EXISTENCE AND MULTIPLE POSITIVE SOLUTIONS TO SYSTEMS OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

We show under some conditions the existence and multiplicity of positive solutions for a system of dierential equations of fractional order, subject to two-point boundary conditions by applying the xed point index theory in cones.

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  • Amann,H., (1976), Fixed point equation and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, pp.620-709.
  • Agarwal,R.P. and O’Regan, D., (2000), Multiple solutions for a couple system of boundary value problems, Dyn. Contin. Discrete Impuls. Syst., 7, pp.97-106.
  • Agarwal,R.P., Zhou,Y., and He,Y., (2010), Existence of fractional neutral functional differential equa- tions, Comput. Math. Appl., 59, pp.1095-3554.
  • Ahmad,B. and Ntouyas,S.K., (2012), A note on fractional differential differential equations with frac- tional separated boundary conditions, Abstr. Appl. Anal., Article ID 818703, pp.1-11.
  • Bai,Z., (2010), On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72, pp.916-924.
  • Bai,Z. and Lu,H., (2005), Positive solutions for boundary value problems of nonlinear fractional dif- ferential equations, J. Math. Anal. Appl., 311, pp.495-505.
  • Balenu, D., Dielheelm,K., Scalas,E., and Trujillo,J.J., (2012), Fractional Calculus Models and Numer- ical Methods, (2008), Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston.
  • Das,S., Functional Fractional Calculus for System Identification and Controls, Springer, New York.
  • Goodrich,C.S., (2010), Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23, pp.1050-1055.
  • Goodrich,C.S., (2011), Existence of a positive solution to systems of differential equations of fractional order, Comp. Math. Appl., 62, pp.1251-1268.
  • Guo,D.J. and Lakshmikantham,V., (1988), Nonlinear Problems in Abstract Cones, Academic Press, New York.
  • Henderson,J. and Luca,R., (2011), Positive solutions for a system of higher-order multipoint boundary value problems, Comput. Math. Appl., 62, pp.3920-3932.
  • Henderson,J. and Luca,R., (2013), Positive solutions for a system of nonlinear fractional boundary value problems, Fract. Calc. Appl. Anal., 16(4), pp.985-1008.
  • Henderson,J. and Luca,R., (2013), Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems, Nonlinear Differ. Equn. Appl., 20(3), pp.1035- 1054.
  • Henderson,J. and Luca,R., (2012), Positive solutions for a system of second-order multi-point bound- ary value problems, Appl. Math. Comput., 218, pp.6083-6094.
  • Henderson,J. and Ntouyas,S.K., (2007), Positive solutions for system of nth order three-point nonlocal boundary value problems, Electron. J. Qual. Theory Differ. Equn., 18, pp.1-12.
  • Henderson,J., Ntouyas,S.K. and Purnaras,I.K., (2008), Positive solutions for systems of second order four-point nonlinear boundary value problems, Comm. Appl. Anal., 12, pp.29-40.
  • Khan,R.A., Rehman,M., and Henderson,J., (2011), Existence and uniqueness of solutions for nonlinear fractional differential equations with integral boundary conditions, Fractional. Diff. Calculus., 1, pp.29- 43.
  • Kilbas,A.A., Srivastava,H.M., and Trujillo,J.J., (2006), Theory and Applications of Fractional Differ- ential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam.
  • Kauffman, E.R. and Mboumi,E., (2008), Positive soltions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equn., 3, pp.1-11.
  • Lakshmikantham,V., Leela, S. and Vasundara Devi, J., (2009), Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge.
  • Liang,S. and Zhang,J., (2009), Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Anal., 71, pp.5545-5550.
  • Miller,K.S. and Ross,B., (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York.
  • Podlubny,I., (1999), Fractional Differential Equations., Academic Press, San Diego
  • Prasad,K.R., Kameswararao,A., and Nageswararao,S., (2012), Existence of positive solutions for the system of higher order two-point boundary value problems, Proc. Indian Acad. Sci., 122(1), pp.139- 152.
  • Zhai,C. and Hao,M., (2014), Multiple positive solution to nonlinear bpundary value problems of a system for fractional differential equations, Scientific. World. J., ID 817542.
  • Zhou,Y. and Xu,Y., (2006), Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations, J. Math. Anal. Appl., 320(2), pp.578-590.
  • Allaka Kameswara Rao for the photography and short autobiography, see TWMS J. App. Eng. Math., V.6, N.2, 2016.