Multiple Unbounded Positive Solutions for Three-Point BVPs with Sign-Changing Nonlinearities on the Positive Half-Line

Abstract

This work is concerned with the existence of unbounded positive solutions for a second-order nonlinear three-point boundary value problem on the positive half-line. The interesting points of the results are that the nonlinearity depends on the solution and its derivative and may change sign. Moreover, it satisfies general polynomial growth conditions. New existence results of nontrivial single and multiple positive solutions are proved using recent fixed point theorems on cones in a special Banach space.

A Novel Approach to Calculation of Reproducing Kernel on Infinite Interval and Applications to Boundary Value Problems

“…According to the former method in [15], which cannot represent reproducing kernel on the infinite interval in polynomial form, the advantages of the present approach are that we use theory of elementary to avoid the complex operation and numerical algorithm will be much more timesaving. On the other hand, the formula in Section 4 can be used to solve multipoint boundary value problems on the positive half-line, such as in [23][24][25].…”

Positive solutions for nonlinear $\phi$-Laplacian third-order impulsive differential equations on infinite intervals

“…Due to some physical laws such that Newton's laws, a large class of such problems are governed by second-order differential equations. This justifies the large amount of research work for second-order bvps that are available in the very recent literature; see, e.g., [2,3,4,5,6] and references therein. For basic mathematical methods to deal with such problems, we refer the reader to [1,7].…”

“…In the last couple of years, some authors investigated the existence of positive solutions for higher order differential equations on finite intervals as well as on infinite intervals of the real line; see, e.g., [8,9,10,11] and the references therein. Notice that positive solutions may refer physically to a position, a temperature, density,... Another interesting direction of the mathematical extension is the natural generalization to problems associated to the p-Laplacian operator s|s| p−2 (p > 1) and more generally to the s → φ -Laplacian operator; see [3,4,5,6,8,12,13,14] and the references. We also mention that many dynamical processes in physics, population dynamics, mechanics, and natural sciences may change state abruptly or be subject to short-term perturbations.…”

Approximate Solution of Nonlinear Multi-Point Boundary Value Problem on the Half-Line

“…Thus, there have been so much work devoted to the existence of solutions for boundary value problem on the half-line, see, for example, [2,7,9] and the references therein. The main methods used to tackle these problems are upper and lower solutions techniques [9,12,14], fixed point theorems in special Banach spaces and recent fixed point theory on cones of Banach spaces [5,6,10,13]. They all discussed the existence and multiplicity of positive solutions of nonlinear differential equations with BVPs on the half-line.…”