2001 An $L_2$-approach to second-order nonlinear functional evolutions involving $M$-accretive operators in Banach spaces
Athanassios G. Kartsatos, Lubomir P. Markov
Differential Integral Equations 14(7): 833-866 (2001). DOI: 10.57262/die/1356123194


The objective of this paper is to initiate the study of second-order nonlinear functional evolutions of the type $$ \begin{cases} & u''(t) \in A(t)u(t) + G(t,u_t),\quad t>0,\\ & u(0)=x, \ \ u_0 = \phi, \ \ \sup_{t \ge 0} \lbrace \|u(t)\| \rbrace < +\infty, \end{cases} \tag*{($P$)} $$ in a real, uniformly smooth Banach space $X$ with strongly monotone duality mapping. The operators $A(t)$ are $m$-accretive and the operators $G$ are Lipschitzian. The problem is lifted into the space $L_2([-r,\infty);X),$ in which it becomes an elliptic-type problem of the type $$\mathcal Au+\mathcal Bu+\mathcal G(\cdot,u_\cdot) \ni 0$$ with $\mathcal A$ and $\mathcal B$ $m$-accretive. Unperturbed results of Xue, Song and Ma are extended to the present case. The main difficulty in the solvability of these problems is due to the presence of a delay and the fact that certain monotonicity properties of some real-valued functions (defined via the duality mapping) which are present in the homogeneous case do not continue to hold in the perturbed case.


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Athanassios G. Kartsatos. Lubomir P. Markov. "An $L_2$-approach to second-order nonlinear functional evolutions involving $M$-accretive operators in Banach spaces." Differential Integral Equations 14 (7) 833 - 866, 2001. https://doi.org/10.57262/die/1356123194


Published: 2001
First available in Project Euclid: 21 December 2012

zbMATH: 1058.34105
MathSciNet: MR1828327
Digital Object Identifier: 10.57262/die/1356123194

Primary: 34K30
Secondary: 34A60 , 34G25 , 35D05 , 35L90 , 35R70 , 47N20

Rights: Copyright © 2001 Khayyam Publishing, Inc.


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Vol.14 • No. 7 • 2001
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