73 research outputs found
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 Ď-Laplacian and functional boundary conditions
We study the existence of solutions of the boundary value problem
Ď(u^(nâ1)(t))Ⲡ+ f (t, u(t), uâ˛(t), . . . , u^(nâ1)(t))= 0, t â (0, 1),
g_i (u, uâ˛, . . . , u^(nâ1), u^(i)(0))= 0, i = 0, . . . , n â 2,
g_nâ1 (u, uâ˛, . . . , u^(nâ1), u^(nâ2)(1))= 0,
where n ⼠2, Ď and g_i, i = 0, . . . , n â 1, are continuous, and f is a CarathĂŠodory 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 Ď-Laplacian BVP with Generalized SturmâLiouville 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 Ď-Laplacian and the nonlinearity f , this paper generalizes several problems due to the
dependence on the (n â 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 th order boundary value problems consisting of an equation having a sign-changing nonlinearity 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 near and 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 . 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 RiemannâStieltjes integral. By relating it to the eigenvalues of a linear SturmâLiouville problem with a twoâpoint 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:Â 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
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