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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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 18 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1492502551

Digital Object Identifier
doi:10.1216/RMJ-2017-47-2-571

Mathematical Reviews number (MathSciNet)
MR3635375

Zentralblatt MATH identifier
1365.35214

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

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

Citation

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. https://projecteuclid.org/euclid.rmjm/1492502551


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References

  • 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.