Geodesic
On the existence and uniqueness of a positive solution to a boundary value problem with integral boundary conditions for one nonlinear ordinary differential equation of the second order
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2024), pp. 12-19 Cet article a éte moissonné depuis la source Math-Net.Ru

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The article considers a boundary value problem with integral boundary conditions for one nonlinear second-order functional differential equation. Using Green's function, the boundary value problem is reduced to the equivalent nonlinear Hammerstein integral equation. Next, having identified the necessary properties of Green's function, we prove that the Hammerstein operator contracts the corresponding cone. The last circumstance, by virtue of the well-known Krasnoselsky theorem, guarantees the existence of at least one positive solution to the boundary value problem. Using a priori estimates and the principle of compressed mappings, sufficient conditions for the uniqueness of a positive solution were obtained. At the end of the article, there is a non-trivial example illustrating the results obtained here.
Mots-clés : positive solution, cone compression.
Keywords: integral boundary value problem, Green's function 
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     title = {On the existence and uniqueness of a positive solution to a boundary value problem with integral boundary conditions for one nonlinear ordinary differential equation of the second order},
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G. È. Abduragimov. On the existence and uniqueness of a positive solution to a boundary value problem with integral boundary conditions for one nonlinear ordinary differential equation of the second order. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2024), pp. 12-19. http://geodesic.mathdoc.fr/item/IVM_2024_12_a1/

[1] Benchohra M., Nieto J.J., Ouahab A., “Second-Order Boundary Value Problem with Integral Boundary Conditions”, Bound Value Probl., 2011 (2011), 1–9 | DOI | MR

[2] Li H., Sun F., “Existence of solutions for integral boundary value problems of second-order ordinary differential equations”, Bound Value Probl., 2012:147 (2012), 1–7 | MR

[3] Cabada A., Iglesias J., “Nonlinear differential equations with perturbed Dirichlet integral boundary conditions”, Bound Value Probl., 2021:66 (2021), 1–19 | MR

[4] Ghanmii A., Jebari R., Zhang Z., “Multiplicity results for a boundary value problem with integral boundary conditions”, SeMA, 76:2 (2019), 365–381 | DOI | MR | Zbl

[5] Djourdem H., Benaicha S., “Positive solutions of nonlinear third-order boundary value problem with integral boundary conditions”, Malaya J. Matem., 7:2 (2019), 269–175 | DOI | MR

[6] Egorov I.E., Efimova E.S., “O razreshimosti kraevoi zadachi s integralnym granichnym usloviem po vremeni dlya uravneniya nechetnogo poryadka s menyayuschimsya napravleniem vremeni”, Matem. zametki SVFU, 26:1 (2019), 6–11

[7] Benaicha S., Bouteraa N., Djourdem H., “Triple positive solutions for a class of boundary value problems with integral boundary conditions”, Bull. Transilvania Univ. Brasov, Ser. III, 13 (62):1 (2020), 51–68 | MR | Zbl

[8] Bugajewska D., Mawhin J., “Boundary value problems with bounded $\varphi$-Laplacian and nonlocal conditions of integral type”, Czech. Math. J., 2023, 1–12 | DOI

[9] Azbelev N.V., Maksimov V.P., Rakhmatullina L.F., Vvedenie v teoriyu funktsionalno-differentsialnykh uravnenii, Nauka, M., 1991 | MR

[10] Bravyi E.I., Kraevye zadachi dlya semeistv lineinykh funktsionalno-differentsialnykh uravnenii, diss. ... dokt. fiz.-matem. nauk, PGTU, Perm, 2017

[11] Abduragimov G.E., “O suschestvovanii i edinstvennosti polozhitelnogo resheniya kraevoi zadachi dlya odnogo nelineinogo differentsialnogo uravneniya vtorogo poryadka s integralnymi granichnymi usloviyami”, Matem. fizika i kompyuter. modelirovanie, 25:4 (2022), 5–14 | MR

[12] Abduragimov G.E., “O suschestvovanii polozhitelnogo resheniya kraevoi zadachi dlya odnogo nelineinogo funktsionalno-differentsialnogo uravneniya vtorogo poryadka”, Itogi nauki i tekhn. Sovrem. matem. i ee pril. Temat. obz., 221, 2023, 3–9 | DOI | MR

[13] Krein S.G., Funktsionalnyi analiz, Nauka, M., 1972 | MR

[14] Krasnoselskii M.A., Zabreiko P.P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975