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On quadratic integral equations in Orlicz spaces. (English) Zbl 1242.45003

The existence of a nondecreasing solution \(x\) in an Orlicz space of the quadratic integral equation \[ x(t)=g(t)+\lambda x(t)\int_a^bK(t,s)f(s,x(s))ds \] is proved for sufficiently small \(\lambda\). By considering Orlicz spaces instead of \(L_p\) spaces, it is possible to consider functions \(f\) which grow faster than polynomials and rather singular kernels \(K\). For both situations corresponding results are described. The other hypotheses are monotonicity assumptions about \(f\), \(g\), and \(K\). The main tool for the proofs is an application of Darbo’s fixed point theorem on a set of monotone functions.

MSC:

45G10 Other nonlinear integral equations
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
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