Differential and Integral Equations

Second-order integrodifferential equation with nonautonomous operators

D. Bahuguna, D. N. Pandey, and A. Ujlayan

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In this paper, we establish some results to show the existence and uniqueness of classical solution of the non-autonomous second-order integrodifferential equation of the type: \begin{eqnarray*} u''(t) & = & A(t)\;[u(t)+\int_0 ^{t} k_{1}(t,s)u(s)ds]'+B(t)\;[u(t)+\int_0 ^{t} k_{2}(t,s)u(s)ds]\\ & & +f(t),\; 0\leq t \leq T, \\ u(0) & = & u_{0} \in E,\quad u'(0)=u_{1} \in E, \end{eqnarray*} on a Banach space $E$, by means of the matrix operator method.

Article information

Differential Integral Equations, Volume 20, Number 6 (2007), 681-692.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45N05: Abstract integral equations, integral equations in abstract spaces
Secondary: 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions 34K30: Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx] 35L15: Initial value problems for second-order hyperbolic equations 35L90: Abstract hyperbolic equations 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx] 47N20: Applications to differential and integral equations


Bahuguna, D.; Pandey, D. N.; Ujlayan, A. Second-order integrodifferential equation with nonautonomous operators. Differential Integral Equations 20 (2007), no. 6, 681--692. https://projecteuclid.org/euclid.die/1356039432

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