Linear operators and equations with partial integrals

A. S. Kalitvin ; V. A. Kalitvin

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 65 (2019) no. 3, pp. 390-433.

Résumé

We consider linear operators and equations with partial integrals in Banach ideal spaces, spaces of vector functions, and spaces of continuous functions. We study the action, regularity, duality, algebras, Fredholm properties, invertibility, and spectral properties of such operators. We describe principal properties of linear equations with partial integrals. We show that such equations are essentially different compared to usual integral equations. We obtain conditions for the Fredholm alternative, conditions for zero spectral radius of the Volterra operator with partial integrals, and construct resolvents of invertible equations. We discuss Volterra–Fredholm equations with partial integrals and consider problems leading to linear equations with partial integrals.

Bibliographie

[1] V. M. Aleksandrov, N. Kh. Arutyunyan, A. V. Manzhirov, “Contact problems of the creepage of nonhomogeneously aging bodies”, Analytic and Numeric Methods of Boundary-Value Problems of Plasticity and Viscoelasticity, UNTS AN SSSR, Sverdlovsk, 1986, 3–13 (in Russian)

[2] V. M. Aleksandrov, E. V. Kovalenko, “Axisymmetric contact problem for linearly deformable groundwork of general kind with abrasion”, Bull. Acad. Sci. USSR. Ser. Contin. Mech., 1978, no. 5, 58–66 (in Russian)

[3] V. M. Aleksandrov, E. V. Kovalenko, “One one class of integral equations in mixed problems of continuum mechanics”, Rep. Acad. Sci. USSR, 252 (1980), 324–328 (in Russian) | Zbl

[4] V. M. Aleksandrov, E. V. Kovalenko, “On the contact interaction of covered bodies with abrasion”, Rep. Acad. Sci. USSR, 275:4 (1984), 827–830 (in Russian) | MR | Zbl

[5] V. M. Aleksandrov, E. V. Kovalenko, Problems of Continuum Mechanics with Mixed Boundary-Value Conditions, Nauka, M., 1986 (in Russian)

[6] S. M. Belotserkovskiy, I. K. Lifanov, Numerical Methods in Singular Integral Equations, Nauka, M., 1985 (in Russian) | MR

[7] A. V. Bitsadze, Boundary-Value Problems for Second-Order Elliptic Equations, Nauka, M., 1966 (in Russian)

[8] V. V. Boltyanskiy, “On solvability of integral equation with partial integrals and a kernel depending on three variables”, Differential Equations, Ryazan', 1981, 3–14 (in Russian) | MR

[9] V. V. Boltyanskiy, L. M. Likhtarnikov, “On one class of integral equation with partial integrals”, Differ. Equ., 18:11 (1982), 1939–1950 (in Russian) | MR

[10] I. N. Vekua, New Methods of Solution of Elliptic Equations, OGIZ Gostekhizdat, M.–L., 1948 (in Russian) | MR

[11] L. Z. Vitova, To the theory of linear integral equations with partial integrals, PhD Thesis, Novgorod, 1977

[12] V. Vol'terra, Theory of Functionals, Integral and Integrodifferential Equations, Nauka, M., 1982 (in Russian)

[13] S. A. Gabov, A. G. Sveshnikov, Linear Problems of Nonstationary Inner Waves, Nauka, M., 1990 (in Russian) | MR

[14] L. A. Galin, I. G. Goryacheva, “Axisymmetric contact problem of elasticity theory with abrasion”, Appl. Math. Mech., 41:5 (1977), 807–812 (in Russian)

[15] I. M. Gel'fand, B. M. Levitan, “On determining of differential equation by its spectral function”, Progr. Math. Sci., 11:1 (1956), 191–198 (in Russian) | MR | Zbl

[16] V. I. Glivenko, Stieltjes Integral, ONTI, M.–L., 1936 (in Russian)

[17] A. A. Govorukhina, N. V. Kovalenko, I. A. Paradoksova, “Two-dimensional integral equations with partial integrals on a plane and a half-plane”, Integral and Differential Equations and Approximate Solutions (Elista, 1985), 23–32 (in Russian) | MR

[18] É. Goursat, Cours d'Analyse Mathématique, Part 2, v. 3, ONTI, 1934 (Russian translation)

[19] P. P. Zabreyko, Investigation of integral operators in ideal spaces, PhD Thesis, Voronezh, 1968 | Zbl

[20] P. P. Zabreyko, “Ideal spaces of functions. I”, Bull Yaroslavl Univ., 8 (1974), 12–52 (in Russian)

