• Journal of the Korean Mathematical Society
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• v.52 no.1
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• pp.43-65
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• 2015

We present a new convergence analysis of popular Broyden's method in the Banach/Hilbert space setting which is applicable to non-smooth operators. Moreover, we do not assume a priori solvability of the equation under consideration. Nevertheless, without these simplifying assumptions our convergence theorem implies existence of a solution and superlinear convergence of Broyden's iterations. To demonstrate practical merits of Broyden's method, we use it for numerical solution of three nontrivial infinite-dimensional problems.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.75-82
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• 2014

In this paper, we consider the periodic boundary value problem with sign changing nonlinearity $$u^{{\prime}{\prime}{\prime}}+{\rho}^3u=f(t,u),\;t{\in}[0,2{\pi}]$$, subject to the boundary value conditions: $$u^{(i)}(0)=u^{(i)}(2{\pi}),\;i=0,1,2$$, where ${\rho}{\in}(o,{\frac{1}{\sqrt{3}}})$ is a positive constant and f(t, u) is a continuous function. Using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The interesting point is the nonlinear term f may change sign.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.697-710
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• 2010

In this paper, we prove existence of infinitely many positive and concave solutions, by means of a simple approach, to $3^{th}$ order three-point singular boundary value problem {$x^{\prime\prime\prime}(t)=\alpha(t)f(t,x(t))$, 0 < t < 1, $x(0)=x'(\eta)=x^{\prime\prime}(1)=0$, (1/2 < $\eta$ < 1). Moreover with respect to multiplicity of solutions, we don't assume any monotonicity on the nonlinearity. We rely on a combination of the analysis of the corresponding vector field on the phase-space along with Knesser's type properties of the solutions funnel and the well-known Krasnosel'ski$\breve{i}$'s fixed point theorem. The later is applied on a new very simple cone K, just on the plane $R^2$. These extensions justify the efficiency of our new approach compared to the commonly used one, where the cone $K\;{\subset}\;C$ ([0, 1], $\mathbb{R}$) and the existence of a positive Green's function is a necessity.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.175-177
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• 1989

The following is the basic problem about the stability in Riemannian Geometry; given a Riemannian manifold N, find all stable complete minimal submanifolds of N. As answers of this problem, do Carmo-Peng [1] and Fischer-Colbrie and Schoen [3] showed that the stable minimal surfaces in R$^{3}$ are planes and Schoen-Yau [5] and Fischer-Colbrie and Schoen [3] gave a solution for the case where the ambient space is a three dimensional manifold with nonnegative scalar curvature. In this paper we will remove the assumption of finite absolute total curvature in [3, Theorem 3].

• Journal of the Chungcheong Mathematical Society
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• v.29 no.2
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• pp.187-220
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• 2016

This paper is concerned with a kind of non-homogeneous boundary value problems for singular second order differential systems with Laplacian operators. Using multiple fixed point theorems, sufficient conditions to guarantee the existence of at least three solutions of this kind of boundary value problems are established. An example is presented to illustrate the main results.

• Journal of applied mathematics & informatics
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• v.25 no.1_2
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• pp.35-50
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• 2007

In this work, optimal harvesting policy for the predator-prey system of three species with age-dependent and diffusion is discussed. Existence and uniqueness of non-negative solution to the system are investigated by using the fixed point theorem. The existence of optimal control strategy is discussed and optimality conditions are obtained. Our results extend some known criteria.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.1
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• pp.81-95
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• 2007

We are concerned with the multiplicity of solutions of the nonlinear biharmonic equation with Dirichlet boundary condition, ${\Delta}^2u+c{\Delta}u=g(u)$, in ${\Omega}$, where $c{\in}R$ and ${\Delta}^2$ denotes the biharmonic operator. We show that there exists at least three solutions of the above problem under the suitable condition of g(u).

• Kyungpook Mathematical Journal
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• v.53 no.2
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• pp.257-271
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• 2013

In this article, we establish the existence of at least three unbounded positive solutions to a boundary-value problem of the nonlinear singular fractional differential equation. Our analysis relies on the well known fixed point theorems in the cones.

• The Pure and Applied Mathematics
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• v.30 no.3
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• pp.249-267
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• 2023

This manuscript is divided into three segments. In the first segment, we prove a unique common fixed point theorem satisfying generalized weak contraction on partially ordered metric spaces and also give an example to support our results presented here. In the second segment of the article, some common coupled fixed point results are derived from our main results. In the last segment, we investigate the solution of integral equation as an application. Our results generalize, extend and improve several well-known results of the existing literature.

• Coupled systems mechanics
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• v.10 no.1
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• pp.21-38
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• 2021

This work is an attempt to design a dynamic model for a non local bio-thermoelastic medium with diffusion. The system of governing equations are formulated in terms of displacement vector field, chemical potential and the tissue temperature in the context of non local dual phase lag (NL DPL) theories of heat conduction and mass diffusion. Based on this considered model, we study the fundamental solution and propagation of plane harmonic waves in tissues. In order to analyze the behavior of the NL DPL model, we construct basic theorem in the terms of elementary function which determine the existence of three longitudinal and one transverse wave. The effects of various parameters on the characteristics of waves i.e., phase velocity and attenuation coefficients are elaborated by plotting various figures of physical quantities in the later part of the paper.