×

Hammerstein-Nemytskii type nonlinear integral equations on half-line in space \(L_1(0,+\infty )\cap L_{\infty}(0,+\infty)\). (English) Zbl 1290.45001

The questions of solvability and uniqueness of the following class of Hammerstein-Nemytskii type nonlinear integral equations as \[ \varphi(x) = A(x,\varphi(x)) + \int\limits_0^\infty k(x-t) B(t,\varphi(t))dt, \, x \geq 0, \] with noncompact integral operator are discussed. Kernel \(K \in L_1(0,\infty) \bigcap L_\infty(0,\infty)\). This class of equations is the natural generalization of Wiener-Hopf type conservative integral equations. Examples are given to illustrate the results.

MSC:

45G05 Singular nonlinear integral equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
PDF BibTeX XML Cite
Full Text: Link

References:

[1] Arabadjyan, L. G., Yengibaryan, N. B.: Convolution equations and nonlinear functional equations. Itogi nauki i teckniki, Math. Analysis 4 (1984), 175-242
[2] Gokhberg, I. Ts., Feldman, I. A.: Convolution Equations and Proections Methods of Solutions. Nauka, Moscow, 1971.
[3] Khachatryan, A. Kh., Khachatryan, Kh. A.: Existence and uniqueness theorem for a Hammerstein nonlinear integral equation. Opuscula, Mathematica 31, 3 (2011), 393-398. · Zbl 1228.45007
[4] Khachatryan, A. Kh., Khachatryan, Kh. A.: On solvability of a nonlinear problem in theory of income distribution. Eurasian Math. Jounal 2 (2011), 75-88. · Zbl 1258.45004
[5] Khachatryan, Kh. A.: On one class of nonlinear integral equations with noncompact operator. J. Contemporary Math. Analysis 46, 2 (2011), 71-86.
[6] Khachatryan, Kh. A.: Some classes of Urysohn nonlinear integral equations on half line. Docl. NAS Belarus 55, 1 (2011), 5-9. · Zbl 1258.45004
[7] Kolmogorov, A. N., Fomin, V. C.: Elements of Functions Theory and Functional Analysis. Nauka, Moscow, 1981 · Zbl 0501.46002
[8] Lindley, D. V.: The theory of queue with a single sever. Proc. Cambridge Phil. Soc. 48 (1952), 277-289.
[9] Milojevic, P. S.: A global description of solution to nonlinear perturbations of the Wiener-Hopf integral equations. El. Journal of Differential Equations 51 (2006), 1-14.
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.