# zbMATH — the first resource for mathematics

On a conservative integral equation with two kernels. (English. Russian original) Zbl 0914.45003
Math. Notes 62, No. 3, 271-277 (1997); translation from Mat. Zametki 62, No. 3, 323-331 (1997).
Summary: We study the solvability of the integral equation $f(x)= g(x)+ \int_0^\infty T_1(x-t)f(t)dt+ \int_{-\infty}^0 T_2(x-t)f(t)dt, \qquad x\in\mathbb{R},$ where $$f\in L_1^{\text{loc}} (\mathbb{R})$$ is the unknown function and $$g$$, $$T_1$$, and $$T_2$$ are given functions satisfying the conditions $g\in L_1(\mathbb{R}), \quad 0\leq T_j\in L_1(\mathbb{R}), \qquad \int_{-\infty}^\infty T_j(t)dt=1, \quad j=1,2.$ Most attention is paid to the nontrivial solvability of the homogeneous equation $s(x)= \int_0^\infty T_1(x-t) s(t)dt+ \int_{-\infty}^0 T_2(x-t)s(t)dt, \quad x\in\mathbb{R}.$

##### MSC:
 4.5e+11 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
Full Text:
##### References:
 [1] N. B. Engibaryan and L. G. Arabadzhyan, ”On some factorization problems for integral operators of convolution type,”Differentsial’nye Uravneniya [Differential Equations],26, No. 8, 1442–1452 (1990). · Zbl 0712.45005 [2] A. A. Arutyunyan, ”On factorization of an integral operator,” in:Differential and Integral Equations [in Russian], Erevan (1979), pp. 43–48. [3] N. B. Engibaryan and A. A. Arutyunyan, ”Integral equations with difference kernels on a half-line and nonlinear functional equations,”Mat. Sb. [Math. USSR-Sb.],97, No. 5, 35–58 (1975). · Zbl 0324.45005 [4] L. G. Arabadzhyan and N. B. Engibaryan, ”Convolution equations and nonlinear functional equations,” in:Itogi Nauki i Tekhniki, Matem. Analiz [in Russian], Vol. 22, VINITI, Moscow (1984), pp. 175–244. · Zbl 0568.45004 [5] L. G. Arabadzhyan, ”On the conservative Wiener-Hopf equation,”Izv. Akad. Armyan. SSR Ser. Mat. [Soviet J. Contemporary Math. Anal.],16, No. 1, 65–80 (1981). · Zbl 0461.45002 [6] R. Bellman and K. Cooke,Differential-Difference Equations, New York (1963). · Zbl 0105.06402 [7] A. N. Kolmogorov and S. V. Fomin,Elements of Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1981); English translation:Introductory Real Analysis, Dover, 1975. [8] L. G. Arabadzhyan, ”On an integral equation of transfer theory in inhomogeneous media,”Differential’nye Uravneniya [Differential Equations],23, No. 9 (1987). · Zbl 0635.45009 [9] S. Karlin, ”On the renewal equation,”Pacific J. Math.,5, No. 2, 229–257 (1955). · Zbl 0067.34902 [10] W. Rudin,Functional Analysis, McGraw-Hill, New York (1973). · Zbl 0253.46001 [11] L. G. Arabadzhyan, ”On the factorization of conservative integral Wiener-Hopf operators,”Mat. Zametki [Math. Notes],46, No. 1, 3–10 (1989). · Zbl 0724.45002
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.