Existence of a positive solution to a first-order -Laplacian BVP on a time scale

Goodrich, Christopher S.

doi:10.1016/j.na.2010.10.062

Cited by 23 publications

(11 citation statements)

References 45 publications

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“…On the other hand, the more general study of nonlocal BVPs has recently seen excellent progress, largely due to the important work of Graef et al [15,16,1,[17][18][19]. Such studies have also occurred in the context of fractional BVPs in both the discrete and the continuous setting as well as in the time scales setting-see, for instance, [20][21][22][23][24] and certain of the references therein.…”

Section: Introductionmentioning

confidence: 99%

On nonlinear boundary conditions satisfying certain asymptotic behavior

Goodrich

2013

Nonlinear Analysis: Theory, Methods & Applications

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Section: Introductionmentioning

confidence: 99%

On nonlinear boundary conditions satisfying certain asymptotic behavior

Goodrich

2013

Nonlinear Analysis: Theory, Methods & Applications

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“…BVPs possessing either nonlinear and/or nonlocal BCs have seen a great deal of study recently, spanning such areas as first-and second-order problems, coupled systems of second-order problems, and higher-order problems -see, for example, recent works by Franco et al [10], Goodrich [11][12][13][14][15][16][17], Graef et al [18], Infante et al [22][23][24][25], Kang et al [26], Webb and Infante [31], Yang [32,33], and the references therein. In addition, the study of BVPs with specifically integral boundary conditions has also seen much attention in recent years, and one may consult [5,19,27] and the references therein for some recent examples.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, the study of BVPs with specifically integral boundary conditions has also seen much attention in recent years, and one may consult [5,19,27] and the references therein for some recent examples. The papers [2,11,28,30] and the references therein provide a broad introduction, nonetheless, to this line of research. The papers [2,11,28,30] and the references therein provide a broad introduction, nonetheless, to this line of research.…”

Section: Introductionmentioning

confidence: 99%

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

Goodrich

2014

APPL ANAL DISCRETE M

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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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“…On the other hand, integer-order p-Laplacian boundary value problems have been widely studied owing to its importance in theory and application of mathematics and physics, see for example, [29][30][31][32][33] and the references therein. Especially, in [29], by using the fixed point index method, Yang and Yan investigated the existence of positive solution for the third-order Sturm-Liouville boundary value problems with pLaplacian operator (φ p (u (t)) + f (t, u(t)) = 0, t ∈ (0, 1),…”

Section: Introductionmentioning

confidence: 99%