Search

In this paper, we introduce the complex valued $ C^{p} $-class function, a type of Geraghty contraction and a type of JS contraction in complete partially ordered complex valued $ G_b $-metric spaces, prove three fixed point theorems in this space, and also we give some examples to support our results.

In this paper, we are concerned with the following modified Schrödinger equation $ \begin{array}{l} -\Delta u+V(|x|)u-\kappa u\Delta(u^2)+ \\ \qquad\qquad\qquad q\frac{h^2(|x|)}{|x|^2}(1+\kappa u^2)u\ + q\left(\int_{|x|}^{+\infty}\frac{h(s)}{s}(2+\kappa u^2(s))u^2(s){\rm{d}}s\right) u = (I_\alpha\ast F(u))f(u), \, \, x\in {\mathbb R}^2, \end{array} $ where $ \kappa $, $ q > 0 $, $ I_\alpha $ is a Riesz potential, $ \alpha\in (0, 2) $ and $ V \in \mathcal{C}({\mathbb R}^2, {\mathbb R}) $, $ F(t) = \int^t_0f(s){\rm{d}}s $. Under appropriate assumptions on $ f $ and $ V(x) $, by using the variational methods, we establish the existence of ground state solutions of the above equation.

In this paper, we deal with the existence of nontrivial solutions for the following Kirchhoff-type equation where $0 < s < 1 < p < \infty$, $sp < N$, $\lambda > 0$ is a real parameter, $(-\Delta)_{p}^{s}$ is the fractional $p$-Laplacian operator, $V:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a potential function, $M$ is a Kirchhoff function, the nonlinearity $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a continuous function and just super-linear in a neighborhood of $u = 0$. By using an appropriate truncation argument and the mountain pass theorem, we prove the existence of nontrivial solutions for the above equation, provided that $\lambda$ is sufficiently large. Our results extend and improve the previous ones in the literature.

In this paper, we study the following quasilinear Schrödinger equation \begin{equation*} \begin{split} -\Delta u&+V(x)u-\kappa u\Delta(u^2)+\mu\frac{h^2(|x|)}{|x|^2}(1+\kappa u^2)u\\ &+\mu\left(\int_{|x|}^{+\infty}\frac{h(s)}{s}(2+\kappa u^2(s))u^2(s)\text{d}s\right)u=f(u)\quad\text{in}~\mathbb{R}^2, \end{split} \end{equation*} where $\kappa>0$, $\mu>0$, $V \in \mathcal{C}^1(\mathbb{R}^2,\mathbb{R})$ and $f \in \mathcal{C}(\mathbb{R},\mathbb{R})$. By using a constraint minimization of Pohožaev–Nehari type and analytic techniques, we obtain the existence of ground state solutions.

In this paper, we consider the following magnetic Laplace nonlinear Choquard equation $ \begin{equation*} -\Delta_A u+V(x)u = (I_{\alpha}*F(|u|))\frac{f(|u|)}{|u|}u, \, \, \text{in}\, \, \mathbb{R}^N, \ \end{equation*} $ where $ u: \mathbb{R}^N\rightarrow C $, $ A: \mathbb{R}^N\rightarrow \mathbb{R}^N $ is a vector potential, $ N\geq 3 $, $ \alpha\, \in\, (N-2, N) $, $ V:\, \mathbb{R}^N \rightarrow \mathbb{R} $ is a scalar potential function and $ I_{\alpha} $ is a Riesz potential of order $ \alpha\, \in\, (N-2, N) $. Under certain assumptions on $ A(x) $, $ V(x) $ and $ f(t) $, we prove that the equation has at least a ground state solution by variational methods.

In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ2u−a+b∫ℝN∇u2dxΔu+Vxu=Iα∗Fufu+λu2∗∗−2u,in ℝN,u∈H2ℝN, where a>0, b≥0, λ is a positive parameter, α∈N−2,N, 5≤N≤8, V:ℝN⟶ℝ is a potential function, and Iα is a Riesz potential of order α. Here, 2∗∗=2N/n−4 with N≥5 is the Sobolev critical exponent, and Δ2u=ΔΔu is the biharmonic operator. Under certain assumptions on Vx and fu, we prove that the equation has ground state solutions by variational methods.

