Focal Boundary Value Problems Research Articles

Iteration with Bisection to Approximate the Solution of a Boundary Value Problem

Due to the restrictive growth and/or monotonicity requirements inherent in their employment, classical iterative fixed-point theorems are rarely used to approximate solutions to an integral operator with Green’s function kernel whose fixed points are solutions of a boundary value problem. In this paper, we show how one can decompose a fixed-point problem into multiple fixed-point problems that one can easily iterate to approximate a solution of a differential equation satisfying one boundary condition, then apply a bisection method in an intermediate value theorem argument to meet a second boundary condition. Error estimates on the iterates are also established. The technique will be illustrated on a second-order right focal boundary value problem, with an example provided showing how to apply the results.

Utilizing an integrating factor to convert a right focal boundary value problem to a fixed point problem

Utilizing an integrating factor to convert a right focal boundary value problem to a fixed point problem

On solvability of focal boundary value problems for higher order functional differential equations with integral restrictions

Sharp conditions are obtained for the unique solvability of focal boundary value problems for higher-order functional differential equations under integral restrictions on functional operators. In terms of the norm of the functional operator, unimprovable conditions for the unique solvability of the boundary value problem are established in the explicit form. If these conditions are not fulfilled, then there exists a positive bounded operator with a given norm such that the focal boundary value problem with this operator is not uniquely solvable. In the symmetric case, some estimates of the best constants in the solvability conditions are given. Comparison with existing results is also performed.

SOLVABILITY OF A CLASS OF N -TH ORDER LINEAR FOCAL PROBLEMS

This paper presents a recursive method which yields necessary and sufficient conditions for the existence of solutions of a class of n-th order linear focal boundary value problems in the interior of a given interval, in the form of integral inequalities. Some results on the sign of the derivatives of the Green functions of n-th order linear focal boundary value problems will also be provided.

The focal boundary value problem for strongly singular higher-order nonlinear functional-differential equations

The a priori boundedness principle is proved for the two-point right-focal boundary value problems for strongly singular higher-order nonlinear functional-differential equations. Several sufficient conditions of solvability of the two-point right-focal problem under consideration are derived from the a priori boundedness principle. The proof of the a priori boundedness principle is based on Agarwal-Kiguradze type theorems, which guarantee the existence of the Fredholm property for strongly singular higher-order linear differential equations with argument deviations under the two-point right-focal boundary conditions.

Positive solutions to a three point fourth order focal boundary value problem

We consider a three point fourth order boundary value problem of focal type. Some upper and lower estimates for positive solutions of the problem are obtained. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are established. An example is included to illustrate the results.

Solvability of a Third-Order Singular Generalized Left Focal Problem in Banach Spaces

We consider the existence of positive solution for a third-order singular generalized left focal boundary value problem with full derivatives in Banach spaces. Green’s function and its properties, explicit a priori, estimates will be presented. By means of the theories of the fixed point in cones, we establish some new and general results on the existence of single and multiple positive solutions to the third-order singular generalized left focal boundary value problem. Our results are generalizations and extensions of the results of the focal boundary value problem. An example is included to illustrate the results obtained.

Computing the positive solutions of the discrete third-order three-point right focal boundary-value problems

The existence of positive solutions to the discrete third-order three-point boundary-value problems (BVPs) was recently established in Ji and Yang [Positive solutions of discrete third-order three-point right focal boundary value problems, J. Differ. Equat. Appl. 15 (2009), pp. 185–195]. In this paper, we propose an algorithm for the computation of such positive solutions. The method is based on the power method for the dominant eigenvalue and the Crout-like factorization algorithm for the sparse system of linear equations. At each iteration of the method, it calls for a linear solver with linear computational complexity. The proposed method is extremely effective for large-scale problems. A numerical example is also included to demonstrate the effectiveness of the algorithm when applied to the third-order three-point BVPs of differential equation.

Solvability of right focal boundary value problems with superlinear growth conditions

Abstract In this paper, we consider n th-order two-point right focal boundary value problems "Equation missing" where f : [ 0 , 1 ] × R n → R is a L p -Carathéodory ( 1 ≤ p < ∞ ) function and satisfies superlinear growth conditions. The existence and uniqueness of solutions for the above right focal boundary value problems are obtained by Leray-Schauder continuation theorem and analytical technique. Meanwhile, as an application of our results, examples are given. MSC:34B15.

The role of concavity in applications of avery type fixed point theorems to higher order differential equations

(Communicated by M. Plum) Abstract. In this article we apply an extension of an Avery type fixed point theorem to a family of boundary value problems for higher order ordinary differential equations. The theorem employs concave and convex functionals defined on a cone in a Banach space. We begin by extending a known application to a right focal boundary value problem for a second order problem to a conjugate boundary value problem for a second order problem. We then extend inductively to a two point boundary value problem for a higher order equation. Concavity of differentiable functions plays a key role in the application to second order equations. A concept of generalized concavity plays the same key role in the application to the higher order equation.

