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Infinitely many solutions for perturbed impulsive fractional differential systems. (English) Zbl 1367.34007

From the introduction and summary: We study the following perturbed impulsive fractional differential system \[ \begin{aligned} _tD^{\alpha_i}_T(a_i(t)_0D^{\alpha_i}_t u_i(t))=\lambda F_{u_i}(t,u)+\mu G_{u_i}(t,u)+ h_i(u_i(t)),\;& t\in (0,T),\;t\neq t_j,\\ \Delta(_tD^{\alpha_i-1}_T(^c_0D^{\alpha_i}_t u_i))(t_j)= I_{ij}(u_i(t_j)),\;& j= 1,2,\dots, m,\\ u_i(0)= u_i(T)=0\end{aligned} \] for \(1\leq i\leq n\), where \(u= (u_1,\dots, u_n)\), \(n\geq 1\), \(0<\alpha_i\leq 1\) for \(1\leq i\leq n\), \(\lambda>0\), \(\mu\geq 0\), \(T>0\), \(a_i\in L^\infty([0,T])\), \(\overline a_i= \text{ess\,inf}_{t\in [0,T]}a_i(t)> 0\), \(_0D^i_t\) and \(_tD^i_T\) denote the left and right Riemann-Liouville fractional derivatives of order \(\iota\), respectively, \(m\geq 1\), \(F,G:[0,T]\times \mathbb{R}^n\to\mathbb{R}\) are measurable with respect to \(t\), for all \(u\in\mathbb{R}^n\), continuously differentiable in \(u\), for almost every \(t\in[0,T]\), such that \(F(t,0,\dots,0)= G(t,0,\dots,0)= 0\) for every \(t\in [0,T]\) and satisfy standard summability condition, \(h_i: \mathbb{R}\to\mathbb{R}\) is a Lipschitz continuous function with the Lipschitz constant \(L_i>0\), satisfying \(h_i(0)= 0\) for \(1\leq i\leq n\), \(I_{ij}\in C(\mathbb{R},\mathbb{R})\) for \(i= 1,\dots, n\), \(j= 1,\dots, m\), \(0= t_0<t_1<t_2<\cdots< t_m< t_{m+1}= T\), the operator \(\Delta\) is defined as \(\Delta(_tD^{\alpha_i-1}_T(^c_0 D^{\alpha_i}_at u))(t_j)= _tD^{\alpha_i-1}_T(^c_0D^{\alpha_i-1}_t u))(t_j)= _tD^{\alpha_i-1}_T(^c_0D^{\alpha_i} u)(t^+_j)- _t D^{\alpha_i-1}_AT(^c_0 D^{\alpha_i}_t u)(t^-_j)\), where \[ _tD^{\alpha_i-1}_T(^c_0 D^{\alpha_i}_at u)(t^+_j)= \lim_{t\to t^+_j} tD^{\alpha_i-1}_T (^c_0D^{\alpha_i}_t u)(t) \] and \[ _tD^{\alpha_i-1}_T(^c_0 D^{\alpha_i}_tu)(t^-_j)= \lim_{t\to t^-_j}{_tD^{\alpha_i-1}}_T(^c_0D^{\alpha_i}_t u)(t) \] and \(^c_0 D^{\alpha_i}_t\) is the left Caputo fractional derivatives of order \(\alpha_i\). Here, \(F_{u_i}\) and \(G_{u_i}\) denote, respectively, the partial derivatives of \(F\) and \(G\) with respect to \(u_i\) for \(1\leq i\leq n\).
The approach is based on variational methods. In addition, examples are presented to illustrate the feasibility and effectiveness of the main results.

MSC:

34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34B37 Boundary value problems with impulses for ordinary differential equations
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