## 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).

### MSC:

 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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### References:

 [1] Futák J.: Existence and boundedness of solutions of the n-th order nonlinear differential equation with delay. Práce a štúdie VŠDS, Žilina, 4(1981), 7-20. [2] Futák J.: On the asymptotic behaviour of the solutions of operator-differential equations. Fasciculi Math., Posnaniae · Zbl 0728.34079 [3] Rosa V.: Existence theorems for initial value problems with nonlinear differential system with delay. Acta Math. Univ. Comen., XL-XLI (1982) , 51-58. · Zbl 0514.34056
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