A Fold Bifurcation Theorem of Degenerate Solutions in a Perturbed Nonlinear Equation
We consider a nonlinear equation , where the parameter is a perturbation parameter, is a differentiable mapping from to and , are Banach spaces. We obtain an abstract bifurcation theorem by using the generalized saddle-node bifurcation theorem.
In [1, 2], Crandall and Rabinowitz proved two celebrated theorems which are now regarded as foundation of the analytical bifurcation theory in infinite-dimensional spaces and both results are based on the implicit function theorem. In , we obtained the generalized saddle-node bifurcation theorem by the generalized inverse. In , we proved a perturbed problem using Morse Lemma. For a more general introduction to bifurcation theory and other related methods in nonlinear analysis, see, for example, [5–7]. On the other hand, [8–11] provide a more detailed introduction to mathematical models in some recent new results in the application of bifurcation theory including chemical reactions, population ecology, and nonautonomous differential equations.
In this paper, we continue the work of  and obtain an abstract bifurcation theorem under the opposite condition in . We consider the solution set of where indicates the perturbation. Fix ; let be a solution of . From the implicit function theorem, a necessary condition for bifurcation is that is not invertible; we call a degenerate solution. In , Shi shows the persistence and the bifurcation of degenerate solutions when varies near by the implicit function theorem and the saddle-node bifurcation theorem. In this paper, we prove a new perturbed bifurcation theorem by the generalized saddle-node bifurcation theorem.
In the paper, we use as the norm of Banach space and as the duality pair of a Banach space and its dual space . For a nonlinear operator , we use as the partial derivative of with respect to argument . For a linear operator , we use as the null space of and as the range of .
Definition 2.1 (see ). Let , be Banach spaces, and let be a linear operator. Then, is called the generalized inverse of if it satisfies(i),(ii).
Lemma 2.3 (see ). Let , then is a generalized regular operator if and only if are topologically complemented in , respectively. In this case, , are bounded linear projectors from , into , , respectively.
We recall the generalized saddle-node bifurcation in  and give an alternate proof here using the generalized Lyapunov-Schmidt reduction.
Theorem 2.4 (generalized saddle-node bifurcation). Let be a neighborhood of . Suppose that (i);(ii) is a generalized regular operator, and
Let , then the subset contains the curve near , where , the mapping is continuously differentiable near , and .
Proof. Since is a generalized regular operator, there exist closed subspaces in , in satisfing , .
Taking an arbitrary , from Lemma 2.3, is equivalent to where , .
Define as because of (iii), then is an isomorphism.
For the equation , by the implicit function theorem, there exist and , with , satisfying From (2.2), we have Differentiating (2.5) with respect to , we have Setting , Thus, since (iii) and we have that is, , we have .
Corollary 2.5. Assume the conditions in Theorem 2.4 are satisfied and then the direction of the solution curves is determined by where , . Furthermore, when is satisfied, , and the solution curve is a parabola-like curve which reaches an extreme point at .
We illustrate our result by a simple example.
Example 2.6. Define where , . From simple calculations, we obtain We analyze the bifurcation at . It is easy to see that , where , , . So, obviously, , , and . From the above calculation, Obviously, and , . Thus, we can apply Corollary 2.5 to (2.12). In fact, all solution curves for all form a surface (see Figure 1).
3. Main Theorems
Applying Theorem 2.4, we discuss the bifurcation of solutions of the perturbed problem. We consider the solution set of where the parameter indicates the perturbation, , , and , are Banach spaces. Let
Suppose that is a solution of . For and by Hahn-Banach theorem, there exists a closed subspace of with codimension 1 such that , where and . Let . Then, is a closed hyperplane of with codimension 1. Since is a closed subspace of and is also a Banach space in the subspace topology, Hence we can regard as a Banach space with product topology. Moreover, the tangent space of is homeomorphic to (see  for more on the setting).
In the following, we will still use the conditions () on defined in .(F1), and ;(F2);(F3);(F4);(F5).
We use the convention that () means that the condition defined in () does not hold.
Theorem 3.1. Let , such that . Suppose that the operator satisfies , , , , and at . One also assumes that where is the unique solution of Then, there exists such that all the solutions of near form two curves: where , ; ; , and where , , , , are the unique solution of respectively.
Proof. We apply Theorem 2.4 to the operator , so we need to verify all the conditions. We define a differential operator ,
(1) . Suppose that and . If , from and , then we have and From , we can define is the unique solution of (3.8). Thus, and .
Next, we consider . Without loss of generality, we assume that . Notice that from , we can define that is unique solution of (3.5). Substituting , into (3.10), we have From (3.4), there exists a unique satisfies (3.9). Then, that is, .
(2) . We only claim that Let and satisfy Using (3.15) and , then and .
Conversely, for any , from , set where . From , we have Set , we obtain that Substituting , into (3.16), we have Using , , then there exists , satisfies Substituting (3.21) into (3.20), we have Applying to (3.22), we have because of the definition of and Thus we can define Therefore, , that is, . Hence, . That is, .
(3) . Since , we need only to show that but that is exactly assumed in . So, the statement of the theorem follows from Theorem 2.4.
4. Calculations of Bifurcation Directions
In Theorem 3.1, we have , , , , , , , , , , .
To completely determine the turning direction of curve of degenerate solutions, we need some calculations.
Let be a curve of degenerate solutions which we obtain in Theorem 3.1. Differentiating with respect to , we obtain Setting in (4.1), we get exactly , (3.5), (3.8), and (3.9). We differentiate (4.1) again, and we have (omit the subscript in the equation)
Setting in (4.2), we obtain And applying to it, we have Using , , . From (4.7), (4.5) implies . Setting in (4.4), Substituting the expression of into (4.9), we have And applying to it, we obtain , that is, Assume and applying to (4.10), We differentiate (4.2) again: Setting in (4.14), we obtain Substituting the expression of into (4.15) and applying to it, we have using (3.4), (4.12), and .
The authors the referee for very careful reading and helpful suggestions on the paper. The paper was partially supported by NSFC (Grant no. 11071051), Youth Science Foundation of Heilongjiang Province (Grant no. QC2009C73), Harbin Normal University academic backbone of youth project, and NCET of Heilongjiang Province of China (1251–NCET–002).
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