This paper deals with the resonance problem for the one-dimensional p-Laplacian with homogeneous Dirichlet boundary conditions and with nonlinear impulses in the derivative of the solution at prescribed points. The sufficient condition of Landesman-Lazer type is presented and the existence of at least one solution is proved. The proof is variational and relies on the linking theorem. MSC: Primary 34A37; 34B37; secondary 34F15; 49K35

We consider the nonlinear first order periodic problem with nonlinear impulses. We apply the Schaeffer fixed point theorem and prove the existence results under Landesman-Lazer type sufficient conditions. We formulate also necessary conditions in some special cases. The impulses can be viewed as a control which compensates the influence of external forces and vice versa. MSC: Primary 34A37; 34B37; secondary 34F15; 47H11

In this paper we consider the periodic problem for the second order equation at resonance with impulses in the derivative. The impulses are considered at fixed times and depend on the actual value of the solution in a nonlinear way. We formulate rather general sufficient condition in terms of the asymptotic properties of both nonlinear restoring force and nonlinear impulses which generalizes classical Landesman-Lazer condition. Moreover, our condition implies existence results for some open problems with vanishing and oscillating nonlinearities.

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We consider a weakly nonlinear boundary-value problem for a system of second-order ordinary differential equations. We find a sufficient condition for the existence of at least one solution of this problem and propose a convergent iterative algorithm for the determination of its solution.Consider the Noether boundary-value probleml is a linear bounded vector functional defined on the space of n-dimensional vector functions continuous on the segment [a, b], l = col (l 1 , l 2 , . . . , l m ), l: C[a, b] → R m , −∞ < a ≤ b < +∞, and α ∈ R m .For (1), the corresponding generating (ε = 0) boundary-value problem has the formThe n-dimensional vector function X(x(t, ε), t, ε) nonlinear in x is continuously differentiable with respect to x in the neighborhood of the solution x 0 of the generating boundary-value problem (2)The nonlinear bounded m-dimensional vector functional J(x(·, ε), ε) is continuously Fréchet differentiable with respect to x and continuous in ε in the neighborhood of the generating solution, and ε is a small nonnegative parameter.Let X(t) = [X 1 (t)X 2 (t)] be an n × 2n fundamental matrix of the linear homogeneous (f (t) = 0) system (2), let X i (t) be n × n matrices whose column vectors are linearly independent solutions of the homogeneous (f (t) = 0) system (2), let D := lX(·) be an m × 2n matrix generated by the action of the functional on the