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Zbl 0924.34056
Balachandran, K.; Chandrasekaran, M.
Nonlocal Cauchy problem for quasilinear integrodifferential equation in Banach spaces. (English)
[J] Dyn. Syst. Appl. 8, No.1, 35-43 (1999). ISSN 1056-2176

The authors consider the quasilinear integrodifferential equation $$ u'(t) + A(t, u)u(t) = \int_0^t k(t, s, u(s)) ds , \quad 0 \le t \le a,$$ in a Banach space $X,$ where each $A(t, u)$ generates a compact evolution operator $U(t, s; u).$ The nonlocal Cauchy problem is: solve the equation with the nonlocal condition $u(0) + g(u) = u_0,$ where $g : C([0, a], X) \to X.$ The results are on existence and uniqueness of mild solutions (that is, continuous solutions to the integrated version) and on existence and uniqueness of strong solutions.
[H.O.Fattorini (Los Angeles)]

MSC 2000:: *34G20 Nonlinear ODE in abstract spaces
47H20 Semigroups of nonlinear operators
47G20 Integro-differential operators
34K30 Functional-differential equations in abstract spaces

Keywords: nonlocal Cauchy problem; quasilinear integrodifferential equations in Banach space

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