A Course of Higher Mathematics. Volume IV by V. I. Smirnov and A. J. Lohwater (Auth.) PDF

By V. I. Smirnov and A. J. Lohwater (Auth.)

ISBN-10: 1483167232

ISBN-13: 9781483167237

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1/^(5), of equation (80), since every other solution is a linear com- 11] FREDHOLM MINORS 41 bination of them. Thus, if condition (79) is fulfilled with ^J(S) = ipk(s) (k = 1, 2, . . , m), it is also fulfilled for any solution ip(s) of equation (80). Note 2. Instead of taking (80) as the adjoint homogeneous equation, we often take b co(s) = λ J K(t, s) o(t) dt. e. if ip(s) is a solution of equation (80), then co(s) = ψ(8) is a solution of (83), and vice versa. If λ is an eigenvalue of the homogeneous equation b φ(8) = λ j " K{8, t)

E. solutions of (61) different from zero: b

Let u(N) be a function continuous in B and v(M) = f K(M; N) u(N) dcoN. (103) The integral written is obviously meaningful in view of (100). We prove that function v(M) is continuous: v{M') - v(M) = j [K{M'; N) - K(M) N)] u{N) άωΝ. B For a continuous function u(N), we have the inequality in B: \u(N) | < Cv (104) where Cx is constant. Further \v(M') - v(M)\

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A Course of Higher Mathematics. Volume IV by V. I. Smirnov and A. J. Lohwater (Auth.)

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