The lower and upper solution method for three-point boundary value problems with integral boundary conditions on a half-line

Abstract

This paper deal with the following second-order three-point boundary value problem with integral boundary condition on a half-line u (x) + q(x) f (x, u(x), u (x)) = 0, x ∈ (0, +∞), u(0) = λ η 0 u(s)ds, u (+∞) = C, where λ > 0, 0 < λη < 1 and f : [0, +∞) × R 2 → R satisfies a Nagumo's condition which plays an important role in the nonlinear term depend on the first-order derivative explicitly. By using Schauder's fixed point theorem, the upper and lower solution method and topological degree theory, first we give sufficient conditions for the existence of at least one solution and next at least three solutions of the above problem. Moreover, an example is included to demonstrate the efficiency of the main results.

Existence of solutions for a third-order boundary value problem with integral boundary conditions

“…Nagumo koşulunu sağlayan yarı sonsuz aralık üzerinde ikinci mertebeden üç noktalı integral koşullu sınır değer problemi, U. Akcan ve E. Çetin tarafından çalışılmıştır [17].…”

Positive solutions of singular Hadamard-type fractional differential equations with infinite-point boundary conditions or integral boundary conditions

“…Due to the fact that fractional-order models are more accurate than integer-order models (that is, there are more degrees of freedom in fractional-order models), the subject of fractional differential equations has recently evolved into an interesting subject for many researchers due to its multiple applications in economics, engineering, physics, chemistry, mechanics. However, most of the results for fractional differential equations are concerned with the Riemann-Liouville fractional derivative or the Caputo fractional derivative (see for example Agarwal et al [1], Akcan and Çetin [4], Bai and Qiu [5], Bai and Sun [6], Callegari and Nachman [9], Chalishajar and Kumar [10], El-Saka et al [11], El-Sayed et al [12], Kosmatov [19], Li and Zhang [21], Liu et al [22], Qiao and Zhou [26], Qiu and Bai [27], Rida et al [28], Song et al [30], Staněk [31], Tian and Chen [33].…”

A Class of Nonlinear Boundary Value Problems for an Arbitrary Fractional-Order Differential Equation with the Riemann-Stieltjes Functional Integral and Infinite-Point Boundary Conditions

“…BVPs with integral BCs arise naturally in semiconductor problems [10], thermal conduction problems [11], hydrodynamic problems [12], population dynamics model [13], and so on (see also [14]). Recently, these BVPs were extensively studied by (among others) Akcan and Çetin [15], Boucherif [16], Benchohra et al [17], Chalishajar and Kumar [18], Dou et al [19], Li and Zhang [20], Liu et al [21], Song et al [22], Tokmagambetov and Torebek [23], Wang et al [24] and Yang and Qin [25] (see also the references to the related earlier works which are cited in each of these investigations).…”