On solutions of hyperbolic differential inclusions with nonconvex right-hand side

The existence of a generalized solution with continuous derivativesu _x ,u _y is proved for the differential inclusionu _xy F(x, y, u) with a nonconvex right-hand side satisfying the Lipschitz conditioninx, y, andu.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 531–534, April, 1995.     

On the smoothness of solutions of differential equations with a discontinuous right-hand side

A method for the investigation of differential equations with a nonclassical right-hand side [1] is applied to the study of the higher-order differentiability of solutions of differential equations with discontinuous right-hand sides with respect to the initial data. We use results from the theory of differential equations with pulse influence [2].Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1587–1594, December, 1993.     

On periodic solutions of ordinary differential equations with discontinuous right-hand side

A new version of the method of translation along trajectories, which does not require the uniqueness of the solution of the Cauchy problem, is applied to the proof of the existence theorem for vector-valued periodic solutions of ordinary differential equations of first and second order. This result is applicable to equations and differential inclusions with discontinuous right-hand side. Several applications of the theorems proved in this paper are considered in cases which are not covered by the classical theory of ordinary differential equations with continuous right-hand side and equations with right-hand side satisfying the Carathéodory conditions.     

On the existence of solutions for fractional differential inclusions with anti-periodic boundary conditions

The existence of solutions of an anti-periodic boundary value problem for fractional differential inclusions of order ????(2,3] is investigated. Several results are obtained by using suitable fixed point theorems when the right hand side has convex or non convex values.     

Classical solutions to differential inclusions with totally disconnected right-hand side

Let F=F(t,x) be a bounded, Hausdorff continuous multifunction with compact, totally disconnected values. Given any y₀∈F(t₀,x₀), we show that the differential inclusion has a globally defined classical solution, with x(t₀)=x₀, .     

Asymptotics of solutions of a second-order linear differential system with a special right-hand side

Odessa. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 2, pp. 89–93, March–April, 1990.     

On the existence of positive solutions of second order differential equations

Using the fixed point method we prove an existence result for positive solutions of nonlinear second order ordinary differential equations. An application to semilinear Schrödinger equations in exterior domains is also presented. Mathematics Subject Classification (2000) 34C10     

On the cauchy problem for a class of degenerate differential equations with Lipschitz right-hand side

This note deals with an application of the fixed point theorem for multivalued contraction mappings to the study of degenerate differential equations with Lipschitz right-hand side and with degeneration determined by a closed surjective linear operator.     

Maximum descent monotone solutions of an ordinary differential equation with a discontinuous right-hand side

We present an existence theorem for absolutely continuous monotone solutions of the Cauchy problem $$\dot x\left( t \right) = proj_{T_{P\left( {x\left( t \right)} \right)} x\left( t \right)} ( - \nabla w(x)(t))),x\left( 0 \right) = x_0 .$$ Moreover, we prove that the limit pointx* of any solution is a minimum forw(·) inP(x*). The results are applied to a decision problem for a firm which wants to satsify twoa priori incompatible criteria: (i) monotonicity with respect to a preorder; and (ii) minimization of costs.