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Nguyen, Huong T. T.; Nguyen, Thang N.; Vu, Luong T.

Asymptotically almost periodic solutions to nonlocal differential equations. (English) Zbl 1516.34113

Rocky Mt. J. Math. 52, No. 6, 2113-2127 (2022).
The paper is focused on the nonlocal fractional differential equation with an initial condition \[ \left\{\begin{array}{l}\displaystyle\frac{d}{dt}\left[k*(u-u_0)\right](t)+Au(t)=f(t,u(t)),\,\,\,t>0,\\ u(0)=u_0, \end{array}\right.\tag{1} \] where \(u:[0,\infty)\to H\) is the unknown function, \(H\) is a separable Hilbert space, the kernel \(k\in L_{\mathrm{loc}}^1(\mathbb{R}^+)\), \(A:D(A)\subset H\to H\) is an unbounded linear operator, the function \(f:\mathbb{R}^+\times H\to H\), and \(k*(u-u_0)\) is the Laplace convolution of functions \(k\) and \(u-u_0\), defined by \((k*(u-u_0))(t)=\int_0^tk(t-s)(u(s)-u_0(s))\,ds\). Under some assumptions on \(k\), \(A\) and \(f\), the authors prove the existence of asymptotically almost periodic mild solutions of problem \((1)\).
Reviewer: Rodica Luca (Iaşi)

MSC:

34K30 Functional-differential equations in abstract spaces
34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
47N20 Applications of operator theory to differential and integral equations

Keywords:

nonlocal differential equations; asymptotic almost periodic solutions; almost periodic solutions to PDEs; fixed-point
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References:

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