×

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
PDF BibTeX XML Cite
Full Text: EuDML

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.