49 (2009) 789-804. · Zbl 1173.34048 |
| [15] | S. Liang, R. Mei, Existence of mild solutions for fractional impulsive neutral evolution equations with nonlocal conditions, Adv. Differential Equations (2014) http://dx.doi.org/10.1186/1687-1847-2014-101. · Zbl 1343.34175 |
| [16] | M. McKibben, Discovering Evolution Equations with Applications, Chapman and Hall/CRC, Boca Raton, 2011. · Zbl 1362.35001 |
| [17] | A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, in: Applied Mathematical Sciences, Springer-Verlag, Berlin, 1983. · Zbl 0516.47023 |
| [18] | M. Pierri, D. O’Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equations with non instantaneous impulses, Appl. Math. Comput. 219 (2013) 6743-6749. · Zbl 1293.34019 |
| [19] | J. Wang, X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput. 46 (2014) 321-334. · Zbl 1296.34036 |
| [20] | D. Yang, J. Wang, Integral boundary value problems for nonlinear non-instataneous impulsive differential equations, J. Appl. Math. Comput. (2016) http://dx.doi.org/10.1007/s12190-016-1025-8. |
| [21] | X. Yu, J. Wang, Periodic boundary value problems for nonlinear impulsive evolution equations on Banach spaces, Commun. Nonlinear Sci. Numer. Simul. 22 (2015) 980-989. · Zbl 1339.34069 |
| [22] | X. Zhang, P. Chen, Fractional evolution equation nonlocal problems with noncompact semigroups, Opuscula Math. 36 (2016) 123-137. · Zbl 1335.34024 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
