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# Article

 Title: Existence of solutions for a coupled system with $\phi$-Laplacian operators and nonlinear coupled boundary conditions (English) Author: Goli, Konan Charles Etienne Author: Adjé, Assohoun Language: English Journal: Communications in Mathematics ISSN: 1804-1388 Volume: 25 Issue: 2 Year: 2017 Pages: 79-87 Summary lang: English . Category: math . Summary: We study the existence of solutions of the system $$\begin {cases} (\phi _1(u_1'(t)))'= f_1(t,u_1(t),u_2(t),u'_1(t),u_2'(t)),\qquad \text {a.e. t\in [0,T]}, (\phi _2(u_2'(t)))'= f_2(t,u_1(t),u_2(t),u'_1(t),u_2'(t)),\qquad \text {a.e. t\in [0,T]}, \end {cases}$$ submitted to nonlinear coupled boundary conditions on $[0,T]$ where $\phi _1,\phi _2\colon (-a, a)\rightarrow \mathbb {R}$, with $0 < a < +\infty$, are two increasing homeomorphisms such that $\phi _1(0) = \phi _2(0) = 0$, and $f_i:[0,T]\times \mathbb {R}^{4}\rightarrow \mathbb {R}$, $i\in \{1,2\}$ are two $L^1$-Carathéodory functions. Using some new conditions and Schauder fixed point Theorem, we obtain solvability result. (English) Keyword: $\phi$-Laplacian; $L^1$-Carath\'eodory function; Schauder fixed-point Theorem. MSC: 34B15 idZBL: Zbl 1391.34052 idMR: MR3745430 . Date available: 2018-02-05T14:37:07Z Last updated: 2020-01-05 Stable URL: http://hdl.handle.net/10338.dmlcz/147058 . Reference: [1] Asif, N. A., Talib, I.: Existence of Solutions to a Second Order Coupled System with Nonlinear Coupled Boundary Conditions.American Journal of Applied Mathematics, Special Issue: Proceedings of the 1st UMT National Conference on Pure and Applied Mathematics (1st UNCPAM 2015), 3, 3-1, 2015, 54-59, MR 3081587 Reference: [2] Asif, N. A., Talib, I., Tunc, C.: Existence of solution for first-order coupled system with nonlinear coupled boundary conditions.Boundary Value Problems, 2015, 1, 2015, 134, Zbl 1342.34035, MR 3377958 Reference: [3] Bereanu, C., Mawhin, J.: Nonhomogeneous boundary value problems for some nonlinear equations with singular $\phi$-Laplacian.J. Math. Anal. Appl., 352, 2009, 218-233, Zbl 1170.34014, MR 2499899, 10.1016/j.jmaa.2008.04.025 Reference: [4] Bergmann, P.G.: Introduction to the Theory of Relativity.1976, Dover Publications, New York, MR 0006876 Reference: [5] Brezis, H., Mawhin, J.: Periodic solutions of the forced relativistic pendulum.Differential Integral Equations, 23, 2010, 801-810, Zbl 1240.34207, MR 2675583 Reference: [6] Franco, D., O'Regan, D.: Existence of solutions to second order problems with nonlinear boundary conditions.Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, 2002, 273-280, MR 2018125 Reference: [7] Franco, D., O'Regan, D.: A new upper and lower solutions approach for second order problems with nonlinear boundary conditions.Arch. Inequal. Appl., 1, 2003, 423-430, Zbl 1098.34520, MR 2020621 Reference: [8] Franco, D., O'Regan, D., Perán, J.: Fourth-order problems with nonlinear boundary conditions.Journal of Computational and Applied Mathematics, 174, 2005, 315-327. Zbl 1068.34013, MR 2106442, 10.1016/j.cam.2004.04.013 .

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