Cone Compression Research Articles

Multiple Solutions of Boundary Value Problems fornth-Order Singular Nonlinear Integrodifferential Equations in Abstract Spaces

The authors discuss multiple solutions for thenth-order singular boundary value problems of nonlinear integrodifferential equations in Banach spaces by means of the fixed point theorem of cone expansion and compression. An example for infinite system of scalar third-order singular nonlinear integrodifferential equations is offered.

Positive solutions of second-order non-local boundary value problem with singularities in space variables

We discuss a non-local boundary value problem of second-order, where the involved nonlinearity depends on the derivative and may be singular. The boundary conditions are given by Riemann-Stieltjes integrals. We establish sufficient conditions for the existence of positive solutions of the considered problem. Our approach is based on the Krasnoselskii-Guo fixed point theorem on cone expansion and compression.

Existence and nonexistence of positive solutions for fractional-order two-point boundary value problems

The purpose of this paper is to establish some results on the existence and nonexistence of positive solutions for a type of nonlinear fractional-order two-point boundary value problems. The main tool is a fixed point theorem of the cone expansion and compression of functional type due to Avery et al. Some examples are presented to illustrate the availability of the main results.

Positive Solutions for a Nonlinear Higher Order Differential System with Coupled Integral Boundary Conditions

We investigate the existence of positive solutions for a nonlinear higher order differential system, where the differential system is coupled not only in the differential system but also through the boundary conditions. By constructing a special cone and using the fixed point theorem of cone expansion and compression of norm type, the existence of single and multiple positive solutions is established. As an application, we give some examples to demonstrate our results.

The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions

We consider the fourth-order difference equation:Δ(z(k+1)Δ3u(k-1))=w(k)f(k,u(k)), k∈{1,2,…,n-1}subject to the boundary conditions:u(0)=u(n+2)=∑i=1n+1g(i)u(i),aΔ2u(0)-bz(2)Δ3u(0)=∑i=3n+1h(i)Δ2u(i-2),aΔ2u(n)-bz(n+1)Δ3u(n-1)=∑i=3n+1h(i)Δ2u(i-2), wherea,b>0andΔu(k)=u(k+1)-u(k)fork∈{0,1,…,n-1}, f:{0,1,…,n}×[0,+∞)→[0,+∞)is continuous.h(i)is nonnegativei∈{2,3,…,n+2};g(i)is nonnegative fori∈{0,1,…,n}. Using fixed point theorem of cone expansion and compression of norm type and Hölder’s inequality, various existence, multiplicity, and nonexistence results of positive solutions for above problem are derived, which extends and improves some known recent results.

Some Existence Results of Positive Solution to Second-Order Boundary Value Problems

We study the existence of positive and monotone solution to the boundary value problemu′′(t)+f(t,u(t))=0,0⩽t⩽1,u(0)=ξu(1),u'(1)=ηu'(0), whereξ,η∈(0,1)∪(1,∞). The main tool is the fixed point theorem of cone expansion and compression of functional type by Avery, Henderson, and O’Regan. Finally, four examples are provided to demonstrate the availability of our main results.

Multiple positive solutions for first-order impulsive singular integro-differential equations on the half line in a Banach space

In this paper, the author discusses the multiple positive solutions for an infinite three-point boundary value problem of first-order impulsive superlinear singular integro-differential equations on the half line in a Banach space by means of the fixed-point theorem of cone expansion and compression with norm type. MSC: 45J05; 34G20; 47H10

Existence of Positive Solution for Fourth Order Superlinear Singular Semipositone Differential System

By transforming the boundary value problem into the corresponding fixed-point problem of a completely continuous operator, the existence is obtained in the paper for two-point boundary value problem of fourth order superlinear singular semipositone differential system via the fixed point theorem concerning cone compression and expansion in norm type.

Solvability of a Class of Integral Inclusions

This paper presents sufficient conditions for the existence of positive solutions for a class of integral inclusions. Our results are obtained via a new fixed point theorem for multivalued operators developed in the paper, in which some nonnegative function is used to describe the cone expansion and compression instead of the classical norm‐type, and lead to new existence principles.

