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Existence of integrals and the solution of integral equations


Author: Jon C. Helton
Journal: Trans. Amer. Math. Soc. 229 (1977), 307-327
MSC: Primary 45A05
DOI: https://doi.org/10.1090/S0002-9947-1977-0445245-1
MathSciNet review: 0445245
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Abstract: Functions are from R to N or $ R \times R$ to N, where R denotes the real numbers and N denotes a normed complete ring. If S, T and G are functions from $ R \times R$ to N, each of $ S({p^ - },p),S({p^ - },{p^ - }),T({p^ - },p)$ and $ T({p^ - },{p^ - })$ exists for $ a < p \leqslant b$, each of $ S(p,{p^ + }),S({p^ + },{p^ + }),T(p,{p^ + })$ and $ T({p^ + },{p^ + })$ exists for $ a \leqslant p < b$, G has bounded variation on [a, b] and $ \smallint _a^bG$ exists, then each of

$\displaystyle \int_a^b S \left[ {G - \int G } \right]T\quad {\text{and}}\quad \int_a^b {S\left[ {1 + G - \prod {(1 + G)} } \right]} \;T$

exists and is zero. These results can be used to solve integral equations without the existence of integrals of the form

$\displaystyle \int_a^b {\left\vert {G - \int G } \right\vert = 0} \quad {\text{and}}\quad \int_a^b {\left\vert {1 + G - \prod {(1 + G)} } \right\vert} = 0.$

This is demonstrated by solving the linear integral equation

$\displaystyle f(x) = h(x) + (LR)\int_a^x {(fG + fH)} $

and the Riccati integral equations

$\displaystyle f(x) = w(x) + (LRLR)\int_a^x {(fH + Gf + fKf)} $

without the existence of the previously mentioned integrals.

References [Enhancements On Off] (What's this?)

