In this Problems of the type presented in this article describe the operation of a harmonic oscillator under the influence of external forces, which is fixed in the extreme left position and has some mechanism at right one, that controls the displacement according to the feedback from devices measuring the displacements along parts of the oscillator. Above in the introduction, references are given to some papers in which integral boundary conditions for differential equations were considered. The paper is organized as follows: the boundary value problem considered in the paper is reduced to an equivalent integral equation, and the existence of a positive solution of the integral equation is established using the fixed point principle of an operator defined on a certain cone. An a priori estimate of such a solution is obtained, which subsequently participates in the proof of the uniqueness of a positive solution. Sufficient conditions for uniqueness follow from the uniqueness principle for u0 convex operators on a cone. At the end of the work, an example is given that demonstrates the results obtained.
Using the fixed point theorem in partially ordered sets, we obtain sufficient conditions for the existence of a unique positive solution to a boundary-value problem of the Sturm-Liouville type for a nonlinear ordinary differential equation, and give an example illustrating the results obtained.
In this article, we consider a two-point boundary value problem for a nonlinear functional differential equation of fractional order with weak nonlinearity on the interval [0,1] with zero Dirichlet conditions on the boundary. The boundary value problem is reduced to an equivalent integral equation in the space of continuous functions. Using special topological tools (using the geometric properties of cones in the space of continuous functions, statements about fixed points of monotone and concave operators), the existence of a unique positive solution to the problem under consideration is proved. An example is given that illustrates the fulfillment of sufficient conditions that ensure the unique solvability of the problem. The results obtained are a continuation of the author’s research (see [Results of science and technology. Ser. Modern mat. and her appl. Subject. review, 2021, vol. 194, pp. 3–7]) devoted to the existence and uniqueness of positive solutions of boundary value problems for non-linear functional differential equations.
Ðàññìàòðèâàåòñÿ äâóõòî÷å÷íàÿ êðàåâàÿ çàäà÷à äëÿ îäíîãî íåëèE íåéíîãî ôóíêöèîíàëüíîEäèôôåðåíöèàëüíîãî óðàâíåíèÿ âòîðîãî ïîðÿäêàF Ñ ïîìîùüþ ñïåöèàëüíûõ òîïîëîãè÷åñêèõ ñðåäñòâ ïîëóE ÷åíû äîñòàòî÷íûå óñëîâèÿD îáåñïå÷èâàþùèå ñóùåñòâîâàíèå ïî êðàéE íåé ìåðå îäíîãî ïîëîaeèòåëüíîãî ðåøåíèÿ èññëåäóåìîé çàäà÷èF Êëþ÷åâûå ñëîâàX ïîëîaeèòåëüíîå ðåøåíèåD êðàåâàÿ çàäà÷àD êîíóñFOn the existence of a positive solution to a boundary value problem for one nonlinear functional -differential equation of the second order A two-point boundary value problem for one nonlinear functional differential equation of the second order is considered. With the help of special topological means sufficient conditions are obtained to ensure the existence at least one positive solution to the problem under study.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.