Abstract

One of the most interesting and important problems in the theory of variational inequalities is the study of efficient iterative schemes for finding approximate solutions and the convergence analysis of algorithms. In this article, we introduce a new inertial hybrid subgradient extragradient method for approximating a common solution of monotone variational inequalities and fixed point problems for an infinite family of relatively nonexpansive multivalued mappings in Banach spaces. In our proposed method, the projection onto the feasible set is replaced with a projection onto certain half spaces, which makes the algorithm easy to implement. We incorporate inertial term into the algorithm, which helps to improve the rate of convergence of the proposed method. Moreover, we prove a strong convergence theorem and we apply our results to approximate common solutions of variational inequalities and zero point problems, and to finding a common solution of constrained convex minimization and fixed point problems in Banach spaces. Finally, we present a numerical example to demonstrate the efficiency and the advantages of the proposed method, and we compare it with some related methods. Our results extend and improve some recent works both in Hilbert spaces and Banach spaces in this direction.

Keywords: Inertial algorithm; hybrid subragradient extragradient method; variational inequality; fixed point problem; multivalued relatively nonexpansive mappings; zero point problems; convex minimization problems.

MSC: 65K15, 47J25, 65J15, 90C33

Abstract

In this paper we find the conditions for the boundedness of Riesz potential $I^{\alpha}$ and its commutators from the generalized weighted Morrey spaces $\mathcal{M}^{p,\varphi_1}_{\omega_1}(\mathbb{R}^n)$ to the generalized weighted Morrey spaces $\mathcal{M}^{q,\varphi_2}_{\omega_2}(\mathbb{R}^n)$, where $0<\alpha

Keywords: Maximal operator; Riesz potential; commutator; weighted Lebegue space; generalized weighted Morrey space; BMO space.

MSC: 42B20, 42B25, 42B35

Abstract

Several upper estimates for the numerical radius of Hilbert space operators are given. Among many other inequalities, it is shown that \begin{align*}{{\omega }^{2}}\left( A \right)\le \frac{1}{4}\left\| {{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}} \right\| +\frac{1}{2}\omega \left( {{A}^{2}} \right)-\frac{1}{2}\underset{\left\| x \right\|=1} {\mathop{\underset{x\in \mathscr H}{\mathop{\inf }}\,}}\,{{\left( \sqrt{\left\langle {{\left| A \right|}^{2}}x,x \right\rangle } -\sqrt{\left\langle {{\left| {{A}^{*}} \right|}^{2}}x,x \right\rangle } \right)}^{2}}.\end{align*}

Keywords: Numerical radius; operator norm; inequality.

MSC: 47A12, 47A30

Abstract

In this work, we investigate a class of nonlinear combined Sturm-Liouville problems with zero Dirichlet boundary conditions. Using the Karamata regular variation theory and the Sch{a}uder fixed point theorem, we prove the existence of a unique positive solution satisfying a precise asymptotic behavior where a competition between singular and non singular terms in the nonlinearity appears.

Keywords: Asymptotic analysis; Sturm-Liouville equation; Dirichlet problem; Green function; Karamata class; Schäuder's fixed point theorem.

MSC: 26A12, 34B16, 34B18, 34B27