Second-order Sturm-Liouville Boundary Value Problems Research Articles

Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero

We deal with the following Sturm–Liouville boundary value problem: − P t x ′ t ′ + B t x t = λ ∇ x V t , x , a.e. t ∈ 0,1 x 0 cos α − P 0 x ′ 0 sin α = 0 x 1 cos β − P 1 x ′ 1 sin β = 0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

Nonlinear Functional Analysis of Boundary Value Problems: Novel Theory, Methods, and Applications

and Applied Analysis 3 1-homogeneous operator in a Banach space and then demonstrate its application in establishing the existence of positive solutions for p-Laplacian boundary value problems under certain conditions. (xi) In the paper titled “Existence of solutions for nonhomogeneous A-harmonic equations with variable growth,” the authors establish a theorem for the existence of weak solutions for nonhomogeneous A-harmonic equations in subspace and then give three examples to demonstrate its application. (xii) In the paper titled “Multiple solutions for degenerate elliptic systems near resonance at higher eigenvalues,” the authors study the degenerate semilinear elliptic system in an open bounded domain with smooth boundary, and some multiplicity results of solutions are obtained for the system near resonance at certain eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory. (xiii) In the paper titled “A regularity criterion for the Navier-Stokes equations in the multiplier spaces,” the authors establish a regularity criterion in terms of the pressure gradient for weak solutions to the NavierStokes equations in a special class. The third set of papers, including four papers, deal with several boundary value problems for highly nonlinear ordinary differential equations. (i) In the paper titled “Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions,” the authors investigate the existence of positive solutions for second-order singular differential equations with a negatively perturbed term, by means of the fixed-point theory in cones. (ii) In the paper titled “Positive solutions for Sturm-Liouville boundary value problems in a Banach Space,” the sufficient conditions for the existence of single and multiple positive solutions for a second-order SturmLiouville boundary value problem are established in a Banach space, by using the fixed-point theorem of strict set contraction operators in the frame of the ODE technique. (iii) In the paper titled “Positive solutions of a nonlinear fourth-order dynamic eigenvalue problem on time scales,” the authors study a nonlinear fourth-order dynamic eigenvalue problem on time scales and obtain the existence and nonexistence of positive solutions when 0 λ, respectively, for some λ, by using the Schauder fixed-point theorem and the upper and lower solution method. (iv) In the paper titled “Bifurcation analysis for a predatorprey model with time delay and delay-dependent parameters,” a class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. By using the normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifur-cating from Hopf bifurcations are obtained. The fourth set of papers focus on finding the approximate and numerical solutions of various complex nonlinear boundary value problems. (i) In the paper titled “On spectral homotopy analysis method for solving linear Volterra and Fredholm integrodifferential equations,” a spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations, and some examples are given to test the efficiency and the accuracy of the proposed method. (ii) In the paper titled “The solution of a class of singularly perturbed two-point boundary value problems by the iterative reproducing kernel method,” the authors establish an iterative reproducing kernel method (IRKM) for solving singular perturbation problems with boundary layers and give two numerical examples to demonstrate the effectiveness of the method. (iii) In the paper titled “A Galerkin solution for Burgers’ equation using cubic B-spline finite elements,” a Galerkin method using cubic B-splines is set up to find the numerical solutions of Burgers’ equation, and the method is shown to be capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to very large. (iv) In the paper titled “Forward-backward splitting methods for accretive operators in Banach spaces,” the authors introduce two iterative forward-backward splitting methods with relaxations to find zeros of the sum of two accretive operators in Banach spaces and prove the weak and strong convergence of these methods under mild conditions, and also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

Positive Solutions for Sturm‐Liouville Boundary Value Problems in a Banach Space

We consider the existence of single and multiple positive solutions for a second‐order Sturm‐Liouville boundary value problem in a Banach space. The sufficient condition for the existence of positive solution is obtained by the fixed point theorem of strict set contraction operators in the frame of the ODE technique. Our results significantly extend and improve many known results including singular and nonsingular cases.

