Existence criteria for integral equations in Banach spaces

A Caratheodory existence theory for nonlinear Volterra and Urysohn integral equations in Banach spaces is presented using a Monch type approach.

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.

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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.

Donal O’Regan,

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

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

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

https://ictp.acad.ro/wp-content/uploads/2022/08/2001e-Precup-Existence-criteria-for-integral-equations-in-Banach-spaces.pdf

http://dx.doi.org/10.1155/S1025583401000066

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MR 2003c:45007, Zbl 0993.45011

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