Existence, Uniqueness, and Approximation for Solutions of a Functional-integral Equation in Lp Spaces

ABSTRACT In this work we consider the general functional-integral equation: y ( t ) = f ( t , ∫ 0 1 k ( t , s ) g ( s , y ( s ) ) d s ) , t ∈ [ 0 , 1 ] , and give conditions that guarantee existence and uniqueness of solution in L p ( [ 0 , 1 ] ), with 1 1 p ∞.We use Banach Fixed Point Theorem and employ the successive approximation method and Chebyshev quadrature for approximating the values of integrals. Finally, to illustrate the results of this work, we provide some numerical examples.

摘要 本研究针对如下一般泛函积分方程：$y(t) = fleft(t, int_{0}^{1} k(t, s) g(s, y(s)) mathrm{d}s
ight), quad t in [0, 1]$，并给出确保该方程在$1 < p < infty$的$L^p([0,1])$空间中解的存在性与唯一性的条件。本研究采用巴拿赫不动点定理（Banach Fixed Point Theorem），结合逐次逼近法与切比雪夫求积法（Chebyshev quadrature）近似计算积分值。最后，本文提供若干数值算例以举例阐明本研究的相关结论。