The Goldie Equation: III. Homomorphisms from functional equations

In this article, we establish some results for the existence of solution of nonlinear functional integral equations by using Darbo’s fixed point theorem in Banach algebra. As an application, we discuss some examples of nonlinear functional integral equations and existence of solutions.

In this paper we establish the general solution of mixed additive and quadratic functional equation \begin{equation*}\label{5.1} f(x + 2y)+ f(x -2y)+8f(y)=2f(x)+4f(2y)\end{equation*} and investigate the generalized Hyers-Ulam-Rassias stability of this equation in non-Archimedean Banach modules over a unital Banach algebra.

The present paper reveals the existence theorem for the first order functional differential equations in Banach algebras is proved under the mixed generalized Lipschitz and Caratheodory conditions. The existence of extremal solutions is also proved under certain monotonicity conditions.

A special solution of Abel's ordinary differential equation of the first kind is considered, which describes the behaviour of a broad spectrum of solutions of partial differential equations with a small parameter in the neighbourhood of cusp points of their slowly varying equilibrium positions. The existence of this special solution is demonstrated; an asymptotic formula for it as , is constructed and substantiated.

Hilbert [13] suggested in connection with his fifth problem that, while the theory of differential equations provides elegant and powerful techniques for solving functional equations, the inherent differentiability assumptions are typically unnatural (see [4]). Such shortcomings can sometimes be overcome by appealing to regularity theorems of the type "continuity implies differentiability" (see e.g. [4], and w of [1]) and of the type "measurability implies continuity" beginning with Fr~chet [9] and continued by scorses of others (see e.g. [4], [5], [16] and [19] for references to further such results). The theory of distribution of Laurent Schwartz has also been used, perhaps first by Feny5 [8], (see also [2], [6], [11], [12], [20] and [25]) to partially overcome this deficiency. The main aim of the present paper is to introduce another distributional method, based on approximating distributional derivatives by differences, which allows one to find the distributional solutions to distributional analogues of many functional equations by reduction to (distributional) differential equations. This method, when combined with regularity theorems, allows one to find all the Continuous (or even locally Lebesgue integrable) solutions of certain functional equations. References to some of the vast literature concerning many of the functional equations considered herein can be found in [1] and [3]. While many of our results concerning functional equations may be considered as bu"t slight generalizations of known results, our main aim is to illustrate a method of solution. A novelty of our approach is that it permits "local" consideration. Unless otherwise indicated, our notation and terminology is that of Rudin [24]. The set of all Schwartz test function on R n will be denoted by 79~. We let D~ denote the set of Schwartz distributions on R n. All measure-theoretic language refers to Lebesgue measure.

In this paper we derive the explicit, closed-form, recursion-free formulae for the arbitrary-order Fr?chet derivatives of the exponential and logarithmic functions in unital Banach algebras (complex or real). These computations are obtained via the Bochner integrals for the Banach algebra valued functions, with respect to the standard Lebesgue measure. As an application, we utilize our results in the approximation schemes of the solutions to stochastic functional differential equations.

It ist shown that the ordinary non‐linear second order differential equation of the phenomenological theory of the one‐dimensional steady shock wave in heat conducting viscous gases can be transformed into the Abelian differential equation of the second class. Among the known special solutions of Abel's equation is one applicable to the shock wave and leads to a hitherto unknown closed form solution.

Following a 70-year old suggestion of Paul Lévy, a condition is formulated for the regularity of growth of real functions. The condition, which is quite explicit, makes use of the iterates of the function and solutions of Abel's functional equation, and is well adapted to a computer testing. Numerous computer experiments reveal interesting properties of the proposed regularity criterion.

We prove the generalized Hyers-Ulam stability of homomorphisms and generalized derivations associated to the following functional equation on Banach algebras.

In this paper, we investigate a general multivariable functional equation. We prove, using the fixed-point method, the generalized Hyers–Ulam stability of this equation in Banach spaces. In this way, we obtain sufficient conditions for the stability of a wide class of functional equations and control functions. We also show, using examples, how some additional assumptions imposed on the function when examining the Hyers–Ulam stability of a functional equation affect the size of the approximating constant and limit the number of considered solutions for this equation. The functional equation studied in this paper has symmetric coefficients (with precision up to the sign), and it is a generalization of an equation characterizing n-quadratic functions, as well as many other functional equations with symmetric coefficients: for example, the multi-Cauchy equation and the multi-Jensen equation. Our results generalize many known outcomes.

Schroder developed several notions which are central to the study of complex dynamics. Despite his failure to rigorously establish his fixed point theorem, it is a fundamental result, and his belief that iteration of an arbitrary function ∅(z) could be reduced to the solution of the so-called Abel and Schroder functional equations was prophetic. Not only was the next phase in the development of complex dynamics ushered in by an interest in the solution of the Schroder and Abel equations, but the study and solution of functional equations is fundamental in many contemporary studies of iteration.

This paper discussed some existence theorems for nonlinear functional integral equations in the space L^1 of Lebesgue integrable functions,by using the Darbo fixed point theorem associated with the Hausdorff measure of noncompactness. Also, as an application, we discuss the existence of solutions for some nonlinear integral equations with fractional order.

In this paper the functional equation¶¶\( F\Bigl(x + G(t(H(y - x))\Bigr) = S\Bigl(A(t)K(x),B(t)K(y)\Bigr), \qquad x,y \in I, x \le y, t \in [0,1] \),¶is solved under monotonicity and injectivity assumptions for the unkown functions involved. This equation turned out to be useful in the axiomatic treatment of various joint-receipt utility structures. After several reduction steps, the solution of the above equation is reduced to that of¶¶\( \Gamma^{-1}\bigl(\Gamma\bigl(\frac{H(u)}{H(u + v)}\bigr) \cdot \frac{H(x)}{H(x + u + v)}\bigr) = \frac{H(u)H(x + u + v)}{H(u + v)H(x + u)},\qquad x,u,v > 0,\,x + u + v\in I \).¶Assuming that H and \( \Gamma \) are strictly monotonic, it is shown that H is a log-concave function, and as a consequence, using Lebesgue's theorem, the everywhere-differentiability of \( \Gamma \) is deduced. Then, differentiating this equation, a functional differential equation for H is derived. The solutions of this new equation are then determined and the form of \( \Gamma \) is described. Thus, the general solution of the original equation is also obtained.

Consider the functional equation ℑ1(f) = ℑ2(f) (ℑ)in a certain general setting. A function g is an approximate solution of (ℑ)if ℑ1(g)and ℑ2(g)are close in some sense. The Ulam stability problem asks whether or not there is a true solution of (ℑ)near g. A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each m = 1,2,3,4, we will find out the general solution of the functional equation f ( ax + y ) + f ( ax - y ) = a m - 2 [ f ( x + y ) + f ( x - y ) ] + 2 ( a 2 - 1 ) [ a m - 2 f ( x ) + ( m - 2 ) ( 1 - ( m - 2 ) 2 ) 6 f ( y ) ] Open image in new window for any fixed integer a with a ≠ 0, ± 1. Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions.

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