Existence of a positive solution to a right focal boundary value problem

In this paper we apply the recent extension of the Leggett-Williams Fixed Point Theorem which requires neither of the functional boundaries to be invariant to the second order right focal boundary value problem. We demonstrate a technique that can be used to deal with a singularity and provide a non-trivial example.

Right focal boundary value problems for difference equations

An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also provided.

Uniqueness implies existence of solutions for nonlinear focal-like boundary value problems

Given an appropriate growth condition for f and a uniqueness assumption on y ( n ) = 0 with respect to certain focal boundary value problems, it is shown that uniqueness of solutions to the nonlinear differential equation y ( n ) = f ( t , y , y ′ , … , y ( n − 1 ) ) , subject to boundary conditions of the form g i j ( y ( t j ) , … , y ( n − 1 ) ( t j ) ) = y i j implies existence of solutions.

Positive solutions of discrete third-order three-point right focal boundary value problems

We consider linear and nonlinear boundary value problems for third-order difference equations with three-point right focal boundary conditions. In the linear case, the existence of positive solutions corresponding to the first eigenvalue of the problem is established and an interval estimate for the first eigenvalue is obtained. In the nonlinear case, sufficient conditions for the existence and nonexistence of positive solutions are obtained. An example is included at the end of the paper to illustrate the sharpness of our results.

On third-order three-point right focal boundary value problems

In this paper, a fixed point theorem in a cone, some inequalities of the associated Green's function and the concavity of solutions are applied to obtain the existence of positive solutions of third-order three-point boundary value problem with dependence on the first-order derivative$\begin{cases}& x'''(t) = f(t, x(t), x'(t)), \quad 0 < t < 1, \\ & x(0) = x'(\eta) = x''(1) = 0, \end{cases}$where $f:[0, 1]\times[0, \infty)\times R\to [0,\infty)$ is a nonnegative continuous function, $\eta\in(1/2, 1).$

Four functionals fixed point theorem

The Four Functionals Fixed Point Theorem is a generalization of the original, as well as the functional generalizations, of the Leggett–Williams Fixed Point Theorem. In the Four Functionals Fixed Point Theorem, neither the upper nor the lower boundary of the underlying set is required to map below or above the boundary in the functional sense. As an application, the existence of a positive solution to a second-order right focal boundary value problem is considered by applying both standard and nonstandard choices of functionals. An extension to multivalued maps is provided for completeness.

Existence of monotone positive solutions to a third order two-point generalized right focal boundary value problem

In this paper, we are concerned with the third order two-point generalized right focal boundary value problem x ‴ ( t ) + f ( t , x ( t ) ) = 0 , a < t < b , x ( a ) = x ′ ( a ) = x ″ ( b ) = 0 . A few new results are given for the existence of at least one, two, three and infinitely many monotone positive solutions of the above boundary value problem by using the Krasnosel’skii fixed-point theorem in cones, and the Leggett–Williams fixed-point theorem.

Positive solutions to an Nth order right focal boundary value problem

The existence of a positive solution is obtained for the n th order right focal boundary value problem y (n) = f(x,y), 0 < x � 1, y (i) (0) = y (n 2) (p) = y (n 1) (1) = 0,i = 0,··· ,n 3, where 1 < p < 1 is fixed and where f(x,y) is singular at x = 0,y = 0, and possibly at y = 1. The method applies a fixed-point theorem for mappings that are decreasing with respect to a cone.

Fixed-sign solutions for a system of singular focal boundary value problems

We consider a system of focal boundary value problems where the nonlinearities may be singular in the independent variable and may also be singular in the dependent arguments. Using Schauder fixed point theorem, we establish criteria such that the system of boundary value problems has at least one fixed-sign solution.

Positive Solutions of Two-point right focal boundary value problems on time scales

We consider the following boundary value problem, ( − 1 ) n − 1 y Δ n ( t ) = ( − 1 ) p + 1 F ( t , y ( σ n − 1 ( t ) ) ) , t ∈ [ a , b ] ∩ T , y Δ i ( a ) = 0 , 0 ≤ i ≤ p − 1 , y Δ i ( σ ( b ) ) = 0 , p ≤ i ≤ n − 1 , where n ≥ 2, 1 ⩽ p ⩽ n - 1 is fixed and T is a time scale. Criteria for the existence of single, double, and multiple positive solutions of the boundary value problem are developed. Upper and lower bounds for these positive solutions are established for two special cases that arise from some physical phenomena. We also include several examples to illustrate the usefulness of the results obtained.