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A quasilinearization method for a class of integro-differential equations with mixed nonlinearities. (English) Zbl 1111.45005
Consider the initial value problem \[ u'(t)=AN\bigl(t,u(t)\bigr)+ B\int^t_{t_0}K\bigl(t,s,u(s)\bigr)\,ds,\;t_0\leq t\leq t_0+T,\quad u(t_0)=u_0,\tag{*} \] where \(A\) and \(B\) are real nonnegative constants and \(N\) has the representation \[ N(t,u)=f(t,u)+ g(t,u)+h(t,u), \] where \(f+\varphi\) is convex for some \(\varphi\), \(g+ \psi\) is concave for some \(\psi\), and \(h\) is Lipschitzian. By means of the method of lower and upper solutions, the author derives conditions such that there exists a monotone sequence of functions converging uniformly and quadratically to a solution of (*).

MSC:
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
45L05 Theoretical approximation of solutions to integral equations
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