Question

Suppose \(f\) is a real function on \((-\infty, \infty) .\) Call \(x\) a fixed point of \(f\) if \(f(x)=x\). (a) If \(f\) is differentiable and \(f^{\prime}(t) \neq 1\) for every real \(t\), prove that \(f\) has at most one fixed point. (b) Show that the function \(f\) defined by $$ f(t)=t+\left(1+e^{t}\right)^{-t} $$ has no fixed point, although \(0

Short answer

Answer: Under the specified conditions, there can be at most one fixed point. Using Banach's Fixed Point Theorem, it can be proved that if the function's derivative meets the given condition, the sequence will converge to the fixed point.

Step-by-step solution

(a) Proving at most one fixed point

Let's assume that there are two distinct fixed points: \(x_1\) and \(x_2\). Thus, we have \(f(x_1) = x_1\) and \(f(x_2) = x_2\). Now, according to the Mean Value Theorem, there exists some \(c\) in the open interval \((x_1, x_2)\) such that \(f'(c) = \frac{f(x_2)-f(x_1)}{x_2-x_1}\). Substituting \(f(x_1) = x_1\) and \(f(x_2) = x_2\), we have \(f'(c) = \frac{x_2-x_1}{x_2-x_1} = 1\). But we are given that \(f'(t) \neq 1\) for every real \(t\). This is a contradiction. Hence, the assumption that there are two distinct fixed points is false. Therefore, there can be at most one fixed point under the given condition.

(b) Showing defined function has no fixed points

We first differentiate the given function \(f(t) = t + (1 + e^t)^{-t}\). We have \[ f'(t) = 1 - \left(\ln(1 + e^t) \right)(1 + e^t)^{-t}. \] Now, we'll analyze the bounds of \(f'(t)\). Since \(\ln(1 + e^t) > 0\) and \((1+e^t)^{-t} < 1\), it follows that \(f'(t) = 1 - (\ln(1 + e^t))(1+e^t)^{-t} < 1\). Moreover, clearly \(f'(t) > 0\) as well. To show that \(f(t)\) has no fixed points, we'll consider the function \(g(t) = f(t) - t\). At a fixed point, we should have \(g(t) = 0\). Now, observe that \(g'(t) = f'(t) - 1 < 0\), therefore \(g(t)\) is a decreasing function. We have \[ g(t) = -\left(\ln(1 + e^t) \right)(1 + e^t)^{-t}. \] As \(t \to -\infty\), the exponent \(-t\) approaches \(\infty\) and \((1 + e^t)^{-t} \to 0\), so \(g(t) \to 0\). On the other hand, as \(t \to \infty\), we have \(\ln(1+e^t) \to \infty\) and \((1+e^t)^{-t} > 0\), so \(g(t) \to -\infty\). Since \(g(t)\) is an increasing function, it cannot have a zero. Hence, function \(f(t)\) has no fixed points.

(c) Proving the existence of fixed points and convergence

We will now use Banach's Fixed Point Theorem (also known as the Contraction Mapping Principle) to prove the existence of fixed points and the convergence of the given sequence to that fixed point. Given that \(|f'(t)| \leq A < 1\), it means \(f\) is Lipschitz continuous with Lipschitz constant \(A\), and this assures that the function has at most one fixed point. Let \(x\) be the fixed point of \(f\), so \(f(x) = x\). Now, consider the sequence \(x_{n+1} = f(x_n)\) with an arbitrary initial point \(x_1\). We have \[ |x_{n+1} - x_n| = |f(x_n) - f(x_{n-1})| \leq A |x_n - x_{n-1}|. \] Iterating this inequality, we get \[ |x_{n+1} - x_n| \leq A^n |x_1 - x_0|. \] Then, using the triangle inequality, we have \[ |x_{n+1} - x| \leq |x_{n+1} - x_n| + |x_n - x| \leq A^n |x_1 - x_0| + |x_n - x|. \] Since \(A < 1\), we have \(A^n |x_1 - x_0| \to 0\) as \(n \to \infty\), and thus \(x_{n+1} \to x\) as \(n \to \infty\). So, \(x = \lim x_n\), which is the fixed point of \(f\).

