• Korean Journal of Mathematics
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• v.20 no.4
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• pp.533-540
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• 2012

We consider the multiplicity of the solutions for the elliptic boundary value problem with $C^1$ nonlinear term decaying at the origin. We get a theorem which shows the existence of the nontrivial solution for the elliptic problem with $C^1$ nonlinear term decaying at the origin. We obtain this result by reducing the elliptic problem with the $C^1$ nonlinear term to the el-liptic problem with bounded nonlinear term and then approaching the variational method and using the mountain pass geometry for the reduced the elliptic problem with bounded nonlinear term.

• Nonlinear Functional Analysis and Applications
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• v.27 no.1
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• pp.189-201
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• 2022

The aim of the current work is to investigate the numerical study of a nonlinear Volterra-Fredholm integro-differential equation with initial conditions. Our approximation techniques modified adomian decomposition method (MADM) and variational iteration method (VIM) are based on the product integration methods in conjunction with iterative schemes. The convergence of the proposed methods have been proved. We conclude the paper with numerical examples to illustrate the effectiveness of our methods.

• 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.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.1-27
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• 2008

We introduce a new concept of convex spaces and a multimap class K having certain KKM property. From a basic KKM type theorem for a K-map defined on an convex space without any topology, we deduce ten equivalent formulations of the theorem. As applications of the equivalents, in the frame of convex topological spaces, we obtain Fan-Browder type fixed point theorems, almost fixed point theorems for multimaps, mutual relations between the map classes K and B, variational inequalities, the von Neumann type minimax theorems, and the Nash equilibrium theorems.

• Honam Mathematical Journal
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• v.31 no.1
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• pp.75-85
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• 2009

We show the existence of the nontrivial periodic solution for a class of the system of the nonlinear suspension bridge equations with Dirichlet boundary condition and periodic condition by critical point theory and linking arguments. We investigate the geometry of the sublevel sets of the corresponding functional of the system, the topology of the sublevel sets and linking construction between two sublevel sets. Since the functional is strongly indefinite, we use the linking theorem for the strongly indefinite functional and the notion of the suitable version of the Palais-Smale condition.

• Geomechanics and Engineering
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• v.11 no.6
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• pp.867-877
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• 2016

Collapse shape of tunnel roof in layered Hoek-Brown rock media is investigated within the framework of upper bound theorem. The traditional collapse mechanism for homogeneous stratum is no longer suitable for the present analysis of roof stability, and it would be necessary to propose a curve failure mode to describe the velocity discontinuity surface in layered media. What is discussed in the paper is that the failure mechanism of tunnel roofs, consisting of two different functions, is proposed for layered rock media. Then it is employed to investigate the impending roof failure. Based on the nonlinear Hoek-Brown failure criterion, the collapse volume of roof blocks are derived with the upper bound theorem and variational principle. Numerical calculations and parametric analysis are carried out to illustrate the effects of different parameters on the shape of failure mechanism, which is of overriding significance to the stability analysis of tunnel roof in layered rock media.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.693-703
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• 2022

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$(1){\hspace{32}}{\ddot{u}}(t)+q(t){\dot{u}}(t)-L(t)u(t)+{\nabla}W(t,u(t))=0,\;{\forall}t{\in}{\mathbb{R}},$$ where q ∈ C(ℝ, ℝ), L ∈ C(ℝ, ${\mathbb{R}}^{N^2}$) is a symmetric and positive definite matix-valued function and W ∈ C¹(ℝ×ℝ^N, ℝ). The novelty of this paper is that, assuming that L is bounded from below unnecessarily coercive at infinity, and W is only locally defined near the origin with respect to the second variable, we show that (1) possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.

• Journal of the Korean Mathematical Society
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• v.39 no.4
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• pp.579-591
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• 2002

New fixed Point results for the (equation omitted) selfmaps ale given. The analysis relies on a factorization idea. The notion of an essential map is also introduced for a wide class of maps. Finally, from a new fixed point theorem of ours, we deduce some equilibrium theorems.

• Journal of applied mathematics & informatics
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• v.36 no.5_6
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• pp.491-504
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• 2018

In this article, we consider a fourth order nonlinear difference system. By making use of the critical point theory, we obtain some new existence theorems of at least one periodic solution with minimal period. Our main approach used in this article is the variational technique and the Saddle Point Theorem.

• Korean Journal of Mathematics
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• v.19 no.4
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• pp.481-493
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• 2011

We prove a theorem which shows the existence of at least three ${\pi}$-periodic solutions of the wave equation with asymptotical linearity. We obtain this result by the finite dimensional reduction method which reduces the critical point results of the infinite dimensional space to those of the finite dimensional subspace. We also use the critical point theory and the variational method.