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On equation \(P(\text{D})u=f(u^{(m)})+g\bigl (t,(u^{(j)})\bigr )\) on the line. (English) Zbl 1116.34023
Let \(D=-i (d/dt)\), \(f\in C(\mathbb R,\mathbb R)\) with \(| f(x)| \leq K| x| \) near \(x=0\). Moreover, let \(g \: \mathbb R\times \mathbb R^k\to \mathbb R\) be Carathéodory continuous such that \(| g(t,y_1,\cdots ,y_k)| \leq h(t)\) for \(h\in L_2(R)\). Additional conditions are considered in order to show the existence of a real solution for a differential equation \[ P(D)u=f(u^{(m)})+g\bigl (t,(u^{(j)})_{j=j_1,\dots ,j_k}\bigr ) \]
in the Sobolev space \(H_n(\mathbb R)\), where \(n\) is the degree of the linear differential operator \(P(\text{D})\).
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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[1] ANDRES J.-GABOR G.-GÓRNIEWICZ L.: Boundary value problems on infinit intervals. Trans. Amer. Math. Soc. 351 (1999), 4861-4903. · Zbl 0936.34023
[2] BERNSTEIN S. N.: Sur les équations du calcul des variations. Ann. Sci. Ecole Norm. Sup. 29 (1912), 431-485. · JFM 43.0460.01
[3] BROWDER F. E.: Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. Contributions to Nonlinear Functional Analysis (E. H. Zarantonello, Academic Press, New York, 1971, pp. 425 500. · Zbl 0267.47038
[4] CONSTANTIN A.: On an infinite interval boundary value problem. Ann. Mat. Pura Appl. (4) 176 (1999), 379-396. · Zbl 0969.34024
[5] FIJAŁKOWSKI P.: On the solvability of nonlinear elliptic equations in Sobolev spaces. Ann. Polon. Math. 56 (1992), 149-156. · Zbl 0776.35017
[6] FIJAŁKOWSKI P.-PRZERADZKI B.: On a boundary value problem for a nonlocal elliptic equation. J. Appl. Anal. · Zbl 1071.35047
[7] GRANAS A.-GUENTHER R.-LEE J.: Nonlinear boundary value problems for ordinary differential equations. Dissertationes Math. (Rozprawy Mat.) 244 (1985). · Zbl 0615.34010
[8] HORMANDER L.: The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coeffìcients. Grundlehren Math. Wiss. 257, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983.
[9] KRASNOSEĽSKII M. A.-PUSTYĽNIK E. I.- SOBOLEVSKII P. E.-ZABREIKO P. P.: Integral Operators of Summable Functions. Nauka, Moscow, 1966.
[10] LLOYD N. G.: Degree Theory. Cambridge Tracts in Math. 73, Cambridge Univ. Press, Cambridge, 1978. · Zbl 0367.47001
[11] YOSIDA K.: Functional Analysis. Springer, Berlin, 1980. · Zbl 0435.46002
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