[21] P. P. Zabreyko, A. S. Kalitvin, E. V. Frolova, “On integral equations with partial integrals in the space of continuous functions”, Differ. Equ., 38:4 (2002), 538–546 (in Russian) | MR | Zbl

[22] P. P. Zabreyko, A. I. Koshelev, M. A. Krasnosel'skiy, S. G. Mikhlin, L. S. Rakovshchik, V. Ya. Stetsenko, Integral Equations, Nauka, M., 1968 (in Russian)

[23] P. P. Zabreyko, A. N. Lomakovich, “Volterra integral operators in spaces of functions of two variables”, Ukr. Math. J., 42:9 (1990), 1187–1191 (in Russian) | MR

[24] A. I. Inozemtsev, A. S. Kalitvin, “On spectrum of operators with multidimensional partial integrals”, Bull. LGPU. Ser. Math. Inform. Phys. Nat. Sci., 2015, no. 2, 8–11 (in Russian)

[25] A. I. Inozemtsev, A. S. Kalitvin, “Operator functions with multidimensional partial integrals”, Sci. Bull. Belgorod State Univ. Math. Phys., 37:25 (2015), 19–29 (in Russian)

[26] V. A. Kakichev, N. V. Kovalenko, “To the theory of two-dimensional integral equations with partial integrals”, Ukr. Math. J., 25:3 (1973), 302–312 (in Russian) | MR | Zbl

[27] A. S. Kalitvin, “On spectrum and eigenfunctions of an operator with partial integrals and the Romanovskiy operator with partial integrals”, Functional Analysis, 22, Ul'yanovsk, 1984, 35–45 (in Russian) | Zbl

[28] A. S. Kalitvin, “On spectrum of some classes of operators with partial integrals”, Operators and Their Applications. Approximation of Functions. Equations, Leningrad, 1985, 27–35 (in Russian) | MR

[29] A. S. Kalitvin, “On multispectrum of linear operators”, Operators and Their Applications. Approximation of Functions. Equations, L., 1985, 91–99 (in Russian) | MR

[30] A. S. Kalitvin, “On spectrum of an operator with partial integrals in spaces with mixed norm”, Partial Differential Equations, L., 1986, 128–131 (in Russian) | MR

[31] A. S. Kalitvin, Study of operators with partial integrals, PhD Thesis, L., 1986

[32] A. S. Kalitvin, “On spectrum of linear operators with partial integrals and positive kernels”, Operators and Their Applications, L., 1988, 43–50 (in Russian) | MR | Zbl

[33] A. S. Kalitvin, “On solvability of some classes of integral equations with partial integrals”, Functional Analysis, 29, Ul'yanovsk, 1989, 68–73 (in Russian) | MR | Zbl

[34] A. S. Kalitvin, “On Volterra equations with partial integrals from the elasticity theory”, Proc. Conf. Math. model. of systems. Methods, applications, and means (Voronezh, 1998), 85–89

[35] A. S. Kalitvin, Linear Operators with Partial Integrals, TsChKI, Voronezh, 2000 (in Russian)

[36] A. S. Kalitvin, “Volterra equations with partial integrals in functional spaces”, Proc. Inst. Math. Nat. Acad. Sci. Belarus, 5 (2000), 72–76 (in Russian) | Zbl

[37] A. S. Kalitvin, “On generalization of one class of equations with partial integrals of contact problems from the theory of creepage of nonhomogeneously aging bodies”, Contemporary Problems of Mechanics and Applied Mathematics, Voronezh, 2000, 189–193 (in Russian)

[38] A. S. Kalitvin, “On generalization of one equation of continuum mechanics”, Bull. Russ. Acad. Nat. Sci. Ser. MMIU, 4:3 (2000), 81–88 (in Russian)

[39] A. S. Kalitvin, “On Volterra–Fredholm equations with partial integrals”, Differ. Equ., 37:10 (2001), 151–152 (in Russian)

[40] A. S. Kalitvin, Nonlinear Operators with Partial Integrals, LGPU, Lipetsk, 2002 (in Russian)

[41] A. S. Kalitvin, Operators and equations with partial integrals and their applications, Doctoral Thesis, Lipetsk, 2003

[42] A. S. Kalitvin, “Integral equations of the third kind with partial integrals”, Contemp. Math. Appl., 36 (2005), 95–99 (in Russian)

[43] A. S. Kalitvin, “On one class of integral equations in the space of continuous functions”, Differ. Equ., 42:9 (2006), 1194–1200 (in Russian) | MR | Zbl

[44] A. S. Kalitvin, Romanovskiy-type Integral Equations with Partial Integrals, LGPU, Lipetsk, 2007 (in Russian)

