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A functional integral inclusion involving Carathéodories. (English) Zbl 1056.47040

This article deals with the existence of minimal and maximal solutions to a nonlinear functional integral inclusion \[ x(t) \in q(t) + \int_0^{\sigma(t)} k(t,s)F(s,x(\eta(s))) \, ds \] where \(\sigma, \eta: [0,1 \to [0,1]\) \(q: [0,1] \to E\), \(k: [0,1] \times [0,1] \to {\mathbb R}\) is a continuous kernel, \(F: [0,1] \times E \to 2^E\) is a Carathéodory nonlinearity that is nondecreasing in \(x\) almost everywhere for \(t \in [0,1]\) and satisfying the Darbo condition with a constant \(\lambda\), \(E\) is an ordered Banach space. As an application, the author considers the initial value problem of type \[ x' \in F(t,x(\eta(t))) \;\text{a.e.} \;t \in J, \quad x(0) = x_0 \in {\mathbb R}, \] and the boundary value problems \[ x''(t) \in F(t,x(\eta(t))), \;\text{a.e.} \;t \in [0,1], \quad x(0) = x(1) = 0 \] and \[ x''(t) \in F(t,x(\eta(t))), \;\text{a.e.} \;t \in [0,1], \quad x(0) = 0, \;x'(1) = 0 \] (\(F: \;[0,1] \times {\mathbb R} \to 2^{\mathbb R}\), \(\eta: \;[0,1] \to [0,1]\)).

MSC:

47H10 Fixed-point theorems
47H04 Set-valued operators
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