Sequences of iterates of random-valued vector functions and continuous solutions of a linear functional equation of infinite order

• Autorzy: Kapica R.
• Języki publikacji: EN

Abstrakty

EN

Given a probability space (Ω, A, P), a separable Banach space X, and measurable functions L: Ω → (0, +∞), M: Ω → X we obtain some theorems on the existence and on the uniqueness of continuous solutions φ: X → R of the equation φ (x) = ∫_Ω φ (L(ω) x + M (ω)) P (dω).

Słowa kluczowe

EN

continuous solutions; iterative functional equations; limit theorems; random-valued vector functions; sequences built of iterates

PL

funkcja wektorowa o losowych wartościach; iteracyjne równanie funkcyjne; rozwiązanie ciągłe; twierdzenie graniczne

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2002
• Tom: Vol. 50, no 4
• Strony: 447--455
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Kapica R.

• Institute of Mathematics, Silesian University, Bankowa 14, 40-007 Katowice, Poland,

Bibliografia

• [1] K. Baron, W. Jarczyk, Recent results on functional equations in a single variable, perspectives and open problems, Aequationes Math., 61 (2001) 1-48.
• [2] K. Baron, W. Jarczyk, Random-valued functions and iterative functional equations, manuscript.
• [3] K. Baron, M. Kuczma, Iteration of random-valued functions on the unit interval, Colloq. Math., 37 (1977) 263-269.
• [4] G. Derfel, The behaviour of solutions of functional and functional-differential equations with certain transformations of the argument (in Russian), Ukrain. Mat. Zh., 34 (1982) 350-356; English translation in: Ukrainian Math. J., 34 (1982) 286-291.
• [5] G. Derfel, A probabilistic method for studying a class of functional-differential equations (in Russian), Ukrain. Mat. Zh., 41 (1989) 1322-1327; English translation in: Ukrainian Math. J., 41 (1989) 1137-1141.
• [6] Ph. Diamond, A stochastic functional equation, Aequationes Math., 15 (1977) 225-233.
• [7] J. Ger, M. Sablik, The law of large numbers and a functional equation, Ann. Polon. Math., 68 (1998) 165-175.
• [8] A. K. Grincevičius, On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines (in Russian), Teor. Veroyatnost. i Primenen., 19 (1974) 163-168; English translation in: Theor. Probab. Appl., 19 (1974) 163-168.
• [9] E. Hille, R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., 31 (1974).
• [10] R. Kapica, Convergence of sequences of iterates of random-valued vector functions, manuscript.
• [11] J. Morawiec, On a linear functional equation, Bull. Pol. Ac.: Math., 43 (1995) 131-142.
• [12] J. Morawiec. Some properties of probability distribution solutions of linear functional equations, Aequationes Math., 56 (1998) 81-90.