On the Principal Eigenvalue of a Biharmonic System

We prove the existence, positivity, simplicity, uniqueness up to nonnegative eigenfunctions, and isolation of the principle eigenvalue of a biharmonic system

On a second order discrete problem

By using variontional methods, we establish criteria for the existence of at least three solutions to a second order discrete problem with two parameters. Applications of the maintheorem to several special cases of the problem are discussed and two examples are included to illustrate the results. This paper extends and complements some of our early work on related problems

Higher order boundary value problems with φ-Laplacian and functional boundary conditions

We study the existence of solutions of the boundary value problem φ(u^(n−1)(t))′ + f (t, u(t), u′(t), . . . , u^(n−1)(t))= 0, t ∈ (0, 1), g_i (u, u′, . . . , u^(n−1), u^(i)(0))= 0, i = 0, . . . , n − 2, g_n−1 (u, u′, . . . , u^(n−1), u^(n−2)(1))= 0, where n ≥ 2, φ and g_i, i = 0, . . . , n − 1, are continuous, and f is a Carathéodory function. We obtain an existence criterion based on the existence of a pair of coupled lower and upper solutions.Wealso apply our existence theorem to derive some explicit conditions for the existence of a solution of a special case of the above problem. In our problem, both the differential equation and the boundary conditions may have dependence on all lower order derivatives of the unknown function, and many boundary value problems with various boundary conditions, studied extensively in the literature, are special cases of our problem. Consequently, our results improve and cover a number of known results in the literature. Examples are given to illustrate the applicability of our theorems

Higher order functional boundary value problems: existence and location results

This paper considers a nth-order phi-laplacian differential equation with functional boundary conditions satisfying certain monotonicity assumptions. We present sufficient conditions on the nonlinearity and the boundary conditions to ensure the existence of solutions. Moreover, from the lower and upper solutions method, some information is given about the location of the solution and its qualitative properties. Due to the functional dependence in the boundary conditions, this work generalizes several results for higher order problems with many types of boundary conditions. The main results are illustrated with examples

Higher Order φ-Laplacian BVP with Generalized Sturm–Liouville Boundary Conditions

This work presents an existence and location result for a higher order boundary value problem with some generalized boundary conditions fof Sturm-Liouville type. In view of the assumptions on φ-Laplacian and the nonlinearity f , this paper generalizes several problems due to the dependence on the (n − 1)-st derivative not only in the differential equation but also in the boundary conditions

Higher order multi-point boundary value problems with sign-changing nonlinearities and nonhomogeneous boundary conditions

We study classes of $n$th order boundary value problems consisting of an equation having a sign-changing nonlinearity $f(t,x)$ together with several different sets of nonhomogeneous multi-point boundary conditions. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. Conditions are determined by the behavior of $f(t,x)/x$ near $0$ and $\pm\infty$ when compared to the smallest positive characteristic values of some associated linear integral operators. This work improves and extends a number of recent results in the literature on this topic. The results are illustrated with examples

Existence of positive periodic solutions for higher order singular functional difference equations

We study a higher order singular functional difference equation on $\mathbb{Z}$. Sufficient conditions are obtained for the existence of at least one positive periodic solution of the equation. Our proof utilizes the nonlinear alternative of Leray-Schauder

Nodal solutions of nonlocal boundary value problems

We study the nonlinear boundary value problem consisting of the second order differential equation on [a, b] and a boundary condition involving a Riemann‐Stieltjes integral. By relating it to the eigenvalues of a linear Sturm‐Liouville problem with a two‐point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes of the existence of different types of nodal solutions as the problem changes. First published online: 14 Oct 201

Existence and uniqueness of solutions for a fractional boundary value problem on a graph

In this paper, the authors consider a nonlinear fractional boundary value problem defined on a star graph. By using a transformation, an equivalent system of fractional boundary value problems with mixed boundary conditions is obtained. Then the existence and uniqueness of solutions are investigated by fixed point theory