×
Liang, Yue; Yang, He; Gou, Kun

Existence of mild solutions for fractional nonlocal evolution equations with delay in partially ordered Banach spaces. (English) Zbl 1422.37056

Adv. Difference Equ. 2017, Paper No. 11, 10 p. (2017); erratum ibid. 2017, Paper No. 40, 1 p. (2017).
Summary: This paper deals with the existence of mild solutions for the abstract fractional nonlocal evolution equations with noncompact semigroup in partially ordered Banach spaces. Under some mixed conditions, a group of sufficient conditions for the existence of abstract fractional nonlocal evolution equations are obtained by using a Krasnoselskii type fixed point theorem. The results we obtained are a generalization and continuation of the recent results on this issue. At the end, an example is given to illustrate the applicability of abstract result.

Cited in 1 Review
Cited in 2 Documents

MSC:

37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
45N05 Abstract integral equations, integral equations in abstract spaces
47H10 Fixed-point theorems
26A33 Fractional derivatives and integrals

Keywords:

partially ordered Banach space; fractional nonlocal evolution equation; positive \(C_{0}\)-semigroup; existence; Krasnoselskii type fixed point theorem
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Dhage, B: A nonlinear alternative with applications to nonlinear perturbed differential equations. Nonlinear Stud. 13, 343-354 (2006) · Zbl 1120.47047
[2] Dhage, B, Dhage, S, Ntouyas, S: Approximating solutions of nonlinear hybrid differential equations. Appl. Math. Lett. 34, 76-80 (2014) · Zbl 1314.34036 · doi:10.1016/j.aml.2014.04.002
[3] Dhage, B: Partially continuous mappings in partially ordered normed linear spaces and applications to functional integral equations. Tamkang J. Math. 45, 397-426 (2014) · Zbl 1343.45004 · doi:10.5556/j.tkjm.45.2014.1512
[4] Guo, D, Lakshmikantham, V, Liu, X: Nonlinear Integral Equations in Abstract Spaces. Mathematics and Its Applications, vol. 373. Kluwer Academic, Dordrecht (1996) · Zbl 0866.45004 · doi:10.1007/978-1-4613-1281-9
[5] Liang, J, Yang, H: Controllability of fractional integro-differential evolution equations with nonlocal conditions. Appl. Math. Comput. 254, 20-29 (2015) · Zbl 1410.93022
[6] Nashine, H, Samet, B: Fixed point results for mappings satisfying (ψ,φ)\((\psi, \varphi )\)-weakly contractive conditions in partially ordered metric spaces. Nonlinear Anal. 74, 2201-2209 (2011) · Zbl 1208.41014 · doi:10.1016/j.na.2010.11.024
[7] Pathak, H, Rodríguez-López, R: Existence and approximation of solutions to nonlinear hybrid ordinary differential equations. Appl. Math. Lett. 39, 101-106 (2015) · Zbl 1321.34018 · doi:10.1016/j.aml.2014.08.018
[8] Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) · Zbl 0516.47023 · doi:10.1007/978-1-4612-5561-1
[9] Sinestrari, E: On the abstract Cauchy problem of parabolic type in space of continuous functions. J. Math. Anal. Appl. 107, 16-66 (1985) · Zbl 0589.47042 · doi:10.1016/0022-247X(85)90353-1
[10] Wang, RN, Xiao, TJ, Liang, J: A note on the fractional Cauchy problems with nonlocal initial conditions. Appl. Math. Lett. 24, 1435-1442 (2011) · Zbl 1251.45008 · doi:10.1016/j.aml.2011.03.026
[11] Xiao, TJ, Liang, J: Coupled second order semilinear evolution equations indirectly damped via memory effects. J. Differ. Equ. 254, 2128-2157 (2013) · Zbl 1264.34121 · doi:10.1016/j.jde.2012.11.019
[12] Zhou, Y, Jiao, F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 11, 4465-4475 (2010) · Zbl 1260.34017 · doi:10.1016/j.nonrwa.2010.05.029
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.