Oscillatory solutions of the system of n integro-differential equations with deviating arguments
Consider a system of n integro-differential equations with advanced arguments of the form \[ x_ k'(t)=\int^{t}_{t_ 0}f_ k(t,s,x_ 1(s),...,x_ n(s),x_ 1(g_ 1(s)),...,x_ n(g_ n(s)))ds, \] \(1\leq k\leq n\), where \(f_ k,g_ k\) are real continuous functions, \(t\geq 0\), \(s\geq 0\), \(g_ k(t)>t\), \(\lim_{t\to \infty}g_ k(t)=+\infty.\) By a solution \(x=(x_ 1,...,y_ n)\) of the system we understand only nontrivial solutions extended to infinity. A solution of the system is called oscillatory (resp. nonoscillatory) if every component \(x_ k(t)\) has an infinite sequence of zeros tending to infinity as \(t\to \infty\) (resp. has a constant sign for sufficiently large \(t\geq T\geq 0).\) Let the functions \(f_ k\) satisfy the following assumptions: \[ (1)\quad f_ k(t,s,x,v)sgn v_{k+1}\geq a_{k+1}(t,s)A_{k+1}(| v_{k+1}|),\quad 1\leq k\leq n-1, \] \[ f_ n(t,s,x,v)sgn v_ 1\leq - a_ 1(t,s)A_ 1(| v_ 1|), \] where \(v=(v_ 1,...,v_ n)\), \(v_ i=x_ i(g_ i(t))\), \(1\leq i\leq n\), \(x_{n+1}=x_ n\), \(a_ k\) are continuous functions and \(A_ k\) are continuous, nondecreasing and nonnegative functions; \[ (2)\quad \lim_{t\to \infty}\int^{t}_{T}\int^{\tau}_{T}a_ k(\tau,s)dsd\tau =\infty,\quad 2\leq k\leq n. \] The author proves three lemmas and two main results. The first one asserts that if (1), (2) and \(\lim_{t\to \infty}\int^{t}_{T}\int^{\tau}_{T}a_ 1(\tau,s)dsd\tau =\infty\) hold, then every bounded solution of the system is oscillatory. The second result asserts that if (1), (2) and a certain condition hold, then every solution of the system is oscillatory for even n, and either oscillatory or tending monotonically to zero as \(t\to \infty\) for odd n.
integro-differential equations with advanced arguments
scientific article; zbMATH DE number 4214691