Sign-changing and multiple solutions of the Sturm–Liouville boundary value problem via invariant sets of descending flow

In this paper, we prove the existence of sign-changing and multiple solutions for the second-order Sturm–Liouville boundary value problem { − L u = f ( x , u ) , x ∈ [ 0 , 1 ] R 1 ( u ) = 0 , R 2 ( u ) = 0 , where L u = ( p ( x ) u ′ ) ′ − q ( x ) u is the Sturm–Liouville operator, R 1 ( u ) = α u ′ ( 0 ) − β u ( 0 ) and R 2 ( u ) = γ u ′ ( 1 ) + σ u ( 1 ) . The technical approach is fully based on minimax methods and invariant sets of descending flow.

Multiple Positive Solutions of Second-Order Sturm-Liouville Boundary Value Problems for Impulsive Differential Equations

Multiple Positive Solutions of Second-Order Sturm-Liouville Boundary Value Problems for Impulsive Differential Equations

MULTIPLE SOLUTIONS FOR A SECOND-ORDER STURM-LIOUVILLE BOUNDARY VALUE PROBLEM

In this paper, we investigate a second-order Sturm-Liouville boundary value problem depending on the parameter $\lambda$. For any $\lambda$ in some explicitly determined open interval, the boundary value problem admits multiple generalized solutions by using very recent three critical points theorems stated in a paper by G. Bonanno (J. Math. Anal. Appl. 29 (2004), 600-614).

On solvability of second-order Sturm-Liouville boundary value problems at resonance

In this paper, based on of the concept q 0 ∈ H 0 ( p , ( 0 , 1 ) , α , β ) q_0\in H_0(p,(0,1),\alpha ,\beta ) , which is a generalized form of the first resonant point π 2 \pi ^2 to the Picard problem x + λ x = 0 x+\lambda x=0 , x ( 0 ) = x ( 1 ) = 0 x(0)=x(1)=0 , we study the solvability of second-order Sturm-Liouville boundary value problems at resonance ( p ( t ) x ′ ) ′ + q 0 ( t ) x + g ( t , x ) = h ( t ) (p(t)x’)’+q_0(t)x+g(t,x)=h(t) , x ( 0 ) cos ⁡ α − p ( 0 ) x ′ ( 0 ) sin ⁡ α = 0 x(0){\cos \alpha }-p(0)x’(0)\sin \alpha =0 , x ( 1 ) cos ⁡ β − p ( 1 ) x ′ ( 1 ) sin ⁡ β = 0 x(1)\cos \beta -p(1)x’(1)\sin \beta =0 , and improve the previous results about problems x + π 2 x + g ( t , x ) = h ( t ) , x ( 0 ) = x ( 1 ) = 0 x+\pi ^2x+g(t,x)=h(t),x(0)=x(1)=0 derived by Chaitan P. Gupta, R. Iannacci and M. N. Nkashama, and Ma Ruyun, respectively.

On Kramer’s Sampling Theorem Associated with General Sturm-Liouville Problems and Lagrange Interpolation

Kramer’s sampling theorem (which is a generalization of Shannon’s sampling theorem) and its relationship with Lagrange interpolation is studied. Kramer’s sampling theorem is extended to the case where the interval $( {a,b} )$ is infinite and it is shown that any function that has a sampling expansion in the scope of Kramer’s theorem also has a Lagrange-type interpolation expansion, provided that the kernel associated with Kramer’s theorem arises from a second-order Sturm-Liouville boundary value problem, whether it is regular or singular. This new technique not only reproduces many of the known sampling expansions such as those for the cosine, the finite Hankel, and the continuous Legendre transforms, but also generates new ones such as those for the continuous Jacobi, Laguerre, and Hermite transforms.

On Lagrange Interpolation and Kramer-Type Sampling Theorems Associated with Sturm–Liouville Problems

This article is devoted to a connection between Kramer’s sampling theorem and sampling expansions generated by Lagrange interpolation. It is shown that any function that has a sampling expansion in the scope of Kramer’s theorem also has a Lagrange-type interpolation expansion provided that the kernel associated with Kramer’s theorem arises from a second-order Sturm–Liouville boundary-value problem. This new approach, which for a variety of regular and singular Sturm–Liouville problems leads to associated sampling theorems, recovers not only many known sampling expansions but also gives new ways to calculate the corresponding sampling functions. New sampling series are included.