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

Year 2017, Volume: 7 Issue: 2, 291 - 302, 01.12.2017

Abstract

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

References

  • 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.
Year 2017, Volume: 7 Issue: 2, 291 - 302, 01.12.2017

Abstract

References

  • 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.
There are 28 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

A. Kameswara Rao This is me

Publication Date December 1, 2017
Published in Issue Year 2017 Volume: 7 Issue: 2

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