Abstract
We use the inequalities of Holder, Gronwall and Wirtinger type to establish sufficient conditions for that the null function is the unique nonnegative solution of some integral inequalities. Such conditions are useful to guarantee compactness properties of Monch and Palais-Smale type for integral operators on spaces of vector-valued functions. An applications to superlinear boundary value problems in Hilbert spaces is also presented.
Authors
Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
Keywords
Inequality involving derivatives; compactness: Papais-Smale condition; nonlinear integral operator; fixed point; critical point theory; boundary value problem.
Paper coordinates
R. Precup, Inequalities and compactness, In: “Inequalities Theory and Applications”, Y.J. Cho, J.K. Kim, S.S. Dragomir eds., Nova Science Publ., Huntington-New York, 2001, 257-271.
About this paper
Journal
Publisher Name
Nova Science Publ.
DOI
Print ISSN
Online ISSN
MR 2065320.
google scholar link
[1[ O. Arama, Sur un probleme d’interpolaiton relatif aux solutions des equations differentielles lineaires du quatrieme ordre, Mathematica (Cluj), 10(33) (1968), 5-15.
[2] D.W. Boyd, Best constant in a class of integral inequalities, Pacific J. Math. 30 (1969), 367-383.
[3] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin-Heidelberg=New York-Tokyo, 1985.
[4] D.Guo, J. Sun and G. Qi, Some extensions of the mountain pass lemma, Differential Ingtegral Equations 1 (1988), 351-358.
[5] D.S. Mitrinovic, Analytic Inequalities, Springer-Verlag, Berlin-Heidelberg, 1970.
[6] H. Monch, Boundary value problems for nonlinear ordinary diferential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), 985-999.
[7] D.O’Regan and R. Precup, Fixed point theorems for set-valued maps and existence principles for ingtegral inclusions, J. Math. Anal. Appl. 245 (2000), 594-612.
[8] D. O’Regan and R. Precup, Existence criteria for integral equations in Banach spaces, J. Inequal. Appl. 6 (2001), 77-97.
[9] D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and B reach, forthcoming.
[10] R. Precup, On the Palais-Smale condition for Hammerstein integral equations in Hilbert spaces, to appear in Nonlinear Anal.
[11] M. Schechter, Linking Methods in Critical Point Theory, Birkhauser, Boston-Basel-Berlin, 1999.