Ordered Banach Spaces Research Articles

Existence of positive strongly decaying solutions of second-order nonlinear q-difference equations

This paper investigates the existence of positive strongly decaying solutions of second-order nonlinear q-difference equation with q-regularly varying coefficients a and b. By means of a fixed point theorem in partially ordered Banach space, necessary and sufficient conditions for the existence of solutions satisfying and as are established. Moreover, with the help of q-regular varying theory, precise asymptotic behaviour of such solutions is described.

Harnack Type Inequalities and Multiple Solutions in Cones of Nonlinear Problems

The paper presents an abstract theory regarding operator equations and systems in ordered Banach spaces. We obtain existence, localization and multiplicity results of positive solutions using Krasnosel'skii's fixed point theorem in cones, and a Harnack type inequality. Concerning systems, the localization is established by the vector version of Krasnosel'skii's theorem, where the compression-expansion conditions are expressed on components. The approach is sufficiently general to cover and unify a large number of results on particular classes of problems. It also can guide future research in this direction.

Existence and uniqueness of monotone positive solutions for fractional higher-order BVPs

We discuss the existence and uniqueness of monotone positive solutions for a class of higher-order nonlinear fractional differential equations infinite-point boundary value problems (for short, BVPs) on cones. The existence and uniqueness of solutions are obtained via applying the properties of the Green function and a fixed point theorem. Our analysis is based on the operator equation T ω + S ω = ω on an ordered Banach space. Finally, a example is given to illustrate our results.

Several fixed point theorems on partially ordered Banach spaces and their applications to integral equations

Several fixed point theorems on partially ordered Banach spaces and their applications to integral equations

Singular irreducible M-operators on ordered Banach spaces

This work continues our earlier work on singular M-operators for operators over ordered Banach spaces, where we show how some interesting results on singular irreducible M-matrices have analogues to operators over ordered Banach spaces.

Almost interior points in ordered Banach spaces and the long-term behaviour of strongly positive operator semigroups

The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalisation of “functions which are strictly positive almost everywhere” on $L^p$-spaces

NONLINEAR ABSTRACT MEASURE HYBRID DIFFERENTIAL EQUATIONS WITH A LINEAR PERTURBATION OF SECOND TYPE

In this paper, an existence result for perturbed abstract measure differential equations is proved via hybrid fixed point theorems of Dhage [4] under the mixed generalized Lipschitz and Carathéodory conditions. The existence of the extremal solutions is also proved under certain monotonicity conditions and using a hybrid fixed point theorem of Dhage [4] on ordered Banach spaces. Our existence results include the existence results of Sharma [23], Joshi [19] and Shendge and Joshi [25] as special cases under weaker continuity condition.

Coincidence point theorems in ordered Banach spaces and applications

In this paper, we study the existence of positive solutions for classes of nonlinear operator equations in Banach spaces ordered by cones. We establish new coincidence point theorems via a generalized monotone iterative method. As applications, we discuss the existence of positive solutions for systems of nonlinear matrix equations and a system of integral equations of Volterra type.

MIXED MONOTONE ITERATIVE TECHNIQUE FOR HILFER FRACTIONAL EVOLUTION EQUATIONS WITH NONLOCAL CONDITIONS

The purpose of this paper is concerned with the existence of mild $ L $-quasi-solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach spaces $ E $. By employing mixed monotone iterative technique, measure of noncompactness and Sadovskii's fixed point theorem, we obtain the existence of mild $ L $-quasi-solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provide to illustrate the feasibility of our main results.

ITERATION PRINCIPLE FOR IVPS OF NONLINEAR FIRST ORDER IMPULSIVE DIFFERENTIAL EQUATIONS

In this paper we prove the existence and approximation theorems for the initial value problems of first order nonlinear impulsive differential equations under certain mixed partial Lipschitz and partial compactness type conditions. Our results are based on the Dhage monotone iteration principle embodied in a hybrid fixed point theorem of Dhage involving the sum of two monotone order preserving operators in a partially ordered Banach space. The novelty of the present approach lies the fact that we obtain an algorithm for the solution. Our abstract main result is also illustrated by indicating a numerical example.

The Existence and Approximation Fixed Point Theorems for Monotone Nonspreading Mappings in Ordered Banach Spaces

In this paper, we proved some existence theorems of fixed points for monotone nonspreading mappings \(T\) in a Banach space \(E\) with the partial order \(\leq\). In order to finding a fixed point of such a mapping \(T\), moreover we proved the convergence theorem of Mann iterative schemes under the condition \(\sum\limits_{n=1}^\infty\beta_n(1-\beta_n)=\infty\), which contain \(\beta_n=\frac1{n+1}\) as a special case.

