Abstract
Goebel’s coincidence theorem, a remarkably simple extension of Banach’s contraction principle widely applied in analysis, has found significant utility in the theory of differential and integral equations. Over the past fifty years, researchers have endeavored to generalize the definition of a metric space, thereby extending the scope of Goebel’s coincidence theorem to diverse settings. This survey paper overviews valuable insights into Goebel’s coincidence theorem’s historical background, its relevance in the field of fixed point theory, and its practical implications in solving problems related to differential and integral equations.
Dedicated to the memory of Professor K. Goebel
Acknowledgements
The author would like to thank the anonymous referees for their valuable suggestions, which helped improve the presentation of the paper.
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Articles in the same Issue
- Frontmatter
- On the Laplace transform
- An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function
- Controllability result in α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces
- Polynomial convergence of iterations of certain random operators in Hilbert space
- Trigonometric approximation of signals (functions) belonging to certain Lipschitz spaces using C δ.C operator
- Almost compactness in neutrosophic topological spaces
- Some results for a weakly coupled system of semi-linear structurally damped σ-evolution equations
- On the stochastic elliptic equations involving fractional derivative
- A note on Köthe–Toeplitz duals and multiplier spaces of sequence spaces involving bicomplex numbers
- A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations
- Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy
- On random pairwise comparisons matrices and their geometry
- Statistically convergent difference sequences of bi-complex numbers
- On some inequalities concerning polynomials with restricted zeros
- New retarded nonlinear integral inequalities of Gronwall–Bellman–Pachpatte type and their applications
- On a new variant of cyclic (noncyclic) condensing operators with existence of optimal solutions to an FDE
- Exponential stability of non-conformable fractional-order systems
Articles in the same Issue
- Frontmatter
- On the Laplace transform
- An uncertainty principle for the windowed Bochner–Fourier transform with the complex-valued window function
- Controllability result in α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces
- Polynomial convergence of iterations of certain random operators in Hilbert space
- Trigonometric approximation of signals (functions) belonging to certain Lipschitz spaces using C δ.C operator
- Almost compactness in neutrosophic topological spaces
- Some results for a weakly coupled system of semi-linear structurally damped σ-evolution equations
- On the stochastic elliptic equations involving fractional derivative
- A note on Köthe–Toeplitz duals and multiplier spaces of sequence spaces involving bicomplex numbers
- A brief survey on the development and applications of Goebel’s coincidence point theorem in differential and integral equations
- Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy
- On random pairwise comparisons matrices and their geometry
- Statistically convergent difference sequences of bi-complex numbers
- On some inequalities concerning polynomials with restricted zeros
- New retarded nonlinear integral inequalities of Gronwall–Bellman–Pachpatte type and their applications
- On a new variant of cyclic (noncyclic) condensing operators with existence of optimal solutions to an FDE
- Exponential stability of non-conformable fractional-order systems