Strict Contraction Research Articles

Locally smooth unitary groups and applications to boundary control of PDEs

Let $\mathcal{P}$ be the projection operator for a closed subspace $\mathcal{S}$ of a Hilbert space $\mathcal{H}$ and let $U$ be a unitary operator on $\mathcal{H}$. We consider the questions  &nbsp 1. Under what conditions is $\mathcal{P}U\mathcal{P}$ a strict contraction?  &nbsp 2. If $g$, $h\in \mathcal{S}$, can we find $f\in \mathcal{H}$ such that $\mathcal{P}f=g$ and $\mathcal{P}Uf=h$?The results are abstract versions and generalisations of results developed for boundary control of partial differential equations. We discuss how these results can be used as tools in the direct construction of boundary controls.

Nonlinear Functional Analysis of Boundary Value Problems: Novel Theory, Methods, and Applications

and Applied Analysis 3 1-homogeneous operator in a Banach space and then demonstrate its application in establishing the existence of positive solutions for p-Laplacian boundary value problems under certain conditions. (xi) In the paper titled “Existence of solutions for nonhomogeneous A-harmonic equations with variable growth,” the authors establish a theorem for the existence of weak solutions for nonhomogeneous A-harmonic equations in subspace and then give three examples to demonstrate its application. (xii) In the paper titled “Multiple solutions for degenerate elliptic systems near resonance at higher eigenvalues,” the authors study the degenerate semilinear elliptic system in an open bounded domain with smooth boundary, and some multiplicity results of solutions are obtained for the system near resonance at certain eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory. (xiii) In the paper titled “A regularity criterion for the Navier-Stokes equations in the multiplier spaces,” the authors establish a regularity criterion in terms of the pressure gradient for weak solutions to the NavierStokes equations in a special class. The third set of papers, including four papers, deal with several boundary value problems for highly nonlinear ordinary differential equations. (i) In the paper titled “Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions,” the authors investigate the existence of positive solutions for second-order singular differential equations with a negatively perturbed term, by means of the fixed-point theory in cones. (ii) In the paper titled “Positive solutions for Sturm-Liouville boundary value problems in a Banach Space,” the sufficient conditions for the existence of single and multiple positive solutions for a second-order SturmLiouville boundary value problem are established in a Banach space, by using the fixed-point theorem of strict set contraction operators in the frame of the ODE technique. (iii) In the paper titled “Positive solutions of a nonlinear fourth-order dynamic eigenvalue problem on time scales,” the authors study a nonlinear fourth-order dynamic eigenvalue problem on time scales and obtain the existence and nonexistence of positive solutions when 0 λ, respectively, for some λ, by using the Schauder fixed-point theorem and the upper and lower solution method. (iv) In the paper titled “Bifurcation analysis for a predatorprey model with time delay and delay-dependent parameters,” a class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. By using the normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifur-cating from Hopf bifurcations are obtained. The fourth set of papers focus on finding the approximate and numerical solutions of various complex nonlinear boundary value problems. (i) In the paper titled “On spectral homotopy analysis method for solving linear Volterra and Fredholm integrodifferential equations,” a spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations, and some examples are given to test the efficiency and the accuracy of the proposed method. (ii) In the paper titled “The solution of a class of singularly perturbed two-point boundary value problems by the iterative reproducing kernel method,” the authors establish an iterative reproducing kernel method (IRKM) for solving singular perturbation problems with boundary layers and give two numerical examples to demonstrate the effectiveness of the method. (iii) In the paper titled “A Galerkin solution for Burgers’ equation using cubic B-spline finite elements,” a Galerkin method using cubic B-splines is set up to find the numerical solutions of Burgers’ equation, and the method is shown to be capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to very large. (iv) In the paper titled “Forward-backward splitting methods for accretive operators in Banach spaces,” the authors introduce two iterative forward-backward splitting methods with relaxations to find zeros of the sum of two accretive operators in Banach spaces and prove the weak and strong convergence of these methods under mild conditions, and also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

Positive Solutions for Sturm‐Liouville Boundary Value Problems in a Banach Space

We consider the existence of single and multiple positive solutions for a second‐order Sturm‐Liouville boundary value problem in a Banach space. The sufficient condition for the existence of positive solution is obtained by the fixed point theorem of strict set contraction operators in the frame of the ODE technique. Our results significantly extend and improve many known results including singular and nonsingular cases.

Schaefer–Krasnoselskii fixed point theorems using a usual measure of weak noncompactness

We present some extension of a well-known fixed point theorem due to Burton and Kirk [T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii–Schaefer type, Math. Nachr. 189 (1998) 423–431] for the sum of two nonlinear operators one of them compact and the other one a strict contraction. The novelty of our results is that the involved operators need not to be weakly continuous. Finally, an example is given to illustrate our results.

Sturm-Liouville BVP in Banach space

We consider the existence of single and multiple positive solutions for fourth-order Sturm-Liouville boundary value problem in Banach space. The sufficient condition for the existence of single and multiple positive solutions is obtained by fixed theorem of strict set contraction operator in the frame of the ODE technique. Our results significantly extend and improve many known results including singular and nonsingular cases.

Existence of positive solutions for nonlocal second-order boundary value problem with variable parameter in Banach spaces

By obtaining intervals of the parameter λ, this article investigates the existence of a positive solution for a class of nonlinear boundary value problems of second-order differential equations with integral boundary conditions in abstract spaces. The arguments are based upon a specially constructed cone and the fixed point theory in cone for a strict set contraction operator.

