Rocky Mountain Journal of Mathematics

Some existence and uniqueness results for nonlinear fractional partial differential equations

H.R. Marasi, H. Afshari, and C.B. Zhai

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In this paper, we study the existence and uniqueness of positive solutions for some nonlinear fractional partial differential equations via given boundary value problems by using recent fixed point results for a class of mixed monotone operators with convexity.

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Rocky Mountain J. Math., Volume 47, Number 2 (2017), 571-585.

First available in Project Euclid: 18 April 2017

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Zentralblatt MATH identifier

Primary: 34B18: Positive solutions of nonlinear boundary value problems

Fractional partial differential equation normal cone boundary value problem mixed monotone operator


Marasi, H.R.; Afshari, H.; Zhai, C.B. Some existence and uniqueness results for nonlinear fractional partial differential equations. Rocky Mountain J. Math. 47 (2017), no. 2, 571--585. doi:10.1216/RMJ-2017-47-2-571.

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  • H. Afshari, S.H. Rezapour and N. Shahzad, Some notes on $(\alpha,\beta)$-hybrid mappings, J. Nonlin. Anal. Optim. 3 (2012), 119–135.
  • R.P. Agarwal, V. Lakshmikanthan and J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlin. Anal. Th. 72 (2010), 2859–2862.
  • B. Ahmad and J.J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal. 2009, article ID 494720, 2009.
  • A.A.M. Arafa, Series solutions of time-fractional host-parasitoid systems, J. Stat. Phys. 145 (2011), 1357-1367.
  • D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional calculus: Models and numerical methods, in Series on complexity, nonlinearity and chaos, World Scientific, Singapore, 2012.
  • D. Baleanu, O.G. Mustafa and R.P. Agarwal, On the solution set for a class of sequential fractional differential equations, J. Phys. Math. Th. 43, article ID 385209, 2010.
  • M. Belmekki, J.J. Nieto and R. Rodriguez-Lopez, Existence of periodic solution for a nonlinear fractional equation, Bound. Value Prob. 2009, article ID 324561, 2009.
  • D. Delbosco and L. Radino, Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl. 204 (1996), 609–625.
  • A.M.A. El-Sayed, S.Z. Rida and A.A.M. Arafa, On the solutions of the generalized reaction-diffusion model for bacterial colony, Acta Appl. Math. 110 (2010), 1501–1511.
  • Y. Fujita, Cauchy problems for fractional order and stable processes, Japan J. Appl. Math. 7 (1990), 459–476.
  • M. Giona and H.E. Roman, Fractional diffusion equation on fractals: One-dimensional case and asymptotic behaviour, J. Phys. 25 (1992), 2093–2105.
  • D. Guo, Fixed points of mixed monotone operators with applications, Appl. Anal. 34 (1988), 215–224.
  • D. Guo and V. Lakskmikantham, Coupled fixed points of nonlinear operators with applications, Nonlin. Anal. Th. 11 (1987), 623–632.
  • I. Hashim, O. Abdulaziz and S. Momani, Homotopy analysis method for fractional IVPs, Comm. Nonlin. Sci. Numer. Simu. 14 (2009), 674–684.
  • J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comp. Meth. Appl. Mech. Eng. 167 (1998), 57-68.
  • H. Jafari and V. Daftardar-Gejji, Positive solution of nonlinear fractional boundary value problems using Adomin decomposition method, J. Appl. Math. Comp. 180 (2006), 700–706.
  • A.A. Kilbas, Partial fractional differential equations and some of their applications, Analysis 30 (2010), 35–66.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo Theory and applications of fractiona differential equations. North-Holland Math. Stud. 204, North-Holland, Amsterdam, 2006.
  • F. Mainardi, The time fractional diffusion-wave equation, Radiofisika 38 (1995), 20–36.
  • H.R. Marasi, H. Afshari, M. Daneshbastam and C.B. Zhai, Fixed points of mixed monotone operators for existence and uniqueness of nonlinear fractional differential equations, J. Contemp. Math. Anal. 52 (2017), 8.
  • H.R. Marasi, H. Piri and H. Aydi, Existence and multiplicity of solutions for nonlinear fractional differential equations, J. Nonlin. Sci. Appl. 9 (2016), 4639–4646.
  • K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.
  • R.R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. 133 (1986), 425–430.
  • K.B. Oldham and J. Spainer, The fractional calculus, Academic Press, New York, 1974.
  • I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
  • A.V. Pskhu, Partial differential equations of fractional order, Nauka, Moscow, 2005.
  • T. Qiu and Z. Bai, Existence of positive solutions for singular fractional equations, Electr. J. Diff. Eq. 146 (2008), 1–9.
  • H.E. Roman and M. Giona, Fractional diffusion equation on fractals: Three-dimensional case and scattering function, J. Phys. 25 (1992), 2107–2117.
  • J. Sabatier, O.P. Agarwal and J.A.T. Machado, Advances in fractional calculus: Theoritical developments and applications in physics and engineering, Springer, Berlin, 2002.
  • S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integral and derivative: Theory and applications, Gordon and Breach, Switzerland, 1993.
  • H. Weitzner and G.M. Zaslavsky, Some applications of fractional equations, Comm. Nonlin. Sci. Numer. Simul. 15 (2010), 935–945.
  • C.B. Zhai, Fixed point theorems for a class of mixed monotone operators with convexity, Fixed Point Th. Appl. 2013 (2013), 119.
  • C.B. Zhai and X.M. Cao, Fixed point theorems for $\tau$-$\varphi$-concave operators and applications, Comput. Math. Appl. 59 (2010), 532–538.
  • C.B. Zhai and C.M. Guo, $\alpha$-convex operators, J. Math. Anal. Appl. 316 (2006), 556–565.
  • C.B. Zhai and M.R. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems, Nonlin. Anal. Th. 75 (2012), 2542–2551.
  • C.B. Zhai and L.L. Zhang, New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems, J. Math. Anal. Appl. 382 (2011), 594–614.
  • S. Zhang, The existence of a positive solution for nonlinear fractional equation, J. Math. Anal. Appl. 252 (2000), 804–812.
  • Y. Zhao, S.H. Sun and Z. Han, The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Comm. Nonlin. Sci. Numer. Simu. 16 (2011), 2086–2097.