[45] A. S. Kalitvin, “Linear equation with partial integrals of the continuum mechanics”, Actual Problems of Natural Sciences and Their Teaching, Lipetsk, 2009, 86–93 (in Russian)

[46] A. S. Kalitvin, “On Volterra operators and equations with partial integrals”, Materials of Int. Conf. S. G. Krein's Voronezh Wintry Math. School-2012 (Voronezh, 2012), 91–94 | MR

[47] A. S. Kalitvin, “On Noether and Fredholm properties and invertibility of linear equations with partial integrals in two classes of ideal spaces”, Proc. Int. Conf. Analytical Methods of Analysis and Differential Equations, AMADE-11, v. 1, Mathematical Analysis, Inst. Math. Nat. Acad. Sci. Belarus, Minsk, 2012, 75–79 (in Russian)

[48] A. S. Kalitvin, “On linear operators with partial equations in spaces of symmetric and skew-symmetric functions”, Bull. LGPU. Ser. Math. Inform. Phys. Nat. Sci., 2012, no. 1, 9–13 (in Russian)

[49] A. S. Kalitvin, “On spectrum of linear operators with partial integrals in the space of vector functions $C(L^2)$”, Materials Int. Conf. S. G. Krein Voronezh Wintry Math. School-2014 (Voronezh, 2014), 157–160 (in Russian)

[50] A. S. Kalitvin, “On spectrum of operators with partial integrals and variable limits of integration”, Materials Regional Sci.-Practic Conf. Actual Problems of Natural Sciences and Their Teaching (Lipetsk, 2014), 91–96 (in Russian) | MR

[51] A. S. Kalitvin, “On multispectrum of linear operators with partial integrals”, Bull. LGPU. Ser. Math. Inform. Phys. Nat. Sci., 2015, no. 1, 7–11 (in Russian)

[52] A. S. Kalitvin, “On Fredholm property of one class of linear equations with partial integrals in the space $L^1(D)$”, Materials Int. Conf. Differential Equations and Dynamical Systems (Suzdal', 2018), 103–104 (in Russian)

[53] A. S. Kalitvin, A. I. Inozemtsev, “On Noether and Fredholm properties and invertibility of linear operators and equations with multidimensional partial integrals”, Sci.-Tech. Bull. Volga Reg., 2018, no. 5, 22–25 (in Russian)

[54] A. S. Kalitvin, V. A. Kalitvin, “Ob uravneniyakh Vol'terra–Fredgol'ma–Romanovskogo s chastnymi integralami”, Proc. Inst. Math. Nat. Acad. Sci. Belarus, 12:1 (2004), 71–75 (in Russian)

[55] A. S. Kalitvin, V. A. Kalitvin, “Volterra integral equations with multidimensional partial integrals”, Proc. XII Int. Symp. Methods of Discrete Singularities in Problems of Mathematical Physics (Khar'kov–Kherson, 2005), 153–156 (in Russian)

[56] A. S. Kalitvin, V. A. Kalitvin, Volterra and Volterra–Fredholm Integral Equations with Partial Integrals, LGPU, Lipetsk, 2007 (in Russian)

[57] A. S. Kalitvin, V. A. Kalitvin, “On Volterra integral equations with multidimensional partial integrals”, Bull. LGPU. Ser. Math. Inform. Phys. Nat. Sci., 2006, no. 1, 20–23 (in Russian)

[58] A. S. Kalitvin, V. A. Kalitvin, “On linear operators and equations with partial integrals and variable limits of integration”, Sci. Bull. Belgorod State Univ. Ser. Math. Phys., 32:19 (2013), 49–56 (in Russian)

[59] A. S. Kalitvin, V. A. Kalitvin, “On one class of mathematical models with partial integrals and multiparameter”, Sci. Bull. Belgorod State Univ. Ser. Math. Phys., 42:6 (2016), 40–44 (in Russian)

[60] A. S. Kalitvin, V. A. Kalitvin, “On linear operators with improper partial integrals”, Sci. Bull. Belgorod State Univ. Ser. Math. Phys., 43:13 (2016), 24–29 (in Russian)

[61] A. S. Kalitvin, V. A. Kalitvin, “On matrix Volterra integral equations with partial integrals in complex domain”, Sci.-Tech. Bull. Volga Reg., 2017, no. 6, 28–30 (in Russian)

[62] A. S. Kalitvin, V. A. Kalitvin, “On operators with partial integrals in spaces of functions of two variables”, Tavricheskiy Bull. Inform. Math., 2017, no. 3, 17–27 (in Russian)

