×

zbMATH — the first resource for mathematics

Existence of solutions for a coupled system with \(\phi\)-Laplacian operators and nonlinear coupled boundary conditions. (English) Zbl 1391.34052
Summary: We study the existence of solutions of the system \[ \begin{cases}(\phi_1(u_1^\prime(t)))^\prime=f_1(t,u_1(t),u_2(t),u_1^\prime(t),u_2^\prime(t)),\quad\mathrm{a.e.}t\in[0,T],\\(\phi_2(u_2^\prime(t)))^\prime=f_2(t,u_1(t),u_2(t),u_1^\prime(t),u_2^\prime(t)),\quad\mathrm{a.e.}t\in[0,T],\end{cases} \] 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.
MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[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, · Zbl 1329.34049
[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
[3] Bereanu, C., Mawhin, J.: Nonhomogeneous boundary value problems for some nonlinear equations with singular \(ϕ\)-Laplacian. J. Math. Anal. Appl., 352, 2009, 218-233, | | · Zbl 1170.34014
[4] Bergmann, P.G.: Introduction to the Theory of Relativity. 1976, Dover Publications, New York,
[5] Brezis, H., Mawhin, J.: Periodic solutions of the forced relativistic pendulum. Differential Integral Equations, 23, 2010, 801-810, | · Zbl 1240.34207
[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,
[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
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.