Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection

The abstract fractional dynamics model is based on a class of bioprocesses of HIV infection.The nonlinear terms and boundary conditions all depend on fractional derivatives of unknown functions.The system is singular and semipositone.The system involves some uncertain parametrical variations λ . Fractional order derivative is nonlocal which exhibits a long time memory behavior. With advantage of these, fractional order dynamic system models are more accurate than integer order ones in understanding the dynamic behavior of bioprocesses such as HIV infection. In this paper, we systematically study the existence of positive solutions of an abstract fractional semipositone differential system involving integral boundary conditions arising from the study of HIV infection models. By using the fixed point theorem in cone, some new results are established and an example is given to demonstrate the application of our main results.

[1]  Henry C Tuckwell,et al.  On the behavior of solutions in viral dynamical models. , 2004, Bio Systems.

[2]  K. Deimling.  Nonlinear functional analysis , 1985 .

[3]  Hongjun Xiang,et al.  Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations , 2010 .

[4]  Xinguang Zhang,et al.  Existence of positive solutions for a singular semipositone differential system , 2008, Math. Comput. Model..

[5]  K. Cole.  ELECTRIC CONDUCTANCE OF BIOLOGICAL SYSTEMS , 1933 .

[6]  Johnny Henderson,et al.  Existence and multiplicity of positive solutions for a system of fractional boundary value problems , 2014, Boundary Value Problems.

[7]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[8]  A. Perelson,et al.  Dynamics of HIV infection of CD4+ T cells. , 1993, Mathematical biosciences.

[9]  Gennaro Infante,et al.  Positive Solutions of Nonlocal Boundary Value Problems: A Unified Approach , 2006 .

[10]  Zhenhai Liu,et al.  Nonlinear boundary value problems of fractional differential systems , 2012, Comput. Math. Appl..

[11]  S. Ruan,et al.  A delay-differential equation model of HIV infection of CD4(+) T-cells. , 2000, Mathematical biosciences.

[12]  Yonghong Wu,et al.  Existence of positive solutions for second-order semipositone differential equations on the half-line , 2007, Appl. Math. Comput..

[13]  Xinguang Zhang,et al.  Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations , 2012, Appl. Math. Lett..

[14]  Lishan Liu,et al.  Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity , 2011 .

[15]  Xinguang Zhang,et al.  Positive solutions of superlinear semipositone singular Dirichlet boundary value problems , 2006 .

[16]  Bashir Ahmad,et al.  The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative , 2014, Appl. Math. Lett..

[17]  J. Webb,et al.  Existence of positive solutions for a thermostat model , 2012 .

[18]  Rui Li,et al.  Unique Solution of a Coupled Fractional Differential System Involving Integral Boundary Conditions from Economic Model , 2013 .

[19]  Wei Wang,et al.  Existence of Positive Solution for Semipositone Fractional Differential Equations Involving Riemann-Stieltjes Integral Conditions , 2012 .

[20]  W. Ames,et al.  Nonlinear problems in abstract cones , 1988 .

[21]  Min Zhao,et al.  Positive solutions for a class of fractional differential equations with integral boundary conditions , 2014, Appl. Math. Lett..

[22]  Lishan Liu,et al.  Positive solutions of nonresonance semipositone singular Dirichlet boundary value problems , 2008 .

[23]  D. O’Regan,et al.  Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions , 2012 .

[24]  Rutherford Aris,et al.  Introduction to the Analysis of Chemical Reactors , 1965 .

[25]  Xinguang Zhang,et al.  Multiple positive solutions of singularly perturbed differential systems with different orders , 2010 .

[26]  Xinguang Zhang,et al.  Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations , 2012 .

[27]  Xingqiu Zhang,et al.  Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter , 2014, Appl. Math. Comput..

[28]  AAM Arafa,et al.  Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection , 2012, Nonlinear biomedical physics.

[29]  Yonghong Wu,et al.  Multiple positive solutions of a singular fractional differential equation with negatively perturbed term , 2012, Math. Comput. Model..

[30]  Patrick W Nelson,et al.  Mathematical analysis of delay differential equation models of HIV-1 infection. , 2002, Mathematical biosciences.

[31]  Yonghong Wu,et al.  Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives , 2012, Appl. Math. Comput..

[32]  Ben Fabry,et al.  Fractional Derivatives Embody Essential Features of Cell Rheological Behavior , 2003, Annals of Biomedical Engineering.

[33]  J. Webb,et al.  Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems , 2009 .

[34]  Hussein A. H. Salem,et al.  On the existence of continuous solutions for a singular system of non-linear fractional differential equations , 2008, Appl. Math. Comput..

[35]  J. Webb.  Nonlocal conjugate type boundary value problems of higher order , 2009 .

[36]  I. Podlubny.  Fractional differential equations , 1998 .