States Of The Solutions Of Some Functional Differential Equations

In this thesis, we first study the stability of some type of the functional differential equation x(t) + p(t)x(t) + q(t)x(t) + c(t)x(t-Ï„) = 0 , where q(t) = q₁(t) + q₂(t) , p(t) = p₁(t) + p₂(t) , q₁(t)>0 , q₁(t) exists and is continuing , and q₁(t) is not a decreasing function . Then we considerthe nonlinear neutral differential equation with positive and negativecoefficients d/(dt) [x(t) - R(t)x(t - r)] + P(t)x{t - r) - Q(t)x(t -δ) + f(t) = 0 Where P,Q,R∈C([t₀,∞),R⁺), Ï„,r, δ∈R⁺ ,τ≥δ . The oscillation of the equation will be mainly discussed in the case of f(t) > 0.In the first part of the thesis, by using a vector we transfer the second differential equation to be a first differential one. And then replace the retarded part to be an integral form with a certain equation. After that, in order to transfer the first differential form to a general one we let the vector be another one. Finally, making use of the second lyapunovmethod , we magnify and transfer the equation and use a Razumikhin-type theorem in a wider way in order to get a theorem , which includes the uniform stability , uniform asymptotic stability of the equation under certain conditions when the coefficient of the retarded part be or not be zero . With special coefficients, we discuss the stability of the equation and get some corollaries of the basic theorem.In the second part, firstly we let the equation be an inequation, then with certain conditions of the coefficients we prove that ,the integral of the left part of the equation y(t) , no matter the eventually positive solution of the inequation is bounded or unbounded , satisfies y(t)≤0 and y(t) > 0. After that, under the condition of another one, we also prove the exactness of y(t)≤0 and y(t)<0. Then with all the things we studied above, we finish the proof of the oscillation of the equation.