Concave Operators Research Articles

A new method for a semi-positone Hadamard fractional boundary value problem

A Hadamard fractional boundary value problem with semi-positone nonlinearity is studied in this paper, and the local existence and uniqueness of solutions are derived by a recent fixed point theorem involving with increasing φ−(h,r)-concave operators defined on ordered spaces. This is an unprecedented approach to solve the semi-positone problems. In practice, this method has a wider range of applicability since it can obtain the local uniqueness of the solutions. Furthermore, we can approximate the unique solution by constructing convergent iterative sequences. In the end, a persuasive example is provided to illustrate that the theoretical results we obtained are applicable and valid.

Unique iterative solution for high-order nonlinear fractional q-difference equation based on psi -(h,r)-concave operators

An objective of this paper is to investigate the boundary value problem of a high-order nonlinear fractional q-difference equation. It was to obtain a unique iterative solution for this problem by means of applying a novel fixed-point theorem of psi -(h,r)-concave operator, in which the operator is increasing and defined in ordered sets. Moreover, we construct a monotone explicit iterative scheme to approximate the unique solution. Finally, we give an example to illustrate the use of the main result.

Weakly concave operators

We study a class of left-invertible operators which we call weakly concave operators. It includes the class of concave operators and some subclasses of expansive strict $m$-isometries with $m > 2$. We prove a Wold-type decomposition for weakly concave operators. We also obtain a Berger–Shaw-type theorem for analytic finitely cyclic weakly concave operators. The proofs of these results rely heavily on a spectral dichotomy for left-invertible operators. It provides a fairly close relationship, written in terms of the reciprocal automorphism of the Riemann sphere, between the spectra of a left-invertible operator and any of its left inverses. We further place the class of weakly concave operators, as the term $\mathcal {A}_1$, in the chain $\mathcal {A}_0 \subseteq \mathcal {A}_1 \subseteq \ldots \subseteq \mathcal {A}_{\infty }$ of collections of left-invertible operators. We show that most of the aforementioned results can be proved for members of these classes. Subtleties arise depending on whether the index $k$ of the class $\mathcal {A}_k$ is finite or not. In particular, a Berger–Shaw-type theorem fails to be true for members of $\mathcal {A}_{\infty }$. This discrepancy is better revealed in the context of $C^*$- and $W^*$-algebras.

Asymptotic properties for compressions of two-isometries

The bounded linear operators T on a Hilbert space which have 2-isometric liftings S on are investigated in this paper. We refer to the structure of those operators in the case of general liftings, as well as for a more special type of such liftings S, namely for which is invariant to the orthogonal projection onto the kernel of S∗S − I. In this special case we describe the canonical triangulation of T induced by S, with entries expressed by operators close to contractions or isometries. Also, in this case we refer to some asymptotic properties of T and of its adjoint T∗ obtained by means of asymptotic limits associated to T∗ and T, relative to a lifting S for T. The canonical triangulations of T in this setting are described in detail, and as an application, the Wold type decomposition of a concave operator is obtained.

О существовании и единственности положительного решения краевой задачи для одного нелинейного функционально-дифференциального уравнения дробного порядка

In this article, we consider a two-point boundary value problem for a nonlinear functional differential equation of fractional order with weak nonlinearity on the interval [0,1] with zero Dirichlet conditions on the boundary. The boundary value problem is reduced to an equivalent integral equation in the space of continuous functions. Using special topological tools (using the geometric properties of cones in the space of continuous functions, statements about fixed points of monotone and concave operators), the existence of a unique positive solution to the problem under consideration is proved. An example is given that illustrates the fulfillment of sufficient conditions that ensure the unique solvability of the problem. The results obtained are a continuation of the author’s research (see [Results of science and technology. Ser. Modern mat. and her appl. Subject. review, 2021, vol. 194, pp. 3–7]) devoted to the existence and uniqueness of positive solutions of boundary value problems for non-linear functional differential equations.

