On the existence of solutions of operator-differential equations. (English) Zbl 0757.34053

The goal of the present paper is to formulate sufficient conditions for the existence of global solutions of the operator-differential equation \(y'(t)=A(t)y(t)+T(y)(t)\) on the real positive axis \(R^ +\). Here \(A: R_ +\to R^{n\times n}\) is a locally integrable matrix function, \(T\) is an operator of Volterra type mapping \(C_{\text{loc}}(R_ +,R^ n)\) in \(L_{\text{loc}}(R_ +,R^ n)\). Under these assumptions the above equation is almost everywhere on \(R^ +\) equivalent to the integral equation \(y(t)=x(t)+\int_ 0^ tX(t,s)\) \(T(y)\) \((s)\) \(ds\), where \(x(t)\) is a solution of the corresponding unperturbed equation \(x'(t)=A(t)x(t)\) and \(X(t,s)\) is its Cauchy matrix coinciding for \(s=t\) with the unit matrix. Under certain conditions for \(X=X(t,s)\) and \(T\) relations between global solutions \(x(t)\) and \(y(t)\) are shown (the assumption of Theorem 2.3 should be represented in a clearer form).


34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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