(d) Visualizing the process

The process described in (c) can be visualized as a zig-zag path moving from the initial point \((x_1, x_2)\) to the fixed point \(x\). The points in the path represent the steps of the iterative procedure: 1. Start at point \((x_1, x_2)\). 2. Move horizontally to \((x_2, x_2)\). 3. Move vertically to \((x_2, x_3)\). 4. Move horizontally to \((x_3, x_3)\). 5. Move vertically to \((x_3, x_4)\). 6. Continue this process indefinitely. As \(n \to \infty\), these points approach the fixed point \(x\). The visualization of this process can help understand the convergence properties and the behavior of the sequence \(x_n\) as it approaches the fixed point.

Key concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental result in calculus that links the behavior of a function over an interval to the function's derivative at some point within that interval. According to the MVT, if a function is continuous on a closed interval \([a, b]\) and differentiable on the open interval \(a, b\), then there exists at least one point \(c\) in \(a, b\) such that:\[f'(c) = \frac{f(b) - f(a)}{b - a}\]This theorem essentially ensures that the tangent to the curve at \(c\) is parallel to the secant line joining the points \((a, f(a))\) and \((b, f(b))\).In our original problem, the MVT was used to determine that a function with no derivative equal to 1 can have at most one fixed point. To apply MVT effectively, simply align the difference in function values to the difference in input values, leading to conclusions about fixed points or roots in calculus problems.

Differentiability

Differentiability is a concept that indicates whether a function has a derivative at every point inside its domain. If a function is differentiable over an interval, it means that graph of the function is smooth without any sharp corners or discontinuities.When analyzing differentiability:

• A differentiable function must be continuous, but a continuous function need not be differentiable.
• The derivative, \( f'(x) \), gives the slope of the function at any given point, providing insight into the function’s rate of change.
• For many problems, like locating fixed points, knowing that the derivative is never equal to some value (like 1, in our case) can lead to significant conclusions.

In our problem, differentiability plays a crucial role because it ensures the function's behavior is regular enough to apply the Mean Value Theorem and other tools of calculus, paving the way to prove results about fixed points.

Contraction Mapping Principle

The Contraction Mapping Principle, also known as Banach's Fixed Point Theorem, is a powerful tool in the context of fixed points and functional analysis. This principle states that if a function brings points closer together, under certain conditions, there is a unique fixed point where the function equals its input.Key characteristics of the Contraction Mapping Principle:

• A function \( f \) is said to be a contraction if there exists a constant \( 0 \leq A < 1 \) such that for all points \( x \) and \( y \), \(|f(x) - f(y)| \leq A|x - y|\).
• The contraction condition ensures that iterations \( x_{n+1} = f(x_n) \) converge to a unique fixed point.
• This principle provides not only the existence but also a method to approximate the fixed point via repeated applications of the function.

In our problem, the proof hinges on this principle to show that if \( f \) is a contraction, starting from any real number and repeatedly applying \( f \) leads the sequence \( x_n \) to converge to the fixed point.

Convergence of Sequences

Convergence of sequences is a fundamental concept in mathematical analysis. A sequence is said to converge if its terms approach a specific value, called the limit, as the sequence progresses towards infinity.In practical terms:

• A sequence \( x_n \) converges to \( L \) if for every small positive number \( \epsilon \), there exists a number \( N \) such that \( |x_n - L| < \epsilon \) for all \( n > N \).
• Convergence is a reassuring property, indicating that a process that begins at an arbitrary point will stabilize at some predictable termination point.
• Understanding convergence is crucial in iterative methods, where the aim is to reach a fixed point or a solution to an equation via successive approximations.

In the context of our problem, the sequence \( x_n \) arising from the iterative process \( x_{n+1} = f(x_n) \) converges to a fixed point thanks to the contraction mapping principle, demonstrating a structured and predictable path to the solution.