Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations

Abstract

Existence of positive solutions for the nonlinear fractional differential equation D ^αu = f(x,u), 0 < α < 1 has been given (S. Zhang. J. Math. Anal. Appl. 252 (2000), 804–812) where D ^α denotes Riemann–Liouville fractional derivative. In the present work we extend this analysis for n-term non autonomous fractional differential equations. We investigate existence of positive solutions for the following initial value problem

$${\user1{L}}(D)u = f(x,u),\quad 0 < x < 1,$$

with initial conditions $u(0) = 0, [D^{\alpha-n+1}u(x)]_{x=0} = b_{n-1} \geq 0,[D^{\alpha-n+j}u(x)]_{x=0} = b_{n-j}, b_{n-j} \geq \sum^{j-1}_{k=1}a_{k}b_{k+n-j}, j = 2,3,\ldots,n-1,n-1\leq\alpha\leq n,n\in\i$ where $\user1{L}(D)=D^{\alpha}-\sum^{n-1}_{j=1}a_jD^{\alpha-j},a_j>0,\forall j,D^{\alpha-j}$ is the standard Riemann–Liouville fractional derivative. Further the conditions on a _j ’s and f, under which the solution is (i) unique and (ii) unique and positive as well, are given