• 대한수학회보
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• 제34권3호
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• pp.491-493
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• 1997

This is to clarify that Taskovic's extension of the Brouwer fixed point theorem is incorrectly stated.

• Nonlinear Functional Analysis and Applications
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• 제28권2호
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• pp.311-335
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• 2023

Fixed point theory is the center of focus for many mathematicians from last few decades. A lot of generalizations of the Banach contraction principle have been established. In this paper, we introduce the concepts of 𝜃-contraction and 𝜃-𝜑-contraction in quasi-metric spaces to study the existence of the fixed point for them.

• Kyungpook Mathematical Journal
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• 제56권2호
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• pp.431-450
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• 2016

In this paper, the existence, uniqueness, stability via continuous dependence and Ulam stabilities of nonlinear integro-differential equations with random impulses are studied under sufficient condition. The results are obtained by using Leray-Schauder alternative fixed point theorem and Banach contraction principle.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제20권1호
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• pp.51-82
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• 2016

In view of ideas for semigroups, fractional calculus, resolvent operator and Banach contraction principle, this manuscript is generally included with existence and controllability (EaC) results for fractional neutral integro-differential systems (FNIDS) with state-dependent delay (SDD) in Banach spaces. Finally, an examples are also provided to illustrate the theoretical results.

• 대한수학회보
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• 제21권2호
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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.

• Kyungpook Mathematical Journal
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• 제59권1호
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• pp.73-82
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• 2019

In this work, we give necessary and sufficient conditions under which every solution of a class of first-order neutral differential equations of the form $$(x(t)+p(t)x({\tau}(t)))^{\prime}+q(t)Hx({\sigma}(t)))=0$$ either oscillates or converges to zero as $t{\rightarrow}{\infty}$ for various ranges of the neutral coefficient p. Our main tools are the Knaster-Tarski fixed point theorem and the Banach's contraction mapping principle.

• 대한수학회보
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• 제55권4호
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• pp.1065-1092
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• 2018

This manuscript is involved with a category of second-order impulsive functional integro-differential equations with nonlocal conditions in Banach spaces. Sufficient conditions for existence and approximate controllability of mild solutions are acquired by making use of the theory of cosine family, Banach contraction principle and Leray-Schauder nonlinear alternative fixed point theorem. An illustration is additionally furnished to prove the attained principles.

• East Asian mathematical journal
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• 제21권1호
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• pp.65-76
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• 2005

In this paper the concept of semi-compatibility has been introduced in Menger space and it has been applied to prove results on existence of unique common fixed point of four self maps satisfying an implicit relation. It results in a generalization of Banach contraction principle established by Sehgal and Bharucha-Reid in [8] All the result presented in this paper are new.

• Nonlinear Functional Analysis and Applications
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• 제29권1호
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• pp.113-130
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• 2024

In this work, we explore the existence and uniqueness results for a class of boundary value issues for implicit Volterra-Fredholm nonlinear integro-differential equations (IDEs) with Atangana-Baleanu-Riemann fractional (ABR-fractional) that have non-instantaneous multi-point fractional boundary conditions. The findings are supported by Krasnoselskii's fixed point theorem, Gronwall-Bellman inequality, and the Banach contraction principle. Finally, a demonstrative example is provided to support our key findings.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제14권2호
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• pp.79-91
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• 2010

In this paper, a class of fractional integrodifferential equations of mixed type with nonlocal conditions is considered. First, using contraction mapping principle and Krasnoselskii's fixed point theorem via Gronwall's inequailty, the existence and uniqueness of mild solution are given. Second, the existence of optimal pairs of systems governed by fractional integrodifferential equations of mixed type with nonlocal conditions is also presented.