Integral and fractional equations, positive solutions, and Schaefer’s fixed point theorem

Abstract

This is the continuation of four earlier studies of a scalar fractional differential equation of Riemann-Liouville type $D^qx(t) = -f(t,x(t)), \quad \lim_{t\to 0^+} t^{1-q} x(t) = x^0 \in\Re \quad (0 \lt q \lt 1), \tag {a}$ in which we first invert it as a Volterra integral equation $x(t)=x^0 t^{q-1} -\frac{1}{\Gamma (q)}\int\limits^t_0 (t-s)^{q-1}f(s,x(s))\,ds \tag {b}$ and then transform it into $\begin{multline}x(t)=x^0t^{q-1}-\int\limits^t_0 R(t-s)x^0s^{q-1}ds\\+\int\limits^t_0R(t-s) \bigg[x(s)-\frac{f(s,x(s))}{J} \bigg] ds, \tag {c}\end{multline}$ where $R$ is completely monotone with $\int^{\infty}_0 R(s)\,ds =1$ and $J$ is an arbitrary positive constant. Notice that when x is restricted to a bounded set, then by choosing J large enough, we can frequently change the sign of the integrand in going from $(b)$ to $(c)$. Moreover, the same kind of transformation will produce a similar effect in a wide variety of integral equations from applied mathematics. Because of that change in sign, we can obtain an a priori upper bound on solutions of $(b)$ with a parameter $\lambda \in (0,1]$ and then obtain an a priori lower bound on solutions of $(c)$. Using this property and Schaefer's fixed point theorem, we obtain positive solutions of an array of fractional differential equations of both Caputo and Riemann-Liouville type as well as problems from turbulence, heat transfer, and equations of logistic growth. Very simple results establishing global existence and uniqueness of solutions are also obtained in the same way.