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## Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2005 | 25 | 1 | 5-18
Tytuł artykułu

### On the semilinear integro-differential nonlocal Cauchy problem

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper, we prove an existence theorem for the pseudo-non-local Cauchy problem $x'(t) + Ax(t) = f(t,x(t),∫_{t₀}^{t} k(t,s,x(s))ds)$, x₀(t₀) = x₀ - g(x), where A is the infinitesimal generator of a C₀ semigroup of operator ${T(t)}_{t > 0}$ on a Banach space. The functions f,g are weakly-weakly sequentially continuous and the integral is taken in the sense of Pettis.
Słowa kluczowe
EN
Kategorie tematyczne
Rocznik
Tom
Numer
Strony
5-18
Opis fizyczny
Daty
wydano
2005
otrzymano
2004-01-01
Twórcy
autor
• Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
autor
• Institute of Mathematics and Physics, University of Technology and Agriculture, Al. Prof. S. Kaliskiego 7, 85-796 Bydgoszcz
Bibliografia
• [1] S. Aizovici and M. McKibben, Existence results for a class of abstract non-local Cauchy problems, Nonlin. Anal. TMA 39 (2000), 649-668.
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• [4] J. Banaś and K. Goebel, Measure of Non-compactness in Banach Spaces, Lecture Notes in Pure and Applied Math. 60, Marcel Dekker, New York-Basel 1980.
• [5] J. Banaś and J. Rivero, On measure of weak non-compactness, Ann. Mat. Pura Appl. 125 (1987), 213-224.
• [6] L. Byszewski, Theorems about the existence of solutions of a semilinear evolution Cauchy problem, J. Math. Anal. Appl. 162 (1991), 494-505.
• [7] L. Byszewski, Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution of non-local Cauchy problem, in: ''Selected Problems of Mathematics'', Cracow University of Technology (1995).
• [8] L. Byszewski, Differential and Functional-Differential Problems with Non-local Conditions, Cracow University of Technology (1995) (in Polish).
• [9] L. Byszewski and V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a non-local abstract Cauchy problem in a Banach space, Applicable Anal. 40 (1990), 11-19.
• [10] L. Byszewski and N.S. Papageorgiou, An application of a non-compactness technique to an investigation of the existence of solutions to a non-local multivalued Darboux problem, J. Appl. Math. Stoch. Anal. 12 (1999), 179-190.
• [11] M. Cichoń, Weak solutions of differential equations in Banach spaces, Discuss. Math. Diff. Incl. 15 (1995), 5-14.
• [12] M. Cichoń and P. Majcher, On some solutions of non-local Cauchy problem, Comment. Math. 42 (2003), 187-199.
• [13] M. Cichoń and P. Majcher, On semilinear non-local Cauchy problems, Atti. Sem. Mat. Fis. Univ. Modena 49 (2001), 363-376.
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• [16] H.-K. Han, J.-Y. Park, Boundary controllability of differential equations with non-local condition, J. Math. Anal. Appl. 230 (1999), 242-250.
• [17] A.R. Mitchell and Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, in: ''Nonlinear Equations in Abstract Spaces'', ed. V. Lakshmikantham, Academic Press (1978), 387-404.
• [18] S.K. Ntouyas and P.Ch. Tsamatos, Global existence for semilinear evolution equations with non-local conditions, J. Math. Anal. Appl. 210 (1997), 679-687.
• [19] N.S. Papageorgiou, On multivalued semilinear evolution equations, Boll. Un. Mat. Ital. (B) 3 (1989), 1-16.
• [20] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York-Berlin 1983.
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Bibliografia
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