Second-order Discrete Boundary Value Problem Research Articles

Existence of multiple solutions to a discrete boundary value problem with mixed periodic boundary conditions

A second-order discrete boundary value problem with mixed periodic boundary conditions is studied. Sufficient conditions on the multiplicity of solutions in a weak sense are obtained by using the critical point theory. An example is given to demonstrate the applications of our results as well.

Nontrivial Solutions for a System of Second-Order Discrete Boundary Value Problems

In this work, we shall study the existence of nontrivial solutions for a system of second-order discrete boundary value problems. Under some conditions concerning the eigenvalues of relevant linear operator, we use the topological degree theory to obtain our main results.

Boundary Value Problem for a Second-Order Difference Equation with Resonance

In this paper, we study the existence and multiplicity of nontrivial solutions of a second-order discrete boundary value problem with resonance and sublinear or superlinear nonlinearity. The main methods are based on the Morse theory and the minimax methods. In addition, some examples are given to illustrate our results.

Positive solutions for a system of second-order discrete boundary value problems

We study the existence and multiplicity of positive solutions for a system of nonlinear second-order difference equations subject to multi-point boundary conditions, under some assumptions on the nonlinearities of the system which contains concave functions. In the proofs of our main results we use some theorems from the fixed point index theory.

Positive Solutions for a System of Discrete Boundary Value Problem

This paper deals with the existence and multiplicity of positive solutions for a system of second-order discrete boundary value problem. The main results are obtained via Jensen’s inequalities, properties of concave and convex functions, and the Krasnosel’skii-Zabreiko fixed point theorem. Furthermore, concave and convex functions are employed to emphasize the coupling behaviors of nonlinear terms \(f\) and \(g\) and we provide two explicit examples to illustrate our main results and the coupling behaviors.

The existence of symmetric positive solutions for a seconder-order difference equation with sum form boundary conditions

Abstract In this paper, we consider the existence of positive solutions for a second-order discrete boundary value problem Δ ( g ( k − 1 ) Δ u ( k − 1 ) ) + w ( k ) f ( k , u ( k ) ) = 0 subject to the boundary conditions: a u ( 0 ) − b g ( 0 ) Δ u ( 0 ) = ∑ i = 1 n − 1 h ( i ) u ( i ) , a u ( n ) + b g ( n − 1 ) Δ u ( n − 1 ) = ∑ i = 1 n − 1 h ( i ) u ( i ) , where a , b > 0 , Δ u ( k ) = u ( k + 1 ) − u ( k ) for k ∈ { 0 , 1 , … , n − 1 } , g ( k ) > 0 is symmetric on { 0 , 1 , … , n − 1 } , w ( k ) is symmetric on { 0 , 1 , … , n } , f : { 0 , 1 , … , n } × [ 0 , + ∞ ) is continuous, f ( k , u ) = f ( n − k , u ) for all ( k , u ) ∈ { 0 , 1 , … , n } × [ 0 , + ∞ ) , and h ( i ) is nonnegative and symmetric on { 0 , 1 , … , n } . By the fixed point theorem and the Hölder inequality, we study the existence of symmetric positive solutions for the above difference equation with sum form boundary conditions.

Generalized Green’s functions for second-order discrete boundary-value problems with nonlocal boundary conditions

In this paper, generalized Green’s functions for second-order discrete boundaryvalueproblems with nonlocal boundary conditions are investigated, where the necessaryand sufficient existence condition of discrete Green’s function is not satisfied and nonlocalboundary conditions are described by linear functionals.

Existence of Two Solutions for a Second-Order Discrete Boundary Value Problem

Abstract The existence of two nontrivial solutions for a class of nonlinear second-order discrete boundary value problems is established. The approach adopted is based on variational methods.

Some new existence results for fractional difference equations

In this paper, we introduce several existence theorems for a discrete fractional boundary value problem with Dirichlet boundary conditions in the case where the order ν of the fractional difference satisfies 1 < ν ≤ 2. We use cone theoretic techniques to deduce the existence of one or more positive solutions. We then deduce uniqueness theorems for the same problem by assuming a Lipschitz condition. We show that many of the classical existence and uniqueness theorems for second-order discrete boundary value problems extend to the fractional-order case.

A Global Description of the Positive Solutions of Sublinear Second-Order Discrete Boundary Value Problems

Let be an integer with , , . We consider boundary value problems of nonlinear second-order difference equations of the form , , , where , and, for , and , , . We investigate the global structure of positive solutions by using the Rabinowitz's global bifurcation theorem.

Uniqueness and Multiplicity of Solutions for a Second-Order Discrete Boundary Value Problem with a Parameter

This paper is concerned with the existence of unique and multiple solutions to the boundary value problem of a second-order difference equation with a parameter, which is a complement of the work by J. S. Yu and Z. M. Guo in 2006.

Existence of Triple Positive Solutions for Second-Order Discrete Boundary Value Problems

By using a new fixed-point theorem introduced by Avery and Peterson (2001), we obtain sufficient conditions for the existence of at least three positive solutions for the equationΔ2x(k−1)+q(k)f(k,x(k),Δx(k))=0, fork∈{1,2,…,n−1}, subject to the following two boundary conditions:x(0)=x(n)=0orx(0)=Δx(n−1)=0, wheren≥3.

Existence theorems for second-order discrete boundary value problems

This article is concerned with the existence of solutions of boundary value problems for nonlinear second-order difference equations of the type Δ [ p n ( Δ x n − 1 ) δ ] + q n x n δ = f ( n , x n ) , where δ > 0 is the ratio of odd positive integers, { p n } and { q n } are real sequences. The authors apply the Linking Theorem and the Mountain Pass Lemma in the critical point theory and give some new results for the existence of solutions.

Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on ƒ involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem y k+1 − 2y k + y k−1 + ƒ(k,y k,v k) = 0, for k = 1,…, n − 1, y 0 = 0 = y n , where ƒ is continuous and v k = y k − y k −1 , for k = 1,…, n. In the special case ƒ(k,t,p) = ƒ(t) ≥ 0, we give growth conditions on ƒ and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue.

Three positive fixed points of nonlinear operators on ordered banach spaces

We generalize the fixed-point theorem of Leggett-Williams, which is a theorem giving conditions that imply the existence of three fixed points of an operator defined on a cone in a Banach space. We then show how to apply our theorem to prove the existence of three positive solutions to a second-order discrete boundary value problem.

A fixed-point approach for nonlinear discrete boundary value problems

Existence results are established for second-order discrete boundary value problems.

Boundary value problems for discrete equations

Two existence results are presented for second-order discrete boundary value problems. The first result is based on the notion of upper and lower solutions and the second result is based on the discrete Gelfand problem.