A new Bihari inequality and initial value problems of first order fractional differential equations
Kunquan Lan, J. R. L. Webb
Abstract
Abstract We prove existence of solutions, and particularly positive solutions, of initial value problems (IVPs) for nonlinear fractional differential equations involving the Caputo differential operator of order $$\alpha \in (0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . One novelty in this paper is that it is not assumed that f is continuous but that it satisfies an $$L^{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -Carathéodory condition for some $$p>\frac{1}{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>α</mml:mi> </mml:mfrac> </mml:mrow> </mml:math> (detailed definitions are given in the paper). We prove existence on an interval [0, T ] in cases where T can be arbitrarily large, called global solutions. The necessary a priori bounds are found using a new version of the Bihari inequality that we prove here. We show that global solutions exist when f ( t , u ) grows at most linearly in u , and also in some cases when the growth is faster than linear. We give examples of the new results for some fractional differential equations with nonlinearities related to some that occur in combustion theory. We also discuss in detail the often used alternative definition of Caputo fractional derivative and we show that it has severe disadvantages which restricts its use. In particular we prove that there is a necessary condition in order that solutions of the IVP can exist with this definition, which has often been overlooked in the literature.