Eckland and Bishop-Phelps variational principles in partially ordered spaces

In this paper, an assertion about the minimum of the graph of a mapping acting in partially ordered spaces is obtained. The proof of this statement uses the theorem on the minimum of a mapping in a partially ordered space from [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Caristi-like condition and the existence of minima of mappings in partially ordered spaces // Journal of Optimization Theory and Applications. 2018. V. 180. Iss. 1, 48–61]. It is also shown that this statement is an analogue of the Eckland and Bishop-Phelps variational principles which are effective tools for studying extremal problems for functionals defined on metric spaces. Namely, the statement obtained in this paper and applied to a partially ordered space created from a metric space by introducing analogs of the Bishop-Phelps order relation, is equivalent to the classical Eckland and Bishop-Phelps variational principles.

On Chaplygin’s theorem for an implicit differential equation of order n

We consider the Cauchy problem for the implicit differential equation of order n g(t,x,(x,) ̇…,x^((n)) )=0,t ∈ [0; T],x(0)= A. It is assumed that A=(A_0,…,A_(n-1) )∈R^n, the function g:[0,T] × R^(n+1)→ R is measurable with respect to the first argument t∈[0,T], and for a fixed t, the function g(t,∙)×R^(n+1)→ R is right continuous and monotone in each of the first n arguments, and is continuous in the last n+1-th argument. It is also assumed that for some sufficiently smooth functions η,ν there hold the inequalities ν^((i) ) (0)≥ A_i ≥ η^((i) ) (0),i= (0,n-1,) ̅ ν^((n) ) (t)≥ η^((n) ) (t),t∈[0; T]; g(t; ν(t),ν ̇(t),...,ν^((n) ) (t) )≥ 0,g(t,η(t),η ̇(t),…,η^((n)) (t))≤ 0,t∈[0; T]. Sufficient conditions for the solvability of the Cauchy problem are derived as well as estimates of its solutions. Moreover, it is shown that under the listed conditions, the set of solutions satisfying the inequalities η^((n) ) (t)≤x^((n) ) (t)≤ν^((n) ) (t), is not empty and contains solutions with the largest and the smallest n -th derivative. This statement is similar to the classical Chaplygin theorem on differential inequality. The proof method uses results on the solvability of equations in partially ordered spaces. Examples of applying the results obtained to the study of second-order implicit differential equations are given.

Some questions of the analysis of mappings of metric and partially ordered spaces

The questions of existence of solutions of equations and attainability of minimum values of functions are considered. All the obtained statements are united by the idea of existence for any approximation to the desired solution or to the minimum point of the improved approximation. The relationship between the considered problems in metric and partially ordered spaces is established. It is also shown how some well-known results on fixed points and coincidence points of mappings of metric and partially ordered spaces are derived from the obtained statements. Further, on the basis of analogies in the proofs of all the obtained statements, we propose a method for obtaining similar results from the theorem being proved on the satisfiability of a predicate of the following form. Let (X,≤) − be a partially ordered space, the mapping Φ:X×X→{0,1} satisfies the following condition: for any x∈X there exists x^'∈X such that x^'≤x and Φ(x^',x)=1. The predicate F(x)=Φ(x,x) is considered, sufficient conditions for its satisfiability, that is, the existence of a solution to the equation F(x)=1. This result was announced in [Zhukovskaya T.V., Zhukovsky E.S. Satisfaction of predicates given on partially ordered spaces // Kolmogorov Readings. General Control Problems and their Applications (GCP–2020). Tambov, 2020, 34-36].

On the extension of Chaplygin’s theorem to the differential equations of neutral type

We consider functional-differential equation x ̇((g(t) )= f(t; x(h(t) ) ),t ∈ [0; 1], where function f satisfies the Caratheodory conditions, but not necessarily guarantee the boundedness of the respective superposition operator from the space of the essentially bounded functions into the space of integrable functions. As a result, we cannot apply the standard analysis methods (in particular the fixed point theorems) to the integral equivalent of the respective Cauchy problem. Instead, to study the solvability of such integral equation we use the approach based not on the fixed point theorems but on the results received in [A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy. Coincidence points principle for mappings in partially ordered spaces // Topology and its Applications, 2015, v. 179, № 1, 13–33] on the coincidence points of mappings in partially ordered spaces. As a result, we receive the conditions on the existence and estimates of the solutions of the Cauchy problem for the corresponding functional-differential equation similar to the well-known Chaplygin theorem. The main assumptions in the proof of this result are the non-decreasing function f(t; •) and the existence of two absolutely continuous functions v,w, that for almost each t ∈ [0; 1] satisfy the inequalities v ̇(g(t) )≥f(t; v(h(t) ) ),w ̇(g(t) )≤f(t;w(h(t) ) ). The main result is illustrated by an example.