### Transversal homoclinics in nonlinear systems of ordinary differential equations.

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The existence of solutions is studied for certain nonlinear differential equations with both linear and nonlinear conditions

Asymptotically quadratic functions defined on Hilbert spaces are studied by using some results of the theory of Morse-Conley index. Applications are given to existence of nontrivial weak solutions for asymptotically linear elliptic partial and ordinary differential equations at resonances.

Existence results for critical points of asymptotically quadratic functions defined on Hilbert spaces are studied by using Morse-Conley index and pseudomonotone mappings. Applications to differential equations are given.

The Nielsen fixed point theory is used to show several results for certain operator equations involving weakly inward mappings.

Ordinary differential inclusions depending on small parameters are considered such that the unperturbed inclusions are ordinary differential equations possessing manifolds of periodic solutions. Sufficient conditions are determined for the persistence of some of these periodic solutions after multivalued perturbations. Applications are given to dry friction problems.

The paper deals with the bifurcation phenomena of heteroclinic orbits for diffeomorphisms. The existence of a Melnikov-like function for the two-dimensional case is shown. Simple possibilities of the set of heteroclinic points are described for higherdimensional cases.

For several specific mappings we show their chaotic behaviour by detecting the existence of their transversal homoclinic points. Our approach has an analytical feature based on the method of Lyapunov-Schmidt.

The existence and multiplicity results are shown for certain types of problems with nonlinear boundary value conditions.

Existence results of nonnegative solutions of asymptotically linear, nonlinear integral equations are studied.

Equivalence and zero sets of certain maps on infinite dimensional spaces are studied using an approach similar to the deformation lemma from the singularity theory.

Iterative methods based on small functions are used both to show local surjectivity of certain operators and a fixed point property of mappings on scales of complete metric spaces.

The existence of nonzero nonnegative solutions are established for semilinear equations at resonance with the zero solution and possessing at most linear growth. Applications are given to nonlinear boundary value problems of ordinary differential equations.

The existence of multiple solutions for perturbed equations is shown near a manifold of solutions of an unperturbed equation via the Nielsen fixed point theory.

Chaos generated by the existence of Smale horseshoe is the well-known phenomenon in the theory of dynamical systems. The Poincaré-Andronov-Melnikov periodic and subharmonic bifurcations are also classical results in this theory. The purpose of this note is to extend those results to ordinary differential equations with multivalued perturbations. We present several examples based on our recent achievements in this direction. Singularly perturbed problems are studied as well. Applications are given...

The existence of classical solutions for some partial differential equations on tori is shown.

We show that certain symmetries of maps imply the existence of their invariant curves.

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