An Arbitrary (Fractional) Orders Differential Equation with Internal Nonlocal and Integral Conditions

ABSTRACT

In this paper we study the existence of solution for the differential equation of arbitrary ( fractional) orders,with the general form of internal nonlocal condition,The problem with nonlocal integral condition will be studied.

In this paper we study the existence of solution for the differential equation of arbitrary ( fractional) orders,with the general form of internal nonlocal condition,The problem with nonlocal integral condition will be studied.

KEYWORDS

Internal Nonlocal Problem, Integral Condition, Fractional Calculus, Existence of Solution, Caratheodory Theorem

Internal Nonlocal Problem, Integral Condition, Fractional Calculus, Existence of Solution, Caratheodory Theorem

Cite this paper

nullA. El-Sayed and E. Bin-Taher, "An Arbitrary (Fractional) Orders Differential Equation with Internal Nonlocal and Integral Conditions,"*Advances in Pure Mathematics*, Vol. 1 No. 3, 2011, pp. 59-62. doi: 10.4236/apm.2011.13013.

nullA. El-Sayed and E. Bin-Taher, "An Arbitrary (Fractional) Orders Differential Equation with Internal Nonlocal and Integral Conditions,"

References

[1] A. Boucherif, “First-Order Differential Inclusions with Nonlo-cal Initial Conditions,” Applied Mathematics Letters, Vol. 15, No. 4, 2002, pp. 409-414. doi:10.1016/S0893-9659(01)00151-3

[2] A. Boucherif, “Nonlocal Cauchy Problems for First-Order Multivalued Dif-ferential Equations,” Electronic Journal of Differential Equa-tions, Vol. 47, 2002, pp. 1-9.

[3] A. Boucherif and R. Precup, “On the Nonlocal Initial Value Problem for First Order Differ-ential Equations,” Fixed Point Theory, Vol. 4, No. 2, 2003, pp. 205-212.

[4] A. Boucherif, “Semilinear Evolution Inclusions with Nonlocal Conditions,” Applied Mathematics Letters, Vol. 22, No. 8, 2009, pp. 1145-1149. doi:10.1016/j.aml.2008.10.004

[5] M. Benchohra, E. P. Gat-sori and S. K. Ntouyas, “Existence Results for Seme-Linear Integrodifferential Inclusions with Nonlocal Conditions,” Rocky Mountain Journal of Mathematics, Vol. 34, No. 3, 2004。

[6] M. Benchohra, S. Hamani and S. K. Ntouyas, “Boundary Value Problems for Differential Equations with Fractional Order and Nonlocal Conditions,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 7-8, 2009, pp. 2391-2396. doi:10.1016/j.na.2009.01.073

[7] R. F. Curtain and A. J. Pritchard, “Functional Analysis in Modern Applied Mathematics,” Academic Press, London, 1977.

[8] K. Deim-ling, “Nonlinear Functional Analysis,” Springer- Verlag, Berlin, 1985.

[9] J. Dugundji and A. Granas, “Fixed Point Theory,” Monografie Mathematyczne, Polska Akademia Nauk, War-szawa, Vol. 1, 1982.

[10] A. M. A. El-Sayed and Sh. A. Abd El-Salam, “On the Stability of a Fractional Order Differential Equation with Nonlocal Initial Condition,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 2009, No. 29, 2008, pp. 1-8.

[11] A. M. A. El-Sayed and E. O. Bin-Taher, “A Nonlocal Problem of an Arbitrary (Fractional) Orders Dif-ferential Equation,” Alexandria Journal of Mathematics, Vol. 1, No. 2, 2010, pp. 1-7.

[12] E. Gatsori, S. K. Ntouyas and Y. G. Sficas, “On a Nonlocal Cauchy Problem for Differential Inclu-sions,” Abstract and Applied Analysis, Vol. 2004, No. 5, 2004, pp. 425-434.

[13] G. M. N’Guérékata, “A Cauchy Problem for Some Fractional Abstract Differential Equation with Non Local Conditions,” Nonlinear Analysis: Theory, Methods & Applica-tions, Vol. 70, No. 5, 2009, pp. 1873-1876. doi:10.1016/j.na.2008.02.087

[14] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, New York and London, 1999.

[15] I. Podlubny and A. M. A. EL-Sayed, “On Two Definitions of Fractional Calculus,” Pre-print UEF 03-96, ISBN 80-7099-252-2, Institute of Experi-mental Physics, Slovak Academy of Science, 1996.

[1] A. Boucherif, “First-Order Differential Inclusions with Nonlo-cal Initial Conditions,” Applied Mathematics Letters, Vol. 15, No. 4, 2002, pp. 409-414. doi:10.1016/S0893-9659(01)00151-3

[2] A. Boucherif, “Nonlocal Cauchy Problems for First-Order Multivalued Dif-ferential Equations,” Electronic Journal of Differential Equa-tions, Vol. 47, 2002, pp. 1-9.

[3] A. Boucherif and R. Precup, “On the Nonlocal Initial Value Problem for First Order Differ-ential Equations,” Fixed Point Theory, Vol. 4, No. 2, 2003, pp. 205-212.

[4] A. Boucherif, “Semilinear Evolution Inclusions with Nonlocal Conditions,” Applied Mathematics Letters, Vol. 22, No. 8, 2009, pp. 1145-1149. doi:10.1016/j.aml.2008.10.004

[5] M. Benchohra, E. P. Gat-sori and S. K. Ntouyas, “Existence Results for Seme-Linear Integrodifferential Inclusions with Nonlocal Conditions,” Rocky Mountain Journal of Mathematics, Vol. 34, No. 3, 2004。

[6] M. Benchohra, S. Hamani and S. K. Ntouyas, “Boundary Value Problems for Differential Equations with Fractional Order and Nonlocal Conditions,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 7-8, 2009, pp. 2391-2396. doi:10.1016/j.na.2009.01.073

[7] R. F. Curtain and A. J. Pritchard, “Functional Analysis in Modern Applied Mathematics,” Academic Press, London, 1977.

[8] K. Deim-ling, “Nonlinear Functional Analysis,” Springer- Verlag, Berlin, 1985.

[9] J. Dugundji and A. Granas, “Fixed Point Theory,” Monografie Mathematyczne, Polska Akademia Nauk, War-szawa, Vol. 1, 1982.

[10] A. M. A. El-Sayed and Sh. A. Abd El-Salam, “On the Stability of a Fractional Order Differential Equation with Nonlocal Initial Condition,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 2009, No. 29, 2008, pp. 1-8.

[11] A. M. A. El-Sayed and E. O. Bin-Taher, “A Nonlocal Problem of an Arbitrary (Fractional) Orders Dif-ferential Equation,” Alexandria Journal of Mathematics, Vol. 1, No. 2, 2010, pp. 1-7.

[12] E. Gatsori, S. K. Ntouyas and Y. G. Sficas, “On a Nonlocal Cauchy Problem for Differential Inclu-sions,” Abstract and Applied Analysis, Vol. 2004, No. 5, 2004, pp. 425-434.

[13] G. M. N’Guérékata, “A Cauchy Problem for Some Fractional Abstract Differential Equation with Non Local Conditions,” Nonlinear Analysis: Theory, Methods & Applica-tions, Vol. 70, No. 5, 2009, pp. 1873-1876. doi:10.1016/j.na.2008.02.087

[14] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, New York and London, 1999.

[15] I. Podlubny and A. M. A. EL-Sayed, “On Two Definitions of Fractional Calculus,” Pre-print UEF 03-96, ISBN 80-7099-252-2, Institute of Experi-mental Physics, Slovak Academy of Science, 1996.