Weak Solutions of Fractional Order Differential Equations via Volterra-Stieltjes Integral Operator

The fractional derivative of the Riemann-Liouville and Caputo types played an important role in the development of the theory of fractional derivatives, integrals and for its applications in pure mathematics([18], [ 21]). In this paper, we study the existence of weak solutions for fractional differential equations of Riemann-Liouville and Caputo types. We depend on converting of mentioned equations to the form of functional integral equations of Voltera-Stieltjes type in reflexive Banach spaces.

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1. J. Banaś, Some properties of Urysohn-Stieltjes integral operators, Internat. J. Math. and Math. Sci. 21 (1998) 79-88.
2. J. Banaś, K. Sadarangani, Solvability of Volterra-Stieltjes operator-integral equations and their applications, Comput. Math. Appl. 41 12 (2001) 1535-1544.
3. J. Banaś, J.C. Mena, Some properties of nonlinear Volterra-Stieltjes integral operators, Comput. Math. Appl. 49 (2005) 1565-1573.
4. J. Banaś, D. O'Regan, Volterra-Stieltjes integral operators, Math. Comput. Modelling. 41 (2005) 335-344.
5. J. Banaś, T. Zając, A new approach to the theory of functional integral equations of fractional order, J. Math. Anal. Appl. 375 (2011) 375-387.
6. M. Benchohra, F. Mostefai, Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces, Opuscula Mathematica 32 1 (2012) 31-40.
7. M. Benchohra, J.R. Graef and F. Mostefai, Weak solutions for nonlinear fractional differential equations on reflexive Banach spaces, Electron. J. Qual. Theory Differ. Equ. 54 (2010) 1-10.
8. C.W. Bitzer, Stieltjes-Volterra integral equations, Illinois J. Math. 14 (1970) 434-451.
9. M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J.R. Astr. Soc. 13 (1967) 529-539.
10. N. Dunford, J.T. Schwartz, Linear Operators, Interscience, Wiley, New York 1958.
11. A.M.A. El-Sayed, W.G. El-Sayed and A.A.H. Abd El-Mowla, Volterra-Stieltjes integral equation in reflexive Banach spaces, Electronic Journal of Mathematical Analysis and Applications 5 1 (2017) 287-293.
12. R.F. Geitz, Pettis integration, Proc. Amer. Math. Soc. 82 (1981) 81-86.
13. E. Hille, R.S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ. Providence, R. I. 1957.
14. H.H.G. Hashem, Weak solutions of differential equations in Banach spaces, Journal of Fractional Calculus and Applications 3 1 (2012) 1-9.
15. T. Margulies, Wave propagation in viscoelastic horns using a fractional calculus rheology model, The Journal of the Acoustical  Society of America 114 2442 (2003), https://doi.org/10.1121/1.4779280.
16. B. Mathieu, P. Melchior, A. Oustaloup and Ch. Ceyral, Fractional differentiation for edge detection, Fractional Signal Processing and Applications 83 (2003) 2285-2480.
17. A.R. Mitchell, Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, Nonlinear Equations in Abstract Spaces (V. Lakshmikantham, ed.) (1978) 387-404.
18. I. Podlubny, Fractional Differential Equations, Academic Press, New York 1999.
19. D. O'Regan, Fixed point theory for weakly sequentially continuous mapping, Math. Comput. Modeling 27 (1998) 1-14.
20. A. Szep, Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces, Studia Sci. Math. Hungar. 6 (1971) 197-203.
21. S.G. Samko, A.A. Kilbas and O. Marichev, Integral and Derivatives of Fractional Orders and Some of Their Applications, Nauka i Teknika, Minsk 1987.
22. H.A.H. Salem, A.M.A. El-Sayed, Weak solution for fractional order integral equations in reflexive Banach spaces, Math. Slovaca 55 (2005) 169-181.
23. H.A.H. Salem, A.M.A. El-Sayed, A note on the fractional calculus in Banach spaces, Studia Sci. Math. Hungar. 42 2 (2005) 115-130.
24. H.A.H. Salem, Quadratic integral equations in reflexive Banach space, Discuss. Math. Differ. Incl. Control Optim. 30 (2010) 61-69.