Opuscula Math. 36, no. 3 (2016), 399-407
http://dx.doi.org/10.7494/OpMath.2016.36.3.399
Opuscula Mathematica
Existence theorems of nonlinear asymptotic BVP for a homeomorphism
Abstract. In this work, we are concerned with the existence of solutions for the following \(\varphi\)-Laplacian boundary value problem on the half-line \[(\varphi (x'))' =f(t,x,x'),\quad x(0)=0,\quad x'(\infty)=0,\] where \(f:\mathbb{R}_+\times\mathbb{R}^k\times\mathbb{R}^k\to\mathbb{R}^k\) is continuous. The results are proved using the properties of the Leray-Schauder topological degree.
Keywords: half-line, nonlinear, asymptotic boundary value problem, \(\varphi\)-Laplacian, Leray-Schauder degree.
Mathematics Subject Classification: 34B15, 34B40.
- J. Andres, G. Gabor, L. Górniewicz, Boundary value problems on infinite intervals, Trans. Amer. Math. Soc. 351 (1999), 4861-4903.
- C.Z. Bai, J.X. Fang, On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals, J. Math. Anal. Appl. 282 (2003), 711-731.
- J.V. Baxley, Existence and uniqueness of nonlinear boundary value problems on infinite intervals, J. Math. Anal. Appl. 147 (1990), 127-133.
- J.W. Bebernes, L.K. Jackson, Infinite interval boundary value problems for \(y^{\prime\prime}=f(x,y)\), Duke Math. J. 34 (1967), 39-47.
- L.E. Bobisud, Existence of positive solutions to some nonlinear singular boundary value problems on finite and infinite intervals, J. Math. Anal. Appl. 173 (1993), 69-83.
- A. Constantin, On an infinite interval boundary value problem, Ann. Mat. Pura Appl. 176 (1999) 4, 379-394.
- S. Djebali, O. Saifi, Positive solutions for singular \(\varphi\)-Laplacian BVPs on the positive half-line, Electron. J. Qual. Theory Differ. Equ. 56 (2009), 24 pp.
- S. Djebali, O. Saifi, Positive solutions for singular BVPs with sign changing and derivative depending nonlinearity on the positive half line, Acta Appl. Math. 110 (2010) 2, 639-665.
- S. Liang, J. Zhang, The existence of countably many positive solutions for one-dimensional p-Laplacian with infinitely many singularities on the half line, Appl. Math. Comput. 201 (2008), 210-220.
- A. Lipowski, B. Przeradzki, K. Szymańska-Dębowska, Periodic solutions to differential equations with a generalized p-Laplacian, Discrete Contin. Dyn. Syst. Ser. B 19 (2014) 8, 2593-2601.
- R. Manásevich, J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Diff. Equ. 145 (1998), 367-393.
- D. O'Regan, B. Yan, R.P. Agarwal, Solutions in weighted spaces of singular boundary value problems on the half-line, J. Comput. Appl. Math. 205 (2007), 751-763.
- P.J. Rabier, C.A. Stuart, A Sobolev space approach to boundary value problems on the half-line, Comm. in Contemp. Math. 7 (2005) 1, 1-36.
- K. Szymańska, On an asymptotic boundary value problem for second order differential equation, J. Appl. Anal. 12 (2006) 1, 109-119.
- Katarzyna Szymańska-Dębowska
- Lodz University of Technology, Institute of Mathematics, 90-924 Łódź, ul. Wólczańska 215, Poland
- Communicated by Giovanni Molica Bisci.
- Received: 2015-05-22.
- Accepted: 2015-11-26.
- Published online: 2016-02-21.
