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Applications of the Leray-Schauder alternative to some Volterra integral and integrodifferential equations. (English) Zbl 0852.45012
The author proves the global existence of solutions for some nonlinear Volterra integral and integrodifferential equations, of the type \[ x(t)=h(t)+\int^t_0 k(t,s)g(s,x(s))ds\tag{1} \] and \[ x'(t)=f\Biggl(t,x(t),\int^t_0 k(t,s)g(s,x(s))ds\Biggr),\qquad x(0)=x_0,\tag{2} \] where \(h\in C([ 0,T ],\mathbb{R}^n)\), \(k\in C([ 0,T ]\times [0, T] \times [0, T],\mathbb{R})\), \(g\in C([0, T] \times\mathbb{R}^n,\mathbb{R}^n)\) and \(f\in C([ 0,T ]\times\mathbb{R}^n \times\mathbb{R}^n,\mathbb{R}^n)\) and \(x_0\) is a given constant. Under some additional growth conditions on the functions \(h\), \(k\), \(g\) and \(f\), he obtains necessary a priori bounds for applying the usual Leray-Schauder degree (for completely continuous perturbations of the identity), to appropriate fixed point problems associated to (1) and (2).

45G10 Other nonlinear integral equations
45J05 Integro-ordinary differential equations
55M25 Degree, winding number