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

Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 9 (2) – Sep 21, 2005
14 pages

/lp/springer_journal/existence-of-positive-solutions-for-n-term-non-autonomous-fractional-vi6b0H6bmt
Publisher
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-005-2715-x
Publisher site
See Article on Publisher Site

### 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

### Journal

PositivitySpringer Journals

Published: Sep 21, 2005

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