Periodic solutions for partial neutral non densely differential equations. (English) Zbl 1465.35386
Summary: This work investigates the existence of periodic solutions for the following partial neutral nonautonomous functional differential equation
\[ \begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{D}u_t = (A+B(t))\mathcal{D}u_t+F(t,u_t),\quad t\geq 0, \\ &u_0=\Phi\in\mathcal{C}=C([-r,0],X), \end{aligned}\tag{1} \]
where the linear operator \(A\) is not necessarily densely defined and satisfies the Hille-Yosida condition, \(B(t)\), \(t\in\mathbb{R}_+\), is a family of bounded linear operators from \(\overline{D(A)}\) into \(X\) and the nonlinear delayed part \(F\) satisfies some locally Lipschitz conditions. More precisely, we study the Massera problem for the existence of a \(\tau \)-periodic solution of (1). Then, we prove for \(\tau=1\), in the dichotomic case, the existence, uniqueness and conditional stability of the periodic solution. Finally, our results are illustrated by an application.
\[ \begin{aligned} &\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{D}u_t = (A+B(t))\mathcal{D}u_t+F(t,u_t),\quad t\geq 0, \\ &u_0=\Phi\in\mathcal{C}=C([-r,0],X), \end{aligned}\tag{1} \]
where the linear operator \(A\) is not necessarily densely defined and satisfies the Hille-Yosida condition, \(B(t)\), \(t\in\mathbb{R}_+\), is a family of bounded linear operators from \(\overline{D(A)}\) into \(X\) and the nonlinear delayed part \(F\) satisfies some locally Lipschitz conditions. More precisely, we study the Massera problem for the existence of a \(\tau \)-periodic solution of (1). Then, we prove for \(\tau=1\), in the dichotomic case, the existence, uniqueness and conditional stability of the periodic solution. Finally, our results are illustrated by an application.
MSC:
| 35R10 | Partial functional-differential equations |
| 35B10 | Periodic solutions to PDEs |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 34G20 | Nonlinear differential equations in abstract spaces |
Keywords:
nonautonomous partial neutral functional differential equation; Hille-Yosida condition; evolution family; periodic solution; Massera problem; dichotomyReferences:
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