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On some fractional functional equations. (English) Zbl 0867.49001
Given two bounded linear operators on $$L^1$$, the authors prove the existence of a monotone $$L^1$$-solution of the operator equation $$x(t)=g(t)+Bf[t,Ax(\varphi(t))]$$. Then main tool is G. Darbo’s fixed point theorem [Rend. Semin. Mat. Univ. Padova 24, 84-92 (1955; Zbl 0064.35704)] for condensing operators.

##### MSC:
 49J05 Existence theories for free problems in one independent variable 26A33 Fractional derivatives and integrals 39B05 General theory of functional equations and inequalities 34A60 Ordinary differential inclusions 34G20 Nonlinear differential equations in abstract spaces 34K05 General theory of functional-differential equations 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)