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DEMONSTRATIO MATIIEMAT1CAVol. XXVIINo 11994Izabela FoltynskaOSCILLATORY P R O P E R T Y FOR T H E S Y S T E MOF N O N L I N E A R I N T E G R O - D I F F E R E N T I A L E Q U A T I O N SWITH DEVIATING ARGUMENTS1. Introduction and preliminariesFor some years increasing intersest has been observed in investigation ofproperties of solutions to the iiitcgro-diiTcrcntial equations. Levin [4], gavesufficient conditions for unbounded solutions of a class of integro-diiTerentialequations to be oscillatory. Asymptotic behaviour of solutions to the integrodiiTerential equation has been considered by S. Nakagiri in [6]. Comparisontheorems for systems intcgro-diiTcrcntial equations with advance argumentsare given in [3].In this paper sufficient conditions for the existence of oscillatory solutionsto the systemt(A)x'k(t) = fMt,s,x1(s),...,xn(s),x1(ff1(s)),...,xn(gn(s)))dsowill be established, where fk : R+ X R+ X Rn X Rn —»• R, gt- : R+ —»• R+ arecontinuous functions, gi:(l) < t, lirnt-.oo gu{t) = oo, k = 1 , . . . , n.The case gic(t) > t, k = 1 , . . . , n for (A) has been considered in [2].A similar problem for the
Demonstratio Mathematica – de Gruyter
Published: Jan 1, 1994
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