Existence and attractivity for fractional order integral equations in Fréchet spaces
Saïd Abbas ; Mouffak Benchohra
Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 33 (2013), p. 47-63 / Harvested from The Polish Digital Mathematics Library

In this paper, we present some results concerning the existence and the attractivity of solutions for some functional integral equations of Riemann-Liouville fractional order, by using an extension of the Burton-Kirk fixed point theorem in the case of a Fréchet space.

Publié le : 2013-01-01
EUDML-ID : urn:eudml:doc:270181
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1141,
     author = {Sa\"\i d Abbas and Mouffak Benchohra},
     title = {Existence and attractivity for fractional order integral equations in Fr\'echet spaces},
     journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
     volume = {33},
     year = {2013},
     pages = {47-63},
     zbl = {1296.26023},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1141}
}
Saïd Abbas; Mouffak Benchohra. Existence and attractivity for fractional order integral equations in Fréchet spaces. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 33 (2013) pp. 47-63. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1141/

[000] [1] S. Abbas, D. Baleanu and M. Benchohra, Global attractivity for fractional order delay partial integro-differential equations, Adv. Difference Equ. 2012 (2012), 62. doi: 10.1186/1687-1847-2012-62 | Zbl 1302.35392

[001] [2] S. Abbas and M. Benchohra, Nonlinear quadratic Volterra Riemann-Liouville integral equations of fractional order, Nonlinear Anal. Forum 17 (2012), 1-9. | Zbl 1293.26008

[002] [3] S. Abbas and M. Benchohra, On the existence and local asymptotic stability of solutions of fractional order integral equations, Comment. Math. 52 (1) (2012), 91-100. | Zbl 1292.26015

[003] [4] S. Abbas, M. Benchohra and J.R. Graef, Integro-differential equations of fractional order, Differ. Equ. Dyn. Syst. 20 (2) (2012), 139-148. doi: 10.1007/s12591-012-0110-1 | Zbl 1266.45011

[004] [5] S. Abbas, M. Benchohra and J. Henderson, On global asymptotic stability of solutions of nonlinear quadratic Volterra integral equations of fractional order, Comm. Appl. Nonlinear Anal. 19 (2012), 79-89. | Zbl 1269.26003

[005] [6] S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012. doi: 10.1007/978-1-4614-4036-9

[006] [7] S. Abbas, M. Benchohra and A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Fract. Calc. Appl. Anal. 15 (2) (2012), 168-182. | Zbl 1302.35395

[007] [8] C. Avramescu, Some remarks on a fixed point theorem of Krasnoselskii, Electron. J. Qual. Theory Differ. Equ. 5 (2003), 1-15.

[008] [9] C. Avramescu and C. Vladimirescu, An existence result of asymptotically stable solutions for an integral equation of mixed type, Electron. J. Qual. Theory Differ. Equ. 25 (2005), 1-6. | Zbl 1104.47063

[009] [10] C. Avramescu and C. Vladimirescu, On the existence of asymptotically stable solutions of certain integral equations, Nonlinear Anal. 66 (2) (2007), 472-483. doi: 10.1016/j.na.2005.11.041 | Zbl 1110.45004

[010] [11] D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific Publishing, New York, 2012. | Zbl 1248.26011

[011] [12] J. Banaś and B.C. Dhage, Global asymptotic stability of solutions of a functional integral equation, Nonlinear Anal. 69 (7) (2008), 1945-1952. doi: 10.1016/j.na.2007.07.038 | Zbl 1154.45005

[012] [13] J. Banaś and B. Rzepka, On existence and asymptotic stability of solutions of a nonlinear integral equation, J. Math. Anal. Appl. 284 (2003), 165-173. doi: 10.1016/S0022-247X(03)00300-7 | Zbl 1029.45003

[013] [14] J. Banaś and T. Zając, Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity, Nonlinear Anal. 71 (2009), 5491-5500. doi: 10.1016/j.na.2009.04.037 | Zbl 1181.45010

[014] [15] J. Banaś and T. Zając, A new approach to the theory of functional integral equations of fractional order, J. Math. Anal. Appl. 375 (2011), 375-387. doi: 10.1016/j.jmaa.2010.09.004 | Zbl 1210.45005

[015] [16] M.A. Darwish, J. Henderson, and D. O'Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional integral equations with linear modification of the argument, Bull. Korean Math. Soc. 48 (3) (2011), 539-553. doi: 10.4134/BKMS.2011.48.3.539 | Zbl 1220.45011

[016] [17] B.C. Dhage, Local asymptotic attractivity for nonlinear quadratic functional integral equations, Nonlinear Anal. 70 (2009), 1912-1922. doi: 10.1016/j.na.2008.02.109 | Zbl 1173.47056

[017] [18] B.C. Dhage, Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem, Nonlinear Anal. 70 (2009), 2485-2493. doi: 10.1016/j.na.2008.03.033 | Zbl 1163.45005

[018] [19] B.C. Dhage, Attractivity and positivity results for nonlinear functional integral equations via measure of noncompactness, Differ. Equ. Appl. 2 (3) (2010), 299-318. | Zbl 1201.45007

[019] [20] M. Frigon and A. Granas, Théorèmes d'existence pour des inclusions différentielles sans convexité, C.R. Acad. Sci. Paris, Ser. I 310 (1990), 819-822.

[020] [21] A.A. Kilbas, Hari M. Srivastava and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006. | Zbl 1092.45003

[021] [22] V. Lakshmikantham, S. Leela and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009. | Zbl 1188.37002

[022] [23] K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. | Zbl 0789.26002

[023] [24] B.G. Pachpatte, On Volterra-Fredholm integral equation in two variables, Demonstratio Math. XL (4) (2007), 839-852. | Zbl 1186.45010

[024] [25] B.G. Pachpatte, On Fredholm type integral equation in two variables, Differ. Equ. Appl. 1 (2009), 27-39. | Zbl 1170.45003

[025] [26] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999.

[026] [27] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.

[027] [28] A.N. Vityuk and A.V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7 (3) (2004), 318-325. doi: 10.1007/s11072-005-0015-9 | Zbl 1092.35500