Positive Solutions of Singular Fourth Order Four Point Boundary Value Problems withp-Laplacian Operator

In this paper, we consider the existence of positive solutions for the singular fourth- order four point boundary value problem with p-Laplacian operator. By using the fixed point theorem of cone expansion and compression, the existence of multiple positive solutions is ob- tained.

Multiple positive solutions for a fourth-order integral boundary value problem on time scales

In this article, we investigate the multiplicity of positive solutions for a fourth-order system of integral boundary value problem on time scales. The existence of multiple positive solutions for the system is obtained by using the fixed point theorem of cone expansion and compression type due to Krasnosel'skill. To demonstrate the applications of our results, an example is also given in the article.

Study on the Existence of Positive Solution of Sub-Linear Fractional Differential Equation

In recent years, the study on nonlinear fractional differential equation has been more concerned as it is widely used in physics, mechanics, geology, automation and many other disciplines and fields. This paper focuses on the sub-linear fractional differential equations, whose nonlinear is constrained by the power function. While in this case, it is possible to have positive solution by using the cone compression fixed point theorem. This study represents analysis on problems related to the fractional differential equations from the above aspects. With further development of this field in theoretical research and application, more explorations are waiting for us to do to lay a good theoretical foundation for its future development, and build up a broader prospect.

Existence of Positive Solution for Multi-Point Sturm-Liouville Boundary Value Problem

Multi-point boundary value problem is studied in this paper. With the condition that nonlinear term is superlinear or sublinear, it is proved that there exists at least one positive solution to multi-point Sturm-Liouville boundary value problem by using the fixed-point theorem concerning cone expansion and compression of norm type.

Results for Twin Singular Nonlinear Problems with Damping Term

By using the functional type cone expansion and compression fixed‐point theorem in cones, some new and general results on the existence of positive solution for twin singular boundary value problems with damping term are obtained. An example is given to illustrate our results.

Fixed points for weakly inward mappings in Banach spaces (II)

S. Hu and Y. Sun[1] defined the fixed point index for weakly inward mappings, investigated its properties and studied fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we further investigate boundary conditions, under which the fixed point index for i(A, Ω, p) is equal to nonzero, where i(A, Ω, p) is the completely continuous and weakly inward mapping. Correspondingly, we can obtain many new fixed point theorems of the completely continuous and weakly inward mapping, which generalize some famous theorems such as Rothe’s theorem, Altman’s theorem, Petryshyn’s theorem etc. in the case of weakly inward mappings. In addition, our conclusions extend the famous fixed point theorem of cone expansion and compression to the case of weakly inward mappings. Moreover, the main results contain and generalize the corresponding results in the recent work[2].

Multiple Positive Solutions for th Order Multipoint Boundary Value Problem

We study the existence of multiple positive solutions for th-order multipoint boundary value problem. , , , , where , , . We obtained the existence of multiple positive solutions by applying the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed-point theorem. The results obtained in this paper are different from those in the literature.

Existence and uniqueness of positive solutions for a third-order three-point problem on time scales

In this paper, a class of third-order three-point boundary value problem on time scales is considered. Using monotone iterative technique and cone expansion and compression fixed point theorem of norm type, we do not only obtain the existence and uniqueness of positive solutions of the problem, but also establish the iterative schemes for approximating the solutions.

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.

Some applications of dominated convergence theorems to a higher-order singular boundary value problem

The existence of positive solution is considered for a singular higher-order boundary value problem, where the nonlinear term is a strong Carathéodory function. Two new existence theorems are proved by applying the Lebesgue dominated convergence theorem, the Fatou lemma and the Krasnosel'skii fixed point theorem of cone expansion or cone compression type.

Positive solutions to singular third-order three-point boundary value problems on time scales

In this paper, we are concerned with singular third-order three-point boundary value problems on time scales. Some theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is also given to illustrate our results.