Semantics of higher-order probabilistic programs with conditioning

We present a denotational semantics for higher-order probabilistic programs in terms of linear operators between Banach spaces. Our semantics is rooted in the classical theory of Banach spaces and their tensor products, but bears similarities with the well-known semantics of higher-order programs a la Scott through the use of ordered Banach spaces which allow definitions in terms of fixed points. Our semantics is a model of intuitionistic linear logic: it is based on a symmetric monoidal closed category of ordered Banach spaces which treats randomness as a linear resource, but by constructing an exponential comonad we can also accommodate non-linear reasoning. We apply our semantics to the verification of the classical Gibbs sampling algorithm.

Upper and lower solution method for Hilfer fractional evolution equations with nonlocal conditions

This paper is concerned with the existence of extremal mild solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach space E. By employing the method of lower and upper solutions, the measure of noncompactness, and Sadovskii’s fixed point theorem, we obtain the existence of extremal mild solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provided to illustrate the feasibility of our main results.

Best Proximity Points for Monotone Relatively Nonexpansive Mappings in Ordered Banach Spaces

In this paper, we give sufficient conditions to ensure the existence of the best proximity point of monotone relatively nonexpansive mappings defined on partially ordered Banach spaces. An example is given to illustrate our results.

Monotone Iterative Technique for Nonlocal Impulsive Finite Delay Differential Equations of Fractional Order

The paper is concerned with the extension of a monotone iterative technique to impulsive finite delay differential equations of fractional order with a nonlocal initial condition in an ordered Banach space. We study the existence of extremal mild solutions with or without assuming the compactness of a semigroup and also prove the uniqueness of the mild solution of the system. The results are obtained with the help of fractional calculus, a measure of non-compactness, the semigroup theory and monotone iterative technique. Finally, an example is provided to show the application of our main.

Existence, Uniqueness and UHR Stability of Solutions to Nonlinear Ordinary Differential Equations with Noninstantaneous Impulses

AbstractWe consider the existence, uniqueness and Ulam–Hyers–Rassias stability of solutions to the initial value problem with noninstantaneous impulses on ordered Banach spaces. The existence and uniqueness of solutions for nonlinear ordinary differential equation with noninstantaneous impulses is obtained by using perturbation technique, monotone iterative method and a new estimation technique of the measure of noncompactness under the situation that the corresponding noninstantaneous impulsive functionsgiare compact and not compact, respectively. Furthermore, the UHR stability of solutions is also obtained, which provides an approach to find approximate solution to noninstantaneous impulsive equations in the sense of UHR stability.

Generalized Dobrushin ergodicity coefficient and uniform ergodicities of Markov operators

In this paper the stability and the perturbation bounds of Markov operators acting on abstract state spaces are investigated. Here, an abstract state space is an ordered Banach space where the norm has an additivity property on the cone of positive elements. We basically study uniform ergodic properties of Markov operators by means of so-called a generalized Dobrushin's ergodicity coefficient. This allows us to get several convergence results with rates. Some results on quasi-compactness of Markov operators are proved in terms of the ergodicity coefficient. Furthermore, a characterization of uniformly $P$-ergodic Markov operators is given which enable us to construct plenty examples of such types of operators. The uniform mean ergodicity of Markov operators is established in terms of the Dobrushin ergodicity coefficient. The obtained results are even new in the classical and quantum settings

Fixed point results in set $P_{h,e}$ with applications to fractional differential equations

In this paper, without assuming operators to be continuous or compact, by employing monotone iterative technique on ordered Banach space, we at first establish new fixed point theorems for some kinds of nonlinear mixed monotone operators in set $P_{h,e}$. Then, we study a new form of fractional two point boundary value problem depending on a certain constant and give the existence and uniqueness of solutions. We also show that the unique solution exists in set $P_{h,e}$ or $P_{h}$ and can be uniformly approximated by constructing two iterative sequences for any initial values. At the end, a concrete example is given to illustrate our abstract results. The conclusions of this article enrich the fixed point theorems and provide new methods to deal with nonlinear differential equations.

Sensitivity analysis for a new class of generalized parametric nonlinear ordered variational inequality problems in ordered Banach spaces

In this study, we introduce a class of new generalized parametric nonlinear ordered variational inequality problems and discuss its existence result. Also, we prove the sensitivity of the solution for the parametric inequality class with the help of B-restricted-accretive method in ordered Banach spaces. Some special cases of the main results are also discussed.

General Basset–Boussinesq–Oseen equation: existence, uniqueness, approximation and regularity of solutions

The Basset–Boussinesq–Oseen equation (BBO equation) is the mathematical formulation of the Lagrangian acceleration of a spherical particle, moving with Lagragian velocity, in an unsteady flow as the sum of the viscous, gravitational, buoyancy, virtual mass, and Basset forces acting respectively on the particle. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving global existence and uniqueness of solutions in an appropriate partially ordered Banach space. The method also gives an approximate solution to the problem with accurate result which is easy to implement. Meanwhile, regularity properties of the solutions are proved under some conditions which gives a representation of the solutions. Illustrative examples exhibit the efficiency of our method.