Boundary growth of Sobolev functions of monotone type for double phase functionals

Our aim in this paper is to deal with boundary growth of spherical means of Sobolev functions of monotone type for the double phase functional \(\Phi_{p,q}(x,t) = t^{p} + (b(x) t)^{q}\) in the unit ball B of \(\mathbb{R}^n\), where \(1 < p < q < \infty\) and \(b(\cdot)\) is a non-negative bounded function on B which is Hölder continuous of order \(\theta \in (0,1]\).

Strong convergence theorems for variational inequalities and fixed point problems in Banach spaces

In this paper, using a sunny generalized non-expansive retraction which is different from the metric projection and generalized metric projection in Banach spaces, we present a retractive iterative algorithm of Krasnosel’skii-type, whose sequence approximates a common solution of a mono-variational inequality of a finite family of η-strongly-pseudo-monotone-type maps and fixed points of a countable family of generalized non-expansive-type maps. Furthermore, some new results relevant to the study are also presented. Finally, the theorem proved complements, improves and extends some important related recent results in the literature.

Simplification method for nonlinear equations of monotone type in Banach space

In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

Asymptotic Properties of Solutions to Discrete Volterra Monotone Type Equations

We investigate the higher order nonlinear discrete Volterra equations. We study solutions with prescribed asymptotic behavior. For example, we establish sufficient conditions for the existence of asymptotically polynomial, asymptotically periodic or asymptotically symmetric solutions. On the other hand, we are dealing with the problem of approximation of solutions. Among others, we present conditions under which any bounded solution is asymptotically periodic. Using our techniques, based on the iterated remainder operator, we can control the degree of approximation. In this paper we choose a positive non-increasing sequence u and use o(un) as a measure of approximation.

A double phase equation with convection

We consider a double phase problem with a gradient dependent reaction (convection). Using the theory of nonlinear operators of monotone type, we show the existence of a nontrivial, positive, bounded solution.

Algorithm for Solutions of Nonlinear Equations of Strongly Monotone Type and Applications to Convex Minimization and Variational Inequality Problems

Real-life problems are governed by equations which are nonlinear in nature. Nonlinear equations occur in modeling problems, such as minimizing costs in industries and minimizing risks in businesses. A technique which does not involve the assumption of existence of a real constant whose calculation is unclear is used to obtain a strong convergence result for nonlinear equations of p,η-strongly monotone type, where η>0,p>1. An example is presented for the nonlinear equations of p,η-strongly monotone type. As a consequence of the main result, the solutions of convex minimization and variational inequality problems are obtained. This solution has applications in other fields such as engineering, physics, biology, chemistry, economics, and game theory.

COMPUTATION OF ZEROS OF MONOTONE TYPE MAPPINGS: ON CHIDUME’S OPEN PROBLEM

For $p\geq 2$, let $E$ be a 2-uniformly smooth and $p$-uniformly convex real Banach space and let $A:E\rightarrow E^{\ast }$ be a Lipschitz and strongly monotone mapping such that $A^{-1}(0)\neq \emptyset$. For given $x_{1}\in E$, let $\{x_{n}\}$ be generated by the algorithm $x_{n+1}=J^{-1}(Jx_{n}-\unicode[STIX]{x1D706}Ax_{n})$, $n\geq 1$, where $J$ is the normalized duality mapping from $E$ into $E^{\ast }$ and $\unicode[STIX]{x1D706}$ is a positive real number in $(0,1)$ satisfying suitable conditions. Then it is proved that $\{x_{n}\}$ converges strongly to the unique point $x^{\ast }\in A^{-1}(0)$. Furthermore, our theorems provide an affirmative answer to the Chidume et al. open problem [‘Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces’, SpringerPlus4 (2015), 297]. Finally, applications to convex minimization problems are given.