Question

Show that if \(\lambda>\frac{1}{2}\) there does not exist a real-valued function \(u\) such that for all \(x\) in the closed interval \(0 \leq x \leq 1\) $$u(x)=1+\lambda \int_{x}^{1} u(y) u(y-x) d y$$

Short answer

If \(\lambda > \frac{1}{2}\), there does not exist a real-valued function satisfying the equation based on the contradiction derived from the assumption that such a function exists.

Step-by-step solution

Assume a Function Exists

To begin with, assuming that there exists a such real-valued function \(u\) that satisfies the given equation.

Apply Integral Properties

Let's examine the integral part of the equation. One aspect of it, \(u(y)\), is simply the function \(u\) evaluated at \(y\). However, the second part, \(u(y-x)\), is the function \(u\) evaluated at \(y - x\). By the properties of integral, the value of the integral doesn't change if the function evaluated at \(y\) is multiplied by a constant — in this case, that constant would be \(u(y - x)\). Therefore, we know that\[\lambda \int_{x}^{1} u(y) u(y-x) dy\]is a large number, which is at least \(\lambda\). Notice that the inequality comes from the assumption that all numbers \(u(y-u(x))\) are smaller than 1.

Substitution into original equation

Substitute the result obtained from step 2 into the original equation:\[u(x)=1+\lambda \int_{x}^{1} u(y) u(y-x) dy \geq 1 + \lambda,\]since we established that the result of the integral is at least \(\lambda\).

The contradiction

Remember that \(u(x)\) is a real-valued function bounded in the interval \([0,1]\). But according to our assumption, \(u(x)\) would have to equal at least \(1 + \lambda\), which is greater than \(1\) if \(\lambda>\frac{1}{2}\). This contradicts with our original premise that \(u(x)\) is bounded within \([0,1]\).

Conclusion

The contradiction that was revealed means that our assumption — that such a function \(u\) that satisfies the equation exists — was wrong. Therefore, no such function exists if \(\lambda > \frac{1}{2}\).

Key concepts

Real-Valued Function

Understanding real-valued functions is essential for solving complex problems in calculus and analysis, such as the one involving integral equations. A real-valued function is a function that maps elements from a domain, which could include numbers, vectors, or anything else, to real numbers. Mathematically, for a function f, if x is in the function's domain, then f(x) is a real number.

An important aspect of real-valued functions is their behavior and properties over an interval. For instance, if you're given a closed interval, say [0,1], you can study the function's values at every point within this range. In the original exercise, the function is supposed to provide outputs within the confines of this interval. However, specific conditions, as posed in the integral equation, make it challenging to find a function that adheres to all the necessary criteria, particularly when real-valued aspects of the function are involved.

Properties of Integrals

In calculus, the properties of integrals play a pivotal role in solving equations and understanding the behavior of functions. One crucial property is linearity, which states that the integral of a sum is equal to the sum of the integrals. Additionally, if a function is multiplied by a scalar (a real number), the integral is also multiplied by that scalar.

The step-by-step solution applies this property to transform the product of functions within the integral. Remember, the integral itself is a measure of the area under the curve of a function—so when we introduce the scalar, we're essentially scaling this area proportionally. Also, integral bounds (the limits within which we're integrating) are significant, as any change in those can dramatically affect the result of the integral. By exploiting these properties, we can manipulate the original integral equation to draw conclusions about the possibility or impossibility of the function's existence, demonstrating the power of these integral properties.

Function Bounds

Function bounds are a way to describe the limitations or restrictions on the range of function values. A 'bounded' function means that the function's values are contained within a specific numerical interval. This is an essential concept in many areas of mathematics, as it can determine the behavior and properties of a function within a given context.

In the exercise at hand, we have a contradiction regarding function bounds. The assumption was that the function's values would not exceed the interval [0,1]. However, the supposed real-valued function u(x) would have values greater than 1 for \(\lambda > \frac{1}{2}\), which directly conflicts with the bounds set by the problem. Identifying such contradictions is crucial in proving or disproving the existence of a function that meets certain conditions.