On the semilinear integro-differential nonlocal Cauchy problem

Majcher, Piotr ; Roszak, Magdalena

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 25 (2005) no. 1, pp. 5-18.

Résumé

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.

Détail

Keywords: integro-differential equations, measure of weak non-compactness, non-local problem

Bibliographie

[1] S. Aizovici and M. McKibben, Existence results for a class of abstract non-local Cauchy problems, Nonlin. Anal. TMA 39 (2000), 649-668.

[2] A. Alexiewicz, Functional Analysis, Monografie Matematyczne 49, Polish Scientific Publishers, Warsaw 1968 (in Polish).

[3] J.M. Ball, Weak continuity properties of mapping and semi-groups, Proc. Royal Soc. Edinbourgh Sect. A 72 (1979), 275-280.

[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.

[14] G. Darbo, Punti uniti in trasformazioni a condominio non compatto, Rend. Sem. Math. Univ. Padova 4 (1955), 84-92.

[15] F.S. DeBlasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259-262.

[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.