# 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: confident suggestions discusses the idea that of a differential equation that brings jointly a suite of extra constraints known as the boundary conditions.

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

To be priceless in purposes, a boundary price challenge could be good posed. which means given the enter to the matter there exists a special answer, 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 bobbing up from medical and engineering purposes are in truth well-posed.

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

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

Sample text

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.

31) Because g(t, 0) = 0 for all t ∈ [0, 1], and g is continuous, it can be shown that there exists a sufficiently small r2 ∈ (0, min{δ1 , 1}) such that g(t, u) ≤ δ1 /m3 for all (t, u) ∈ [σ , 1 − σ ] × [0, r2 ]. Hence, for any u ∈ B¯ r2 ∩ P we obtain 1−σ σ G2 (s, τ )g(τ , u(τ )) dτ ≤ 1−σ σ J2 (τ )g(τ , u(τ )) dτ ≤ δ1 , ∀ s ∈ [σ , 1−σ ]. 6, we deduce that for any u ∈ B¯ r2 ∩ P we have (Au)(t) ≥ 1−σ σ ≥ C9 ≥ C0 1−σ σ 1−σ σ ≥ C0 ν2 1−σ G1 (t, s)f s, G1 (t, s) G1 (t, s) 1−σ σ G1 (t, s) σ 1−σ σ 1−σ σ G2 (s, τ )g(τ , u(τ )) dτ ds G2 (s, τ )g(τ , u(τ )) dτ ds G2 (s, τ )u(τ ) dτ 1−σ σ J2 (τ )u(τ ) dτ ds ds =: (Lu)(t), t ∈ [0, 1], where the linear operator L : P → P is defined by (Lu)(t) = C0 ν2 1−σ σ 1−σ J2 (τ )u(τ ) dτ σ G1 (t, s) ds , t ∈ [0, 1].

1, we conclude that there exists δ1 ∈ (0, 1) such that q2 (x) ≤ C2 xr2 , ∀ x ∈ [0, δ1 ]. 6, for any u ∈ ∂Bδ1 ∩ P0 and s ∈ [0, 1] we obtain 1 1 G2 (s, τ )g(τ , u(τ )) dτ ≤ C2 0 J2 (τ )p2 (τ ) dτ · u = C2 β0 δ1r2 ≤ δ1r2 < 1. 46) and (L5), for any u ∈ ∂Bδ1 ∩ P0 and t ∈ [0, 1], we deduce 1 (Du)(t) ≤ C1 0 G1 (t, s)p1 (s) r1 G2 (s, τ )g(τ , u(τ )) dτ r1 G2 (s, τ )p2 (τ )(u(τ ))r2 dτ ds 0 1 0 1 G1 (t, s)p1 (s) C2 0 ≤ C1 ds 0 1 ≤ C1 1 1 J1 (s)p1 (s) ds · C2 0 r1 J2 (τ )p2 (τ ) dτ · u r1 r2 ≤ u . Systems of second-order ordinary differential equations 43 Therefore, Du ≤ u , ∀ u ∈ ∂Bδ1 ∩ P0 .