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Solutions of some functional-integral equations in Banach algebra. (English) Zbl 1053.45007

The authors study the existence of a solution to the functional-integral equation

x(t)=ft, 0 t v(t,s,x(s))ds,x(α(t))·gt, 0 a u(t,s,x(s))ds,x(β(t)),t[0,a]·

As special cases this equation contains integral equations of both Volterra and Urysohn type. The proof depends on measures of noncompactness in the space of continuous functions. The assumptions include the existence of a constant m such that |f(t,y,0)|m and |g(t,y,0)|m and the fact that both f and g are Lipschitz continuous with respect to their last variable with a constant k that satisfies 4km<1.

45N05Abstract integral equations, integral equations in abstract spaces
47H09Mappings defined by “shrinking” properties
45G10Nonsingular nonlinear integral equations
47N20Applications of operator theory to differential and integral equations