# Existence criteria for integral equations in Banach spaces

## Abstract

The paper contains a few existence results concerning nonlinear integral equations of Volterra and Uryson type in Banach spaces. It is assumed that the kernel $$f(t,s,x)$$ of both integral equations satisfies Carathéodory-type conditions and $$\mu (f(t,s,M))\leq \omega (t,s,\mu(M))$$ for all $$t,s\in[0,T]$$ and for any bounded subset M of a Banach space E. Here μ denotes the Kuratowski or Hausdorff measure of noncompactness. Special cases of the mentioned integral equations are also studied in detail.

xxx

The paper contains a few existence results concerning nonlinear integral equations of Volterra and Uryson type in Banach spaces. It is assumed that the kernel \(\backslash(f(t,s,x)\backslash))\ of both integral equations satisfies
Carath\'{e}odory-type conditions and \(\backslash(\backslash mu(f(t,s,M))\backslash leq\backslash omega(t,s,\backslash mu(M))\backslash))\
for all \(\backslash(t,s\backslash in[0,T]\backslash))\~and for any bounded subset~\(M)\~of a Banach space~\(E)\. Here~%
%TCIMACRO{\U{3bc}}%
%BeginExpansion
\(\mu)\%
%EndExpansion
~denotes the Kuratowski or Hausdorff measure of noncompactness. Special cases of the mentioned integral equations are also studied in detail.

## Authors

Donal O’Regan,

Babeş-Bolyai University, Department of Mathematics, Cluj-Napoca, Romania

## Keywords

Volterra and Urysohn integral equations in abstract spaces; Measure of noncompactness; Continuation method; Fixed point

##### Cite this paper as:

D. O’Regan, R. Precup, Existence criteria for integral equations in Banach spaces, J. Inequal. Appl. 6 (2001), 77-97, http://dx.doi.org/10.1155/S1025583401000066

## pdf

##### Journal

Journal of Inequalities and Applications

Springer

##### Print ISSN

Not available yet.

##### Online ISSN

??

MR 2003c:45007, Zbl 0993.45011

## References

[1] R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina and B.N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhäuser, Basel, 1992. MR1153247
[2] J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, 1980. MR0591679
[3] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. MR0787404
[4] D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer Academic Publishers, Dordrecht, 1996. MR1418859
[5] H.P. Heinz, On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Anal.7 (1983), 1351–1371. MR0726478
[6] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal.4 (1980), 985–999. MR0586861
[7] H. Mönch and G.F. von Harten, On the Cauchy problem for ordinary differential equations in Banach spaces, Arch. Math. (Basel)39 (1982), 153–160. MR0675655
[8] D. O’Regan, Volterra and Urysohn integral equations in Banach spaces, J. Appl. Math. Stochastic Anal.11 (1998), 449–464. MR1663604
[9] D. O’Regan, Multivalued integral equations in finite and infinite dimensions, Comm. Appl. Anal.2 (1998), 487–496. MR1636992
[10] D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and Breach (forthcoming). cf. MR1937722
[11] R. Precup, Discrete continuation method for boundary value problems on bounded sets in Banach spaces, J. Comput. Appl Math. (to appear). cf. MR1735829
[12] R. Precup, Discrete continuation method for nonlinear integral equations in Banach spaces, Pure Math. Appl. (to appear). cf. MR1839944
[13] S. Szufla, On the differential equation x(m)=f(t,x) in Banach spaces, Funkcial. Ekvac.41(1998), 101–105. MR1627349
[14] K. Yosida, Functional Analysis, Springer, Berlin, 1978. MR0500055