[63] A. S. Kalitvin, V. A. Kalitvin, “Linear equations with partial integrals and variable limits of integration”, Abstracts Int. Conf. XXIX Crimean Autumnal Math. School on Spectral and Evolution Problems, Sec. 1–3 (Simferopol', 2018), 70–72 (in Russian)

[64] A. S. Kalitvin, E. V. Frolova, “On Volterra equations with partial integrals in the space of continuous and bounded functions on a half-strip”, Proc. Inst. Math. Nat. Acad. Sci. Belarus, 9 (2001), 68–72 (in Russian)

[65] A. S. Kalitvin, E. V. Frolova, Linear Equations with Partial Integrals. $C$-Theory, LGPU, Lipetsk, 2004 (in Russian)

[66] A. S. Kalitvin, E. V. Yankelevich, “Operators with partial integrals in the space of continuous functions. I”, Bull. Chelyabinsk State Univ. Ser. Math. Mech., 1994, no. 1, 61–67 (in Russian) | MR

[67] V. A. Kalitvin, Operator methods of study of Volterra–Fredholm equations with partial integrals, Lipetsk, 2003

[68] L. V. Kantorovich, G. P. Akilov, Functional Analysis, Nauka, M., 1977 (in Russian)

[69] T. Kato,, Perturbation Theory for Linear Operators, Mir, M., 1972 (Russian translation)

[70] I. S. Kats, “Behavior of solutions of a second-order linear differential equation (concerning one paper by E. Hille)”, Math. Digest, 62:4 (1963), 476–495 (in Russian) | Zbl

[71] I. S. Kats, M. G. Kreyn, “Criterion for discreteness of spectrum of a singular string”, Bull. Higher Edu. Inst. Ser. Math., 2 (1958), 136–153 (in Russian)

[72] E. V. Kovalenko, “Study of axisymmetric contact problem on wearing of a pair circular stamp – elastic rugged half-space”, Appl. Math. Mech., 49:5 (1985), 836–843 (in Russian) | MR | Zbl

[73] N. V. Kovalenko, “On solution of two-dimensional integral equation with partial integrals in the space $L_{2}$”, Talks at 2 Conf. Rostov Sci. Math. Soc. (Rostov, 1968), 41–49 (in Russian)

[74] N. V. Kovalenko, “On one homogeneous integral equation with partial integrals”, Phys.-Math. Investigations, Rostov-na-Donu, 1972, 3–7 (in Russian)

[75] V. B. Korotkov, Integral Operators, Nauka, Novosibirsk, 1983 (in Russian) | MR

[76] M. A. Krasnosel'skiy, P. P. Zabreyko, E. I. Pustyl'nik, P. E. Sobolevskiy, Integral Operators in Spaces of Summable Functions, Nauka, M., 1966 (in Russian) | MR

[77] S. G. Kreyn, Linear Equations in Banach Space, Nauka, M., 1971 (in Russian)

[78] V. L. Levin, “Tensor products and functors in categories of Banach spaces defined by KV-lineals”, Proc. Moscow Math. Soc., 20, 1969, 43–82 (in Russian) | Zbl

[79] B. M. Levitan, I. S. Sargsyan, Sturm–Liouville and Dirac Operators, Nauka, M., 1988 (in Russian) | MR

[80] L. M. Likhtarnikov, “On one operator equations with two parameters in Hilbert space”, Functional Anal., 3, Ul'yanovsk, 1974, 92–95 (in Russian)

[81] L. M. Likhtarnikov, “On spectrum of one class of linear integral equations with two parameters”, Differ. Equ., 11:6 (1975), 1108–1117 (in Russian) | MR | Zbl

[82] L. M. Likhtarnikov, L. Z. Vitova, “On spectrum of integral operator with partial integrals”, Lit. Math. Digest, 15:2 (1975), 41–47 (in Russian) | MR | Zbl

[83] L. M. Likhtarnikov, L. Z. Vitova, “On solvability of a linear integral equation with partial integrals”, Ukr. Math. J., 28:1 (1976), 83–87 (in Russian) | MR | Zbl

[84] L. M. Likhtarnikov, L. M. Morozova, “On one method of study of integral equations with partial integrals”, Functional Anal., 21, Ul'yanovsk, 1983, 108–112 (in Russian) | MR | Zbl

[85] A. V. Manzhirov, “Axisymmetric contact problems for nonhomogenously aging viscoelastic foliated bases”, Appl. Math. Mech., 47:4 (1983), 684–694 (in Russian)

[86] A. V. Manzhirov, “On one method of solution for two-dimensional integral equations of axisymmetric contact problems for bodies with complex rheology”, Appl. Math. Mech., 49:6 (1985), 1019–1025 (in Russian) | MR | Zbl

