Topological structure of solution set for a class of fractional neutral evolution equations on the half-line

Le Hoan Hoa, Nguyen Ngoc Trong, Le Xuan Truong



A topological structure of the set of all mild solutions of fractional neutral evolution equations with finite delay on the half-line is investigated. We show that the solution set is an R$_\delta$-set. It is proved on compact intervals by establishing a result on topological structure of fixed point set of Krasnosel'skiĭ type operators. Next, using the inverse limit method, we obtain the same result on the half-line.


Topology structure; solution set; fractional differential equation; Krasnosel'skiĭ type operator

Full Text:



J. Andres and L. Górniewicz, Topology Fixed Point Priciples for Boundary Value Problems, Topol. Fixed Point Theory Appl., Kluwer, Dordrecht, 2003.

P.N. Aronszajn, Le correspondant topologique de l’unicité dans la theorie des equations différentielles, Ann. Math. 43 (1942), 730–78.

F.E. Browder and C.P. Gupta, Topological degree and nonlinear mappings of analytic type in Banach space, J. Math. Anal. Appl. 26 (1969), 390–402.

D. Baleanu and O.G. Mustafa, On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl. 59 (5) (2010), 835–1841.

D. Baleanu, O.G. Mustafa and R.P. Agarwal, An existence result for a superlinear fractional differential equation, Appl. Math. Lett. 23 (9) (2010), 1129–1132.

D. Bugajewska, On the structure of solution sets of differential equations in Banach spaces, Math. Slovaca 50 (2000), 463–471.

M. Cichoń and I. Kubiaczyk, Some remarks on the structure of the solution set for differential inclusions in Banach spaces, J. Math. Anal. Appl. 233 (1999), 597–606.

F.S. De Blasi and J. Myjak, On the solution sets for differential inclusion, Bull. Pol. Acad. Sci. Math. 12 (1985), 17–23.

S.D. Eidelman and A.N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations 199 (2004), 211–255.

M.M. El-Borai, Semigroups and some nonlinear fractional differential equations, Appl. Math. Comput. 149 (2004), 823–831.

M.M. El-Borai, Some probability densities and fundamental solutions of fractional evolution equations, Chaos Solitons Fractals 149 (2004), 823–831.

X. Fu and K. Ezzinbi, Existence of solutions for neutral differential evolution equations with nonlocal conditions, Nonlinear Anal. 54 (2003), 215–227.

G. Gabor, On the acyclicity of fixed point sets of multivalued maps, Topol. Methods Nonlinear Anal. 14 (1999), 327–343.

L. Górniewicz, Topological structure of solution sets: current results, Arch. Math. (Brno) 36 (2000), 343–382.

L. Górniewicz, Solving equations by topological methods, Opuscula Math. 25 (2005), 195–225.

L.H. Hoa and K. Schmitt, Fixed point theorems of Krasnosel’skii type in locally convex space and applications to integral equation, Results Math. 25 (1994), 291–313.

L.H. Hoa and K. Schmitt, Periodic solutions of functional-differential equations of retarded and neutral types in Banach spaces, Bound. Value Probl. Funct. Differential Equations (1995), 177–185.

S. Ji and G. Li and M. Wang, Controllability of impulsive differential systems with nonlocal conditions, Appl. Math. Comput. 217 (2011), 6981–6989.

O.K. Jardat, A. Al-Omari and S. Momani, Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal. 69 (9) (2008), 3153–3159.

W. Kryszewski, Topological structure of solution sets of differential inclusions: the constrained case, Abstr. Appl. Anal. (2003), 325–351.

Z. Kubacek, On the structure of fixed point sets of some compact maps in the Fréchet space, Math. Bohem. 118 (1993), 343–358.

J.J. Nieto, Aronszajn’s theorem for some nonlinear Dirichlet problems with unbounded nonlinearities, Proc. Edinburgh Math. Soc. 31 (1988), 345–351.

J.P.C. Santos, M.M. Arjunan and C. Cuevas, Existence results for fractional neutral integro-differential equations with state dependent delay, Comput. Math. Appl. 62 (2011), 1275–1283.

J.P.C. Santos, V. Vijayakumar and R. Murugesu, Existence of mild solutions for nonlocal Cauchy problem for fractional neutral integro-differential equation with unbounded delay, Commun. Math. Anal. 14 (2013), 59–71.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.

J.R. Wang, Z. Fan and Y. Zhou, Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces, J. Optim. Theory Appl. 154 (2012), 292–302.

X. Zheng and X. Yang, The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces, E? Sci. China Ser. A 51 (2008), 1243–1256.

Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl. 59 (2010), 1063–1077.


  • There are currently no refbacks.

Partnerzy platformy czasopism