Journal of Integral Equations and Applications

Weak solutions for partial Pettis Hadamard fractional integral equations with random effects

Saïd Abbas, Wafaa Albarakati, Mouffak Benchohra, and Yong Zhou

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In this article, we apply M\"onch and Engl's fixed point theorems associated with the technique of measure of weak noncompactness to investigate the existence of random solutions for a class of partial random integral equations via Hadamard's fractional integral, under the Pettis integrability assumption.

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J. Integral Equations Applications, Volume 29, Number 4 (2017), 473-491.

First available in Project Euclid: 10 November 2017

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Primary: 26A33: Fractional derivatives and integrals 45G05: Singular nonlinear integral equations 45N05: Abstract integral equations, integral equations in abstract spaces

Random functional integral equation partial Pettis Hadamard fractional integral measure of weak noncompactness random solution


Abbas, Saïd; Albarakati, Wafaa; Benchohra, Mouffak; Zhou, Yong. Weak solutions for partial Pettis Hadamard fractional integral equations with random effects. J. Integral Equations Applications 29 (2017), no. 4, 473--491. doi:10.1216/JIE-2017-29-4-473.

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  • S. Abbas, M. Benchohra and G.M. N'Guérékata, Topics in fractional differential equations, Springer, New York, 2012.
  • ––––, Advanced fractional differential and integral equations, Nova Science Publishers, New York, 2015.
  • S. Abbas, M. Benchohra and B.A. Slimani, Existence and Ulam stabilities for partial fractional random differential inclusions with nonconvex right hand side, Pan American Math. J. 25 (2015), 95–110.
  • S. Abbas, M. Benchohra and A.N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Fract. Calc. Appl. Anal. 15 (2012), 168–182.
  • R.R. Akhmerov, M.I. Kamenskii, A.S. Patapov, A.E. Rodkina and B.N. Sadovskii, Measures of noncompactness and condensing operators, Birkhauser Verlag, Basel, 1992.
  • J.C. Alvàrez, Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces, Rev. Real. Acad. Cienc. Exact. 79 (1985), 53–66.
  • J. Bana\`s and K. Goebel, Measures of noncompactness in Banach spaces, Marcel Dekker, New York, 1980.
  • A. Benaissa and M. Benchohra, Functional differential equations with state-dependent delay and random effects, Rom. J. Math. Comp. Sci. 5 (2015), 84–94.
  • M. Benchohra, J. Graef and F-Z. Mostefai, Weak solutions for boundary-value problems with nonlinear fractional differential inclusions, Nonl. Dynam. Syst. Th. 11 (2011), 227–237.
  • M. Benchohra, J. Henderson and F-Z. Mostefai, Weak solutions for hyperbolic partial fractional differential inclusions in Banach spaces, Comp. Math. Appl. 64 (2012), 3101–3107.
  • M. Benchohra, J. Henderson, S.K. Ntouyas and A. Ouahab, Existence results for functional differential equations of fractional order, J. Math. Anal. Appl. 338 (2008), 1340–1350.
  • M. Benchohra, J. Henderson and D. Seba, Measure of noncompactness and fractional differential equations in Banach spaces, Comm. Appl. Anal. 12 (2008), 419–428.
  • D. Bugajewski and S. Szufla, Kneser's theorem for weak solutions of the Darboux problem in a Banach space, Nonlin. Anal. 20 (1993), 169–173.
  • P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), 1–27.
  • P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002), 1–15.
  • F.S. De Blasi, On the property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. 21 (1977), 259–262.
  • B.C. Dhage, S.V. Badgire and S.K. Ntouyas, Periodic boundary value problems of second order random differential equations, Electr. J. Qual. Th. Diff. Eq. 21 (2009), 1–14.
  • H.W. Engl, A general stochastic fixed-point theorem for continuous random operators on stochastic domains, J. Math. Anal. Appl. 66 (1978), 220–231.
  • D. Guo, V. Lakshmikantham and X. Liu, Nonlinear integral equations in abstract spaces, Kluwer Academic Publishers, Dordrecht, 1996.
  • J. Hadamard, Essai sur l'étude des fonctions données par leur développment de Taylor, J. Pure Appl. Math. 4 (1892), 101–186.
  • X. Han, X. Ma and G. Dai, Solutions to fourth-order random differential equations with periodic boundary conditions, Electr. J. Diff. Eq. 235 (2012), 1–9.
  • R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
  • S. Itoh, Random fixed point theorems with applications to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), 261–273.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science, Amsterdam, 2006.
  • K.S. Miller and B. Ross, An introduction to the fractional calculus and differential equations, John Wiley, New York, 1993.
  • A.R. Mitchell and Ch. Smith, Nonlinear equations in abstract spaces, in An existence theorem for weak solutions of differential equations in Banach spaces, V. Lakshmikantham, ed., Academic Press, New York, 1978.
  • D. O'Regan, Fixed point theory for weakly sequentially continuous mapping, Math. Comp. Mod. 27 (1998), 1–14.
  • ––––, Weak solutions of ordinary differential equations in Banach spaces, Appl. Math. Lett. 12 (1999), 101–105.
  • B.J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277–304.
  • S. Pooseh, R. Almeida and D. Torres, Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim. 33 (2012), 301–319.
  • S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives. Theory and applications, Gordon and Breach, Yverdon, 1993.
  • V.E. Tarasov, Fractional dynamics: Application of fractional calculus to dynamics of particles, fields and media, Springer, Heidelberg, 2010.
  • J. Wang, M. Feckan and Y. Zhou, On the new concept of solutions and existence results for impulsive fractional evolution equations, Dynam. Part. Diff. Eq. 8 (2011), 345–361.
  • J. Wang, A.G. Ibrahim and M. Feckan, Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces, Appl. Math. Comp. 257 (2015), 103–118.
  • J. Wang and Y. Zhang, On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives, Appl. Math. Lett. 39 (2015), 85–90.
  • J. Wang and Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlin. Anal. 12 (2011), 3642–3653.
  • Y. Zhou, Basic theory of fractional differential equations, World Scientific, Singapore, 2014.
  • ––––, Fractional evolution equations and inclusions: Analysis and control, Academic Press, San Diego, 2016.
  • Y. Zhou, V. Vijayakumar and R. Murugesu, Controllability for fractional evolution inclusions without compactness, Evol. Eq. Contr. Th. 4 (2015), 507–524.