Topological Chaos for Differential Inclusions with Multivalued Impulses on Tori

A multivalued version of the Ivanov inequality for the lower estimate of topological entropy of admissible maps is applied to differential inclusions with multivalued impulses on tori via the associated Poincaré translation operators along their trajectories. The topological chaos in the sense of a positive topological entropy is established in terms of the asymptotic Nielsen numbers of the impulsive maps being greater than 1. This condition implies at the same time the existence of subharmonic periodic solutions with infinitely many variety of periods. Under a similar condition, the coexistence of subharmonic periodic solutions of all natural orders is also carried out.

Investigation of the Fractional Strongly Singular Thermostat Model via Fixed Point Techniques

Our main purpose in this paper is to prove the existence of solutions for the fractional strongly singular thermostat model under some generalized boundary conditions. In this way, we use some recent nonlinear fixed-point techniques involving α-ψ-contractions and α-admissible maps. Further, we establish the similar results for the hybrid version of the given fractional strongly singular thermostat control model. Some examples are studied to illustrate the consistency of our results.

On Some Interpolative Contractions of Suzuki Type Mappings

The goal of this study is to propose a new interpolative contraction mapping by using an interpolative approach in the setting of complete metric spaces. We present some fixed point theorems for interpolative contraction operators using w -admissible maps which satisfy Suzuki type mappings. In addition, some results are given. These results generalize several new results present in the literature. Moreover, examples are provided to show the suitability of our given results.

Coincidence Theory in the Coercive Case and Minimax Inequalities

We established coincidence results between maps with continuous selections and admissible maps. Both the compact and coercive cases were considered, and our argument relied on new coincidence ideas established recently by the author. Using our coincidence theory, we established new analytic alternatives, which then generate new minimax inequalities of the Neumann–Sion type.

Chaos for Differential Equations with Multivalued Impulses

The deterministic chaos in the sense of a positive topological entropy is investigated for differential equations with multivalued impulses. Two definitions of topological entropy are examined for three classes of multivalued maps: [Formula: see text]-valued maps, [Formula: see text]-maps and admissible maps in the sense of Górniewicz. The principal tool for its lower estimates and, in particular, its positivity are the Ivanov-type inequalities in terms of the asymptotic Nielsen numbers. The obtained results are then applied to impulsive differential equations via the associated Poincaré translation operators along their trajectories. The main theorems for chaotic differential equations with multivalued impulses are formulated separately on compact subsets of Euclidean spaces and on tori. Several illustrative examples are supplied.