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Some results for fractional impulsive boundary value problems on infinite intervals. (English) Zbl 1240.26011
The authors study the following problem: \[ \begin{aligned} &D^{\alpha }_{0^{+}} u(t)+ f(t,u(t))=0,\,t\in (0,+\infty ),\;t\neq t_k,\;k=1,2,\dots,m,\\ &u(t_k^{+})-u(t_k^{-} )= - I_k(u(t_k)),\;k=1,2,\dots,m,\\ &u(0) = 0,\;D^{\alpha -1}_{0^{+}}u(\infty ) =0,\end{aligned}\tag{P} \] where \(0<\alpha \leq 2\) \(D^{\alpha }_{0^{+}}\) is the classical Riemann-Liouville fractional derivative, \( t_0 = 0\), \(1< t_1 < t_2<\dots <t_m <\infty\), \(u(t_k^{+}) = \lim _{h\to 0^{\pm }}u(t_k \pm h)\), \(D^{\alpha -1}_{0^{+}}u(\infty ) = \lim _{t\to +\infty }D^{\alpha -1}_{0^{+}}u(t).\)
Given \(F(t,u)=f(t,(1+t^{\alpha }u),\) the authors consider the following conditions:
(i) \(F\:[0,+\infty )\times [0,+\infty ) \to [0,+\infty )\) is continuous;
(ii) \(| F(t,u)| \leq \phi (t) \kappa (| u| ),\) where \( \kappa \in C([0,+\infty ),[0,+\infty ))\) is nondecreasing and \(0< \int _0^{+\infty }\phi (t)\, dt <+\infty ;\)
(iii) \(I_k\:[0,+\infty )\to [0,+\infty )\), \(k=1,2,\dots ,m \) are continuous;
(iv) there exist \(c_k\in \mathbb {R}\) such that \(| I_k(u)| \leq c_k\), \(k=1,2,\dots ,m.\)
One of the most important results proved is the following theorem: If (i),(ii),(iii) and (iv) hold and
(1) there exist \(a,b,c \in {\mathbb {R}}\) such that \(0<\frac {1}{\Gamma (\alpha )}\sum _{k=1}^m\frac {c_k}{t_k^{\alpha -1}- t_k^{\alpha -2}}<a<b<c\) and \(\kappa (u) < N_1(a-\sum _{k=1}^m\frac {c_k}{t_k^{\alpha -1}- t_k^{\alpha -2}})\) for all \(u\in [0,a]\);
(2) \(F(t,u) \geq N_2 b\) for all \((u,t)\in [k,l]\times [b,c]\);
(3) \(\kappa (u) \leq N_1(c-\sum _{k=1}^m\frac {c_k}{t_k^{\alpha -1}- t_k^{\alpha -2}})\) for all \(u\in [a,c]\);
then the problem (P) has at least three positive solutions \(u_1,u_2, u_3 \) such that \[ \sup_{t\in [0,+\infty )}\left | \frac {u_1(t)}{1+t^{\alpha }}\right | < a,\;b < \min_{t\in [k,l]}\left | \frac {u_3(t)}{1+t^{\alpha }}\right | < \sup_{t\in [0,+\infty )}\left | \frac {u_2(t)}{1+t^{\alpha }}\right | \leq c, \]
\[ a < \sup_{t\in [0,+\infty )}\left | \frac {u_3(t)}{1+t^{\alpha }}\right | \leq c,\, \min_{t\in [k,l]}\left | \frac {u_3(t)}{1+t^{\alpha }}\right | <b. \]
The existence of successive iteration solutions to problem (P) is proved and some examples are presented.

MSC:
26A33 Fractional derivatives and integrals
34A37 Ordinary differential equations with impulses
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