Positive Solutions Of Singular Fractional Differential Systems With Multi-point And Riemann-stieltjes Integral Boundary Conditions

In this thesis,we first study the existence of positive solutions to the following Rieman-Liouville derivative,λ and μ are positive parameter.f1:(0,1)×(0,+∞)â†'[0,+∞),g1:[0,1]×[0,+∞)â†'(0,+∞)are continuous and f1(t,v)may be singular at t=0.1and v=0.We assume(1.1)satisfies the following conditions:(H1)f1∈C((0,1)×(0,+∞),[0,+∞))is decreasing for v,g1∈C([0,1]×[0,+∞),(0,+∞)) is increasing for u.(H2)For any l,μ>0, Then singular fractional differential system with multi-point boundary conditions exists at least a positive solution.Next we are concerned with the existence and uniqueness of positive solutions for the following high order singular fractional differential system involving Riemann-Stieltjes integral conditions arising in the abstract model for HIV infection of CD4+T-cells and HCV infection are functions of bounded variation,∫01u(s)dA(s)and∫01v(s)dB(s)denote the Riemann-Stieltjes integrals of u,v with respect to A and B,f2∈C((0,1)×(0,+∞),[0,+∞)), g2∈C((0,1)×[0,+∞),[0,+∞)),f2and g2may be singular at t=0and t=1,moreover f2can be singular at v=0.We assume(1.2)satisfies the following conditions:(H3)A,B are functions of bounded variation such that GA≥0,GB≥0for s∈[0,1] and0≤A,B<1ï¼›(H4)f2∈C((0,1)×(0,+∞),[0,+∞)),f2(t,v)is decreasing in v and for any r∈(0,1), there exists a constant λ<0such that,for any(t,v)∈(0,1)×(0,+∞),(H5)g2∈C((0,1)×[0,+∞),[0,+∞))is increasing in u and for any r∈(0,1),there exists a constant λ2>0satisfying一1<λ1λ2<0and for any(t,u)∈(0,1)×[0,+∞),The main results of this thesis are listed as follows:Theorem1Suppose(H1)一(H2)hold,then for any(λ,μ)∈(0,+∞)×(0,+∞),the system(1.1)has at least one positive solution(u*,v*),and there exist positive constants ri*,r2*,d1*,d2*such that then(i)the high order singular fractional differential system(1.2)has a unique positive solution(u**,v**)in C[0,1]×C[0,1]ï¼›(ii)for any initial value x0∈P2,the successive iterative sequences{(xn,yn),n 1,2,…}generated byconverge to the unique positive solution(u**,v**)uniformly on[0,1],i.e.(iii)the error between xn and u**can be estimated bywhile the rate of convergence iswhere0<κ<1is a positive constant which depends on the initial value x0ï¼›(iv)there exist constants0<l1<1,0<l2<1suchh that...