Existence Of Solutions For Several Singular Nonlinear Fractional Differential Equations

Fractional differential equations have attracted more and more atten-tion from the research communities due to their numerous applications in many fields of science and engineering. On the other hand, boundary value problems with integral boundary conditions for ordinary differ-ential equations and boundary value problems for a coupled system of nonlinear differential equations arise in many fields of applied mathe-matics and physics such as heat conduction, chemical engineering and plasma physics. The existence of positive solutions for such problems have become an important area of investigation in recent years.In this paper, by using the properties of the fixed point theory in cones and mixed monotone method, we obtain the existence, uniqueness and multiplicity results of the problem. And we give some examples to demonstrate the application of our theoretical results.The thesis is divided into two chapters according to contents.In Chapter 1, by using the properties of the Green function and the fixed point theory in cones, we obtain some results on the existence of positive solutions for the following singular nonlinear fractional differen-tial equations with Riemann-Stieltjes integral boundary conditions: where n-1<α≤n,n≥2, p, q∈C(0,1), p(t) and q(t) are allowed to be singular at t=0 or t = 1,f, g:[0,1] × (0, ∞) → [0,∞) are continuous and f(t,x),g(t,x) may be singular at x=0; h:(0,1) → [0,∞) is continuous with h ∈ L1(0,1);∫01 h(s)u(s)dA(s) denotes the Riemann-Stieltjes integral with a signed measure, in which A:[0,1] → R is a function of bounded variation.In Chapter 2, we study the existence and uniqueness of positive solu- tions for a class of singular fractional differential systems with coupled integral boundary value problems. By using the properties of the Green function, mixed monotone method and the fixed point theory in cones, we obtain the existence and uniqueness results of the problem, where n-1<α≤n, m-1<β≤m, n,m∈N, n, m≥2, D0α+ and D0β+ denote the Riemann-Liouville derivatives of orders α and β, respectively. Pi,qi ∈ C((0,1),[0,∞)), a,b ∈ C([0,1],[0,∞)), fi ∈ C((0,1) × [0,∞)× (0,∞),[0,∞),gi ∈ C((0,1) × (0,∞) × [0,∞),[0,∞)) and fi(t,x,y) may be singular at t=0,1 and y=0 and gi(t,x,y) may be singular at t= 0,1 and x=0, i= 1,2. ∫01 a {s)v{s)dA{s), ∫01 b(s)u(s)dB(s) denote the Riemann-Stieltjes integral with a signed measure, that is, A, B:[0,1] → [0,∞) are functions of boundary variation. By a positive solution of BVP(2.1.1), we mean a pair of functions (u, v) ∈ C[0,1] × C[0,1] satisfying BVP(2.1.1) with u(t)> 0 and v(t)> 0 for all t ∈ (0,1].