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

By Johnny Henderson, Rodica Luca

Boundary price difficulties for platforms of Differential, distinction and Fractional Equations: optimistic ideas 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 publication will offer a well timed presentation at the subject. difficulties concerning the wave equation, reminiscent of the choice of standard modes, are frequently acknowledged as boundary price difficulties.

To be worthy in purposes, a boundary price challenge will be good posed. which means given the enter to the matter there exists a special answer, which relies continually 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 functions are actually 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 matched necessary boundary conditions

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

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If u ∈ M, then there exists λ ≥ 0 such that u(t) = (Au)(t) + λu0 (t), t ∈ [0, 1]. Hence, we have u(t) = (Au)(t) + λ(B y0 )(t) = B (Fu(t)) + λ(B y0 )(t) = B (Fu(t) + λy0 (t)) ∈ P0 , where F: P → P is defined by (Fu)(t) = f t, M ⊂ P0 , and 1 0 G2 (t, s)g(s, u(s)) ds . Therefore, 28 Boundary Value Problems for Systems of Differential, Difference and Fractional Equations u ≤ 1 ν inf t∈[σ ,1−σ ] ∀ u ∈ M. 15) From (1) of assumption (H6), we deduce that there exist C1 , C2 > 0 such that f (t, u) ≥ C1 up − C2 , If i f˜∞ ∀ (t, u) ∈ [σ , 1 − σ ] × [0, ∞).

We introduce m1 = min m1 , m1 , minu+v∈[r3 ,r ] mint∈[σ ,1−σ ] 3 we obtain f (t, u, v) ≥ m1 (u + v), f (t,u,v) u+v > 0. Then ∀ u, v ≥ 0, t ∈ [σ , 1 − σ ]. 1−σ We define λ˜ 0 = νν11m1 A > 0, where A = σ J1 (s)p(s) ds. We shall show that for every λ > λ˜ 0 and μ > 0 problem (S)–(BC) has no positive solution. Let λ > λ˜ 0 and μ > 0. We suppose that (S)–(BC) has a positive solution (u(t), v(t)), t ∈ [0, 1]. Then we obtain Systems of second-order ordinary differential equations 1 u(σ ) = Q1 (u, v)(σ ) = λ 23 G1 (σ , s)p(s)f (s, u(s), v(s)) ds 0 ≥λ 1−σ σ ≥ λm1 G1 (σ , s)p(s)f (s, u(s), v(s)) ds 1−σ σ G1 (σ , s)p(s)(u(s) + v(s)) ds 1−σ ≥ λm1 ν1 σ J1 (s)p(s)ν( u + v ) ds = λm1 νν1 A (u, v) Y.

Then we obtain Systems of second-order ordinary differential equations 1 u(σ ) = Q1 (u, v)(σ ) = λ 23 G1 (σ , s)p(s)f (s, u(s), v(s)) ds 0 ≥λ 1−σ σ ≥ λm1 G1 (σ , s)p(s)f (s, u(s), v(s)) ds 1−σ σ G1 (σ , s)p(s)(u(s) + v(s)) ds 1−σ ≥ λm1 ν1 σ J1 (s)p(s)ν( u + v ) ds = λm1 νν1 A (u, v) Y. > λ˜ 0 m1 νν1 A (u, v) Y, Therefore, we deduce u ≥ u(σ ) ≥ λm1 νν1 A (u, v) Y Y = (u, v) and so (u, v) Y = u + v ≥ u > (u, v) Y , which is a contradiction. Therefore, the boundary value problem (S)–(BC) has no positive solution.

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