Measure of Noncompactness in Integral Equations

The measure of noncompactness has emerged as a fundamental analytical tool for studying the existence, uniqueness and stability of solutions to integral equations in infinite‐dimensional settings. Originally introduced to characterise compactness properties in Banach spaces, this concept assigns to each bounded set a nonnegative scalar quantifying its “degree” of noncompactness. In the context of integral equations—ranging from classical Volterra and Hammerstein formulations to systems of fractional and hybrid inclusions—measures of noncompactness facilitate the application of fixed point theorems under conditions too weak to guarantee compactness. By estimating kernels, nonlinearities and feedback terms in suitable function or sequence spaces, researchers obtain condensing operators whose fixed points correspond to solutions of the underlying equations. This approach accommodates fractional‐order integrals defined via Riemann–Liouville or Hadamard operators, infinite systems in ℓp and BK spaces, and stability analyses on semi‐infinite intervals. Concrete examples include asymptotic stability of delayed cubic inclusions with fractal feedback and solvability of quadratic Hammerstein systems. Beyond pure analysis, the technique has found applications in mathematical biology, control theory and physics, where integral formulations model memory effects, spatial diffusion and nonlocal interactions. The global significance of this framework lies in its unifying treatment of compactness deficiencies, enabling rigorous proofs of solution existence and long‐term behaviour in models that resist classical compact‐operator methods.

Research from Nature Portfolio

No recent Nature Portfolio content available.

Measure of Noncompactness in Integral Equations publication trend

The graph below shows the total number of articles in measure of noncompactness in integral equations across all publications each year (not limited to Nature Index journals).

Technical terms

Measure of noncompactness: A functional assigning to each bounded set in a Banach space a nonnegative value that quantifies how far the set is from being relatively compact.

Banach space: A complete normed vector space serving as the ambient setting for many integral equations and fixed point arguments.

Darbo fixed point theorem: An extension of Schauder’s theorem that guarantees fixed points for condensing operators defined by measures of noncompactness.

Hammerstein integral equation: A nonlinear integral equation of the form u(x)=∫K(x,y) f(y,u(y)) dy, often modelling feedback or source terms.

Riemann–Liouville operator: A classical fractional-order integral operator that generalises the n-fold integral to non-integer orders, introducing memory effects.

References

1. A study on the solvability of fractional integral equation in a Banach algebra via Petryshyn's fixed point theorem. Journal of Taibah University for Science (2024).
2. Asymptotically Stable Solutions of Infinite Systems of Quadratic Hammerstein Integral Equations. Symmetry (2024).
3. Solvability of Implicit Fractional Order Integral Equation in ℓp(1 ≤ p. Journal of Function Spaces (2022).
4. Investigating Asymptotic Stability for Hybrid Cubic Integral Inclusion with Fractal Feedback Control on the Real Half-Axis. Fractal and Fractional (2023).