• Bulletin of the Korean Mathematical Society
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• v.21 no.2
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• pp.55-60
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• 1984

In [2], D. Downing and W.A. Kirk obtained a number of fixed point theorems for set-valued maps in matric and Banach spaces. The authors considered maps which are more general than the contractions with nonempty and closed mapping values, and obtain results for maps satisfying certain "inwardness" conditions. A key aspect of their approach is the application of a general fixed point theorem due to Caristi [1]. On the other hand, in [6], the present author obtained a number of equivalent formulations of the well-known result of I. Ekeland [3, 4] on the variational principle for approximate solutions of minimization problems. Some of such formulations include sharpened forms of the Caristi theorem. In this paper, using one of such formulations, we show that Theorems 1-3 and Corollaries 1-5 of [2] are substantially improved by giving geometric estimations of fixed points.ed points.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.987-996
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• 2010

In this paper, we investigate the Cauchy-Rassias stability in Banach spaces and also the Cauchy-Rassias stability using the alternative fixed point for the functional equation: $$f(\frac{sx+ty}{2}+rz)+f(\frac{sx+ty}{2}-rz)+f(\frac{sx-ty}{2}+rz)+f(\frac{sx-ty}{2}-rz)=s^2f(x)+t^2f(y)+4r^2f(z)$$ for any fixed nonzero integers s, t, r with $r\;{\neq}\;{\pm}1$.

• Kyungpook Mathematical Journal
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• v.50 no.4
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• pp.557-564
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• 2010

By using an idea of C$\u{a}$dariu and Radu [4], we prove the generalized Hyers-Ulam stability of the functional equation f(x + y,z - w) + f(x - y,z + w) = 2f(x, z) + 2f(y, w). The quadratic form $f\;:\;\mathbb{R}\;{\times}\;\mathbb{R}{\rightarrow}\mathbb{R}$ given by f(x, y) = $ax^2\;+\;by^2$ is a solution of the above functional equation.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.2
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• pp.261-269
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• 2009

We adopt the idea of $C\check{a}dariu$ and Radu to prove the stability of Jordan triple linear derivations and we take account of the Jacobson radical range problem for linear derivations.

• Journal of the Chungcheong Mathematical Society
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• v.33 no.1
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• pp.77-87
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• 2020

In this paper, we investigate the stability of a functional equation f(x + 3y) - 5f(x + 2y) + 10f(x + y) - 8f(x) + 5f(x - y) - f(x - 2y) - 2f(-x) - f(2x) + f(-2x) = 0 by using the fixed point theory in the sense of L. Cǎdariu and V. Radu.

• East Asian mathematical journal
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• v.35 no.5
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• pp.559-568
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• 2019

In this paper, we investigate the stability problems for a functional equation f(x + 2y)+f(x - 2y) - 4f(x + y) - 4f(x - y) + 6f(x) - 2f(2y) + 12f(y) - 4f(-y) = 0 by using the fixed point theory in the sense of L. C˘adariu and V. Radu.

• Nonlinear Functional Analysis and Applications
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• v.26 no.5
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• pp.1059-1075
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• 2021

In this paper, we unify and enrich the well-known classical metrical coincidence theorems on a complete metric space due to Machuca, Goebel and Jungck. We further extend our newly proved results on a subspace Y of metric space X, wherein X need not be complete. Finally, we slightly modify the existing results involving (E.A)-property and (CLR_g)-property and apply these results to deduce our coincidence and common fixed point theorems.

• Nonlinear Functional Analysis and Applications
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• v.28 no.3
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• pp.851-863
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• 2023

We study the appropriate conditions for the findings of uniqueness and existence for a group of boundary value problems for impulsive Ψ-Caputo fractional nonlinear Volterra-Fredholm integro-differential equations (V-FIDEs) with symmetric boundary non-instantaneous conditions in this paper. The findings are based on the fixed point theorem of Krasnoselskii and the Banach contraction principle. Finally, the application is provided to validate our primary findings.

• Nonlinear Functional Analysis and Applications
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• v.29 no.2
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• pp.545-558
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• 2024

In this study, a class of nonlinear boundary fractional Caputo Volterra-Fredholm integro-differential equations (CV-FIDEs) is taken into account. Under specific assumptions about the available data, we firstly demonstrate the existence and uniqueness features of the solution. The Gronwall's inequality, a adequate singular Hölder's inequality, and the fixed point theorem using an a priori estimate procedure. Finally, a case study is provided to highlight the findings.

• The Pure and Applied Mathematics
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• v.14 no.4
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• pp.283-287
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• 2007

An extension of the contraction mapping theorem is provided in a Banach space setting to approximate fixed points of operator equations. Our approach is justified by numerical examples where our results apply whereas the classical contraction mapping principle cannot.