The nonlocal boundary value problem for the parabolic differential equation v'(t) + A(t)v(t) = f(t) (0 ≤ t ≤ T), v(0) = v(λ) + φ, 0 < λ ≤ T in an arbitrary Banach space E with the dependent linear positive operator A(t) is investigated. The well-posedness of this problem is established in Banach spaces C 0 β,γ(Eα−β) of all Eα−β-valued continuous functions φ(t) on [0, T] satisfying a Hölder condition with a weight (t + τ)γ. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.
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机译：抛物型方程v'（t）+ A（t）v（t）= f（t）（0≤t≤T），v（0）= v（λ）+φ，0的非局部边值问题研究了具有相关线性正算子 A （ t ）的任意Banach空间 E 中的<λ≤T。该问题的适定性在Banach空间 C 0 β，γ （ E α-β）的所有 E α-β- [0， T ]上的值连续函数φ（ t ）满足Hölder条件且权重（ t + τ）γ 。建立了Hölder准则中新的Schauder型精确估计，用于求解两个具有相关系数的抛物方程的非局部边值问题。
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