Article
Full entry |
PDF
(0.1 MB)
Feedback
PDF
(0.1 MB)
Feedback
Keywords:
reflection; Brouwer fixed point; Kirchhoff equation
reflection; Brouwer fixed point; Kirchhoff equation
Summary:
We study the nonlinear boundary value problem involving reflection of the argument \[ -M\Big (\int _{-1}^1\vert u^{\prime }(s)\vert ^2\,ds\Big )\,u^{\prime \prime }(x) = f\big (x,u(x),u(-x)\big ) \quad \quad x \in [-1,1]\,, \] where $M$ and $f$ are continuous functions with $M>0$. Using Galerkin approximations combined with the Brouwer’s fixed point theorem we obtain existence and uniqueness results. A numerical algorithm is also presented.
References:
[1] Arosio A., Panizzi S.: On the well-posedness of the Kirchhoff string. Trans. Amer. Math. Soc. 348 (1996), 305–330. MR 1333386 | Zbl 0858.35083
[2] Chipot M., Rodrigues J. F.: On a class of nonlinear nonlocal elliptic problems. RAIRO Modél. Math. Anal. Numér. 26 (1992), 447–467. MR 1160135
[3] Gupta C. P.: Existence and uniqueness theorems for boundary value problems involving reflection of the argument. Nonlinear Anal. 11 (1987), 1075–1083. MR 0907824 | Zbl 0632.34069
[4] Hai D. D.: Two point boundary value problem for differential equations with reflection of argument. J. Math. Anal. Appl. 144 (1989), 313–321. MR 1027038 | Zbl 0699.34017
[5] Kesavan S.: Topics in Functional Analysis and Applications. Wiley Eastern, New Delhi, 1989. MR 0990018 | Zbl 0666.46001
[6] Ma T. F.: Existence results for a model of nonlinear beam on elastic bearings. Appl. Math. Lett. 13 (2000), 11–15. MR 1760256 | Zbl 0965.74030
[7] O’Regan D.: Existence results for differential equations with reflection of the argument. J. Austral. Math. Soc. Ser. A 57 (1994), 237–260. MR 1288675 | Zbl 0818.34037
[8] Sharma R. K.: Iterative solutions to boundary-value differential equations involving reflection of the argument. J. Comput. Appl. Math. 24 (1988), 319–326. MR 0974020 | Zbl 0664.65080
[9] Wiener J., Aftabizadeh A. R.: Boundary value problems for differential equations with reflection of the argument. Int. J. Math. Math. Sci. 8 (1985), 151–163. MR 0786960 | Zbl 0583.34055
Search
Partner of