In this research paper, we concentrate on the existence and uniqueness of common fixed points of various mappings in b2 metric space under generalized (ϕ,f)λ-expansive conditions and implicit contractive conditions. Additionally, we derive a progression of noteworthy approaches and ideas extending the conclusions on the real metric space to the b2 metric space, which advances the investigation of the b2 metric space will be able to advance.

Abstract We study an approximate coincidence point and a common fixed point problem for a hybrid pair of mappings with constraints in Menger PM-spaces, and obtain some new results. We derive interesting consequences of the main results by using the properties of a Menger–Hausdorff metric, and analogous results based on graphs instead of partial orders can be similarly formulated. Moreover, we construct two examples to reveal that the main results are valid, and show that the main results can be used to explore the existence of solutions to a system of nonlinear integral equations.

Abstract The forward–backward splitting algorithm is a popular operator-splitting method for solving monotone inclusion of the sum of a maximal monotone operator and an inverse strongly monotone operator. In this paper, we present a new convergence analysis of a variable metric forward–backward splitting algorithm with extended relaxation parameters in real Hilbert spaces. We prove that this algorithm is weakly convergent when certain weak conditions are imposed upon the relaxation parameters. Consequently, we recover the forward–backward splitting algorithm with variable step sizes. As an application, we obtain a variable metric forward–backward splitting algorithm for solving the minimization problem of the sum of two convex functions, where one of them is differentiable with a Lipschitz continuous gradient. Furthermore, we discuss the applications of this algorithm to the fundamental of the variational inequalities problem, constrained convex minimization problem, and split feasibility problem. Numerical experimental results on LASSO problem in statistical learning demonstrate the effectiveness of the proposed iterative algorithm.

In this paper, we first propose a new concept of Z-Z-B spaces, which is a generalization of Z-C-X spaces. Meanwhile, the new concept of the superior cone is introduced. Secondly, we study some new problems for semi-closed 1-set-contractive operators in the Z-Z-B space and obtain some new results. These new theorems are proven by combining partial order theory with fixed point index theory. Regarding these theorems, in the latter part of the paper, the proofs are omitted since the methods of proving these theorems are similar. Moreover, two important inequality lemmas are proven.

ABSTRACT In filamentous fungi, 1,8-dihydroxynaphthalene (DHN) melanin is a major component of the extracellular matrix, endowing fungi with environmental tolerance and some pathogenic species with pathogenicity. However, the subcellular location of the melanin biosynthesis pathway components remains obscure. Using the gray mold pathogen Botrytis cinerea, the DHN melanin intermediate scytalone was characterized via phenotypic and chemical analysis of mutants, and the key enzymes participating in melanin synthesis were fused with fluorescent proteins to observe their subcellular localizations. The Δbcscd1 mutant accumulated scytalone in the culture filtrate rather than in mycelium. Excessive scytalone appears to be self-inhibitory to the fungus, leading to repressed sclerotial germination and sporulation in the Δbcscd1 mutant. The BcBRN1/2 enzymes responsible for synthesizing scytalone were localized in endosomes and found to be trafficked to the cell surface, accompanied by the accumulation of BcSCD1 proteins in the cell wall. In contrast, the early-stage melanin synthesis enzymes BcPKS12/13 and BcYGH1 were localized in peroxisomes. Taken together, the results of this study revealed the subcellular distribution of melanin biosynthetic enzymes in B. cinerea, indicating that the encapsulation and externalization of the melanin synthetic enzymes need to be delicately orchestrated to ensure enzymatic efficiency and protect itself from the adverse effect of the toxic intermediate metabolite. IMPORTANCE The devastating gray mold pathogen Botrytis cinerea propagates via melanized conidia and sclerotia. This study reveals that the sclerotial germination of B. cinerea is differentially affected by different enzymes in the melanin synthesis pathway. Using gene knockout mutants and chemical analysis, we found that excessive accumulation of the melanin intermediate scytalone is inhibitory to B. cinerea. Subcellular localization analysis of the melanin synthesis enzymes of B. cinerea suggested two-stage partitioning of the melanogenesis pathway: the intracellular stage involves the steps until the intermediate scytalone was translocated to the cell surface, whereas the extracellular stage comprises all the steps occurring in the wall from scytalone to final melanin formation. These strategies make the fungus avert self-poisoning during melanin production. This study opens avenues for better understanding the mechanisms of secondary metabolite production in filamentous fungi.

