Abduragimov G.E. On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Symmetric Boundary Conditions for One Nonlinear Fourth-Order Ordinary Differential Equation
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https://doi.org/10.15688/mpcm.jvolsu.2024.2.1
Gusen E. Abduragimov
Candidate of Sciences (Physics and Mathematics), Associate Professor, Department of Applied Mathematics,
Dagestan State University
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Dzerzhinskogo St, 12, 367025 Makhachkala, Russian Federation
Abstract. The article deals with the boundary value problem
where g(t) is a non-negative and continuous function on the interval [0, 1] such that
The function f(t, u) is assumed to be non-negative and continuous on [0, 1] × × [0,∞), and f(·, 0) ≡ 0. Using the Green’s function, this problem is reduced to an equivalent integral equation and, based on some properties of the Green’s function, a cone with respect to which the corresponding integral operator is invariant is selected in a special way. Further, under power-law restrictions on the function f, using the well-known Go-Krasnoselsky theorem, the presence of at least one fixed point of the specified operator is established. The last circumstance is equivalent to the existence of at least one positive solution to the boundary value problem under consideration. Sufficient conditions for the uniqueness of a positive solution were obtained only in the sublinear case. For this purpose, the principle of compressed mappings was used. The paper concludes with an example illustrating the results obtained.
Key words: boundary value problem, positive solution, Green’s function, cone, fixed point.
On the Existence and Uniqueness of a Positive Solution to a Boundary Value Problem with Symmetric Boundary Conditions for One Nonlinear Fourth-Order Ordinary Differential Equation by Abduragimov G.E., is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 27 No. 2 2024, pp. 5-13
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