Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843 - 7265 (print)
 Volume 2 (2007), 145 - 156

UPPER AND LOWER BOUNDS OF SOLUTIONS FOR FRACTIONAL INTEGRAL EQUATIONS

Rabha W. Ibrahim and Shaher Momani

Abstract. In this paper we consider the integral equation of fractional order in sense of Riemann-Liouville operator

um(t) = a(t) Iα [b(t)u(t)]+f(t)
with m ≥ 1, t ∈ [0, T], T
2000 Mathematics Subject Classification: 34G10; 26A33; 34A12; 42B05.
Keywords: Riemann-Liouville operators; Upper and lower bound of solution; Volterra integral equation.

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References

  1. K. Balachandran and J. P. Dauer, Elements of Control Theory, New Delhi, Narosa Publishing House, 1999. Zbl 0965.93002

  2. P. Butzer and L. Westphal, An introduction to fractional calculus. Hilfer, R. (ed.), Applications of fractional calculus in physics. Singapore: World Scientific. (2000), 1-85. MR1890105(2003g:26007). Zbl 0987.26005

  3. K. Deimling, Nonlinear Functional Analysis, Berlin, Springer-Verlag, 1985. MR0787404(86j:47001). Zbl 0559.47040.

  4. R. Gorenflo and S. Vessella, Abel integral equations. Analysis and applications. Lecture Notes in Mathematics, 1461 Springer-Verlag, Berlin, 1991. MR1095269(92e:45003). Zbl 0717.45002

  5. E. Hille and J. Tamarkin, On the theory of linear integral equations, Ann. of Math. (2) 31 (1930), 479-528. MR1502959. JFM 56.0337.01.

  6. V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics Series, 301. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. MR1265940(95d:26010). Zbl 0882.26003.

  7. K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, Inc., 1993. MR1219954(94e:26013). Zbl 0789.26002.

  8. I. Podlubny, Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering, 198. Academic Press, 1999. MR1658022(99m:26009). Zbl 0924.34008.

  9. M. R. Rao, Ordinary Differential Equations. Theory and applications, New Delhi-Madras: Affiliated East-West Press, 1980. Zbl 0482.34001.

  10. B. Ross and B. K. Sachdeva, The solution of certain integral equations by means of operators of arbitrary order, Amer. Math. Monthly 97 (1990), 498-502. MR1055906(91e:45005). Zbl 0723.45002.

  11. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gorden and Breach, New York, 1993. MR1347689(96d:26012).

  12. D. R. Smart, Fixed Point Theorems, Cambridge University Press, 1980. Zbl 0427.47036.

  13. H. M. Srivastava and R. G. Buschman, Theory and Applications of Convolutions Integral Equations, Kluwer Acad., Dordrecht, 1992. MR1205580(94a:45002). Zbl 0755.45002.

Rabha W. Ibrahim Shaher Momani
P.O. Box 14526, Sanaa, Department of Mathematics, Mutah University,
Yemen. P.O. Box 7, Al-Karak, Jordan.
e-mail: rabhaibrahim@yahoo.com e-mail: shahermm@yahoo.com

http://www.utgjiu.ro/math/sma