Second-order abstract Cauchy problem of conformable fractional type

Document Type : Research Paper

Authors

1 Department of Mathematics, Jordan University, Amman, Jordan

2 Department of Mathematics, Yarmouk University, Irbid, Jordan

3 Department of Mathematics, Lusail University, Doha, Qatar

Abstract

    In this paper, we discuss atomic solutions of the second-order abstract Cauchy problem of conformable fractional type 
\begin{eqnarray*}
u^{(2\alpha )}(t)+Bu^{(\alpha )}(t)+Au(t) &=&f(t) \\
u(0) &=&u_{0}, \\
u^{(\alpha )}(0) &=&u_{0}^{(\alpha )},
\end{eqnarray*}%
where $A,B$ are closed linear operators on a Banach space $X,$ $f$ $ :[0,\infty )\rightarrow X$ \ is continuous and $u$ is a continuously differentiable function on $[0,\infty )$. Some new results on atomic solutions using tensor product technique are obtained.

Keywords

[1] M. Abu Hammad and R. Khalil, Systems of linear fractional differential equations, Asian J. Math. Comput. Res.
12 (2016), no. 2, 120–126.
[2] T. Abdeljawad, M. Al Horani and R. Khalil, Conformable fractional semigroups of operators, J. Semigroup Theory
Appl. 2015 (2015), (1-9) Article ID 7.
[3] M. Al Horani, M. Abu Hammad and R. Khalil, Variation of parameters for local nonhomogeneous linear differential equations, J. Math. Comp. Sci. 16 (2016), 147–153.
[4] M. Al Horani, An identification problem for some degenerate differential equations, Le Matematiche 57 (2002),
217–227.
[5] Sh. Al-Sharif, A. Malkawi, Modification of conformable fractional derivative with classical properties. Ital. J. Pure
Appl. Math. 44 ( 2020), 30–39.
[6] D. Anderson and D. Ulness . Newly defined conformable derivatives, Adv. Dyn. Syst. Appl. 10(2015), no. 2,
109–137.
[7] E. Boyce and C. Di Prima, Elementary differential equations and boundary value problems, 9th ed. John Wiley
and Sons, Inc , 2008.
[8] R.W. Carroll and R.E. Showalter, Singular and Degenerate Cauchy Problem, Academic Press, New York, San
Francisco, London, 1976.
[9] W. Deeb and R. Khalil, Best approximation in L(X,Y), Math. Proc. Camb. Phil. Soc. 104 (1988), 527–531.
[10] W. Deeb and R. Khalil, Best approximation in L
p
(I, X), 0 < p < 1, J. Approx. Theory 58 (1989), no. 1, 68–77.
[11] N. Dunford and J. Schwartz, Linear operators, Part I, General theory, Interscience Pub. New York, 1958.
[12] A. Favini and A. Yagi, Degenerate differential equations in Banach spaces, Dekker, New York, Basel-Hong Kong,
1999.
[13] D. Hussein and R. Khalil, Best approximation in tensor product space, Soochow J. Math. 18 (1982), no. 4,
397–407.
[14] R. Khalil and L. Abdullah, Atomic solution of certain inverse problems, Eur. J. Pure Appl. Math. 3 (2010), no.
4, 725–729.
[15] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl.
Math. 264 (2014), 65–70.
[16] S. Khamis, M. Al Horani and R. Khalil, Rank two solutions of the abstract Cauchy problem, J. Semigroup Theory
Appl. 2018 (2018), Article ID 3.
[17] A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, Math. Studies
204, North-Holland, New York, 2006.
[18] W. Light and E. Cheney, Approximation theory in tensor product spaces, Lecture Notes in Mathematics, 1169,
Springer Verlag, Berlin, New York, 1985.
[19] G. Samko, A. Kilbas and A. Marichev, Fractional integrals and derivatives, theory and applications, Gordon and
Breach, Yverdon, 1993.
[20] B. Thaller and S. Thaller, Factorization of degenerate Cauchy problem, the linear case, J. Oper. Theory 36 (1996),
121–146.
[21] A. Ziqan, M. Al Horani and R. Khalil, Tensor product technique and the degenerate homogeneous abstract Cauchy
problem, J. Appl. Funct. Anal. 5 (2010), no. 1, 121–138.
[22] A. Ziqan, M. Al Horani and R. Khalil, Tensor product technique and non-homogeneous degenerate abstract Cauchy
problem, Int. J. Appl. Math. Res. 23 (2010), no. 1, 137–158.
Volume 13, Issue 2
July 2022
Pages 1143-1150
  • Receive Date: 01 June 2020
  • Revise Date: 10 August 2020
  • Accept Date: 28 September 2020