## 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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