### Existence and Uniqueness of the Mild Solution of an Abstract Semilinear Fractional Differential Equation with State Dependent Nonlocal Condition.

Download PDF

**Authors:**M. A. E. HERZALLAH, A. H. A. RADWAN

**DOI:**10.46793/KgJMat2106.909H

**Abstract:**

The purpose of this paper is to investigate the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract fractional diﬀerential equation with state dependent nonlocal condition. Continuous dependence of solutions on initial conditions and local ????-approximate mild solution of the considered problem will be discussed.

**Keywords:**

Caputo derivative, state dependent nonlocal condition, C

_{0}-semigroups, continuous dependence, ????-approximate solution, Krasnoselskii’s ﬁxed point theorem.

**References:**

[1] L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162(2) (1991), 494–505.

[2] K. Deng, Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl. 179(2) (1993), 630–637.

[3] A. M. A. El-Sayed, E. M Hamdallah and Kh. W. Elkadeky,
Solutions of a class of deviated-advanced nonlocal problems for the
diﬀerential inclusion x^{′}(t) ∈ F(t,x(t)), Abstr. Appl. Anal. 2011
(2011), Paper ID 476392, 9 pages, DOI 10.1155/2011/476392.

[4] C. L. Evans, Partial Diﬀerential Equations, American Mathematical Society, Providence, Rhode Island, 1998.

[5] I. Farmakis and M. Moskowitz, Fixed Point Theorems and There Applications, World Scientiﬁc, New York, 2013.

[6] J. A. Goldstein, Semigroups of Linear Operators and Applications, Second Edition, Oxford University Press, Oxford, USA, 1985.

[7] E. Hernndez, On abstract diﬀerential equations with state dependent non-local conditions, J. Math. Anal. Appl. 466(1) (2018), 408–425.

[8] E. Hernndez and D. ORegan, On state dependent non-local conditions, Appl. Math. Lett. 83 (2018), 103–109.

[9] M. Kamenskii, V. Obukhovskii, G. Petrosyan and J. Yao, On approximate solutions for a class of semilinear fractional-order diﬀerential equations in Banach spaces, Fixed Point Theory Appl. 2017(28) (2017), DOI 10.1186/s13663-017-0621-0.

[10] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Diﬀerential Equations, Elsevier, Amsterdam, 2006.

[11] F. Mainardi, P. Paradisi and R. Gorenﬂo, Probability distributions generated by fractional diﬀusion equations, in: J. Kertesz, I. Kondor (Eds.), Econophysics: An Emerging Science, Kluwer, Dordrecht, 2000.

[12] C. F. Lorenzo and T. T. Hartley, The Fractional Trigonometry with Applications to Fractional Diﬀerential Equations and Science, John Wiley and Sons, Inc., Hoboken, New Jersey, 2017.

[13] A. Pazy, Semigroups of Linear Operators and Applications to Partial Diﬀerential Equations, Springer, Berlin, 1983.

[14] I. Podlubny, Fractional Diﬀerential Equations, Academic Press, San Diego, CA, 1999.

[15] Y. Povstenko, Fractional thermoelasticity, Solid Mech. Appl. 219 (2015), DOI 10.1007/978-3-319-15335-3.

[16] V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Higher Education Press, Heidelberg, 2010.

[17] J. Wang and Y. Zhou, A class of fractional evolution equations and optimal controls, Nonlinear Anal. Real World Appl. 12 (2011), 262–272.

[18] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional diﬀerential equation, J. Math. Anal. Appl. 328 (2007), 1075–1081.

[19] Y. Zhou, Basic Theory of Fractional Diﬀerential Equations, World Scientiﬁc, Singapore, 2014.

[20] Y. Zhou and F. Jiao, Nonlocal Cauchy problem for fractional evolution equations, Nonlinear Anal. Real World Appl. 11 (2010), 4465–4475.

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