On a second-order delay integro-differential equation in Banach spaces

Gheorghe Moroşanu; Adrian Petruşel

doi:10.3934/dcdss.2024156

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2025, Volume 18, Issue 8: 2083-2096. Doi: 10.3934/dcdss.2024156

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On a second-order delay integro-differential equation in Banach spaces

Gheorghe Moroşanu^1, and
Adrian Petruşel^{1, 2, ,}

1.
Babeş-Bolyai University, Cluj-Napoca and Academy of Romanian Scientists, Bucharest, Romania
2.
China Medical University, Taichung, Taiwan

^*Corresponding author: Adrian Petruşel
^*Corresponding author: Adrian Petruşel

Received: June 2024

Early access: August 30, 2024

Published online: August 30, 2024

The authors are supported by [Babeş-Bolyai University Institutional Development Fund].

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Abstract

Consider in a Banach space $ (X, \Vert \cdot \Vert) $ the following delay integro-differential equation

$ \begin{align} u^{\prime \prime}(t) = Au(t)+\int_{-\infty}^0g(s,u_t(s))\, ds +f(t), \ \ 0\le t\le T, \end{align} $

where $ A:D(A)\subset X \to X $ is a densely defined, closed linear operator generating a strongly continuous cosine family, while $ g $ is an unbounded nonlinear function which is dominated in some sense by $ A $. Global existence and uniqueness are derived for the initial value problem associated with the above equation by using a fixed point approach combined with properties of the cosine and sine families. Continuity of solutions with respect to the initial data and some stability properties will be also addressed. We illustrate our results with an application to the wave equation with nonlinear memory.

Keywords:

Semilinear integro-differential equation,
delay,
nonlinear memory,
cosine and sine families of linear operators,
fixed point approach,
stability properties,
wave equation with memory.

Mathematics Subject Classification: Primary: 34K40, 35L71.Secondary: 47D09, 47H10, 47J25.

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