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A Survey of Useful Inequalities in Fractional Calculus

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Abstract

We present a survey on inequalities in fractional calculus that have proven to be very useful in analyzing differential equations. We mention in particular, a “Leibniz inequality” for fractional derivatives of Riesz, Riemann-Liouville or Caputo type and its generalization to the d-dimensional case that become a key tool in differential equations; they have been used to obtain upper bounds on solutions leading to global solvability, to obtain Lyapunov stability results, and to obtain blowing-up solutions via diverging in a finite time lower bounds. We will also mention the weakly singular Gronwall inequality of Henry and its variants, principally by Medved, that plays an important role in differential equations of any kind. We will also recall some “traditional” inequalities involving fractional derivatives or fractional powers of the Laplacian.

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References

  1. A.A. Alikhanov, A priori estimates for solutions of boundary value problems for equations of fractional order. Differ. Equ. 46 (2010), 660–666.

    MathSciNet  MATH  Google Scholar 

  2. M. Al-Refai, On the fractional derivatives at extreme points, Electronic J. of Qualitative Theory of Differential Equations, 55 (2012), 1–5.

    MathSciNet  MATH  Google Scholar 

  3. M. Al-Refai and Y. Luchko, Maximum principle for the fractional diffusion equations with the Riemann-Liouville fractional derivative and its applications. Fract. Calc. Appl. Anal. 17, No 2 (2014), 483–498; DOI: 10.2478/s13540-014-0181-5; https://www.degruyter.com/view/j/fca.2014.17.issue-2/issue-files/fca.2014.17.issue-2.xml.

    MathSciNet  MATH  Google Scholar 

  4. A. Alsaedi, B. Ahmad and M. Kirane, Maximum principle for certain generalized time and space-fractional diffusion equations. Quart. Appl. Math. 73 (2015), 163–175.

    MathSciNet  MATH  Google Scholar 

  5. A. Alsaedi, B. Ahmad and M. Kirane, Nonexistence of global solutions of nonlinear space-fractional equations on the Heisenberg group. Electron. J. Differential Equations 2015 (2015), Art. ID No 227, 1–10.

    MATH  Google Scholar 

  6. G.A. Annastassiou, J.J. Koliha and J. Pečarić, Opial inequalities for fractional derivatives. Dynam. Systems Appl. 10 (2001), 395–406.

    MathSciNet  MATH  Google Scholar 

  7. V.V. Arestov, Inequalities for fractional derivatives on the half-line, Approximation theory. Banach Center Publications 4 (1979), 19–34.

    Google Scholar 

  8. A. Babakhani, H. Agahi, R. Mesiar, A (*, *)-based Minkowskis inequality for Sugeno fractional integral of order α > 0. Fract. Calc. Appl. Anal. 18, No 4 (2015), 862–874; DOI: 10.1515/fca-2015-0052; https://www.degruyter.com/view/j/fca.2015.18.issue-4/issue-files/fca.2015.18.issue-4.xml.

    MathSciNet  MATH  Google Scholar 

  9. V. F. Babenko and M. S. Churilova, On the Kolmogorov type inequalities for fractional derivatives. East J. of Approximations 8, No 4 (2002), 537–446.

    MathSciNet  Google Scholar 

  10. V.F. Babenko, M.S. Churilova, N.V. Parfinovych and D.S. Skorokhodov, Kolmogorov type inequalities for the Marchaud fractional derivatives on the real line and the half-line. J. of Inequalities and Applications 2014 (2014), 504.

    MathSciNet  MATH  Google Scholar 

  11. S. Belarbi and Z. Dahmani, On some new fractional integral inequalities. J. Inequal. Pure Appl. Math. 10, No 3 (2009), Art. ID 86, 5 pp.

    Google Scholar 

  12. L.A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. of Math. 171 (2010), 1903–1930.

    MathSciNet  MATH  Google Scholar 

  13. P. Constantin, Euler equations. Navier-Stokes equations and turbulence. Mathematical foundation of turbulent viscous flows. In: Lecture Notes in Math. 1871, Springer, Berlin (2006), 117, 1–43.

    MathSciNet  Google Scholar 

  14. P. Constantin and V. Vicol, Nonlinear maximum principle for dissipative linear nonlocal operators and applications. Geom. Funct. Anal. 22 (2012), 1289–1321.

    MathSciNet  MATH  Google Scholar 

  15. A. Cordoba and D. Cordoba, A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249 (2004), 511–528.

    MathSciNet  MATH  Google Scholar 

  16. J.I. Diaz, T. Pierantozzi and L. Vazquez, On the finite time extinction phenomenon for some nonlinear fractional evolution equations. Symposium on Applied Fractional Calculus, Badajoz (2007).

    Google Scholar 

  17. M.A. Duarte-Mermoud, N. Aguila-Camacho and J.A. Gallegos, Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2951–2957.

    MathSciNet  MATH  Google Scholar 

  18. S. Eilertsen, On weighted positivity and the Wiener regularity of a boundary point for the fractional Laplacian. Ark. Mat. 38 (2000), 53–75.

    MathSciNet  MATH  Google Scholar 

  19. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida (1992).

    MATH  Google Scholar 

  20. F. Ferrari and B. Franchi, Harnack inequality for fractional sub-Laplacian in Carnot groups. Math. Z. 279 (2015), 435–458.

    MathSciNet  MATH  Google Scholar 

  21. R.A.C. Ferreira, Some discrete fractional Lyapunov-type inequalities. Fract. Differ. Calc. 5, No 1 (2015), 87–92; DOI: 10.7153/fdc-05-08.