In this paper, we consider a new kind of Kirchhoff-type equation which is stated in the introduction. Under certain assumptions on potentials, we prove by variational methods that the equation has at least a ground state solution and investigate the asymptotic behavior of solutions.

In this paper, we consider the following Kirchhoff-type equation: {u∈H1(RN),−(a+b∫RN|∇u|2dx)Δu+V(x)u=(Iα∗F(u))f(u)+λg(u),inRN, where a>0, b≥0, λ>0, α∈(N−2,N), N≥3, V:RN→R is a potential function and Iα is a Riesz potential of order α∈(N−2,N). Under certain assumptions on V(x), f(u) and g(u), we prove that the equation has at least one nontrivial solution by variational methods.

Some new coupled coincidence point and coupled fixed point theorems are established in partially ordered metric-like spaces, which generalize many results in corresponding literatures. An example is given to support our main results. As an application, we discuss the existence of the solutions for a class of nonlinear integral equations.

The resolvent is a fundamental concept in studying various operator splitting algorithms. In this paper, we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators. First, we discuss several explicit solutions of this resolvent operator by taking into account additional constraints on the linear operator. Second, we propose a fixed point approach for computing this resolvent operator in a general case. Based on the Krasnoselskii⁻Mann algorithm for finding fixed points of non-expansive operators, we prove the strong convergence of the sequence generated by the proposed algorithm. As a consequence, we obtain an effective iterative algorithm for solving the scaled proximity operator of a convex function composed by a linear operator, which has wide applications in image restoration and image reconstruction problems. Furthermore, we propose and study iterative algorithms for studying the resolvent operator of a finite sum of maximally monotone operators as well as the proximal operator of a finite sum of proper, lower semi-continuous convex functions.

In this paper, we introduce the concept of partial cone b-metric spaces as a generalization of partial metric, cone metric and b-metric spaces and establish some topological properties of partial cone b-metric spaces. Moreover, we also prove some common fixed point theorems for cyclic contractive mappings in such spaces. Our results generalize and extend the main results of Huang and Zhang [Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332:1468–1476, 2007], Stanić et al. [Common fixed point under contractive condition of Ćirić's type on cone metric type spaces, Fixed Point Theory Appl., 2012:35, 2012] and Latif et al. [Fixed point results for generalized (α,ψ)‐Meir–Keeler contractive mappings and applications, J. Inequal. Appl., 2014:68, 2014]. Some examples and an application are given to support the usability of the obtained results.

In this work, the Cauchy initial value problem is discussed for a class of fractional impulsive differential equations with delay, and the criteria on existence and uniqueness are obtained. Finally, an example is also provided to illustrate the effectiveness of our main results.

The aim of this paper is to present the concept of binary comparable operators in partially ordered Banach spaces and prove several fixed point theorems under some contractive conditions. The results of this paper can be used to investigate a large class of nonlinear problems. As an application, we study the existence of solution of a nonlinear integral equation.

We introduce the concepts of (H,ψ,Φ)-contraction and probabilistic (α,φ)-contraction mappings in generalized probabilistic metric spaces and prove some fixed point theorems for such two types of mappings in generalized probabilistic metric spaces. Our results generalize and extend many comparable results in existing literature. Some examples are also given to support our results. Finally, an application to the existence of solutions for a class of integral equations is presented by utilizing one of our main results.

We extend a common fixed point theorem of Radenovic and Rhoades for four non-self-mappings in cone metric spaces.

Based on previous notions of expansive mapping, n times reasonable expansive mapping is defined. The existence of fixed point for n times reasonable expansive mapping is discussed and some new results are obtained.

In this paper, some fixed point results under the Meir-Keeler S type contraction conditions on S-metric spaces and some fixed point theorems with F control functions on S-metric spaces are proved. Moreover, the relationships among G-metric and S-metric are investigated and an example of discontinuity at a fixed point is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the complex valued Cp-class function, a type of Geraghty contraction and a type of JS contraction in complete partially ordered complex valued Gb-metric spaces, prove three fixed point theorems in this space, and also we give some examples to support our results. [ABSTRACT FROM AUTHOR]