[87] V. A. Morozov, “Application of regularization method to solution of one ill-posed problem”, Bull. MSU, 1:4 (1965), 13–25 (in Russian)

[88] G. Myuntts, Integral Equations, v. 1, GTTI, L.–M., 1934 (in Russian)

[89] A. M. Nakhushev, Equations of Mathematical Biology, Vysshaya shkola, M., 1995 (in Russian)

[90] O. P. Okolelov, “To the theory of two-dimensional integral equations with partial integrals”, Materials 6th Inter-Univ. Sci. Conf. Far East. Differ. and Integral Equ. (Khabarovsk, 1967), v. 3, 142–149 (in Russian) | MR

[91] O. P. Okolelov, Investigation of equations with partial integral operators, PhD Thesis, Irkutsk, 1967

[92] V. S. Pilidi, “On one class of linear operator equations”, Math. Anal. Appl., 7 (1975), 34–42 (in Russian) | Zbl

[93] E. V. Frolova, “On one operator of continuum mechanics”, Proc. Conf. Math. Modelling of Systems. Methods, Applications, and Means (Voronezh, 1998), 183–187 (in Russian)

[94] E. V. Frolova, Linear operators with partial integrals, PhD Thesis, Lipetsk, 2000

[95] Appell J., Frolova E. V., Kalitvin A. S., Zabrejko P. P., “Partial integral operators on $C([a,b]\times [c,d])$”, Integr. Equ. Oper. Theory, 27 (1997), 125–140 | DOI | MR | Zbl

[96] Appell J., Kalitvin A. S., Nashed M. Z., “On some partial integral equations arising in the mechanics of solids”, ZAMM Z. Angew. Math. Mech., 79:2 (1999), 703–713 | 3.0.CO;2-W class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[97] Appell J., Kalitvin A. S., Zabrejko P. P., “Partial integral operators in Orlich spaces with mixed norms”, Collect. Math., 78:2 (1998), 293–306 | MR | Zbl

[98] Appell J., Kalitvin A. S., Zabrejko P. P., Partial integral operators and integro-differential equations, Marcel Dekker, New York–Basel, 2000 | MR | Zbl

[99] Fenyö S., “Beiträge zur Theorie der linearen partiellen Integralgleichungen”, Publ. Math., 4:1 (1955), 98–103 | MR | Zbl

[100] Frolova E. V., Kalitvin A. S., Zabrejko P. P., “Operator-functions with partial integrals on $C$ and $L_p$”, J. Electrotech. Math. Pristina, 6 (2001), 29–50

[101] Ichinose T., “Operational calculus for tensor products of linear operators in Banach spaces”, Hokkaido Math. J., 4 (1975), 306–334 | DOI | MR | Zbl

[102] Ichinose T., “Spectral properties of tensor products of linear operators. I”, Trans. Am. Math. Soc., 235 (1978), 75–113 | DOI | MR | Zbl

[103] Ichinose T., “Spectral properties of tensor products of linear operators. II”, Trans. Am. Math. Soc., 237 (1978), 223–254 | MR | Zbl

[104] Kalitvin A. S., “Spectral properties of partial integral operators of Volterra and Volterra–Fredholm type”, Z. Anal. Anwend., 17:2 (1998), 297–309 | DOI | MR | Zbl

[105] Kalitvin A. S., “On a class of integral equations in the space of continuous functions”, Differ. Equ., 42:9 (2006), 1262–1268 | DOI | MR | Zbl

[106] Kalitvin A. S., Zabrejko P. P., “On the theory of partial integral operators”, J. Integral Equ. Appl., 3:3 (1991), 351–382 | DOI | MR | Zbl

[107] Kantorovitz S., “A note on partial linear integral equations”, Bull. Res. Council Israel, 7:4 (1957), 181–186 | MR

[108] Kantorovitz S., “On the integral equation $\varphi(x,y) - \lambda a(x,y)\int\limits \varphi (x,y)dx - \mu b(x,y)\int\limits \varphi (x,y)dy = c(x,y)$”, Riveon le Matematika, 12 (1958), 24–26 | MR

[109] Mauro P., “Su un'equazione integrale lineare di tipo non ancora considerato”, Rend. Accad. Naz. Sci. XL, 5:1 (1976), 55–59

[110] Salam A., “Fredholm solution of partial integral equations”, Math. Proc. Cambridge Philos. Soc., 49 (1953), 213–217 | DOI | MR | Zbl

[111] Volterra V., Lecons sur les equations integrales et les equations integro-differentielles, Gauthier-Villars, Paris, 1913 | Zbl