Discrete Boundary Value Problems Research Articles

Positive Solutions for a Class of Discrete Mixed Boundary Value Problems with thep,q-Laplacian Operator

In this paper, we consider the existence of solutions for the discrete mixed boundary value problems involvingp,q-Laplacian operator. By using critical points theory, we obtain the existence of at least two positive solutions for the boundary value problem under appropriate assumptions on the nonlinearity.

Large Constant-Sign Solutions of Discrete Dirichlet Boundary Value Problems with p-Mean Curvature Operator

In this paper, we consider the existence of infinitely many large constant-sign solutions for a discrete Dirichlet boundary value problem involving p -mean curvature operator. The methods are based on the critical point theory and truncation techniques. Our results are obtained by requiring appropriate oscillating behaviors of the non-linear term at infinity, without any symmetry assumptions.

Existence of two solutions for a fourth-order difference problem with p(k) exponent

The existence of nontrivial solutions for a fourth-order discrete anisotropic boundary value problem involving the p(k)-Laplacian operator with the Dirichlet and the Neumann boundary value conditions is investigated. Variational approach based on a new critical point theorem is applied. An example is inserted to illustrate main results.

Existence of solutions to discrete boundary value problem of fractional difference equations

In this paper, we obtained the sufficient conditions for the existence of solutions to the discrete boundary value problems of fractional difference equation depending on parameters. We use Krasnoselskii fixed point theorem to establish the existence results.

Multiple and particular solutions of a second order discrete boundary value problem with mixed periodic boundary conditions

Multiple and particular solutions of a second order discrete boundary value problem with mixed periodic boundary conditions

Computational recovery of time-dependent volatility from integral observations in option pricing

In this paper robust algorithms for numerical identification of time dependent volatility by integral observations of one- and two-asset Black–Scholes models are developed. An average linearization in time of diffusion terms of the discrete initial boundary value problems is used. Then, a decomposition with respect to the volatility of the approximate solution is applied so that the transition to a new time layer is carried out by solving standard discrete elliptic problems. Numerical experiments using simulated as well as real data confirm the effectiveness of the present approach.

Existence of solutions for a second order discrete boundary value problem with mixed periodic boundary conditions

In this paper, a second order discrete boundary value problem with a pair of mixed periodic boundary conditions is considered. Sufficient conditions on the existence and multiplicity of solutions are obtained by using variational methods. A particular Banach space and an associated functional are presented to overcome the asymmetry of the mixed periodic boundary conditions. Examples are also given to illustrate the applications of the main result.

Hyers - Ulam Stability Results for Discrete Antiperiodic Boundary Value Problem with Fractional Order

In this present work, we investigate Ulam stability for the following nonlinear discrete antiperiodic boundary value problem with fractional order of the form 0 ( ) C ( ) = 1, ( 1) , k v k k v k   + − + − for   0 k[0,L + 2] = 0,1,...,L + 2 , with boundary conditions v( −3) = −v( + L) , v( − 3) = −v( + L) , 2v( −3) = −2v( + L) , where 2 :[ 2, L]    −  − +  → is a continuous and 0 C k   is the Caputo fractional difference operator with order 2 <3   . Finally, the main results are illustrated by some examples.

Existence results for a fourth-order discrete boundary value problem with a parameter

The discrete boundary value problem for a fourth-order difference equation is discussed in this paper. The existence result of multiple solutions is established by restricting the algebraic assumptions on the nonlinear term and confirming the parameter μ in an interval related to eigenvalues. The main methods adopted are the critical point theorems applicable for the finite dimensional Banach space recently proposed by Bonanno et al. (2014).

Existence of positive solutions for a semipositone discrete boundary value problem

We investigate the existence of positive solutions for a nonlinear second-order difference equation with a linear term and a sign-changing nonlinearity, supplemented with multi-point boundary conditions. In the proof of our main results, we use the Guo–Krasnosel'skii fixed point theorem.

An application of a nonstandard cone to discrete boundary value problems with unbounded indefinite forcing

ABSTRACTWe consider the summation equation in the setting in which f may change sign. In fact, by means of the nonlocal element in the above summation equation, we are able to guarantee existence of at least one positive solution even in the case where for each and, furthermore, where . We obtain this result by applying a nonstandard order cone and attendant open set. Finally, by choosing the maps γ and G in particular ways, we are able to relate positive solutions of the summation equation to positive solutions of discrete boundary value problems.

Existence of positive solutions for a system of semipositone coupled discrete boundary value problems

ABSTRACTWe study the existence of positive solutions for a system of nonlinear second-order difference equations with parameters and sign-changing nonlinearities, subject to multi-point coupled boundary conditions. In the proof of our main theorems we use the nonlinear alternative of Leray-Schauder type and the Guo-Krasnosel'skii fixed point theorem.

Infinitely Many Positive Solutions for a Coupled Discrete Boundary Value Problem

In this paper, we obtain some results for the existence of infinitely many positive solutions for a coupled discrete boundary value problem. The approach is based on variational methods.

A note on homoclinic solutions of (p,q)-Laplacian difference equations

We prove the existence of at least two positive homoclinic solutions for a discrete boundary value problem of equations driven by the -Laplace operator. The properties of the nonlinearity ensure that the energy functional, corresponding to the problem, satisfies a mountain pass geometry and a Palais–Smale compactness condition.

Positive solutions for a semipositone nonlocal discrete boundary value problem

We investigate the existence of positive solutions for a system of nonlinear second-order difference equations with a parameter and a sign-changing nonlinearity, subject to multi-point boundary conditions. In the proof of our main results we apply the nonlinear alternative of Leray–Schauder type and the Guo–Krasnosel’skii fixed point theorem.

Multiple positive solutions of second-order nonlinear difference equations with discrete singular φ-Laplacian

ABSTRACTWe consider the discrete boundary value problems with mean curvature operator in the Minkowski space where is a parameter, n>4 and q>1. Using upper and lower solutions, topological methods and Szulkin's critical point theory for convex, lower semicontinuous perturbations of -functionals, we show that there exists such that the above problem has zero, at least one or two positive solutions according to , or . Moreover, Λ is strictly decreasing with respect to n.

To the theory of discrete boundary value problems

We consider discrete analogues of pseudo-differential operators and related discrete equations and boundary value problems. Existence and uniqueness results for special elliptic discrete boundary value problem and comparison for discrete and continuous solutions are given for certain smooth data in discrete Sobolev–Slobodetskii spaces.

Existence of solutions to discrete and continuous second-order boundary value problems via Lyapunov functions and a priori bounds

Existence of solutions to discrete and continuous second-order boundary value problems via Lyapunov functions and a priori bounds

POSITIVE SOLUTIONS FOR A NONLINEAR DISCRETE FRACTIONAL BOUNDARY VALUE PROBLEM WITH A <inline-formula><tex-math id="M1">$ P $</tex-math></inline-formula>-LAPLACIAN OPERATOR

In this paper using the monotone iterative technique we establish the existence and uniqueness of positive solutions for a nonlinear discrete fractional boundary value problem with a $p$-Laplacian operator. Also we discuss an iterative sequence which yields the approximate solution for this problem.

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.