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The investigation on the oscillatory and asymptotic properties of differential equations plays an important role in many applications, such as engineering, physics and medicine. In this work, we establish some new sufficient conditions for both the oscillatory and asymptotic behavior of the solutions to a second-order neutral impulsive differential system (IDS, for short) with deviating argument and fixed moments of impulse effect under both canonical and non-canonical operators with various values of the neutral coefficient. Also, we obtain the condition for the existence of a positive solution of the system under consideration. Finally, two examples are provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of a positive solution for a nonlinear Dirichlet problem driven by the p -Laplacian and a reaction which has the competing effects of a singular term and of a convection (gradient dependent) term. Using the "frozen" variable method and eventually the Leray-Schauder Alternative Theorem, we show that the problem has a positive smooth solution. We mention that on the gradient dependent perturbation x ↦ f (z , x , y) , we do not impose any bilateral growth condition. [ABSTRACT FROM AUTHOR]

We consider a nonlinear elliptic problem driven by the p-Laplacian plus and indefinite potential term. The reaction is (p - 1)-linear and resonant and the boundary term is concave. The problem is nonparametric. Using variational tools, together with truncation and perturbation techniques and critical groups, we show that the problem has at least three nontrivial smooth solutions. [ABSTRACT FROM AUTHOR]

We consider a nonlinear eigenvalue problem for equations driven by a weighted (p , q) -Laplacian and a superlinear reaction. We prove a global (with respect to the parameter λ > 0) existence and multiplicity result. We also generate nodal (sign-changing) solutions. Finally, we determine the topological properties of the solution set and prove the continuity properties of the solution multifunction. [ABSTRACT FROM AUTHOR]

We consider nonlinear Robin problems driven by the p-Laplacian plus an indefinite potential. In the reaction, we have the competing effects of a parametric concave (that is, (p − 1) -sublinear) term and of a convex (that is, (p − 1) -superlinear) term which need not satisfy the Ambrosetti–Rabinowitz condition. We prove a "bifurcation-type" theorem describing in a precise way the dependence the dependence of the set of positive solutions on the parameter λ > 0 . In addition, we show the existence of a smallest positive solution u λ * and determine the monotonicity and continuity properties of the map λ ↦ u λ * . [ABSTRACT FROM AUTHOR]