Common fixed points of strict contractions in Menger spaces

The aim of this paper is to prove some common fixed point theorems under certain strict contractive conditions for mappings sharing the common property (E.A) in Menger spaces. As applications to our results, we obtain the corresponding common fixed point theorems under strict contraction in metric spaces. Thus, our results generalize many known results in Menger as well as metric spaces. Some related results are also derived besides presenting several illustrative examples.

Some Fixed Point Theorems on Meir Keeler Type under Strict Contractions

This paper explores the common fixed point theorems involving two pairs of weakly compatible mappings. Also the property (E.A) is proved under a new contractive condition which is independent of the previous known contractive definitions.

Multiple positive solutions for singular [formula omitted]th-order nonlocal boundary value problems in Banach spaces

In this paper, we consider a class of singular n th-order nonlocal boundary value problems in Banach spaces. The existence of multiple positive solutions for the problem is obtained by using the fixed point index theory of strict set contraction operators. To demonstrate the applications of our results, two examples are also given in the paper.

A Lipschitz constant formula for vector addition in cones with applications to Stein-like equations

For a cone C equipped with the Thompson metric and \({a\in C}\), we show that the translation map \({x\mapsto x+a}\) is a strict contraction on any lower (or initial) set \({C\cap x-C}\) of the cone and derive an explicit formula for the Lipschitz constant. We apply our results to Stein equations, Riccati equations, and Ferrante–Levy equations on normal cones of Banach spaces to establish the existence, uniqueness and continuous dependence on parameters of positive solutions.

Multiple Positive Solutions for Singular Periodic Boundary Value Problems of Impulsive Differential Equations in Banach Spaces

By means of the fixed point theory of strict set contraction operators, we establish a new existence theorem on multiple positive solutions to a singular boundary value problem for second‐order impulsive differential equations with periodic boundary conditions in a Banach space. Moreover, an application is given to illustrate the main result.

Fixed Point Theorems in Symmetric Spaces and Invariant Approximations

We present common fixed point theory for generalized weak contractive condition in symmetric spaces under strict contractions and obtain some results on Invariant Approximations.

Common fixed points of nonlinear hybrid mappings under strict contractions in semi-metric spaces

In this paper, we extend the notion of common property (E.A) to pairs of hybrid mappings defined on an arbitrary set with values in a semi-metric space and use it to prove some coincidence and common fixed point theorems in semi-metric (potent semi-metric) spaces under strict contractions. Our results generalize several previously known fixed point theorems of the existing literature.

Positive Fixed Point of Strict Set Contraction Operators on Ordered Banach Spaces and Applications

The fixed point theorem of cone expansion and compression of norm type for a strict set contraction operator is generalized by replacing the norms with a convex functional satisfying certain conditions. We then show how to apply our theorem to prove the existence of a positive solution to a second‐order differential equation with integral boundary conditions in an ordered Banach space. An example is worked out to demonstrate the main results.

Multiple positive solutions for second order impulsive boundary value problems in Banach spaces

By means of the fixed point index theory of strict set contraction operators, we establish new existence theorems on multiple positive solutions to a boundary value problem for second-order impulsive integro-differential equations with integral bound- ary conditions in a Banach space. Moreover, an application is given to illustrate the

Relaxed commutant lifting and a relaxed Nehari problem: Redheffer state space formulas

AbstractThe description of all solutions to the relaxed commutant lifting problem in terms of an underlying contraction, obtained earlier in joint work of the author with A. E. Frazho and M. A. Kaashoek, is transformed into a linear fractional Redheffer state space form. Under certain additional conditions the coefficient functions in this representation are described explicitly in terms of the original data. The main theorem is a generalization of the Redheffer description of all solutions to the classical commutant lifting problem. To illustrate the result a relaxed version of the Nehari extension problem is considered, and an explicit Redheffer description of all its solutions is given, assuming that a certain truncated Hankel operator is a strict contraction (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Existence and nonexistence of positive solutions for a class of third order BVP with integral boundary conditions in Banach spaces

This paper investigates the existence, nonexistence, and multiplicity of positive solutions for a class of nonlinear boundary-value problems of third order differential equations with integral boundary conditions in ordered Banach spaces by means of fixed-point principle in cone and the fixed-point index theory for strict set contraction operator. In addition, an example is worked out to demonstrate the main results.

Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces

This paper investigates the existence of solutions for a class of second-order boundary-value problems with integral boundary conditions of nonlinear impulsive integro-differential equations in Banach spaces. The arguments are based upon the fixed point theorem of strict set contraction operators. Meanwhile, an example is worked out to demonstrate the main results.

Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition

In this work, we establish new coincidence and common fixed point theorems for hybrid strict contraction maps by dropping the assumption “ f is T -weakly commuting” for a hybrid pair ( f , T ) of multivalued maps in Theorem 3.10 of T. Kamran (2004) [8]. As an application, an invariant approximation result is derived.

Krasnosel’skii type fixed point theorems under weak topology features

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let M be a nonempty bounded closed convex subset of a Banach space X . Suppose that A : M → X and B : X → X are two weakly sequentially continuous mappings satisfying: (i) A M is relatively weakly compact; (ii) B is a strict contraction; (iii) ( x = B x + A y , y ∈ M ) ⇒ x ∈ M . Then A + B has at least one fixed point in M . This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.