• Communications of the Korean Mathematical Society
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• v.38 no.1
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• pp.137-151
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• 2023

In this paper we prove the existence of nontrivial weak solutions to the boundary value problem -G₁u = u³ + f(x, y, u) in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, where Ω is a bounded domain with smooth boundary in ℝ³, G₁ is a Grushin type operator, and f(x, y, u) is a lower order perturbation of u³ with f(x, y, 0) = 0. The nonlinearity involved is of critical exponent, which differs from the existing results in [11, 12].

• Nonlinear Functional Analysis and Applications
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• v.28 no.2
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• pp.557-570
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• 2023

The conceptions of generalized b-metric spaces or G_b-metric spaces and a generalized Ω-distance mappings play a key role in proving many important theorems in existence and uniqueness of fixed point theory. In this manuscript, we establish a new type of contraction namely, Ω_b(𝓗, 𝜃, s)-contraction, this contraction based on the concept of a generalized Ω-distance mappings which established by Abodayeh et.al. in 2017 together with the concept of 𝓗-simulation functions which established by Bataihah et.al [10] in 2020. By utilizing this new notion we prove new results in existence and uniqueness of fixed point. On the other hand, examples and application were established to show the importance of our results.

• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.989-1004
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• 2023

In this paper, we are motivated to evaluate and investigate the necessary conditions for the fractional Volterra Fredholm integro-differential equation involving the ς-Hilfer fractional derivative. The given problem is converted into an equivalent fixed point problem by introducing an operator whose fixed points coincide with the solutions to the problem at hand. The existence and uniqueness results for the given problem are derived by applying Krasnoselskii and Banach fixed point theorems respectively. Furthermore, we investigate the convergence of approximated solutions to the same problem using the modified Adomian decomposition method. An example is provided to illustrate our findings.

• Nonlinear Functional Analysis and Applications
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• v.28 no.4
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• pp.957-988
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• 2023

We define new classes of expansive-type mappings in the setting of modular 𝜔^G-metric spaces and prove the existence of common unique fixed point for these classes of expansive-type mappings on 𝜔^G-complete modular 𝜔^G-metric spaces. The results established in this paper extend, improve, generalize and compliment many existing results in literature. We produce some examples to validate our results.

• Bulletin of the Korean Mathematical Society
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• v.41 no.3
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• pp.483-492
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• 2004

Let : [0, 1] $\times$ [0, $\infty$) $\longrightarrow$ [0, $\infty$) be continuous and a ${\in}$ C([0, 1], [0, $\infty$)),and let ${\xi}_{i}$ $\in$ (0, 1) with 0 < {\xi}$_1$ < ${\xi}_2$ < … < ${\xi}_{m-2}$ < 1, $a_{i}$, $b_{i}$ ${\in}$ [0, $\infty$) with 0 < $\Sigma_{i=1}$ /$^{m-2}$ $a_{i}$ < 1 and $\Sigma_{i=1}$$^{m-2}$ < l. This paper is concerned with the following m-point boundary value problem: $\chi$″(t)＋a(t) (t.$\chi$(t))＝0,t ${\in}$(0,1), $\chi$'(0)＝$\Sigma_{i=1}$ $^{m-2}$ /$b_{i}$$\chi$'(${\xi}_{i}$),$\chi$(1)＝$\Sigma_{i=1}$$^{m-2}$$a_{i}$$\chi$(${\xi}_{i}$). The existence, multiplicity and uniqueness of positive solutions of this problem are discussed with the help of two fixed point theorems in cones, respectively.

• Bulletin of the Korean Mathematical Society
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• v.37 no.4
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• pp.729-741
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• 2000

In this paper, we study in Banach spaces the existence of fixed points of asymptotically regular mapping T satisfying: for each x, y in the domain and for n=1, 2,…, $$\parallelT^nx-T^ny\parallel\leq$\leq$a_n\parallelx-y\parallel+b_n (\parallelx-T^nx\parallel+\parallely-T^ny\parallely)$$ where $a_n,\; b_n,\; C_n$ are nonnegative constants satisfying certain conditions. We also establish some fixed point theorems for these mappings in a Hibert space, in L(sup)p spaces, in Hardy space H(sup)p, and in Soboleve space $H^{k,p} for 1<\rho<\infty \; and \; k\geq0$. We extend results from papers [10], [11], and others.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.493-507
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• 1995

The Hartman-Stampacchia variational inequality was first suggested and studied by Hartman and Stampacchia [8] in finite dimensional spaces during the time establishing the base of variational inequality theory in 1960s [4]. Then it was generalized by Lions et al. [6], [9], [10], Browder [3] and others to the case of infinite dimensional inequality [3], [9], [10], and the results concerning this variational inequality have been applied to many important problems, i.e., mechanics, control theory, game theory, differential equations, optimizations, mathematical economics [1], [2], [6], [9], [10]. Recently, the Browder-Hartman-Stampaccnia variational inequality was extended to the case of set-valued monotone mappings in reflexive Banach sapces by Shih-Tan [11] and Chang [5], and under different conditions, they proved some existence theorems of solutions of this variational inequality.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.61-73
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• 2008

In this paper common fixed point theorems dealing with compatible mappings of type (P) are established. As a application, the existence and uniqueness of common solution for a system of functional equations arising in dynamic programming is given. The results presented in this paper improve, generalize and unify the corresponding results in this field.

• Journal of applied mathematics & informatics
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• v.36 no.1_2
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• pp.39-50
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• 2018

In this paper, we deal with the uniqueness problem for the power of p-adic meromorphic functions. The results obtained in this paper are the p-adic analogues and supplements of the theorems given by Yang and Zhang [Non-existence of meromorphic solution of a Fermat type functional equation, Aequationes Math. 76(2008), 140-150], Chen, Chen and Li [Uniqueness of difference operators of meromorphic functions, J. Ineq. Appl. 2012(2012), Art 48], Zhang [Value distribution and shared sets of differences of meromorphic functions, J. Math. Anal. Appl. 367(2010), 401-408].

• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.789-807
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• 2015

We propose coincidence and common fixed point results for a quadruple of self mappings satisfying common limit range property and weakly compatibility under generalized ${\Phi}$-contractive conditions i Non-Archimedean Menger PM-spaces. As examples we exhibit different types of situations where these conditions can be used. A common fixed point theorem for four finite families of self mappings is presented as an application of the proposed results. The existence and uniqueness of solutions for certain system of functional equations arising in dynamic programming are also presented as another application.