Conformable Fractional Cosine Families of Operators

Yıl 2019, Cilt: 2 Sayı: 2, 112 - 116, 30.08.2019

Elomari M'hamed , Said Melliani L. S. Chadli

https://doi.org/10.33187/jmsm.435481

Öz

In this paper we are concerned with the problem \begin{eqnarray*}\begin{cases} u^{(\alpha)}(t)=Au(t)+f(t,u(t))& t\in [0,T]\\ u(0)=u_0, D^{\alpha}u(0)=u_1\end{cases}\end{eqnarray*}  \begin{eqnarray*}     \begin{cases}     u^{(\alpha)}(t)=Au(t)+f(t,u(t))& t\in [0,T]\\     u(0)=u_0, D^{\alpha}u(0)=u_1     \end{cases}   \label{pb1} \end{eqnarray*}   Where $\alpha\in (1,2]$, and we use the conformable derivative. We give the notion of $\alpha$-Cosine families and proveded the existence and uniqueness of the problem 0.1.

Anahtar Kelimeler

$\alpha$-cosine families, Conformable derivative, Mild solution

Kaynakça

• [1] Agarwal, RP, Zhou, H, He, Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59(3), 1095-1100 (2010)
• [2] Abdeljawad, T., On Conformable Fractional Calculus,Journal of Computational ad Applied Mathematics,Vol. 279, 1 May 2015, 57-66, arXiv: 1402.6892v1 [math D, S] 27 Feb 2014.
• [3] Balakrishnan, A. V., Fractional Powers Of Closed Operators And The Semigroups Generated By Them, Pacific Journal of Mathematics 10, pp. 419-439, 1960.
• [4] TRAVIS. C. C and WEBB. G. F. Cosine Families and abstract nonlinear second order differential equations. Acta Mathematica Academiae Seientiarum Hungaricae Tomus 32 (3–4), (1978), 75–96.
• [5] Mohammed AL Horani. Roshdi Khalil and Thabet Abdeljawad. Conformable Fractional Semigroups of Operators. arXiv:1502.06014v1 [math.FA] 21 Nov 2014 Conformable.
• [6] Khalil, R., Al Horani, M., Yousef. A. and Sababheh, M., A new Definition Of Fractional Derivative, J. Comput. Appl. Math. 264. pp. 65?0, 2014.
• [7] Kilbas, AA, Srivastava, HH, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
• [8] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differntial Equations, Springer-Verlag, 1983.