Fractional BVPs with strong time singularities and the limit properties of their solutions (Q403185)

This paper is interesting as the author establishes a relation between the solutions of singular BVP of the ordinary differential equations of the type \[ U''+ \frac{a}{t} u'=f(t,u,u')\tag{1} \] satisfying the boundary conditions \[ u(0)=A,~ u(1)=B,~\text{ and }~u'(0)=0, \] and a sequence of solutions of singular BVPs of fractional differential equations of the corresponding type \[ \frac{d}{dt} {^{c}D^{\alpha _n} } u +\frac{a}{t} D^{\alpha _n}u = f(t,u, ^{c}D^{\beta _n}u)\tag{2} \] with the boundary conditions given by \[ u(0)=A,~ u(1)=B~\text{ and }~^{c}D^{\alpha _n}u(t) |_{t=0}=0 \] with \(a<0,~0< \beta _n \leq \alpha _n <1\) and \(\lim_{n \rightarrow \infty} \beta_n=1.\) They show that the sequence of solutions of (2) will converge to a solution of the ordinary BVP (1) under certain conditions. The first part of the paper deals with the existence of solutions of the singular BVP of the form \[ \frac{d}{dt} {^{c}D^{\alpha}}u +\frac{a}{t} D^{\alpha }u = H(u)\tag{3} \] with the corresponding boundary condition \[ u(0)=A,~ u(1)=B,~\text{ and }~{^{c}D^{\alpha}}u(t) |_{t=0}=0, \] with \(a<0,~\alpha \in (0,1).\) Observe that equation (3) is a generalization of (2). The existence result is proved using the nonlinear Leray-Schauder Alternative. The second part of the paper deals with criteria and results concerning the convergence of sequence of solutions of (2) to a solution of (1).