Multiple positive solutions to singular positone and semipositone m-point boundary value problems of nonlinear fractional differential equations

摘要

In this paper, we consider the properties of Greena??s function for the nonlinear fractional differential equation boundary value problem $$egin{aligned} &mathbf{D}_{0+}^{lpha}u(t)+figl(t,u(t)igr)=0,quad 0 t 1, &u(0)=0, qquad u(1)= {sum_{i=1}^{m-2}} eta_{i} u(eta_{i}), end{aligned}$$ where (1lpha2), (0eta_{i}1), (i=1,2,ldots, m-2), (0eta_{1}eta _{2}cdotseta_{m-2}1), (sum_{i=1}^{m-2}eta_{i}eta_{i}^{lpha-1}1), (mathbf{D}_{0+}^{lpha}) is the standard Riemanna??Liouville derivative. Here our nonlinearity f may be singular at (u=0). As an application of Greena??s function, we give some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leraya??Schauder nonlinear alternative and a fixed point theorem on cones.KeywordsMultiple positive solutions??Singular fractional differential equation??Semipositone??m-point boundary value problem??MSC34B15??1 IntroductionIn this paper, we consider the existence and multiplicity of positive solutions of the nonlinear fractional differential equation semipositone boundary value problem: $$egin{aligned} egin{aligned}& mathbf{D}_{0+}^{lpha}u(t)+figl(t,u(t)igr)=0,quad 0 t 1, &u(0)=0, qquad u(1)= {sum_{i=1}^{m-2}} eta_{i} u(eta_{i}), end{aligned} end{aligned}$$ (1.1) where (1lpha2, 0eta_{i}1, i=1,2,ldots, m-2, 0eta_{1}eta _{2}cdotseta_{m-2}1, sum_{i=1}^{m-2}eta_{i}eta_{i}^{lpha-1}1), (mathbf{D}_{0+}^{lpha}) is the standard Riemanna??Liouville derivative. Here our nonlinearity f may be singular at (u=0). The nonlinear fractional differential equation for the multi-point boundary value problem has been studied extensively. For details, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and the references therein.For (m=3), Bai [15] investigated the existence and uniqueness of positive solutions for a nonlocal boundary value problem of the fractional differential equation $$ egin{aligned} &mathbf{D}_{0+}^{lpha}u(t)+figl(t,u(t)igr)=0, quad 0 t 1, &u(0)=0,qquad u(1)=eta u(eta), end{aligned} $$ (1.2) via the contraction map principle and fixed point index theory, where (1lphaleq2, 0etaeta^{lpha-1}1, 0eta1), (mathbf{D}_{0+}^{lpha}) is the standard Riemanna??Liouville derivative. The function f is continuous on ([0,1]imes [0,infty)). Wang, Xiang and Liu [16] investigated the existence and uniqueness of a positive solution to nonzero three-point boundary values problem for a coupled system of fractional differential equations. Ahmad and Nieto [17] considered the three point boundary value problems of the fractional order differential equation. By using some fixed point theorems, they obtained the existence and multiplicity result of positive solution to this problem. They considered the case when f has no singularities. Xu, Jiang et al. [18] deduced some new properties of Greena??s function of (1.2). By using some fixed point theorems, they obtained the existence, uniqueness and multiplicity of positive solutions to singular positone and semipositone problems. Hussein A.H. Salem [1] investigated the existence of pseudo-solutions for the nonlinear m-point boundary value problem of the fractional case, $$ egin{aligned}& mathbf{D}_{0+}^{lpha}x(t)+q(t)figl(t,x(t)igr)=0,quad mbox{a.e. on }[0,1], lphain(n-1, n], ngeq2, &x(0)=x'(0)=cdots=x^{(n-1)}=0,qquad x(1)= sum _{i=1}^{m-2}zeta_{i} x(eta_{i}), end{aligned} $$ (1.3) where (0eta_{1}eta_{2}cdotseta_{m-2}1, zeta_{i}0) with (sum_{i=1}^{m-2}zeta_{i}eta_{i}^{lpha-1}1). It is assumed that q is a real-valued continuous function and f is a nonlinear Pettis integrable function.However, no paper to date has discussed the multiplicity for the semipositone singular problem. This paper attempts to fill this gap in the literature, and as a corollary, we give a result for singular positone problems.