  • [1] D. P. Puig Adam, Los fracciones continuous de cocientes incompletos differenciales y sus aplicaciones, Revista Mat. Hisp.-Amer. (4) 11 (1951), 180-190. (Spanish) MR 13, 540. MR 0045175 (13:540f)
  • [2] W. D. L. Appling, Interval functions and real Hilbert spaces, Rend. Circ. Mat. Palermo (2) 11 (1962), 154-156. MR 27 #4040. MR 0154081 (27:4040)
  • [3] C. W. Bitzer, Stieltjes-Volterra integral equations, Illinois J. Math. 14 (1970), 434-451. MR 0415227 (54:3318)
  • [4] -, Convolution, fixed point, and approximation in Stieltjes-Volterra integral equations, J. Austral. Math. Soc. 14 (1972), 182-199. MR 47 #5528. MR 0316980 (47:5528)
  • [5] J. R. Dorroh, Integral equations in normed abelian groups, Pacific J. Math. 13 (1963), 1143-1158. MR 28 #1469. MR 0158243 (28:1469)
  • [6] F. R. Gantmacher, The theory of matrices, Vol. 2, GITTL, Moscow, 1953; Chelsea, New York, 1959. MR 21 #6372c.
  • [7] B. W. Helton, Integral equations and product integrals, Pacific J. Math. 16 (1966), 297-322. MR 32 #6167. MR 0188731 (32:6167)
  • [8] -, A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493-500. MR 40 #1562. MR 0248310 (40:1562)
  • [9] -, Solutions of $ f(x) = f(a) + (RL)\smallint _a^x(fH + fG)$ for rings, Proc. Amer. Math. Soc. 25 (1970), 735-742. MR 41 #4159. MR 0259521 (41:4159)
  • [10] -, The solution of a nonlinear Gronwall inequality, Proc. Amer. Math. Soc. 38 (1973), 337-342. MR 46 #9287. MR 0310185 (46:9287)
  • [11] -, A product integral solution of a Riccati equation, Pacific J. Math. 56 (1975), 113-130. MR 0414975 (54:3067)
  • [12] -, A special integral and a Gronwall inequality, Trans. Amer. Math. Soc. 217 (1976), 163-181. MR 0407215 (53:10998)
  • [13] J. C. Helton, An existence theorem for sum and product integrals, Proc. Amer. Math. Soc. 39 (1973), 149-154. MR 47 #5596. MR 0317048 (47:5596)
  • [14] -, Mutual existence of sum and product integrals, Pacific J. Math. 56 (1975), 495-516. MR 0405098 (53:8894)
  • [15] -, Product integrals and the solution of integral equations, Pacific J. Math. 58 (1975), 87-103. MR 0385480 (52:6341)
  • [16] -, Solution of integral equations by product integration, Proc. Amer. Math. Soc. 49 (1975), 401-406. MR 0405048 (53:8844)
  • [17] -, Mutual existence of product integrals in normed rings, Trans. Amer. Math. Soc. 211 (1975), 353-363. MR 0387536 (52:8376)
  • [18] J. V. Herod, Solving integral equations by iteration, Duke Math. J. 34 (1967), 519-534. MR 36 #625. MR 0217536 (36:625)
  • [19] -, Multiplicative inverses of solutions for Volterra-Stieltjes integral equations, Proc. Amer. Math. Soc. 22 (1969), 650-656.
  • [20] -, A pairing of a class of evolution systems with a class of generators, Trans. Amer. Math. Soc. 157 (1971), 247-260. MR 43 #6778. MR 0281059 (43:6778)
  • [21] -, A product integral representation for an evolution system, Proc. Amer. Math. Soc. 27 (1971), 549-556. MR 43 #1001. MR 0275244 (43:1001)
  • [22] T. H. Hildebrandt, On systems of linear differentio-Stieltjes-integral equations, Illinois J. Math. 3 (1959), 352-373. MR 21 #4339. MR 0105600 (21:4339)
  • [23] D. B. Hinton, A Stieltjes-Volterra integral equation theory, Canad. J. Math. 18 (1966), 314-331. MR 32 #6169. MR 0188733 (32:6169)
  • [24] D. L. Lovelady, Addition in a class of nonlinear Stieltjes integrators, Israel J. Math. 10 (1971), 391-396. MR 45 #7548. MR 0298496 (45:7548)
  • [25] -, Algebraic structure for a set of nonlinear integral operations, Pacific J. Math. 37 (1971), 421-427. MR 46 #2485. MR 0303348 (46:2485)
  • [26] -, Multiplicative integration of infinite products, Canad. J. Math. 23 (1971), 692-698. MR 44 #4539. MR 0287333 (44:4539)
  • [27] -, Perturbations of solutions of Stieltjes integral equations, Trans. Amer. Math. Soc. 155 (1971), 175-187. MR 0433175 (55:6154)
  • [28] -, Product integrals for an ordinary differential equation in a Banach space, Pacific J. Math. 48 (1973), 163-168. MR 49 #3283. MR 0338519 (49:3283)
  • [29] J. S. Mac Nerney, Stieltjes integrals in linear spaces, Ann. of Math. (2) 61 (1955), 354-367. MR 16, 716. MR 0067354 (16:716d)
  • [30] -, Continuous products in linear spaces, J. Elisha Mitchell Sci. Soc. 71 (1955), 185-200. MR 18, 54. MR 0079234 (18:54c)
  • [31] -, Determinants of harmonic matrices, Proc. Amer. Math. Soc. 7 (1956), 1044-1046. MR 18, 906. MR 0084704 (18:906f)
  • [32] -, Integral equations and semigroups, Illinois J. Math. 7 (1963), 148-173. MR 26 #1726. MR 0144179 (26:1726)
  • [33] -, A linear initial-value problem, Bull. Amer. Math. Soc. 69 (1963), 314-329. MR 26 #4133. MR 0146613 (26:4133)
  • [34] -, A nonlinear integral operation, Illinois J. Math. 8 (1964), 621-638. MR 29 #5082. MR 0167815 (29:5082)
  • [35] P. R. Masani, Multiplicative Riemann integration in normed rings, Trans. Amer. Math. Soc. 61 (1947), 147-192. MR 8, 321. MR 0018719 (8:321c)
  • [36] J. W. Neuberger, Continuous products and nonlinear integral equations, Pacific J. Math. 8 (1958), 529-549. MR 21 #1509. MR 0102723 (21:1509)
  • [37] -, Concerning boundary value problems, Pacific J. Math. 10 (1960), 1385-1392. MR 23 #A2012. MR 0124701 (23:A2012)
  • [38] -, A generator for a set of functions, Illinois J. Math. 9 (1965), 31-39. MR 30 #2369. MR 0172143 (30:2369)
  • [39] J. A. Reneke, A product integral solution of a Stieltjes-Volterra integral equation, Proc. Amer. Math. Soc. 24 (1970), 621-626. MR 40 #6214. MR 0252999 (40:6214)
  • [40] -, Product integral solutions for hereditary systems, Trans. Amer. Math. Soc. 181 (1973), 483-493. MR 48 #2692. MR 0324340 (48:2692)
  • [41] -, Stieltjes integral equations in partially ordered sets, J. Math. Anal. Appl. 50 (1975), 288-302. MR 0367597 (51:3839)
  • [42] H. S. Wall, Concerning continuous continued fractions and certain systems of Stieltjes integral equations, Rend. Circ. Mat. Palermo (2) 2 (1953), 73-84. MR 15, 533. MR 0059460 (15:533d)
  • [43] -, Concerning harmonic matrices, Arch. Math. 5 (1954), 160-167. MR 15, 801. MR 0061268 (15:801a)
  • [44] G. F. Webb, Nonlinear evolution equations and product integration in Banach spaces, Trans. Amer. Math. Soc. 148 (1970), 273-282. MR 42 #901. MR 0265992 (42:901)
  • [45] -, Product integral representation of time dependent non-linear evolution equations in Banach spaces, Pacific J. Math. 32 (1970), 269-281. MR 41 #2483. MR 0257834 (41:2483)
  • [46] -, Nonlinear evolution equations and product stable operators on Banach spaces, Trans. Amer. Math. Soc. 155 (1971), 409-426. MR 34 #2582. MR 0276842 (43:2582)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0445245-1
Keywords: Sum integral, product integral, subdivision-refinement integral, existence, interval function, normed complete ring, linear integral equation, Riccati integral equation
Article copyright: © Copyright 1977 American Mathematical Society

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