Ordered Banach Spaces Research Articles

Existence and uniqueness results for positive solutions of Hadamard type fractional BVP

In this paper, we investigate the existence and multiplicity results of positive solutions for a fractional boundary value problem associated with integral boundary condition via Hadamard Fractional operator. We first derive Green¢s function associated with the BVP and prove its properties. The results for existence and uniqueness of solutions are derived by applying two fixed point theorems of a sum operator in partial ordering Banach space. The main results obtained can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it.

Dhage iterative principle for quadratic perturbation of fractional boundary value problems with finite delay

In this article, by employing Dhage iterative method embodied in current hybrid fixed point theorem (HFPT) of Dhage, we derive an algorithm for the numerical solutions via construction of a sequence of successive approximations for a fractional order boundary value problem (FBVP) with finite delay. By using this technique, we obtain existence as well as approximation of solutions under weaker partial Lipschitz and partial compactness type conditions in a partially ordered Banach space. Additionally, we prove an existence and uniqueness theorem under a weaker partial nonlinear Lipschitz condition. The assumptions and main outcomes are also illustrated by two examples.

Monotone Iterative Technique for Periodic Boundary Value Problem of Fractional Differential Equation in Banach Spaces

Abstract In this paper, we are concerned with the periodic boundary value problem of fractional differential equations on ordered Banach spaces. By introducing a concept of upper and lower solutions, we construct a new monotone iterative technique for the periodic boundary value problems of fractional differential equation, and obtain the existence of solutions between lower and upper solutions.

Convergence Analysis of an Accelerated Iteration for Monotone Generalizedα-Nonexpansive Mappings with a Partial Order

In this paper, we introduce a new accelerated iteration for finding a fixed point of monotone generalizedα-nonexpansive mapping in an ordered Banach space. We establish some weak and strong convergence theorems of fixed point for monotone generalizedα-nonexpansive mapping in a uniformly convex Banach space with a partial order. Further, we provide a numerical example to illustrate the convergence behavior and effectiveness of the proposed iteration process.

Common Fixed Point in Cone Metric Space for $mathbf{s}-mathbf{varphi}$-contractive

Huang and Zhang cite{Huang} have introduced the concept of cone metric space where the set of real numbers is replaced by an ordered Banach space. Shojaei cite{shojaei} has obtained points of coincidence and common fixed points for s-Contraction mappings which satisfy generalized contractive type conditions in a complete cone metric space.In this paper, the notion of complete cone metric space has been introduced. We have defined $s-phi$-contractive and obtained common fixed point theorem for a mapping $f,s$ which satisfies $s-phi$-contractive.

$M$-operators on partially ordered Banach spaces

‎For a matrix $A \in \mathbb{R}^{n \times n}$ whose off-diagonal entries are nonpositive‎, ‎there are several well-known properties that are equivalent to $A$ being an invertible $M$-matrix‎. ‎One of them is the positive stability of $A$‎. ‎A generalization of this characterization to partially ordered Banach spaces is considered in this article‎. ‎Relationships with certain other equivalent conditions are derived‎. ‎An important result on singular irreducible $M$-matrices is generalized using the concept of $M$-operators and irreducibility‎. ‎Certain other invertibility conditions of $M$-operators are also investigated‎.

Nonlinear sum operator equations and applications to elastic beam equation and fractional differential equation

In this paper, by studying the solutions of the abstract operator equation A(x,x)+B(x,x)+e=x on ordered Banach spaces, where A, B are two mixed monotone operators and ein P with theta leq eleq h, we prove a class of boundary value problems on elastic beam equation to have a unique solution. Furthermore, we also apply our abstract result to establish the existence and uniqueness theorem of nontrivial solutions for nonlinear fractional boundary value problems. The iterative sequences to approximate unique solutions for the above two classes of problems are also obtained.

Long-term analysis of positive operator semigroups via asymptotic domination

We consider positive operator semigroups on ordered Banach spaces and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup $$\mathcal {T}$$ are inherited by other semigroups which are asymptotically dominated by $$\mathcal {T}$$ . Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity. Our theorems are applicable to time-discrete as well as time-continuous semigroups. They generalise several results from the literature to considerably larger classes of ordered Banach spaces.

Existence of mild solutions for impulsive fractional evolution equations with periodic boundary conditions

In this paper, we concern on the periodic boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space E. First, we establish the existence results of mild solutions for the associated linear periodic boundary value problem. Next, we obtain the existence results of mild solutions by using the monotone iterative technique with L-quasi-upper and lower solutions, the results are new and extend some previously known results. Finally, two examples are also given to illustrate the main results.

Multiplicity Results for Operator Systems via Fixed Point Index

We establish existence, localization and multiplicity results of positive solutions for general operator systems in ordered Banach spaces. Our main tool is the fixed point index in cones which we compute in suitable relatively open sets. In this context, each component of the fixed point operator can satisfy either the expansion condition or the compression condition. If some component of the operator is expansive, then we obtain multiplicity results. As an application, new results concerning systems of Hammerstein equations and systems of $$\phi $$ -Laplace equations are deduced.

