On Global Solvability of Nonlinear Equations with Parameters

We consider smooth mappings acting from one Banach space to another and depending on a parameter belonging to a topological space. Under various regularity assumptions, sufficient conditions for the existence of global and semilocal continuous inverse and implicit functions are obtained. We consider applications of these results to the problem of continuous extension of implicit functions and to the problem of coincidence points of smooth and continuous compact mappings.

Given Banach spaces X and Y, a topological space Σ, and a continuous mapping we consider the equation (1) where is the unknown and is a parameter. A continuous function defined on Σ or on a subset of and satisfying the identity is called an implicit function.
This paper presents global implicit function theorems, i.e., conditions under which, for all values of , Eq. (1) is solvable and its solution depends continuously on σ. As a special case, we obtain a global (and semilocal) inverse function theorem providing sufficient conditions for the existence of a continuous right inverse of a given smooth mapping defined on a given ball in Y. As applications of the implicit function theorem, we derive results concerning a continuous extension of an implicit function, an ε-implicit function, and the existence of coincidence points.
A global inverse function theorem goes back to Hadamard, who proved in [1] (see also [2]) that, if and a continuously differentiable mapping is uniformly nonsingular, i.e., the lin- ear operator is invertible for each and the function is bounded on X, then F is a diffeomorphism. Here and below, denotes the norm of a linear bounded operator and the norm in the spaces X and Y.
Equation (1) is of particular interest when X and Y are not linearly topologically isomorphic spaces, for example, when , and In this case, global implicit and inverse function theorems were obtained in [3] assuming that the mapping f is smooth in x. A global inverse function theorem for locally Lipschitz mappings was derived in [4].
Let us extend the result of [3] to Hilbert spaces. For this purpose, we first introduce the necessary concepts.
As usual, is the space of linear bounded operators A acting from a Banach space X to a Banach space Y, and is the set of all surjective operators By , we denote a closed ball in X of radius centered at the point and the same notation is used for balls in Given a linear operator , we define The Banach open mapping theorem means that > 0 if and only if The Michael selection theorem (see [5]) guarantees that, if Theorem 1 is proved by applying methods of the theory of ordinary differential equations; more specifically, the proof consists in an analysis of solutions to some Cauchy problem based on the mapping f (see [3]).
Of particular interest and complexity is the situation when X and Y are Banach spaces. In this case, the above-described method of reducing Eq. (1) to a Cauchy problem cannot be applied, since there exist Banach spaces X and Y and a linear operator such that A does not have a continuous linear (or even nonlinear, but locally Lipschitz) right inverse mapping (for more details, see [6]). Accordingly, the problem of a global implicit function in Banach spaces requires a different approach.
As such an approach, it is possible to use a method based on sufficient conditions for the existence of a minimum of lower semicontinuous functionals (see [7,8]). By applying the results of [7,8], the following assertion can be proved.
Let be a given continuous function (where is the set of nonnegative real numbers). For  : If assumptions (2) and (3) are both violated, they can be satisfied by decreasing the set Σ (and, accordingly, reducing the left-hand side of (3) and increasing the right-hand side of (3)). In this context, the following natural question arises: under what conditions does the set exist? Smoothness and continuity considerations imply that, if there is a point × Σ such that and then there exists a neighborhood of the point in which (3) holds for By using this argument, it is easy to derive local implicit function theorems from Theorems 1 and 2.
A simple sufficient condition under which relation (2) holds is (7) This condition corresponds to the assumptions of the Hadamard theorem, but it is more burdensome than condition (2), which was, in fact, used in [9,10].
Below, we give an example in which condition (2) holds for any bounded function ϕ, while the condition (7) is violated. Let be an arbitrary continuous even function such that u decreases on , and Note that u can be specified, for example, as u(t) = where Define Obviously, the mapping f is continuously differentiable.
From Theorem 2, we can derive its semilocal version.

Corollary 1. Suppose that X and Y are Banach spaces and the mapping f satisfies Assumptions A1-A3.
Then, given any continuous mapping and any for which For an inverse function theorem, conditions (2) and (3) take a simpler form. Below is a corresponding inverse function theorem following from Theorem 2.
Let X and Y be Banach spaces and be a given continuously differentiable mapping.

Theorem 3. Assume that the derivative is bounded
on any bounded set and the mapping F is uniformly differentiable, i.e., Then, given any point and any R > 0 satisfying (8) for any there exists a continuous function such that for all In particular, if (11) then there exists a continuous function satisfying conditions (9) and (10) for all Theorem 3 is semilocal, since it guarantees the existence of a continuous inverse function G defined on a given ball Similar to Theorem 3, an assertion for Hilbert spaces can be derived from Theorem 1.
Corollary 1 implies the following version of Theorem 3, which is also a semilocal inverse function result.
Corollary 2. Assume that the mapping F is uniformly differentiable and the derivative is bounded on any bounded set. Then, given any point and any r > 0 for which Under the assumptions of Theorem 3, an inverse G of F that is smooth or at least satisfies the Lipschitz condition in a neighborhood of the point may not exist even if F is infinitely differentiable. This conclusion is explained by the fact that, as was noted above, there exist linear operators from one Banach space to another that do not have a Lipschitz right inverse. For some classes of Banach spaces X and Y, including the class of all Hilbert spaces, global theorems on smooth and locally Lipschitz inverse and implicit functions were obtained in [11].
An important feature of the above-presented results is a priori estimates (5), (6), and (9) obtained for implicit and inverse functions. These estimates have various applications.
We begin with the following theorem on continuous extension of an implicit function.
Theorem 4. Suppose that the topological space Σ is Hausdorff and paracompact and the mapping f satisfies Assumptions A1-A3 and (7).
Then, for any closed subset and any continuous function for which there exists a continuous function such that Another application of Theorem 2 is the following theorem on an approximate implicit function.
Given a continuous mapping is called an ε-implicit function if In other words, ϕ is an "implicit function up to ε." Proposition 1. Suppose that a mapping f satisfies Assumptions A1-A3 and (7). Then, for any ε-implicit function ϕ, there exists a continuous function such that The last property is known as Ulam-Hyers stability (see [12]).
The following result is a corollary to Theorem 3 and represents a generalization of the Brouwer and Schauder fixed point theorems to the coincidence point problem. Recall that a point is called a coincidence point of two mappings if Proposition 2. Suppose that a mapping satisfies the assumptions of Theorem 3.