Existence and uniqueness of a positive solutions for the product of operators

<abstract><p>In this paper, we prove the existence of a positive solution for some equations involving multiplication of concave (possibly nonlinear) operators. Also, we provide a successively sequence to approximate the solution for such equations. This kind of the solution is necessary for quadratic differential and integral equations.</p></abstract>

λ-Interval of Triple Positive Solutions for the Perturbed Gelfand Problem

We study a two-point Boundary Value Problem depending on two parameters that represents a mathematical model arising from the combustion theory. Applying fixed point theorems for concave operators, we prove uniqueness, existence, upper, and lower bounds of positive solutions. In addition, we give an estimation for the value of λ* such that, for the parameter λ∈[λ*,λ*], there exist exactly three positive solutions. Numerical examples are presented to illustrate various cases. The results complement previous work on this problem.

On the Equivalence between Two Fixed Point Theorems for Concave-Type Operators

In this note, we prove the equivalence between a fixed point theorem for φ – h , e –concave operators and a previous one for generalized concave operators.

Concave Multifunctions and the Hammerstein Integral inclusion Problem

In this paper, we shell generalize concave operators to multifunction versions. Then we obtain some fixed point results for such multifunctions in partially ordered spaces.

Hilbert space operators with two-isometric dilations

A continuous linear Hilbert space operator S is said to be a 2-isometry if the operator S and its adjoint S∗ satisfy the relation S∗2S2−2S∗S+I=0. We study operators having liftings or dilations to 2-isometries. The adjoint of an operator which admits such liftings is the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators and to operators similar to contractions. Two types of liftings to 2-isometries, as well as the extensions induced by them, are constructed and isomorphic minimal liftings are discussed.

A fractional q-difference equation eigenvalue problem with p(t)-Laplacian operator

This article is devoted to studying a nonhomogeneous boundary value problem involving Stieltjes integral for a more general form of the fractional q-difference equation with p(t)-Laplacian operator. Here p(t)-Laplacian operator is nonstandard growth, which has been used more widely than the constant growth operator. By using fixed point theorems of φ – (h, e)-concave operators some conditions, which guarantee the existence of a unique positive solution, are derived. Moreover, we can construct an iterative scheme to approximate the unique solution. At last, two examples are given to illustrate the validity of our theoretical results.

The obstacle problem for the Monge–Ampère equation with the lower obstacle

In this paper, we study the existence and optimal regularity of the solution and the regularity of the free boundary of the obstacle problem for the Monge–Ampère equation with the lower obstacle function, which arises in the prescribed Gauss curvature with an obstacle.The main feature of this paper is that we consider the obstacle problem for Monge–Ampère operator, which is a log-concave operator, with the lower obstacle function. Generally, the problem for a convex operator with a lower obstacle or a concave operator with an upper obstacle is considered due to the definition of the solutions and the classification of the global solutions. In this paper, difficulties caused by the incongruousness of the operator and the location of the obstacle are considered in many parts such as the penalization problem, classification of the global solutions, and the directional monotonicity.

Krasnoselskii-type fixed point theorems using $$\alpha $$-concave operators

In this paper, we discuss the fixed point theorems for a combination of $$\alpha $$-concave and compact operators. The results can be seen as variants of Krasnoselskii’s theorem using $$\alpha $$-concave operators.

Existence and uniqueness of periodic solutions for a system of differential equations via operator methods

This article investigates the existence and uniqueness of periodic solutions for a new system of differential equations. By employing fixed point theorems for increasing φ-(h,tau )-concave operators, we establish the existence of unique periodic solution for our differential system and then give a monotone iterative scheme to approximate the unique periodic solution. Some examples are presented in the end to illustrate the validity of our main results.

Existence and monotone iteration of unique solution for tempered fractional differential equations Riemann\u2013Stieltjes integral boundary value problems

The primary objective of this research paper is to investigate two kinds of high-order tempered fractional differential equations integral boundary value problems. By means of the mixed monotone operators fixed point theorems with perturbation and the increasing varphi -(h,sigma )-concave operators fixed point theorems, we can not only guarantee the existence-uniqueness of solution, but also construct successively sequences for approximating the unique solution. In addition, we demonstrate the effectiveness of the main result by using two examples.