Boundary value problems with solutions in convex sets

By means of the continuation method for contractions we prove the existence of solutions of Dirichlet boundary value problems in convex sets. As an application we prove the existence of concave solutions of certain boundary value problems in ordered Banach spaces.

Fixed Point Theorems for Monotone Mappings in Ordered Banach Spaces Under Weak Topology Features

Fixed Point Theorems for Monotone Mappings in Ordered Banach Spaces Under Weak Topology Features

MIXED MONOTONE ITERATIVE TECHNIQUE FOR SEMILINEAR IMPULSIVE FRACTIONAL EVOLUTION EQUATIONS

In this paper, we deals with the existence of mild $ L $-quasi-solutions to the boundary value problem for a class of semilinear impulsive fractional evolution equations in an ordered Banach space $ E $. Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive fractional evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. As some application that illustrate our results, An example is also given.

Fixed Points of Non-Smooth Functions on Finite Dimensional Ordered Banach Spaces via Clarke Generalized Jacobian

Considering Lipschitz functions which are not necessarily Fr´echet differentiable, we obtain a non-smooth version of Lakshmikantham’s theorem in finite dimensional ordered Banach spaces . We also present an application of the obtained result in dynamical Coulomb friction problem.

System of generalized mixed nonlinear ordered variational inclusions

In this paper, we consider a system of generalized mixed nonlinear ordered variational inclusions in partially ordered Banach spaces and suggest an algorithm for a solution of the considered system. We prove an existence and convergence result for the solution of the system of generalized mixed nonlinear ordered variational inclusions.

A new existence theory for periodic solutions to evolution equations

ABSTRACT This paper deals with a new existence theory for periodic solutions to a broad class of evolution equations. We first establish new fixed point theorems for affine maps in locally convex spaces and ordered Banach spaces. Our new fixed point results extend, encompass and complement a number of well-known theorems in the literature, including the famous Chow and Hale fixed point theorem. With these obtained fixed point results, we investigate the existence of periodic solutions for some class of nonhomogeneous linear systems in Banach spaces with lack of compactness. Some illustrative examples are also given.

The Solvability of Nonlinear Split Ordered Variational Inequality Problems in Partially Ordered Vector Spaces

The concept of nonlinear split ordered variational inequality problems on partially ordered Banach spaces extends the concept of the linear split vector variational inequality problems on Banach spaces, while the latter is a natural extension of vector variational inequality problems on Banach spaces. In this article, we prove the solvability of some nonlinear split vector variational inequality problems by using fixed-point theorems on partially ordered Banach spaces. It is important to notice that in the results obtained in this article, the considered mappings are not required to have any type of continuity and they just satisfy some order-monotonic conditions. Consequently, both the solvability of linear split vector variational inequality problems and vector variational inequality problems will be immediately obtained from the solvability of nonlinear split vector variational inequality problems. We will apply these results to solving nonlinear split vector optimization problems. The underlying spaces of the considered variational inequality problems may just be vector spaces which do not have topological structures, the considered mappings are not required to satisfy any continuity conditions, which just satisfy some order-increasing conditions.

M-ideals and split faces of the quasi state space of a non-unital ordered Banach space

We characterize M-ideals in order smooth $$\infty $$-normed spaces by extending the notion of split faces of the state space to those of the quasi-state space. We also characterize approximate order unit spaces as those order smooth $$\infty $$-normed spaces V that are M-ideals in $$\tilde{V}$$. Here $$\tilde{V}$$ is the order unit space obtained by adjoining an order unit to V. To prove these results, we develop an order theoretic version of the “Alfsen-Efffros’ cone decomposition theorem” for order smooth 1-normed spaces. (As a quick application of this result, we sharpen a result on the extension of bounded positive linear functionals on subspaces of order smooth $$\infty $$-normed spaces).

Fixed point theorems for a class of nonlinear sum-type operators and application in a fractional differential equation

In this paper, we consider the fixed point for a class of nonlinear sum-type operators ‘A+B+C’ on an ordered Banach space, where A, B are two mixed monotone operators, C is an increasing operator. Without assuming the existence of upper-lower solutions or compactness or continuity conditions, we prove the unique existence of a positive fixed point and also construct two iterative schemes to approximate it. As applications, we research a nonlinear fractional differential equation with multi-point fractional boundary conditions. By using the obtained fixed point theorems of sum-type operator, we get the sufficient conditions which guarantee the existence and uniqueness of positive solutions. At last, a specific example is provided to illustrate our result.

Positive solutions for periodic boundary value problem of fractional differential equation in Banach spaces

This paper discusses the existence and uniqueness of positive solutions for a periodic boundary value problem of a fractional differential equation in an ordered Banach space E. The existence and uniqueness results of solutions for the associated linear periodic boundary value problem of the fractional differential equation are established, and the norm estimation of resolvent operator is accurately obtained. With the aid of this estimation, the existence and uniqueness results of positive solutions are obtained by using a monotone iterative technique.