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Applicable Analysis
An International Journal
Volume 87, 2008 - Issue 7
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Original Articles

Existence results for boundary value problems with non-linear fractional differential equations

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Pages 851-863 | Received 28 Apr 2008, Accepted 19 Jun 2008, Published online: 29 Sep 2008

References

  • Diethelm , K and Freed , AD . 1999 . “ On the solution of non-linear fractional order differential equations used in the modeling of viscoplasticity ” . In Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties , Edited by: Keil , F , Mackens , W , Voss , H and Werther , J . 217 – 224 . Heidelberg : Springer-Verlag .
  • Gaul , L , Klein , P and Kempfle , S . 1991 . Damping description involving fractional operators . Mech. Systems Signal Processing , 5 : 81 – 88 .
  • Glockle , WG and Nonnenmacher , TF . 1995 . A fractional calculus approach of self-similar protein dynamics . Biophys. J. , 68 : 46 – 53 .
  • Hilfer , R . 2000 . Applications of Fractional Calculus in Physics , Singapore : World Scientific .
  • Mainardi , F . 1997 . “ Fractional calculus: Some basic problems in continuum and statistical mechanics ” . In Fractals and Fractional Calculus in Continuum Mechanics , Edited by: Carpinteri , A and Mainardi , F . 291 – 348 . Wien : Springer-Verlag .
  • Metzler , F . 1995 . Relaxation in filled polymers: A fractional calculus approach . J. Chem. Phys. , 103 : 7180 – 7186 .
  • Oldham , KB and Spanier , J . 1974 . The Fractional Calculus , New York, London : Academic Press .
  • Kilbas , AA , Srivastava , HM and Trujillo , JJ . 2006 . “ Theory and Applications of Fractional Differential Equations ” . In North-Holland Mathematics Studies , Vol. 204 , Amsterdam : Elsevier Science B.V. .
  • Miller , KS and Ross , B . 1993 . An Introduction to the Fractional Calculus and Differential Equations , New York : John Wiley .
  • Samko , SG , Kilbas , AA and Marichev , OI . 1993 . “ Fractional Integrals and Derivatives ” . In Theory and Applications , Yverdon : Gordon and Breach .
  • Agarwal , RP , Benchohra , M and Hamani , S . 2008 . Boundary value problems for differential inclusions with fractional order . Adv. Stud. Contemp. Math , 16 : 181 – 196 .
  • Benchohra , M and Hamani , S . Nonlinear boundary value problems for differential inclusions with Caputo fractional derivative . Topol. Meth. Nonlinear Anal , (to appear)
  • Delbosco , D and Rodino , L . 1996 . Existence and uniqueness for a non-linear fractional differential equation . J. Math. Anal. Appl. , 204 : 609 – 625 .
  • Diethelm , K and Ford , NJ . 2002 . Analysis of fractional differential equations . J. Math. Anal. Appl. , 265 : 229 – 248 .
  • Diethelm , K and Walz , G . 1997 . Numerical solution of fractional order differential equations by extrapolation . Numer. Algorithms , 16 : 231 – 253 .
  • El-Sayed , AMA . 1995 . Fractional order evolution equations . J. Fract. Calc. , 7 : 89 – 100 .
  • El-Sayed , AMA . 1996 . Fractional order diffusion-wave equations . Intern. J. Theo. Physics , 35 : 311 – 322 .
  • El-Sayed , AMA . 1998 . Nonlinear functional differential equations of arbitrary orders . Nonlinear Anal. , 33 : 181 – 186 .
  • Kaufmann , ER and Mboumi , E . 2007 . Positive solutions of a boundary value problem for a non-linear fractional differential equation . Electron. J. Qual. Theory Differ. Equ. , 3 : 11
  • Kilbas , AA and Marzan , SA . 2005 . Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions . Differ. Equ. , 41 : 84 – 89 .
  • Momani , SM and Hadid , SB . 2003 . Some comparison results for integro-fractional differential inequalities . J. Fract. Calc. , 24 : 37 – 44 .
  • Momani , SM , Hadid , SB and Alawenh , ZM . 2004 . Some analytical properties of solutions of differential equations of noninteger order . Int. J. Math. Math. Sci. , 2004 : 697 – 701 .
  • Podlubny , I . 2002 . Analogue realizations of fractional-order controllers. Fractional order calculus and its applications . Nonlinear Dynam. , 29 : 281 – 296 .
  • Yu , C and Gao , G . 2005 . Existence of fractional differential equations . J. Math. Anal. Appl. , 310 : 26 – 29 .
  • Lakshmikantham , V and Vatsala , AS . 2008 . Basic theory of fractional differential equations . Nonlinear Anal. , 69 : 2677 – 2681 .
  • Lakshmikantham , V and Vatsala , AS . 2007 . Theory of fractional differential inequalities and applications . Commun. Appl. Anal. , 11 : 395 – 402 .
  • Lakshmikantham , V and Vatsala , AS . 2008 . General uniqueness and monotone iterative technique for fractional differential equations . Appl. Math. Letters , 21 : 828 – 834 .
  • Belarbi , A , Benchohra , M and Ouahab , A . 2006 . Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces . Appl. Anal. , 85 : 1459 – 1470 .
  • Benchohra , M . 2008 . Existence results for fractional order functional differential equations with infinite delay . J. Math. Anal. Appl. , 338 : 1340 – 1350 .
  • Belarbi , A . 2007 . Perturbed functional differential equations with fractional order . Commun. Appl. Anal. , 11 : 429 – 440 .
  • Benchohra , M , Hamani , S and Ntouyas , SK . 2008 . Boundary value problems for differential equations with fractional order . Surv. Math. Appl. , 3 : 1 – 12 .
  • Belarbi , A , Benchohra , M and Dhage , BC . 2006 . Existence theory for perturbed boundary value problems with integral boundary conditions . Georgian Math. J. , 13 : 215 – 228 .
  • Benchohra , M , Hamani , S and Henderson , J . 2007 . Functional differential inclusions with integral boundary conditions . Electron. J. Qual. Theory Differ. Equ. , 15 : 1 – 13 .
  • Benchohra , M , Hamani , S and Nieto , JJ . The method of upper and lower solutions for second order differential inclusions with integral boundary conditions . Rocky Mountain J. Math , (to appear)
  • Heymans , N and Podlubny , I . 2006 . Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives . Rheologica Acta. , 45 : 765 – 772 .
  • Podlubny , I . 2002 . Geometric and physical interpretation of fractional integration and fractional differentiation . Fract. Calculus Appl. Anal. , 5 : 367 – 386 .
  • Blayneh , KW . 2002 . Analysis of age structured host-parasitoid model . Far East J. Dyn. Syst. , 4 : 125 – 145 .
  • Adomian , G and Adomian , GE . 1985 . Cellular systems and aging models . Comput. Math. Appl. , 11 : 283 – 291 .
  • Podlubny , I . 1999 . Fractional Differential Equation , San Diego : Academic Press .
  • Zhang , S . 2006 . Positive solutions for boundary-value problems of non-linear fractional differential equations . Electron. J. Differ. Equ. , 36 : 1 – 12 .
  • Granas , A and Dugundji , J . 2003 . Fixed Point Theory , New York : Springer-Verlag .
  • Burton , TA and Kirk , C . 1998 . A fixed point theorem of Krasnoselskii-Schaefer type . Math. Nachr. , 189 : 23 – 31 .

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