Yanping Guo

Positive solutions to (n-1,1) m-point boundary value problems with dependence on the first order derivative

Using the extension of Krasnoselskii’s fixed point theorem in a cone, we prove the existence of at least one positive solution to the nonlinear nth order m-point boundary value problem with dependence on the first order derivative. The associated Green’s function for the nth order m-point boundary value problem is given, and growth conditions are imposed on the nonlinear term f which ensures the existence of at least one positive solution. A simple example is presented to illustrate applications of the obtained results.

Existence of Solutions for m-point Boundary Value Problems on a Half-Line

By using the Leray-Schauder continuation theorem, we establish the existence of solutions for -point boundary value problems on a half-line , where and are given.

Positive solutions for second-order quasilinear multi-point boundary value problems☆

Abstract In this paper a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for second-order quasilinear m -point boundary value problem ( ϕ p ( x ′ ( t ) ) ) ′ + a ( t ) f ( t , x , x ′ ) = 0 , 0 ⩽ t ⩽ 1 , x ′ ( 0 ) = 0 , x ( 1 ) = ∑ i = 1 m - 2 α i x ( ξ i ) , where f is a nonnegative continuous function and ϕ p ( x ) = ∣ x ∣ p - 2 x , p > 1 . A simple example is presented to illustrate the applications of the obtained results.

Existence of Triple Positive Solutions for Second-Order Discrete Boundary Value Problems

By using a new fixed-point theorem introduced by Avery and Peterson (2001), we obtain sufficient conditions for the existence of at least three positive solutions for the equation Δ2x(k−1)

Triple Positive Solutions of a Nonlocal Boundary Value Problem for Singular Differential Equations with p-Laplacian

We establish the existence of triple positive solutions of an m-point boundary value problem for the nonlinear singular second-order differential equations of mixed type with a p-Laplacian operator by Leggett-William fixed point theorem. At last, we give an example to demonstrate the use of the main result of this paper. The conclusions in this paper essentially extend and improve the known results.

Solvability of second-order m-point difference equation boundary value problems on infinite intervals

In this paper, we study second-order m-point difference boundary value problems on infinite intervals  ∆2x(k− 1) + f(k, x(k),∆x(k− 1)) = 0, k ∈ N, x(0) = m−2 ∑ i=1 αix(ηi), lim k→∞∆x(k) = 0, where N = {1, 2, · · · }, f : N× R2 → R is continuous, αi ∈ R, m−2 ∑ i=1 αi 6= 1, ηi ∈ N, 0 < η1 < η2 < · · · <∞ and ∆x(k) = x(k+ 1) − x(k), the nonlinear term is dependent in a difference of lower order on infinite intervals. By using Leray-Schauder continuation theorem, the existence of solutions are investigated. Finally, we give one example to demonstrate the use of the main result. c ©2017 All rights reserved.

Computation of positive solutions for nonlinear fractional q-difference equations with nonlocal conditions

In this paper, we discuss the existence of positive solutions for a nonlinear fractional q-difference equations boundary value problem {(D^α_qu)(t) + a(t)f(u(t))=0, 0<;t<;1, u(0) = (D_qu)(0) = 0, (D_qu)(1) = βu(η), where 0 <; q <; 1, 2 <; α ≤ 3, 0 <; η <; 1, 0 <; βη^α-1 <; [α-1]_q, by applying a fixed point theorem on cone. At last, we give an example to illuminate the results by computation.

Positive Solutions for a Class of Singular Boundary Value Problems with Fractional -Difference Equations

We discuss a class of singular boundary value problems of fractional -difference equations. Some existence and uniqueness results are obtained by a fixed point theorem in partially ordered sets. Finally, we give an example to illustrate the results.

The existence of symmetric positive solutions for a seconder-order difference equation with sum form boundary conditions

In this paper, we consider the existence of positive solutions for a second-order discrete boundary value problem Δ(g(k−1)Δu(k−1))+w(k)f(k,u(k))=0 subject to the boundary conditions: au(0)−bg(0)Δu(0)=∑i=1n−1h(i)u(i), au(n)+bg(n−1)Δu(n−1)=∑i=1n−1h(i)u(i), where a,b>0, Δu(k)=u(k+1)−u(k) for k∈{0,1,…,n−1}, g(k)>0 is symmetric on {0,1,…,n−1}, w(k) is symmetric on {0,1,…,n}, f:{0,1,…,n}×[0,+∞) is continuous, f(k,u)=f(n−k,u) for all (k,u)∈{0,1,…,n}×[0,+∞), and h(i) is nonnegative and symmetric on {0,1,…,n}. By the fixed point theorem and the Hölder inequality, we study the existence of symmetric positive solutions for the above difference equation with sum form boundary conditions.

The Existence of Positive Solutions for a Fourth-Order Difference Equation with Sum Form Boundary Conditions

We consider the fourth-order difference equation: ,   subject to the boundary conditions: , , , where and for ,   is continuous. is nonnegative ; is nonnegative for . Using fixed point theorem of cone expansion and compression of norm type and Holder’s inequality, various existence, multiplicity, and nonexistence results of positive solutions for above problem are derived, which extends and improves some known recent results.