Nonlinear Sturm-Liouville Problems Research Articles

Global Bifurcation from Infinity in Some Nonlinear Sturm-Liouville Problems

Global Bifurcation from Infinity in Some Nonlinear Sturm-Liouville Problems

Newton-Kantorovich Method for Solving One of the Non-Linear Sturm-Liouville Problems

Due to its importance in physics and applied mathematics, the non-linear Sturm-Liouville problems witnessed massive attention since 1960. A powerful Mathematical technique called the Newton-Kantorovich method is applied in this work to one of the non-linear Sturm-Liouville problems. To the best of the authors’ knowledge, this technique of Newton-Kantorovich has never been applied before to solve the non-linear Sturm-Liouville problems under consideration. Accordingly, the purpose of this work is to show that this important specific kind of non-linear Sturm-Liouville differential equations problems can be solved by applying the well-known Newton-Kantorovich method. Also, to show the efficiency of applying this method to solve these problems, a comparison is made in this paper between the Newton-Kantorovich method and the Adomian decomposition method applied to the same non-linear Sturm-Liouville problems under consideration in this work. As a result of this comparison, the results of the Newton-Kantorovich method agreed with the results obtained by applying Adomian’s decomposition method.

Global bifurcation from zero in nonlinear Sturm-Liouville equation with a spectral parameter in the boundary condition

In this paper, we consider the nonlinear Sturm-Liouville problem with a spectral parameter in the boundary condition. We show the existence of two families of unbounded continua of nontrivial solutions of this problem corresponding to the nodal properties that having the eigenfunctions of the corresponding linear problem and emanating from bifurcation points and intervals of the line of trivial solutions.

Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels

<abstract><p>This article presents the Laplace-Adomian decomposition method (LADM), which produces a fast convergence series solution, for two types of nonlinear fractional Sturm-Liouville (SL) problems. The fractional derivatives are defined in the Caputo, conformable, Caputo-Fabrizio in the sense of Caputo (CFC), Caputo type Atangana-Baleanu (ABC) senses. With the help of this method, approximate solutions of the investigated problems were obtained. The solutions generated from the Caputo and ABC derivatives are represented by the Mittag-Leffler function, which is intrinsic to fractional derivatives, and the solution obtained using the conformable and CFC derivatives generate the hyperbolic sine and cosine functions. Thus, we derive some novel solutions for fractional-order versions of nonlinear SL equations. The fractional calculus provides more data than classical calculus and has been widely used in mathematical modeling with memory effect. Finally, we analyzed and compared these novel solutions of the considered problems by graphs under different values of $ p $, $ \lambda $ and different orders of $ \alpha $.</p></abstract>

On nonlinear discrete Sturm–Liouville boundary value problems with unbounded nonlinearities

This paper is devoted to the study of nonlinear Sturm–Liouville problems in the discrete setting. We consider problems for which nonlinearities in both the dynamics and boundary conditions may be unbounded. For such problems, we provide a criteria for the existence of solutions.

On a Class of Elliptic (p, q)-Kirchhoff Type Systems with Multiple Parameters

This article concerns the existence of positive solutions for elliptic (p, q)-Kirchhoff type systems with multiple parameters. Our approach is based on the method of sub and super-solutions. The concepts of sub- and super-solution were introduced by Nagumo (Proc Phys-Math Soc Jpn19:861–866, 1937) in 1937 who proved, using also the shooting method, the existence of at least one solution for a class of nonlinear Sturm-Liouville problems. In fact, the premises of the sub- and super-solution method can be traced back to Picard. He applied, in the early 1880s, the method of successive approximations to argue the existence of solutions for nonlinear elliptic equations that are suitable perturbations of uniquely solvable linear problems. This is the starting point of the use of sub- and super-solutions in connection with monotone methods. Picard’s techniques were applied later by Poincare (J Math Pures Appl 4:137–230, 1898) in connection with problems arising in astrophysics. We refer to Rădulescu (Qualitative analysis ofnonlinear elliptic partial differential equations: monotonicity, analytic, and variational methods, contemporary mathematics and its applications, Hindawi Publishing Corporation, New York, 2008).

Bifurcation from zero or infinity in nonlinearizable Sturm–Liouville problems with indefinite weight

In this paper, we consider bifurcation from zero or infinity of nontrivial solutions of the nonlinear Sturm–Liouville problem with indefinite weight. This problem is mainly important because of it is related with a selection-migration model in genetic population. We show the existence of four families of unbounded continua of nontrivial solutions to this problem bifurcating from intervals of the line of trivial solutions or the line R × {∞} (these intervals are called bifurcation intervals). Moreover, these global continua have the usual nodal properties in some neighborhoods of bifurcation intervals.

Existence of solutions to nonlinear Sturm-Liouville problems with large nonlinearities

Existence of solutions to nonlinear Sturm-Liouville problems with large nonlinearities

Qualitative properties and global bifurcation of solutions for a singular boundary value problem

This paper deals with a singular, nonlinear Sturm–Liouville problem of the form {A(x)u 0 (x)} 0 + λu(x) = f(x, u(x), u 0 (x)) on (0, 1) where A is positive on (0, 1] but decays quadratically to zero as x approaches zero. This is the lowest level of degeneracy for which the problem exhibits behaviour radically different from the regular case. In this paper earlier results on the existence of bifurcation points are extended to yield global information about connected components of solutions.

