Some conditions for the existence of the total differential

We give conditions for the almost everywhere existence of the total differential of the functions of the weighted classes ACL_p^μ in a domain. The proofs rest on the technique of the weighted modulus of a family of curves.     

Conditions for the existence of continuous schedules of duration five

Each device must perform one operation with each of two demands assigned to it. The assignments are such that the maximum number of operations for one demand equals five, the partial precedence limitations are missing, simultaneous servicing of two or more demands by the same device or of one demand by two or more devices is forbidden, and the duration of each operation equals a unit. The necessary and sufficient conditions are obtained for the existence of a schedule of duration five conforming to the specified assignments and such that each device performs its operations during two consecutive intervals of unit duration. Hence follows the polynomial solvability of the problem under discussion.     

Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Let E be a finite-dimensional Banach space, let C⁰(R; E) be a Banach space of functions continuous and bounded on R and taking values in E; let K:C ⁰(R ,E) → C ⁰(R, E) be a c-continuous bounded mapping, let A: E → E be a linear continuous mapping, and let h ∈ C ⁰(R, E). We establish conditions for the existence of bounded solutions of the nonlinear equation

Conditions for the existence and uniqueness of bounded solutions of nonlinear differential equations

We establish conditions required for the existence and uniqueness of bounded solutions of the nonlinear differential equation f₁( \fracdx(t)dt ) = f₂( x(t) ) {f_1}\left( {\frac{{dx(t)}}{{dt}}} \right) = {f_2}\left( {x(t)} \right) , t ∈ ℝ.     

Conditions for the existence of nonoscillatory solutions of a second-order nonlinear differential equation

For the nonlinear differential equation

Conditions for the existence of a solution of a system of differential equations close to an approximate solution

We consider nonautonomous systems of differential equations and state conditions for the existence of an exact solution in a neighborhood of an approximate one by analyzing the linear system of the first approximation in a neighborhood of the constructed approximate solution. We present conditions for the existence of a bounded solution of a linear inhomogeneous system of differential equations.     

Conditions for the existence of bounded solutions of one class of nonlinear differential equations

For systems of nonlinear differential equations (dx/dt) = A(x)x + f(t) in a Banach space, we establish sufficient conditions for the existence of their solutions bounded on the entire real axis R. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 317–325, March, 2006.     

Convergence theorems for continuous functions on an arbitrary interval

In this paper, we introduce a new iterative scheme for finding a fixed points of continuous functions on an arbitrary interval. The convergence theorems are also established. Further, the numerical examples comparing with Mann, Ishikawa and Noor iterations are demonstrated. Main results generalize and unify the corresponding ones announced in the literature.     

On the existence of solutions for an elliptic system of equations with arbitrary order growth

Let BR be the ball centered at the origin with radius R in RN ( N ≥2). In this paper we study the existence of solution for the following elliptic systemu -△u+λu=p/(p + q)κ(| x |)) u(p-1)vq1,x ∈BR1,-△u+λu=p/(p + q)κ(| x |)) upv(q-1)1,x ∈BR1,u > 01,v > 01,x ∈ BR1,(u)/(v)=01,(v)/(v)=01,x ∈BRwhereλ > 0 , μ > 0 p ≥ 2, q ≥ 2,ν is the unit outward normal at the boundary BR . Under certainassumptions on κ ( | x | ), using variational methods, we prove the existence of a positive and radially increasing solution for this problem without growth conditions on the nonlinearity.     

Proof of the favard theorem on the existence of almost-periodic solution for an arbitrary Banach space

The well-known Favard-Amerio theorem on the existence of an almost-periodic solution of a linear equation is based on the geometry of a uniformly convex space, since the almost-periodic solution is found by the minimax condition. In the present note an essentially different method for finding the almost-periodic solution is developed, which enables us to prove the Favard-Amerio theorem for an arbitrary Banach space.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 121–126, January, 1978.     

一个常微分方程整体解的存在性结果

研究常微分方程d2u/dx2 K(x)e2u=0在(-∞, ∞)上整体解的存在性问题.此方程是熟知的在R2上预定高斯曲率方程的一个特例.本文证明了一个存在性定理.     

Conditions for the existence of limit cycles of the second kind for a system of differential equations: II

For a system of differential equations with a cylindrical phase space, we obtain conditions for the existence of several limit cycles of the second kind. These results are applied to phase synchronization systems.     

Conditions for the existence of limit cycles of the second kind for a system of differential equations: I

For a system of differential equations with a cylindrical phase space, we obtain conditions for the existence of several limit cycles of the second kind. These results are applied to phase synchronization systems.     

On the existence of an invariant manifold for weakly nonlinear matrix differential equations

For a system of nonlinear matrix differential equations with small parameter, we establish conditions of the existence of an invariant manifold in terms of the Green function.     

Coefficient conditions for existence of an optimal control for systems of differential equations

For systems of ordinary differential equations, we obtain some sufficient conditions of existence of an optimal control in terms of their right-hand sides and the cost functional with the use of compactness methods. In the theorems proven, either the time interval is finite and a solution belongs to some domain or the time interval coincides with the semiaxis.