Positivity (2005) 9:193–206 © Springer 2005
Existence of Positive Solutions for N-term
Non-autonomous Fractional Differential
A. BABAKHANI and VARSHA DAFTARDAR-GEJJI
Department of Mathematics, University of Pune, Ganeshkhind, Pune-411007, India.
E-mails: email@example.com (A. Babakhani) and firstname.lastname@example.org
Abstract. Existence of positive solutions for the nonlinear fractional differential equation
u = f (x, u), 0 <α<1 has been given (S. Zhang, J. Math. Anal. Appl. 252 (2000), 804–812)
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:
(D)u = f (x, u), 0 <x<1,
with initial conditions u(0) = 0, [D
,j = 2, 3,... ,n − 1,n − 1 <α<n,n∈ IN where
(D) = D
> 0, ∀j,D
is the standard Riemann–Liouville fractional derivative.
Further the conditions on a
’s and f, under which the solution is (i) unique and (ii)
unique and positive as well, are given.
Key words: Riemann–Liouville fractional derivatives and integrals, semi-ordered Banach
space, normal cone, completely continuous operator, equicontinuous set.
Fractional calculus, which deals with differentiation and integration to
an arbitrary order, has gained considerable interest through the pioneer-
ing works of Leibniz, Bernoulli, Euler, Lagrange, Abel, Fourier, Riemann,
Liouville and many others. In the present decade notable contributions
to both theory and applications of this subject have been carried out [1–
3]. Recent investigations have shown that many physical systems can be
represented more accurately through fractional derivative formulation .
Fractional differential equations, therefore ﬁnd numerous applications in
the ﬁeld of visco-elasticity, feed back ampliﬁers, electrical circuits, electro
analytical chemistry, fractional multipoles, neuron modelling encompassing
different branches of Physics, Chemistry and Biological sciences .