First-order Periodic Boundary Value Problems Research Articles

Geraghty Type Generalized F-Contractions and Related Applications in Partial b-Metric Spaces

The purpose of this paper is to introduce new concepts of (α,β)-admissible Geraghty type generalized F-contraction and to prove that some fixed point results for such mappings are in the perspective of partial b-metric space. As an application, we inaugurate new fixed point results for Geraghty type generalized graphic F-contraction defined on partial metric space endowed with a directed graph. On the other hand, one more application to the existence and uniqueness of a solution for the first-order periodic boundary value problem is also provided. Our findings encompass various generalizations of the Banach contraction principle on metric space, partial metric space, and partial b-metric space. Moreover, some examples are presented to illustrate the usability of the new theory.

Positive solutions to PBVPs for nonlinear first-order impulsive dynamic equations on time scales

By using the classical fixed point theorem for operators on a cone, in this paper, some results of one and two positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained.

Positive solutions to nonlinear first-order impulsive dynamic equations on time scales

By using the classical fixed point theorem for operators on a cone, in this paper, some results of single and multiple positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained. It is worth noticing that the nonlinearity f and the pulse function in this paper are not positive.

On first-order periodic boundary value problems and distributional Henstock-Kurzweil integrals

This paper is devoted to the study of existence and dependence of solutions of the first-order periodic boundary value problems involving the distributional Henstock-Kurzweil integral. The methods used are mainly the method of upper and lower solutions and a fixed point theorem.

A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems

The objective of this paper is to present a numerical iterative method for solving systems of first-order ordinary differential equations subject to periodic boundary conditions. This iterative technique is based on the use of the reproducing kernel Hilbert space method in which every function satisfies the periodic boundary conditions. The present method is accurate, needs less effort to achieve the results, and is especially developed for nonlinear case. Furthermore, the present method enables us to approximate the solutions and their derivatives at every point of the range of integration. Indeed, three numerical examples are provided to illustrate the effectiveness of the present method. Results obtained show that the numerical scheme is very effective and convenient for solving systems of first-order ordinary differential equations with periodic boundary conditions.

Fixed point results for G-α-contractive maps with application to boundary value problems.

We unify the concepts of G-metric, metric-like, and b-metric to define new notion of generalized b-metric-like space and discuss its topological and structural properties. In addition, certain fixed point theorems for two classes of G-α-admissible contractive mappings in such spaces are obtained and some new fixed point results are derived in corresponding partially ordered space. Moreover, some examples and an application to the existence of a solution for the first-order periodic boundary value problem are provided here to illustrate the usability of the obtained results.

Generalized nonexpansive mappings in ordered metric spaces and their (common) fixed points with applications

We introduce a generalized nonexpansive condition for a map in an ordered metric space and derive fixed point theorems for a nondecreasing mapping. Moreover, common fixed point results are obtained for a pair of mappings satisfying an appropriate nonexpansive condition. To illustrate our results, we give throughout the paper some examples. Applications of the obtained results to integral equations and first-order periodic boundary value problems are presented.

Nontrivial solutions of Hammerstein integral equations with reflections

Using the theory of fixed point index, we establish new results for the existence of nonzero solutions of Hammerstein integral equations with reflections. We apply our results to a first-order periodic boundary value problem with reflections. MSC: Primary 34K10; secondary 34B15; 34K13

Periodic boundary value problems for nonlinear first-order impulsive dynamic equations on time scales

By using the classical fixed point theorem for operators on cone, in this article, some results of one and two positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained. Two examples are given to illustrate the main results in this article.Mathematics Subject Classification: 39A10; 34B15.

Maximum principles for the periodic boundary value problem for impulsive integro-differential equations

This paper studies a first-order periodic boundary value problem for impulsive integro-differential equations and establishes some new maximum principles. Some points of a recent paper are also corrected and extended.

Monotone Iterative Technique for First-Order Nonlinear Periodic Boundary Value Problems on Time Scales

We investigate the following nonlinear first-order periodic boundary value problem on time scales: , , . Some new existence criteria of positive solutions are established by using the monotone iterative technique.

Multiple Positive Solutions for Nonlinear First-Order Impulsive Dynamic Equations on Time Scales with Parameter

By using the Leggett-Williams fixed point theorem, the existence of three positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales with parameter are obtained. An example is given to illustrate the main results in this paper.

Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales

In this paper, existence criteria of positive solutions to a class of nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales are obtained. The main tool used in this paper is the well-known Guo–Krasnoselskii fixed-point theorem.

Positive solutions to nonlinear first-order PBVPs with parameter on time scales

In this paper we consider the following nonlinear first-order periodic boundary value problem with parameter on a time scale T { x Δ ( t ) + p ( t ) x ( σ ( t ) ) = λ f ( x ( t ) ) , t ∈ [ 0 , T ] T , x ( 0 ) = x ( σ ( T ) ) , where λ > 0 is a parameter. For suitable λ > 0 , some existence, multiplicity and nonexistence criteria of positive solutions are established by using well-known results from the fixed-point index.

Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales

In this paper, we consider the following nonlinear first-order periodic boundary value problems on time scales { x Δ ( t ) + p ( t ) x ( σ ( t ) ) = f ( x ( t ) ) , t ∈ [ 0 , T ] T , x ( 0 ) = x ( σ ( T ) ) . Some new existence and multiplicity criteria of positive solutions are established by using several well-known fixed point theorems.

Existence of solutions to nonlinear first-order PBVPs on time scales

In this paper we are concerned with the following nonlinear first-order periodic boundary value problem on time scales { x Δ ( t ) + p ( t ) x ( σ ( t ) ) = g ( t , x ( σ ( t ) ) ) , t ∈ [ 0 , T ] T , x ( 0 ) = x ( σ ( T ) ) . Some new existence criteria of at least one solution are established by using novel inequalities and the well-known Schaefer fixed point theorem.

Existence of positive solutions of PBVPs for first-order difference equations

We consider the existence of positive solutions for the following first-order periodic boundary value problem:x(n+1)=x(n)−f(n,x(n)),0≤n≤ω−1,x(0)=x(ω). Some criteria for existence of positive solutions of the above difference boundary problem are established by using Krasnosel'skiĭ's fixed point theorem, and some multiplicity results of positive solutions are also derived.

Existence of solutions to first-order periodic boundary value problems

This article investigates the existence of solutions to boundary value problems (BVPs) involving systems of first-order ordinary differential equations and two-point, periodic boundary conditions. The methods involve novel differential inequalities and fixed-point theory to yield new theorems guaranteeing the existence of at least one solution.

Remarks on the periodic boundary value problems for first-order differential equations

This paper is concerned with the existence of solutions and the monotone method of first-order periodic boundary value problems when the lower and upper solutions α and β violate the boundary conditions α(0) ≤ α(2 π) and β(0) ≥ β(2 π). Using the topological degree theory, two existence theorems are established under weaker conditions than the one-side Lipschitz conditions. An example is given, which illustrates that PBVP may not have solutions between α and β without further restrictions to f( t, u). The monotone method is also discussed with some new results.

Cubic and Quartic Convergence for First-Order Periodic Boundary-Value Problems

In this paper, the results of Lakshmikantham et al. (Ref. 1) for first-order periodic boundary-value problems are extended, by using the extended method of quaislinearization and rapid convergence for initial-value problems of Mohapatra et al. (Ref. 2). Also, it is shown that monotone sequences converge cubically to the unique solution when the forcing function in the differential equation is 2–hyperconvex and converge quartically when the forcing function is 3–hyperconvex. Several other generalizations of the problem are also presented.