On Some Qualitative Properties of Integrable Solutions for Cauchy-type Problem of Fractional Order

The paper discusses the existence of solutions for Cauchy-type problem of fractional order in the space of Lebesgue integrable functions on bounded interval. Some qualitative properties of solutions are presented such as monotonicity, uniqueness and continuous dependence on the initial data. The main tools used are measure of weak (strong) noncompactness, Darbo fixed poit theorem and fractional calculus.

Full text (pdf)

1. A. Aghajani, M. Aliaskari and A. Shole Haghighi, Some existence theorems for systems of equations involving condensing operators and applications, Mediterr. J. Math. (2017) 14-47.
2. J. Appell, P.P. Zabrejko, Nonlinear Superposition Operators, Cambridge Tracts in Mathematics 95, Cambridge University Press, Cambridge 1990.
3. I.K. Argyros, On a class of quadratic integral equations with perturbations, Functiones et Approximatio 20 (1992) 51-63.
4. J. Banaś, K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes in Math. 60, M. Dekker, New York-Basel 1980.
5. J. Banaś, L. Olszowy, Measures of noncompactness related to monotonicity, Comment. Math. Prace Matem. 41 (2001) 13-23.
6. J. Banaś, K. Sadarangani, Solutions of some functional-integral equations in Banach algebras, Math. Comput. Model. 38 (2003) 245-250.
7. J. Caballero, A.B. Mingarelli and K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electr. Jour. Differ. Equat. 57 (2006) 1-11.
8. M. Cichoń, M. Metwali, On a fixed point theorem for the product of operators, J. Fixed Point Theory Appl. 18 (2016) 753-770.
9. M. Cichoń, M. Metwali, On the existence of solutions for quadratic integral equations in Orlicz spaces, Math. Slovaca 66 (2016) 1413-1426.
10. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin 1985.
11. A. Dishlev, D. Bainov, Continuous dependence on the initial condition of the solution of a system of differential equations with variable structure and with impulses, Putt. RIMS, Kyoto Univ. 23 (1987)  923-936.
12. A.M.A. El-Sayed, Sh.A. Abd El-Salam, Weighted Cauchy-type problem of a functional differ-integral equation, Electronic Journal of Qualitative Theory of Differential Equations 30 (2007) 1-9.
13. A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary order, Nonlin. Anal. 33 (1998) 181-186.
14. N. Erzakova, Compactness in measure and measure of noncompactness, Siberian Math. J. 38 (1997) 926-928.
15. K.M. Furati, N.E. Tatar, Power-type estimates for a nonlinear fractional differential equation, Nonlin. Anal. 62 (2005) 1025-1036.
16. M.A.E. Herzallah, D. Baleanu, On fractional order hybrid differential equations, Abstr. Appl. Anal., Volume 2014 (2014), Article ID 389386, 7 pages, doi:10.1155/2014/389386.
17. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science Inc., New York, NY, USA 2006.
18. J. Krzyż, On monotonicity-preserving transformations, Ann. UMCS 6 (1952) 91-111.
19. H. Lu, S. Sun, D. Yang and H. Teng, Theory of fractional hybrid differential equations with linear perturbations of second type, Bound. Value Probl. (2013) 2013:23 16 pages.
20. M. Metwali, On a class of quadratic Urysohn-Hammerstein integral equations of mixed type and initial value problem of fractional order, Mediterr. J. Math. 13 (2016) 2691-2707.
21. L. Mishra, M. Sen and R. Mohapatra, On Existence theorems for some generalized nonlinear functional-integral equations with applications, Filomat 31 (2017) 2081-2091.
22. I. Podlubny, Fractional Differential Equations, Acad. Press, San Diego-New York-London 1999.
23. Yu. Rossikhin, M.V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev. 50 (1997) 15-67.
24. D. Somjaiwang, P. Sa Ngiamsunthorn,  Existence and approximation of solutions to fractional order hybrid differential equations, Advances in Difference Equations (2016) 2016-278.
25. T.A. Surguladze, On certain applications of fractional calculus to viscoelasticity, J. Math. Sci. 112 (2002) 4517-4525.
26. M. Väth, Volterra and Integral Equations of Vector Functions, Marcel Dekker, New York-Basel 2000.
27. P.P. Zabrejko, A.I. Koshlev, M.A. Krasnoselskii, S.G. Mikhlin, L.S. Rakovshchik and V.J. Stecenko, Integral Equations, Noordhoff, Leyden 1975.