    MathSciNet  MATH  Google Scholar 

  22. R. Ferreira, Lyapunov-type inequality for an anti-periodic fractional boundary value problem. Fract. Calc. Appl. Anal. 20, No 1 (2017), 284–291; DOI: 10.1515/fca-2017-0015; https://www.degruyter.com/view/j/fca.2017.20.issue-1/issue-files/fca.2017.20.issue-1.xml..

    MathSciNet  MATH  Google Scholar 

  23. S.P. Geisberg, An extension of the Hadamard inequality. Sb. Nauch. Tr. LOMI 50 (1965), 42–54.

    MathSciNet  Google Scholar 

  24. G.H. Hardy, E. Landau and J.E. Littlewood, Some inequalities satisfied by the integrals or derivatives of real or analytic functions. Mathematische Zeitschrift 39 (1935), 677–695.

    MathSciNet  MATH  Google Scholar 

  25. N. Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Comm. Math. Phys. 251 (2004), 365–376.

    MathSciNet  MATH  Google Scholar 

  26. R.J. Hughes, Hardy-Landau-Littlewood inequalities for fractional derivatives in weighted Lp spaces. J. London Math. Soc. 32-35 (1987), 489–498.

    MATH  Google Scholar 

  27. R.J. Hughes, On fractional integrals and derivatives in Lp. Indiana Univ. Math. J. 26 (1977), 325–328.

    MathSciNet  MATH  Google Scholar 

  28. T. Kato, G. Ponce, Cummutator estimatesand the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988), 891–907.

    MathSciNet  MATH  Google Scholar 

  29. T.D. Ke, N. Van Loi, V. Obukhovskii, Decay solutions for a class of fractional differential variational inequalities. Fract. Calc. Appl. Anal. 18, No 3 (2015), 531–553; DOI: 10.1515/fca-2015-0033; https://www.degruyter.com/view/j/fca.2015.18.issue-3/issue-files/fca.2015.18.issue-3.xml.

    MathSciNet  MATH  Google Scholar 

  30. C.E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46 (1993), 527–620.

    MathSciNet  MATH  Google Scholar 

  31. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam (2006).

    MATH  Google Scholar 

  32. M. Kirane and N. Tatar, Global existence and stability of some semilinear problems. Arch. Math. (Brno) 36 (2000), 33–44.

    MathSciNet  MATH  Google Scholar 

  33. S.G. Krein, Linear Differential Equations in Banach Space. AMS Translations of Math. Monographs, Vol. 29, Providence, R.I. (1971).

  34. S.-Y. Lin, Generalized Gronwall inequalities and their applications to fractional differential equations. J. Inequal. Appl. 2013 (2013), Art. ID 549, 9 pp.

    MathSciNet  MATH  Google Scholar 

  35. Q.-H. Ma and J. Pecaric, Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations. J. Math. Anal. Appl. 341 (2008), 894–905.

    MathSciNet  MATH  Google Scholar 

  36. M. Medved, A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. Appl. 214 (1997), 349–366.

    MathSciNet  MATH  Google Scholar 

  37. E. Mitidieri and S.I. Pohozaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. Proc. Steklov Inst. Math. 234 (2001), 1–362.

    MATH  Google Scholar 

  38. D. Mitrovic, On a Leibnitz type formula for fractional derivatives. Filomat 27, No 6 (2013), 1141–1146.

    MathSciNet  MATH  Google Scholar 

  39. A.M. Nakhushev, Fractional Calculus and its Applications. Fizmatlit, Moskva (2003) (in Russian).

    MATH  Google Scholar 

  40. S.K. Ntouyas, S.D. Purohit and J. Tariboon, Certain Chebyshev type integral inequalities involving Hadamard’s fractional operators. Abstr. Appl. Anal. 2014 (2014), Art. ID 249091, 7 pp.

    MathSciNet  MATH  Google Scholar 

  41. Y.J. Park, Fractional Gagliardo-Nirenberg inquality. J. Chungcheong Mathematical Society 24 (2011), 583–586.

    Google Scholar 

  42. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Science Publ. (1993).

    MATH  Google Scholar 

  43. H. Tanabe, Equations of Evolution. Monographs and Studies in Mathematics, No. 6, Pitman, London-San Francisco-Melbourne (1979).

    MATH  Google Scholar 

  44. P. Thiramanus, J. Tariboon and S. K. Ntouyas, Henry-Gronwall integral inequalities with maxima and their applications to fractional differential equations. Abstr. Appl. Anal. 2014 (2014), Art. ID 276316, 10 pp.

    MathSciNet  MATH  Google Scholar 

  45. Z. Ye and X. Xu, Global well-posedness of the 2-D Boussinesq equations with fractional Laplacian dissipation. J. Differential Equations 260 (2016), 67166744.

  46. J. Wu, Lower bound for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces. Commun. Math. Phys. 263 (2006), 803–831.

    MathSciNet  MATH  Google Scholar 

  47. R. Zacher, Global strong solvability of a quasilinear subdiffusion problem. J. Evol. Equ. 12, No 4 (2012), 813–831.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. NAAM Research Group, Department of Mathematics Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Ahmed Alsaedi & Bashir Ahmad

  2. LaSIE, Pôle Sciences et Technologies, Université de La Rochelle Avenue M. Crépeau, 17042, La Rochelle Cedex, France

    Mokhtar Kirane

  3. NAAM Research Group, Department of Mathematics Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Mokhtar Kirane

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  1. Ahmed Alsaedi
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  2. Bashir Ahmad
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Correspondence to Ahmed Alsaedi.

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Alsaedi, A., Ahmad, B. & Kirane, M. A Survey of Useful Inequalities in Fractional Calculus. FCAA 20, 574–594 (2017). https://doi.org/10.1515/fca-2017-0031

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