We prove a quasiconformal analogue of Koebe’s theorem related to the average Jacobian and use a normal family argument here to prove a quasiregular analogue of this result in certain domains in $n$-dimensional space. As an application, we establish that Lipschitz-type properties are inherited by a quasiregular function from its modulo. We also prove some results of Hardy-Littlewood type for Lipschitz-type spaces in several dimensions, give the characterization of Lipschitz-type spaces for quasiregular mappings by the average Jacobian, and give a short review of the subject.

Let $f(x)$ be a smooth strictly convex solution of $\text{det}({\partial }^{2}f/\partial x_i\partial x_j)=\text{exp}\left\{(1/2){\sum }_{i=1}^n{x}_i(\partial f/\partial x_i)-f\right\}$ defined on a domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$; then the graph $M_{\nabla f}$ of $\nabla f$ is a space-like self-shrinker of mean curvature flow in Pseudo-Euclidean space ${\mathbb{R}}_n^{\mathrm{2}n}$ with the indefinite metric $\sum d{x}_{i}d{y}_i$. In this paper, we prove a Bernstein theorem for complete self-shrinkers. As a corollary, we obtain if the Lagrangian graph $M_{\nabla f}$ is complete in $R_n^{\mathrm{2}n}$ and passes through the origin then it is flat.

We introduce the notion of generalized weaker $(\alpha \text{-}\varphi \text{-}\phi )$-contractive mappings in the context of generalized metric space. We investigate the existence and uniqueness of fixed point of such mappings. Some consequences on existing fixed point theorems are also derived. The presented results generalize, unify, and improve several results in the literature.

The aim of this paper is to define weak $\alpha$-$\psi$-$\phi$-contractive mappings and to establish coupled and tripled coincidence point theorems for such mappings defined on $G_b$-metric spaces using the concept of rectangular $G$-$\alpha$-admissibility. As an application, we derive new coupled and tripled coincidence point results for weak $\psi$-$\phi$-contractive mappings in partially ordered $G_b$-metric spaces. Our results are generalizations and extensions of some recent results in the literature. We also present an example as well as an application to nonlinear Fredholm integral equations in order to illustrate the effectiveness of our results.

We obtained some generalized common fixed point results in the context of complex valued metric spaces. Moreover, we proved an existence theorem for the common solution for two Urysohn integral equations. Examples are presented to support our results.

We consider a nonlinear integral equation which can be interpreted as a generalization of Theodorsen’s nonlinear integral equation. This equation arises in computing the conformal mapping between simply connected regions. We present a numerical method for solving the integral equation and prove the uniform convergence of the numerical solution to the exact solution. Numerical results are given for illustration.

We construct an implicit sequence suitable for the approximation of solutions of K-positive definite operator equations in real Banach spaces. Furthermore, implicit error estimate is obtained and the convergence is shown to be faster in comparsion to the explicit error estimate obtained by Osilike and Udomene (2001).

We discuss Caristi’s fixed point theorem for mappings defined on a metric space endowed with a graph. This work should be seen as a generalization of the classical Caristi’s fixed point theorem. It extends some recent works on the extension of Banach contraction principle to metric spaces with graph.

By using the method of differential subordinations, we derive some properties of multivalent analytic functions. All results presented here are sharp.

A new existence result of $\epsilon$-vector equilibrium problem is first obtained. Then, by using the existence theorem of $\epsilon$-vector equilibrium problem, a weakly $\epsilon$-cone saddle point theorem is also obtained for vector-valued mappings.

The aim of this paper is to introduce new concepts of $\alpha$-$\eta$-complete metric space and $\alpha$-$\eta$-continuous function and establish fixed point results for modified $\alpha$-$\eta$-$\psi$-rational contraction mappings in $\alpha$-$\eta$-complete metric spaces. As an application, we derive some Suzuki type fixed point theorems and new fixed point theorems for $\psi$-graphic-rational contractions. Moreover, some examples and an application to integral equations are given here to illustrate the usability of the obtained results.

We introduce new implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inclusions in a real Hilbert space. Under suitable control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets.