Some results for fractional impulsive boundary value problems on infinite intervals
type: Item
instance of: scholarly article
title: Some results for fractional impulsive boundary value problems on infinite intervals
zbMATH Open document ID: 1240.26011
author: Xiangkui Zhao
author: Weigao Ge
published in: Applications of Mathematics
publication date: 2011-10-25T00:00:00Z
full work available at URL: https://eudml.org/doc/116545
full work available at URL: http://hdl.handle.net/10338.dmlcz/141600
review text: 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…
reviewed by: Vanna Zanelli
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 34A37
zbMATH DE Number: 5963756
zbMATH Keywords: fractional derivative
zbMATH Keywords: impulsive equation
zbMATH Keywords: monotone iterative method
MaRDI profile type: Publication
OpenAlex ID: W1968771316
cites work: Multiple nonnegative solutions for second order impulsive differential equations.
cites work: Positive solutions for boundary value problem of nonlinear fractional differential equation
cites work: Periodic boundary value problem for the second order impulsive functional differential equations
cites work: Approximate analytical solution for seepage flow with fractional derivatives in porous media
cites work: Monotone-iterative techniques of V. Lakshmikantham for a boundary value problem for systems of impulsive differential-difference equations
cites work: Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations
cites work: Periodic boundary value problems for a class of second-order impulsive integro-differential equations in Banach spaces
cites work: Fractals and fractional calculus in continuum mechanics
cites work: Some results for fractional boundary value problem of differential inclusions
cites work: Boundary value problem for a coupled system of nonlinear fractional differential equations
cites work: Periodic boundary value problems for second order impulsive integrodifferential equations of mixed type in Banach spaces
cites work: Boundary value problems on the half-line with impulses and infinite delay
cites work: Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line
DOI: 10.1007/S10492-011-0021-4
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description: scientific article; zbMATH DE number 5963756