Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Solvability of a fractional Cauchy problem based on modified fixed point results of non-compactness measures

1 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, TN, India
2 Department of Mathematics and Applied Mathematics, University of Johannesburg, Kingsway Campus, Auckland Park 2006, South Africa
3 Cloud Computing Center, University Malaya, Malaysia

Special Issues: Initial and Boundary Value Problems for Differential Equations

We study the solvability of a fractional Cauchy problem based on new development of fixed point theorem, where the operator is suggested to be non-compact on its domain. Moreover, we shall prove that the solution is bounded by a fractional entropy (entropy solution). For this purpose, we establish a collection of basic fixed point results, which generalizes and modifies some well known results. Our attention is toward the concept of a measure of non-compactness to generalize µ-set contractive condition, using three control functions.
  Figure/Table
  Supplementary
  Article Metrics

References

1.R. P. Agarwal, M. Benchohra and D. Seba, On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math., 55 (2009), 221-230.    

2.R. P. Agarwal, M. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001.

3.A. Aghajani, J. Banas and N. Sabzali, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin., 20 (2013), 345-358.

4.A. Aghajani, J. Banas and Y. Jalilian, Existence of solution for a class of nonlinear Volterra singular integral equations, Comp. Math. Appl., 62 (2011), 1215-1227.    

5.A. Aghajani and N. Sabzali, A coupled fixed point theorem for condensing operators with application to system of integral equations, J. Nonlinear Convex Anal., 15 (2014), 941-952.

6.A. Aghajani, R. Allahyari and M. Mursaleen, A generalization of Darbo's theorem with application to the solvability of systems of integral equations, J. Comput. Appl. Math., 260 (2014), 68-77.    

7.R. Arab, Some fixed point theorems in generalized Darbo fixed point theorem and the existence of solutions for system of integral equations, J. Korean Math. Soc., 52 (2015), 125-139.    

8.R. Arab, The existence of fixed points via the measure of noncompactness and its application to functional-integral equations, Mediterr. J. Math., 13 (2016), 759-773.    

9.J. Banas, Measures of noncompactness in the space of continuous tempered functions, Demonstr. Math., 14 (1981), 127-133.

10.J. Banas and K. Goebel, Measures of noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, New York: Dekker, 1980.

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

12. G. Darbo, Punti uniti in transformazion a condominio non compatto, Rend. Sem. Math. Univ. Padova, 24 (1955), 84-92.

13.M. M. El-Borai, Semigroups and some nonlinear fractional differential equations, Appl. Math. Comput., 149 (2004), 823-831.

14.M. A. A. El-Sayeed, Fractional order diffusion wave equation, Int. J. Theor. Phys., 35 (1996), 311-322.    

15.D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373 of Mathematics and Its Applications, Dordrecht, The Netherlands : Kluwer Academic Publishers, 1996.

16.R. W. Ibrahim and H. A. Jalab, Existence of entropy solutions for nonsymmetric fractional systems, Entropy, 16 (2014), 4911-4922.    

17.R. W. Ibrahim and H. A. Jalab, Existence of Ulam stability for iterative fractional differential equations based on fractional entropy, Entropy, 17 (2015), 3172-3181.    

18.O. K. Jaradat, A. Al-Omari and S. Momani, Existence of the mild solution for fractional semilinear initial value problem, Nonlinear Anal., 69 (2008), 3153-3159.    

19.A. Kilbas, and S. Marzan, Cauchy problem for differential equation with Caputo derivative, Fract. Calc. Appl. Anal., 7 (2004), 297-321.

20.M. Mursaleen and S. A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in lp spaces, Nonlinear Anal. Theory, Methods Appl., 75 (2012), 2111-2115.    

21.M. Mursaleen and A. Alotaibi, Infinite system of differential equations in some BK spaces, Abstr. Appl. Anal., 2012 (2012), Article ID 863483.

22.C. Tsallis, Generalized entropy-based criterion for consistent testing, Phys. Rev. E, 58 (1998), 1442.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved