Pure mathematics

Boundary value problem

Mathematics

Biology

Computer science

Differential equation

Fractional calculus

Integral equation

Quantum mechanics

Singular integral

Compact space

Type (biology)

Singularity

Ordinary differential equation

Parametrix

Artificial intelligence

Embedding

Microlocal analysis

Mathematical analysis

Singular solution

Ecology

Fourier integral operator

Physics

Nonlinear system

Kernel (algebra)

Journal of Mathematical Analysis and Applications

This paper establishes compactness of nonlinear integral operators in the space of continuous functions. One result deals with operators whose kernel can have jumps across a finite number of curves, which typically arise from the study of ordinary differential equations with boundary conditions of local or nonlocal type. Several other results deal with operators whose kernels have a singularity, which arise from the study of fractional differential equations. We motivate the study of these integral equations by discussing some initial value problems for fractional differential equations of Caputo and Riemann-Liouville type. We prove a compact embedding theorem for fractional integrals in order to give a new treatment for the singular kernel case.

Compactness of nonlinear integral operators with discontinuous and with singular kernels