Semilinear inclusions with nonlocal conditions without compactness in non-reflexive spaces
KeywordsWeak measure of noncompactness, weakly condensing map, semilinear differential inclusion, nonlocal boundary condition, fixed point theorems with weak topology, containment result
AbstractAn existence result for an abstract nonlocal boundary value problem $x'\in A(t)x(t)+F(t,x(t))$, $Lx\in B(x)$, is given, where $A(t)$ determines a linear evolution operator, $L$ is linear, and $F$ and $B$ are multivalued. To avoid compactness conditions, the weak topology is employed. The result applies also in nonreflexive spaces under a hypothesis concerning the De Blasi measure of noncompactness. Even in the case of initial value problems, the required condition is essentially milder than previously known results.
R.P. Agarwal, D. O’Regan and M.-A. Taoudi, Fixed point theory for multivalued weakly convex-power condensing mappings with applications to integral inclusions, Mem. Differential Equations Math. Phys. 57 (2012), 17–40.
R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina and B.N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhäuser, Basel, Boston, Berlin, 1992.
J.N. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), 370–373.
M.M. Basova and V.V. Obukhovskii, On some boundary-value problems for functionaldifferential inclusions in Banach spaces, J. Math. and Sci. 149 (2008), 1376–1384.
I. Benedetti, N.V. Loi and L. Malaguti, Nonlocal problems for differential inclusions in Hilbert spaces, Set-Valued Var. Anal. 22 (2014), 639–656.
I. Benedetti, L. Malaguti land V. Taddei, Semilinear evolution equations in abstract spaces and applications, Rend. Istit. Mat. Univ. Trieste 44 (2012), 371–388.
I. Benedetti, L. Malaguti land V. Taddei, Nonlocal semilinear evolution equations without strong compactness: Theory and applications, Bound. Value Probl. 22 (2013), 1–18.
L. Byszewski, Theorems about the existence and uniquencess of a solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl. 162 (1991), 495–505.
T. Cardinali and P. Rubbioni, Multivalued fixed point theorems in terms of weak topology and measure of weak noncompactness, J. Math. Anal. Appl. 405 (2013), 409–415.
B. Cascales, V.M. Kadets and J. Rodrı́guez, Measurability and selections of multifunctions in Banach spaces, J. Convex Anal. 17 (2010), 229–240.
C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lect. Notes Math. vol. 558, Springer, Berlin, New York, 1977.
M. Cichoń, On bounded weak solutions of a nonlinear differential equation in Banach spaces, Funct. Approx. Comment. Math. 21 (1992), 27–35.
F.S. De Blasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 21 (1977), 259–262.
J. Diestel, W.M. Ruess and W. Schachermayer, Weak compactness in L1 (µ, X), Proc. Amer. Math. Soc. 118 (1993), 447–453.
N. Dunford and J.T. Schwartz, Linear Operators I, Int. Publ., New York, 3rd edition, 1966.
K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121–126.
I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of Linear Operators, Vol. I, Birkhäuser, Basel, Boston, Berlin, 1990.
A.M. Gomaa, On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces, Ann. Soc. Math. Polon. Ser. I Comment. Math. Prace Mat. 47 (2007), 179–191.
C.J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72.
M.I. Kamenskii, V.V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter, Berlin, 2001.
S.S. Khurana and J. Vielma, Weak sequential convergence and weak compactness in spaces of vector-valued continuous functions, J. Math. Anal. Appl. 195 (1995), 251–260.
S.G. Kreǐn, Linear Differential Equations in Banach Space, Nauka, Moscow, 1967 (in Russian); English transl.: Providence, R.I., Amer. Math. Soc. 1971.
M. Kunze and G. Schlüchtermann, Strongly generated Banach spaces and measures of noncompactness, Math. Nachr. 191 (1998), 197–214.
V.V. Obukhovskii, On the topological degree for a class of noncompact multivalued mappings, Funkt. Anal., Ul’yanovsk 23 (1984), 82–93 (in Russian).
D. Ozdarska and S. Szufla, Weak solutions of a boundary value problem for nonlinear ordinary differential equations of second order in Banach spaces, Math. Slovaca 43 (1993), 301–307.
N.S. Papageorgiou, Existence of solutions for boundary value problems of semilinear evolution inclusions, Indian J. Pure Appl. Math. 23 (1992), 477–488.
B.N. Sadovskii, Limit-compact and condensing operators, Uspekhi Mat. Nauk 27 (1972), 81–146 (in Russian); Engl. transl.: Russian Math. Surveys 27 (1972), no. 1, 85–155.
M. Väth, Ideal Spaces, Lect. Notes in Math., vol. 1664, Springer, Berlin, Heidelberg, 1997.
M. Väth, Fixed point theorems and fixed point index for countably condensing maps, Topol. Methods Nonlinear Anal. 13 (1999), 341–363.
M. Väth, An axiomatic approach to a coincidence index for noncompact function pairs, Topol. Methods Nonlinear Anal. 16 (2000), 307–338.
M. Väth, Coincidence points of function pairs based on compactness properties, Glasgow Math. J. 44 (2002), 209–230.
M. Väth, Integration Theory. A Second Course, World Scientific Publ., Singapore, New Jersey, London, Hong Kong, 2002.
M. Väth, Continuity of single- and multivalued superposition operators in generalized ideal spaces of measurable vector functions, Nonlinear Functional Anal. Appl. 11 (2006), 607–646.
M. Väth, Topological Analysis. From the Basics to the Triple Degree for Nonlinear Fred holm Inclusions, de Gruyter, Berlin, New York, 2012.
H. Vogt, An Eberlein–Šmulian type result for the weak∗ topology, Arch. Math. (Basel) 95 (2010), 31–34.
I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Pitman Publ., Oxford, 2nd edition, 1987.
I.I. Vrabie, C0 -Semigroups and Applications, Elsevier, Amsterdam, 2nd edition, 2003.
R. Whitley, An elementary proof of the Eberlein–Šmulian theorem, Math. Ann. 172 (1967), 116–118.
How to Cite
Number of views and downloads: 0
Number of citations: 0