UNIQUE MARKOV EQUILIBRIUM UNDER LIMITED COMMITMENT

AbstractI develop a recursive method for the characterization of competitive equilibrium under limited commitment with not‐too‐tight solvency constraints. The reputational mechanism is fragile, as it sustains constrained efficiency as well as a large set of constrained inefficient equilibria. However, I establish that the only strongly ergodic Markov equilibrium with trade is constrained efficient. The method relies on a planning program along with the theory of monotone concave operators. It is suitable for other applications to macroeconomics and dynamic contracts.

Existence of solutions for subquadratic convex or $B$-concave operator equations and applications to second order Hamiltonian systems

Existence of solutions for subquadratic convex or $B$-concave operator equations and applications to second order Hamiltonian systems

Properties of positive solutions for a fractional boundary value problem involving fractional derivative with respect to another function

In this article, we develop the existence and uniqueness of positive solutions to a class of fractional boundary value problems involving fractional order derivative with respect to another function for any given parameter. The analysis is based upon the fixed point theorems of concave operators in partial ordering Banach spaces. For the sake of discussing the existence of solutions for the problem, we first construct Green function and study its properties. Furthermore, some properties of positive solutions to the boundary value problem are proved under the different parameters. Examples illustrating the results are also presented.

Solutions to fractional differential equations involving integral boundary conditions

ABSTRACT In this article, we discuss the existence and uniqueness of solutions to a class of integral boundary value problem of fractional differential equations. Different fractional order derivations are included in the nonlinear term and boundary conditions. The unique solution is derived by a new fixed point theorem of increasing -concave operators defined on ordered sets. Further, we construct a monotone iterative scheme to approximate the unique solution. An example is given to illustrate the validity of our main results.

Unique solution for a new system of fractional differential equations

In this article, we discuss a new system of fractional differential equations \t\t\t{D0+s1u(t)+f(t,u(t),v(t))=z1(t),0<t<1,D0+s2v(t)+g(t,u(t),v(t))=z2(t),0<t<1,u(0)=u(1)=u′(0)=u′(1)=0,D0+β1u(0)=0,D0+β1u(1)=b1D0+β1u(η1),v(0)=v(1)=v′(0)=v′(1)=0,D0+β2v(0)=0,D0+β2v(1)=b2D0+β2v(η2),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} D_{0^{+}}^{s_{1}} u(t) + f(t,u(t),v(t))=z_{1}(t), \\quad 0< t< 1, \\\\ D_{0^{+}}^{s_{2}} v(t) + g(t,u(t),v(t))=z_{2}(t), \\quad 0< t< 1, \\\\ u(0)=u(1)=u^{\\prime }(0)=u^{\\prime }(1)=0, \\qquad D_{0^{+}}^{\\beta _{1}} u(0)=0, \\qquad D_{0^{+}}^{\\beta _{1}} u(1)= b_{1} D_{0^{+}}^{\\beta _{1}} u(\\eta _{1}), \\\\ v(0)=v(1)=v^{\\prime }(0)=v^{\\prime }(1)=0, \\qquad D_{0^{+}}^{\\beta _{2}} v(0)=0, \\qquad D_{0^{+}}^{\\beta _{2}} v(1)= b_{2} D_{0^{+}}^{\\beta _{2}} v(\\eta _{2}), \\end{cases} $$\\end{document} where s_{i}=alpha _{i}+beta _{i}, alpha _{i} in (1,2], beta _{i} in (3,4], z_{i} :[0,1]rightarrow [0,+infty ) is continuous, D_{0^{+}}^{alpha _{i}} and D_{0^{+}}^{beta _{i}} are the standard Riemann–Liouville derivatives, eta _{i} in (0,1), b_{i} in (0, {eta _{i}}^{1-alpha _{i}}), i=1,2, and f,gin C([0,1]times mathbf{R}^{2} , mathbf{R}). We establish the existence and uniqueness of solutions for the problem by a recent fixed point theorem of increasing Ψ-(h,e)-concave operators defined on ordered sets. Furthermore, the results obtained are well proven by means of a specific example.