Some theorems on fractional semilinear evolution equations

• Autorzy: Tidke H. L.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we investigate the existence, uniqueness and other properties of solutions of fractional semilinear evolution equations in Banach spaces. The results are obtained by using fractional calculus, the well-known Banach fixed point theorem coupled with Bielecki type norm and the integral inequality established by E. Hernandez.

Słowa kluczowe

EN

existence of solution; evolution equation; fractional calculus; continuous dependence; Hernandez's inequality

PL

istnienie rozwiązania; równania ewolucyjne; pochodna ułamkowa; zależność ciągła; nierówność Hernandeza

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2012
• Tom: Vol. 18, nr 2
• Strony: 209--224
• Opis fizyczny: Bibliogr. 38 poz.

Twórcy

autor

Tidke H. L.

• Department of Mathematics, School of Mathematical Sciences, North Maharashtra University, Jalgaon 425 001, India,

Bibliografia

• [1] S.A. Abd El-Salam and A. M. A. El-Sayed, On the stability of some fractional order non-autonomous systems, Electron. J. Qual. Theory Differ. Equ. (2007), paper no. 6, 1-14.
• [2] R. P. Agarwal, M. Benchohra and S. Hamani, Boundary value problems for fractional differential equations, Adv. Stud. Contemp. Math. 16 (2008), 181-196.
• [3] A. Anguraj, P. Karthikeyan and G. M. N’Guerekata, Nonlocal Cauchy problem for some fractional abstract differential equations in Banach spaces, Comm. Math. Analysis 6 (2009), 31-35.
• [4] K. Assaleh and W. M. Ahmad, Modeling of speech signals using fractional calculus, in: 9th International Symposium on Signal Processing and Its Applications (ISSPA 2007), IEEE Conference Publications (2007), 1-4.
• [5] K. Balachandran and S. Kiruthika, Existence of solutions of abstract fractional impulsive semilinear evolution equations, Electron. J. Qual. Theory Differ. Equ. (2010), paper no. 4, 1-12.
• [6] K. Balachandran and J. Y. Park, Nonlocal Cauachy problem for abstract fractional semilinear evolution equations, Nonlinear Anal. 71 (2009), 4471-475.
• [7] M. Benchohra and B. A. Slimani, Existence and uniqueness of solutions to impulsive fractional differential equations, Electron. J. Differential Equations (2009), paper no. 10, 1-11.
• [8] A. Bielecki, Une remarque sur la methode de Banach-Cacciopoli-Tikhonov dans la theorie des equations differentilles ordinaires, Bull. Acad. Polon. Sci. 4 (1956), 261— 264.
• [9] B. Bonilla, M. Rivero, L. Rodriguez-Germa and J. J. Trujillo, Fractional differential equations as alternative models to nonlinear equations, Appl. Math. Comput. 187 (2007), 79-88.
• [10] M. Caputo, Diffusion with space memory modelled with distributed order space fractional differential equations, Annals of Geophysics 46 (2003), 223-234.
• [11] C. Corduneanu, Bielecki method in the theory of integral equations, Ann. Univ. Mariae Curie-Sklodowskia, Section A 38 (1984), 23-t0.
• [12] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, 1991.
• [13] D. Craiem and R. L. Magin, Perspectives: Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics, Phys. Biol. 7(2010),013001.
• [14] S. Das, Functional Fractional Calculus, Springer, Berlin, 2011.
• [15] D. Delbosco and L. Rodino, Existence and uniqueness for a fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609-625.
• [16] K. Diethelm and A. D. Freed, On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, in: Scientifice Computing in Chemical Engineering II - Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, Springer, Heidelberg (1999), 217-224.
• [17] J. F. Douglas, Some applications of fractional calculus to polymer science, in: Advances in Chemical Physics 102, John Wiley and Sons (1997), 121-191.
• [18] A. M. A. El-Sayed and S. A. Abd El-Salam, On the stability of a fractional order differential equation with nonlocal initial condition, Electron. J. Qual. Theory Differ. Equ. (2008), paper no. 29, 1-8.
• [19] L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Syst. Sig. Process. 5 (1991), 81-88.
• [20] W. G. Glockle and T. F. Nonnenmacher, A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68 (1995), 46-53.
• [21] E. Hernanadez, Existence result for partial neutral functional integrodifferential equations with unbounded delay, J. Math. Anal. Appl. 292 (2004), 194-210.
• [22] R. Hilfer, Application of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [23] O. K. Jaradat, A. Al-Omari and S. Momani, Existence of the mild solution for fractional semilinear initial value problems, Nonlinear Anal. 69 (2008), 3153-3159.
• [24] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.
• [25] V. V. Kulish and J. L. Lage, Application of fractional calculus to fluid mechanics, J. Fluids Eng. 124 (2002), 803-806.
• [26] V. Lakshmikantham, Theory of fractional differential equations, Nonlinear Anal. 60 (2008), 3337-3343.
• [27] V. Lakshmikantham and A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. 69 (2008), 2677-2682.
• [28] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien (1997), 291-348.
• [29] F. Metzler, W. Schick, H. G. Kilian and T. F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995), 7180-7186.
• [30] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.
• [31] G. M. N’Guerekata, A Cauchy problem for some fractional abstract differential equation with nonlocal conditions, Nonlinear Anal 70 (2009), 1873-1876.
• [32] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
• [33] A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
• [34] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering 198, Academic Press, San Diego, 1999.
• [35] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calculus Appl. Anal. 5 (2002), 367-386.
• [36] J. Sabatier, O. P. Agrawal and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, 2007.
• [37] N. Sebaa, Z. E. A. Fellah, W. Lauriks and C. Depollier, Application of fractional calculus to ultrasonic wave propagation in human cancellous bone, Signal Process. Arch. 86 (2006), 2668-2677.
• [38] A. J. Turski, B. Atamaniuk and E. Turska, Application of fractional derivative operators to anomalous diffusion and propagation problems, preprint (2007), http: //arxiv.org/abs/math-ph/0701068.