Finite Element Approximation of the Minimal Eigenvalue and the Corresponding Positive Eigenfunction of a Nonlinear Sturm—Liouville Problem

The problem of finding the minimal eigenvalue and the corresponding positive eigenfunction of the nonlinear Sturm—Liouville problem for the ordinary differential equation with coefficients nonlinear depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. A sufficient condition for the existence of a minimal eigenvalue and the corresponding positive eigenfunction of the nonlinear Sturm— Liouville problem is established. The original differential eigenvalue problem is approximated by the finite element method with Lagrangian finite elements of arbitrary order on a uniform grid. The error estimates of the approximate eigenvalue and the approximate positive eigenfunction to exact ones are proved. Investigations of this paper generalize well known results for the Sturm—Liouville problem with linear entrance on the spectral parameter.

Existence theory for nonlinear Sturm-Liouville problems with non-local boundary conditions

Existence theory for nonlinear Sturm-Liouville problems with non-local boundary conditions

On the existence of infinitely many eigenvalues in a nonlinear Sturm–Liouville problem arising in the theory of waveguides

We consider a nonlinear eigenvalue problem of the Sturm–Liouville type on an interval with boundary conditions of the first kind. The problem describes the propagation of polarized electromagnetic waves in a plane two-layer dielectric waveguide. The cases of a homogeneous and an inhomogeneous medium are studied. The existence of infinitely many positive and negative eigenvalues is proved.

Eigenvalues for iterative systems of Sturm-Liouville fractional order two-point boundary value problems

Abstract In this paper, we determine the eigenvalue intervals of λ 1, λ 2, ..., λ n for which the iterative system of nonlinear Sturm-Liouville fractional order two-point boundary value problem possesses a positive solution by an application of Guo-Krasnosel’skii fixed point theorem on a cone.

Half-linear Sturm-Liouville problem with weights: Asymptotic behavior of eigenfunctions

We consider a nonlinear Sturm-Liouville problem with weights. Under suitable assumptions on weights we prove an asymptotic estimate for the decay of the eigenfunctions. An application to the radial problem on the entire ℝN is given.

Existence theory for nonlinear Sturm-Liouville problems with unbounded nonlinearities

Existence theory for nonlinear Sturm-Liouville problems with unbounded nonlinearities

Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

This paper is devoted to prove the existence of solutions of the nonlinear Sturm-Liouville boundary value problem on time scales in Banach spaces. We obtain the sufficient conditions for the existence of solutions in terms of Kuratowski measure of noncompactness. Mönch's fixed point theorem is used to prove the main result. By the unification property of time scales, our result is valid for Sturm-Liouville differential equations and difference equations, but more interestingly by the extension property, it is also valid for Sturm-Liouville $q$-difference equation.

Global bifurcation and nodal solutions for a Sturm–Liouville problem with a nonsmooth nonlinearity

In this paper, we shall establish the unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity of a class of nonlinear Sturm–Liouville problems with nondifferentiable nonlinearity, respectively. As applications of the above results, we shall determine the interval of r, in which there exist nodal solutions for the following problem{u″(t)+ra(t)f(u)=0,t∈(0,1),u(0)=u(1)=0, where a(t) is a positive continuous function on [0,1], f∈C(R,R) but is not necessarily differentiable at the origin and infinity. Moreover, as a special case of the above problem, we shall prove more details about the existence of nodal solutions for a class of half-linear eigenvalue problems.

Positive solutions for systems of a nonlinear fourth-order singular semipositone Sturm-Liouville boundary value problem

This paper concerns the existence of positive solutions for systems of a fourth-order singular semipositone Sturm-Liouville boundary value problem. By applying the fixed point index theorem, some sufficient conditions for positive solutions are established. An example is given to demonstrate the application of our main results.

Fixed point theorems of convex-power 1-set-contraction operators in Banach spaces

In this article, we give the definition of a class of new operators, namely, convex-power 1-set-contraction operators in Banach spaces, and study the existence of fixed points of this class of operators. By using methods of approximation by operators, we obtain fixed point theorems of convex-power 1-set-contraction operators, which generalize fixed point theorems of 1-set-contraction operators in Banach spaces. By using the fixed point theorem, the existence of solutions of nonlinear Sturm-Liouville problems in Banach spaces is investigated under more general conditions than those used in former literatures.Mathematics Subject Classification 2010: 47H10.

A nonlinear integral transform and a global inverse bifurcation theory

We consider a nonlinear integral transform and show that the transform acts as a homeomorphism between certain metric spaces of positive functions. We apply the result to the inverse bifurcation problem of determining the nonlinear term of a certain nonlinear Sturm-Liouville problem from its first bifurcating branch, and we establish the well-posedness of the inverse problem. An application to an inverse problem of determining a restoring force from a timemap is also given. Mathematics Subject Classification (2010): 44A15 (primary); 34A55, 45P05 (secondary). 1. The main result This paper studies the nonlinear integral transform K defined by (K f )(x) = ∫ 1 0 dt (∫ 1 t sβ−1 f (xs)ds )1−δ , x ∈ I, (1.1) where 0 0, and I is a bounded, closed interval containing 0. Our objective is to show that the transform K is a homeomorphism of an appropriate metric space onto a twin metric space reflecting the smoothing property of K and, as its application, that an inverse problem to determine a nonlinear term of a certain nonlinear Sturm-Liouville problem from its first bifurcating branch is globally wellposed. Throughout the paper, θ denotes the Euler differential operator θ = x d dx . To state the main result explicitly, for a nonnegative integer k, a number α ∈ (0, 1], Supported by Grant-in-Aid for Scientific Research No. 21540165, Japan Society for Promotion of Science. Received February 12, 2009; accepted in revised form July 29, 2010.