On a first-order semipositone boundary value problem on a time scale

Abstract

We consider the existence of a positive solution to the first-order dynamic equation y ∆ (t)+p(t)y σ (t) = λf (t, y σ (t)) , t ∈ (a, b) T , subject to the boundary condition y(a) = y(b) + τ 2 τ 1 F (s, y(s)) ∆s for τ1, τ2 ∈ [a, b] T . In this setting, we allow f to take negative values for some (t, y). Our results generalize some recent results for this class of problems, and because we treat the problem on a general time scale T we provide new results for this problem in the case of differential, difference, and q-difference equations. We also provide some discussion of the applicability of our results.= e ⊖p (t, s);4. e p (t, s)e p (s, r) = e p (t, r); 5. if p is nonnegative and a ≤ t ≤ b, then e p (a, t) ≤ e p (b, t); and 6. if p is nonnegative and a ≤ t ≤ b, then e p (a, t) ≤ e p (a, a) = 1.Let us next state the cone in which we search for positive solutions of problem (1). In particular, let B be the Banach space C rd ([a, b] T ) when equipped with the usual supremum norm · . We then define the cone K ⊆ B by K := {y ∈ B : y(t) ≥ e p (a, b) y for each t ∈ [a, b] T } .

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Existence of solutions for a first-order nonlocal boundary value problem with changing-sign nonlinearity

“…In a recent paper [8], using Krasnosel'skiȋ's fixed point theorem, Goodrich studied the existence of a positive solution to the first-order problem given by (if T = R)…”

“…In [8], Goodrich found the upper and lower bounds for Green's function on the general time scales in Lemma 2.4. Since the following lemma can be proven in a similar way, we give the lemma without the proof.…”

“…In this paper, we are interested in the existence of positive solutions for the following first-order m-point nonlocal boundary value problem: First-order equations with various boundary conditions, including multipoint and nonlocal conditions, are of recent interest; see [3][4][5][6][7][8][9][10][11]13] and the references therein.…”

Countably Infinitely Many Positive Solutions for Even Order Boundary Value Problems with Sturm-Liouville Type Integral Boundary Conditions on Time Scales

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