On involutive division on monoids

We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the SetMM element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set XX. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.

Solvability for a Fully Elastic Beam Equation with Left-End Fixed and Right-End Simply Supported

The aim of the present paper is to consider a fully elastic beam equation with left-end fixed and right-end simply supported, i.e., u 4 t = f t , u t , u ′ t , u ″ t , u ‴ t , t ∈ 0,1 u 0 = u ′ 0 = u 1 = u ″ 1 = 0 , where f : 0,1 × ℝ 4 ⟶ ℝ is a continuous function. By applying Leray–Schauder fixed point theorem of the completely continuous operator, the existence and uniqueness of solutions are obtained under the conditions that the nonlinear function satisfies the linear growth and superlinear growth. For the case of superlinear growth, a Nagumo-type condition is introduced to limit that f t , x 0 , x 1 , x 2 , x 3 is quadratical growth on x 3 at most.

TO EXISTENCE OF COMPLETELY CONTINUOUS DIFFERENTIAL REALIZATION SECOND ORDER BILINEAR SYSTEM

For a continuous nonlinear infinite-dimensional behavioristic system (dynamical system of J. Willems), a functional-geometric study of the necessary and sufficient conditions for the existence of six non-stationary coefficients-operators of the model of bilinear differential realization of this system in the class of second-order differential equations in a separable Hilbert space is carried out. The case is investigated when the simulated operators are burdened with a condition that ensures the complete continuity of the integral form of the equations of realization in the entropy setting.

Periodicity and positivity in nonlinear neutral integro-dynamic equations with variable delay

Let T be a periodic time scale. We use Krasnoselskii's fixed point theorem for a sum of two operators to show new results on the existence of periodic and positive periodic solutions of a nonlinear neutral integro-dynamic equation with variable delay. We invert this equation to construct a sum of a contraction and a completely continuous map which is suitable for applying Krasnoselskii's theorem. The uniqueness results of this equation are studied by the contraction mapping principle.

Best Proximity Point for the Sum of Two Non-Self-Operators

In the present paper, we focus our attention on the existence of the fixed point for the sum of the cyclic contraction and the noncyclic accretive operator. Also, we study the best proximity point for the sum of two non-self-mappings. Moreover, we provide the existence of the best proximity point for the cyclic contraction through the notion of the nonlinear D-set contraction. Finally, we give the existence of the best proximity point for the sum of the nonlinear D-set contraction mapping and partially completely continuous mapping in the setting of the partially ordered complete normed linear space.

Thermal Conductivity of Silicone Rubber Filled with Al2O3

Crosslinker (polyvinylsilicone oil, C gum) and curing agent (2, 5-bis (tert-butyl peroxy)-2, 5-dimethyl hexane, DBPMH) were used to change the crosslink density of the thermal conductivity of silicone rubber filled with Al2O3. The results show that the thermal conductivity of silicone rubber changed with its degree of crosslinking. When the Al2O3 loading was 5 vol. % and a completely continuous conducting network had not formed, the thermal conductivity of the vulcanizates decreased with increasing crosslink density. The thermal conductivity of the vulcanizate with a suitable amount of C gum increased to 53%, and the tensile strength increased by 0.8 MPa compared to the vulcanizate without C gum. When the Al2O3 loading was 30 vol. % and a completely continuous conducting network had formed, the crosslink density of vulcanizates changed as the amount of DBPMH changed. The thermal conductivity of vulcanizates first decreased and then increased with increasing crosslink density. There was a valley value in the thermal conductivity–crosslink density curve.

Perturbation of the fixed point problem for continuous mappings

We consider the problem of a double fixed point of pairs of continuous mappings defined on a convex closed bounded subset of a Banach space. It is shown that if one of the mappings is completely continuous and the other is continuous, then the property of the existence of fixed points is stable under contracting perturbations of the mappings. We obtain estimates for the distance from a given pair of points to double fixed points of perturbed mappings. We consider the problem of a fixed point of a completely continuous mapping on a convex closed bounded subset of a Banach space. It is shown that the property of the existence of a fixed point of a completely continuous map is stable under contracting perturbations. Estimates of the distance from a given point to a fixed point are obtained. As an application of the obtained results, the solvability of a difference equation of a special type is proved.