Positive Solutions for Second Order Singular Boundary Value Problems with Derivative Dependence on Infinite Intervals

Abstract

The existence of at least one positive solution and the existence of multiple positive solutions are established for the singular second-order boundary value problemusing the fixed point index, where f may be singular at x = 0 and px = 0.

Unbounded Solutions for Functional Problems on the Half-Line

“…Proof. Let , be, respectively, lower and upper solutions of (1), (2) verifying (16). Consider the modified problem…”

“…Lower and upper solutions method is a very adequate technique to deal with boundary value problems as it provides not only the existence of bounded or unbounded solutions but also their localization and, from that, some qualitative data about solutions, their variation and behavior (see [12][13][14]). Some results are concerned with the existence of bounded or positive solutions, as in [15,16], and the references therein. For problem (1), (2) we prove the existence of two types of solution, depending on : if ̸ = 0 the solution is unbounded and if = 0 the solution is bounded.…”

Existence of nonnegative solution for a boundary value problem via upper and lower method on the half-line

“…In this work, we prove the existence results for the following boundary value problem (bvp in short) ( −u 00 = q(t)f (t, u(t), u 0 (t)), t ∈ (0, +∞), u(0) = 0, u 0 (∞) = 0. Boundary value problems posed on the half-line arise in different areas of physics, mechanics, epidemilogy and more generally in applied mathematics (see, e.g., [1], [4], [7], [8], [10], [12], [13], [15], [16], [17]). In these works authors applied some fixed point theorems, a monotone iterative technique, upper and lower solution method.…”

Singular $\phi $-Laplacian third-order BVPs with derivative dependance

“…In this paper, we are concerned with the existence of positive solutions to the following third-order boundary value problem for a φ-Laplacian operator: Many applied problems modeling various phenomena in physics, epidemiology, combustion theory, mechanics (see, e.g., [2] and the references therein) are governed by boundary value problems (bvps for short) posed on the half-axis [0, +∞); we quote for instance the propagation of a flame in a long tube. A large amount of research papers have been devoted to these problems, in particular for the second-order boundary value problems; we refer the reader to [4], [5], [6], [7], [8], [9], [12], [15], [16], [17], and the references therein. However problems with higher-order differential equations on [0, +∞) have not been so extensively investigated; we can only cite [13], [14], [18], and [19].…”