Differential and Integral Equations

An $L_2$-approach to second-order nonlinear functional evolutions involving $M$-accretive operators in Banach spaces

Athanassios G. Kartsatos and Lubomir P. Markov

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Abstract

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.

Article information

Source
Differential Integral Equations, Volume 14, Number 7 (2001), 833-866.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356123194

Mathematical Reviews number (MathSciNet)
MR1828327

Zentralblatt MATH identifier
1058.34105

Subjects
Primary: 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx]
Secondary: 34A60: Differential inclusions [See also 49J21, 49K21] 34G25: Evolution inclusions 35D05 35L90: Abstract hyperbolic equations 35R70: Partial differential equations with multivalued right-hand sides 47N20: Applications to differential and integral equations

Citation

Kartsatos, Athanassios G.; Markov, Lubomir P. An $L_2$-approach to second-order nonlinear functional evolutions involving $M$-accretive operators in Banach spaces. Differential Integral Equations 14 (2001), no. 7, 833--866. https://projecteuclid.org/euclid.die/1356123194


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