Boundary value problems for systems of differential, by Johnny Henderson, Rodica Luca

By Johnny Henderson, Rodica Luca

Boundary worth difficulties for structures of Differential, distinction and Fractional Equations: optimistic strategies discusses the concept that of a differential equation that brings jointly a collection of extra constraints referred to as the boundary conditions.

As boundary worth difficulties come up in different branches of math given the truth that any actual differential equation may have them, this publication will supply a well timed presentation at the subject. difficulties regarding the wave equation, reminiscent of the choice of ordinary modes, are frequently acknowledged as boundary price difficulties.

To be precious in functions, a boundary worth challenge may be good posed. which means given the enter to the matter there exists a distinct resolution, which relies consistently at the enter. a lot theoretical paintings within the box of partial differential equations is dedicated to proving that boundary worth difficulties coming up from medical and engineering purposes are in reality well-posed.

  • Explains the structures of moment order and better orders differential equations with fundamental and multi-point boundary conditions
  • Discusses moment order distinction equations with multi-point boundary conditions
  • Introduces Riemann-Liouville fractional differential equations with uncoupled and matched fundamental boundary conditions

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Extra info for Boundary value problems for systems of differential, difference and fractional equations : positive solutions

Example text

1. The results presented in this section were published in Henderson and Luca (2012e). 3 with a(t) = c(t) = 1 and b(t) = d(t) = 0 for all t ∈ [0, 1], α = α˜ = 1, β = β˜ = 0, γ = γ˜ = 1, δ = δ˜ = 0, H1 and K1 are constant functions, and H2 and K2 are step functions—that is, the integral boundary conditions become multipoint boundary conditions. Namely, we consider here the system of nonlinear second-order ordinary differential equations u (t) + f (t, v(t)) = 0, t ∈ (0, T), v (t) + g(t, u(t)) = 0, t ∈ (0, T), (S0 ) with the multipoint boundary conditions ⎧ m−2 ⎪ ⎪ ⎪ bi u(ξi ), ⎪ ⎨ u(0) = 0, u(T) = ⎪ ⎪ ⎪ ⎪ ⎩ v(0) = 0, v(T) = i=1 n−2 (BC0 ) ci v(ηi ), i=1 where m, n ∈ N, m, n ≥ 3, bi , ξi ∈ R for all i = 1, .

1−σ We define μ˜ 0 = νν21m2 C > 0, where C = σ J2 (s)q(s) ds. We shall show that for every μ > μ˜ 0 and λ > 0 problem (S)–(BC) has no positive solution. Let μ > μ˜ 0 and λ > 0. We suppose that (S)–(BC) has a positive solution (u(t), v(t)), t ∈ [0, 1]. Then we obtain 1 v(σ ) = Q2 (u, v)(σ ) = μ G2 (σ , s)q(s)g(s, u(s), v(s)) ds 0 ≥μ 1−σ σ ≥ μm2 G2 (σ , s)q(s)g(s, u(s), v(s)) ds 1−σ σ ≥ μm2 ν2 G2 (σ , s)q(s)(u(s) + v(s)) ds 1−σ σ J2 (s)q(s)ν( u + v ) ds = μm2 νν2 C (u, v) Y. 24 Boundary Value Problems for Systems of Differential, Difference and Fractional Equations Therefore, we deduce v ≥ v(σ ) ≥ μm2 νν2 C (u, v) Y > μ˜ 0 m2 νν2 C (u, v) Y = (u, v) Y, and so (u, v) Y = u + v ≥ v > (u, v) Y , which is a contradiction.

58) satisfies u(t) ≥ 0 for all t ∈ [0, T]. 4 (Henderson and Luca, 2012e). If α ≥ 0, β ≥ 0, ai ≥ 0 for all i = 1 1, . . 58) satisfies αT + β d αξj + β u(ξj ) ≥ d T u(t) ≤ (T − s)y(s) ds, 0 ≤ t ≤ T, (T − s)y(s) ds, ∀ j = 1, . . , m. 5 (Li and Sun, 2006). Assume that α ≥ 0, β ≥ 0, ai ≥ 0 for all m i = 1, . . , m, 0 < ξ1 < · · · < ξm < T, T > i=1 ai ξi > 0, d > 0, and y ∈ 1 C(0, T) ∩ L (0, T), y(t) ≥ 0 for all t ∈ (0, T). 58) satisfies inft∈[ξ1 ,T] u(t) ≥ r1 supt ∈[0,T] u(t ), where r1 = min m ai (T − ξi ) ξ1 , i=1 m , T T − i=1 ai ξi m i=1 ai ξi T , s−1 i=1 ai ξi + T− m i=s ai (T m i=s ai ξi − ξi ) , s = 2, .

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