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Compactness conditions in the study of functional, differential, and integral equations. (English) Zbl 1266.45008
Summary: We discuss some existence results for various types of functional, differential, and integral equations which can be obtained with the help of argumentations based on compactness conditions. We restrict ourselves to some classical compactness conditions appearing in fixed point theorems due to Schauder, Krasnosel’skii-Burton, and Schaefer. We present also the technique associated with measures of noncompactness and we illustrate its applicability in proving the solvability of some functional integral equations. Apart from this, we discuss the application of the mentioned technique to the theory of ordinary differential equations in Banach spaces.
MSC:
45G10Nonsingular nonlinear integral equations
45N05Abstract integral equations, integral equations in abstract spaces
47H08Measures of noncompactness and condensing mappings, K-set contractions, etc.
34G20Nonlinear ODE in abstract spaces