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

By Johnny Henderson, Rodica Luca

Boundary price difficulties for structures of Differential, distinction and Fractional Equations: optimistic strategies discusses the idea that of a differential equation that brings jointly a suite of extra constraints known as the boundary conditions.

As boundary price difficulties come up in numerous branches of math given the truth that any actual differential equation can have them, this publication will supply a well timed presentation at the subject. difficulties concerning the wave equation, similar to the decision of ordinary modes, are frequently acknowledged as boundary worth difficulties.

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

  • Explains the platforms of moment order and better orders differential equations with quintessential 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 essential boundary conditions

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

Example text

5, we deduce that problem (S )–(BC) has at least one positive solution. Here, f and g are both sublinear. 4. Let a(t) = 1, b(t) = 4, c(t) = 1, and d(t) = 1 for all t ∈ [0, 1], α = 1, β = 3, γ = 1, δ = 1, α˜ = 3, β˜ = 2, γ˜ = 1, δ˜ = 3/2, H1 (t) = t2 , ⎧ t ∈ [0, 1/3), ⎨ 0, H2 (t) = ⎩ 7/2, t ∈ [1/3, 2/3), 11/2, t ∈ [2/3, 1], K1 (t) = ˆ 0, t ∈ [0, 1/2), 4/3, t ∈ [1/2, 1], ˆ K2 (t) = t3 , f (t, x) = a(xαˆ + xβ ), and g(t, x) = b(xγˆ + xδ ) for all t ∈ [0, 1], x ∈ 1 [0, ∞), with a, b > 0, αˆ > 1, βˆ < 1, γˆ > 2, and δˆ < 1.

Is relatively compact, there exists a subsequence of (Dupk )k which converges in P to some u∗ ∈ P . Without loss of generality, we assume that (Dupk )k itself converges to u∗ —that is, limk→∞ Dupk − u∗ = 0. From the above relation, we deduce that (Dupk )(t) → u∗ (t), as k → ∞ for all t ∈ [0, 1]. 6, we obtain G2 (s, τ )g(τ , upk (τ )) ≤ J2 (τ )p2 (τ )q2 (upk (τ )) ≤ M5 J2 (τ )p2 (τ ) for all s, τ ∈ [0, 1], where M5 = supx∈[0,M4 ] q2 (x) < ∞. 43) where M6 = supx∈[0,β0 M5 ] q1 (x). 43), and Lebesgue’s dominated convergence theorem, we obtain u∗ (t) = lim (Dupk )(t) = (Du)(t), ∀ t ∈ [0, 1]; k→∞ u∗ that is, = Du.

Let v1 (t) = 0 G2 (t, s)g(s, u1 (s)) ds. Then (u1 , v1 ) ∈ P × P is a solution of (S )–(BC). By using (H5), we also have v1 > 0. If we suppose that v1 (t) = 0 for all t ∈ [0, 1], then by using (H5), we have f (s, v1 (s)) = f (s, 0) = 0 for all s ∈ [0, 1]. This implies u1 (t) = 0 for all t ∈ [0, 1], which contradicts u1 > 0. 4 is completed. 5. Assume that (H1)–(H5) hold. If the functions f and g also satisfy the following conditions (H8) and (H9), then problem (S )–(BC) has at least one positive solution (u(t), v(t)), t ∈ [0, 1]: Systems of second-order ordinary differential equations 31 (H8) There exist α1 , α2 > 0 with α1 α2 ≤ 1 such that f (t, u) s (1) f˜∞ = lim sup sup ∈ [0, ∞) α u→∞ t∈[0,1] u 1 and (2) g˜ s∞ = lim sup sup u→∞ t∈[0,1] g(t, u) = 0.

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