Patricia J. Y. Wong

Positive solutions of differential, difference and integral equations

Preface. Ordinary Differential Equations. 1. First Order Initial Value Problems. 2. Second Order Initial Value Problems. 3. Positone Boundary Value Problems. 4. Semi-positone Boundary Value Problems. 5. Semi-Infinite Interval Problems. 6. Mixed Boundary Value Problems. 7. Singular Boundary Value Problems. 8. General Singular and Nonsingular Boundary Value Problems. 9. Quasilinear Boundary Value Problems. 10. Delay Boundary Value Problems. 11. Coupled System of Boundary Value Problems. 12. Higher Order Sturm-Liouville Boundary Value Problems. 13. (n,p) Boundary Value Problems. 14. Focal Boundary Value Problems. 15. General Focal Boundary Value Problems. 16. Conjugate Boundary Value Problems. Difference Equations. 17. Discrete Second Order Boundary Value Problems. 18. Discrete Higher Order Sturm-Liouville Boundary Value Problems. 19. Discrete (n,p) Boundary Value Problems. 20. Discrete Focal Boundary Value Problems. 21. Discrete Conjugate Boundary Value Problems. Integral and Integrodifferential Equations. 22. Volterra Integral Equations. 23. Hammerstein Integral Equations. 24. First Order Integrodifferential Equations. References. Authors Index. Subject Index.

Sturm-Liouville eigenvalue problems on time scales

For Sturm-Liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish Rayleighs principle. Our results not only unifly the corresponding theories for differential and difference equations, but are also new in the discrete case.

Error inequalities in polynomial interpolation and their applications

1. Lidstone Interpolation. 2. Hermite Interpolation. 3. Abel-Gontscharoff Interpolation. 4. Miscellaneous Interpolation. 5. Piecewise-Polynomial Interpolation. 6. Spline Interpolation.

Stability analysis of fractional differential system with Riemann-Liouville derivative

In this paper we focus on establishing stability theorems for fractional differential system with Riemann-Liouville derivative, in particular our analysis covers the linear system, the perturbed system and the time-delayed system.

Eigenvalues of Lidstone boundary value problems

We consider the following boundary value problem: (-1)^ny^(^2^n^)=@lF(t,y),n>=1,t@?(0,1),y^(^2^i^)(0)=y^(^2^i^)(1)=0,0=0. The values of @l are characterized so that the boundary value problem has a positive solution. In addition, we derive explicit intervals of @l such that for any @l in the interval, existence of a positive solution of the boundary value problem is guaranteed. Several examples are also included to dwell upon the importance of the results obtained.

The oscillation and asymptotically monotone solutions of second-order quasilinear differential equations

Abstract We offer sufficient conditions for the oscillation of all solutions of the perturbed quasilinear differential equation (a(t)|y′| α−1 y′)′ + Q(t, y) = P(t, y, y′) as well as for the existence of a positive monotone solution of the damped differential equation (a(t)|y′| α−1 y′)′ + b(t)|y′| α−1 y′ + H(t, y) = 0, where α > 0. Examples that dwell upon the importance of our results are also included.

Positive solutions and eigenvalues of conjugate boundary value problems

We consider the following boundary value problem where λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

Oscillation criteria for nonlinear partial difference equations with delays

Abstract We offer sufficient conditions for the oscillation of all solutions of the partial difference equations y(m+1,n)+β(m,n)y(m,n+1)−δ(m,n)y(m,n)+P(m,n,y(m−k,n−l)) = Q (m,n,y(m − k,n −l)) , and y(m+1,n)+β(m,n)y(m,n+1)−δ(m,n)y(m,n)+ ∑ i=1 τ P i (m,n,y(m − k i , n − l i )) = ∑ i=1 τ Q i (m,n,y(m − k i , n − l i )) . Several examples, which dwell upon the importance of our results, are also included.

Double positive solutions of (n,p) boundary value problems for higher order difference equations

Abstract We shall provide existence criteria for double positive solutions of the (n,p) boundary value problem δ n y+F(k,y,δy,…,δ n-2 y)=G(k,y,δy…,δ n−1 y) , n−1⩽k⩽N , δ i y(0)=0 , 0⩽i⩽n−2 , δ p y(N+n−p)=0 , where n ≥ 2 and 0 ≤ p ≤ n − 1 is fixed. Upper and lower bounds for the two positive solutions are also established for a particular boundary value problem when n = 2. Several examples are included to dwell upon the importance of the results obtained.

EIGENVALUE CHARACTERIZATION FOR (n; p) BOUNDARY-VALUE PROBLEMS

We consider the.n; p/ boundary value problem y .n/ CH.t; y/DK.t; y/; n 2; t2.0; 1/; y . p/ .1/D y .i/ .0/D 0; 0 i n 2; where >0 and 0 p n 1 is fixed. We characterize the values of such that the boundary value problem has a positive solution. For the special case D 1, we also offer sufficient conditions for the existence of positive solutions of the boundary value problem.