# Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Existence Theorem of Bifurcation Points and Manifolds of Nonlinear Equations

**Definition**

**1.**

**Corollary**

**1.**

**Definition**

**2.**

**Corollary**

**2.**

**Theorem**

**1.**

**Proof.**

**Condition**

**1.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Definition**

**3.**

**Lemma**

**1.**

**Corollary**

**4.**

**Proof.**

**Condition**

**2.**

**Lemma**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

**Corollary**

**5.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Lemma**

**3.**

**Proof.**

**Example**

**4.**

## 3. Solutions Parametrization and Iterations in Branch Points Neighborhood

**Condition**

**3.**

**Definition**

**4.**

- $(\xi ,\alpha )\ge \theta $ for $\xi \in \mathrm{s}upp\phantom{\rule{0.166667em}{0ex}}L,$
- $l\cap supp\phantom{\rule{0.166667em}{0ex}}L\ne 0.$

**Condition 4.**Let an algebraic system

**Lemma**

**4.**

**Theorem**

**4.**

## 4. N-Step Iteration Scheme for Construction of The Solution of Equation (13)

**full rank**solution. Let $r=min({\alpha}_{1},\cdots ,{\alpha}_{n+1}).$ Solution $x\left(\epsilon \right),\phantom{\rule{0.166667em}{0ex}}\lambda \left(\epsilon \right)$ of Equation (13) we seek in the form

## 5. Remarks, Regularization and Generalizations

**Example**

**5.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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N | $\mathit{\epsilon}$ | $\mathit{\lambda}$ | ${\mathit{x}}_{+}\left(0\right)$ | ${\mathit{x}}_{+}(\mathit{\pi}/2)$ | ${\mathit{x}}_{+}\left(\mathit{\pi}\right)$ |
---|---|---|---|---|---|

6 | ${10}^{-3}$ | $-0.2197\times {10}^{-6}$ | $0.5652\times {10}^{-3}$ | $2\times {10}^{-9}$ | $-0.5632\times {10}^{-3}$ |

5 | ${10}^{-2}$ | $-0.2194\times {10}^{-6}$ | $0.5737\times {10}^{-3}$ | $2\times {10}^{-9}$ | $-0.5751\times {10}^{-3}$ |

5 | ${10}^{-1}$ | $-2.1331\times {10}^{-3}$ | $56.5654\times {10}^{-3}$ | 0 | $-56.6065\times {10}^{-3}$ |

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**MDPI and ACS Style**

Sidorov, N.; Sidorov, D.; Dreglea, A.
Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator. *Symmetry* **2020**, *12*, 912.
https://doi.org/10.3390/sym12060912

**AMA Style**

Sidorov N, Sidorov D, Dreglea A.
Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator. *Symmetry*. 2020; 12(6):912.
https://doi.org/10.3390/sym12060912

**Chicago/Turabian Style**

Sidorov, Nikolai, Denis Sidorov, and Aliona Dreglea.
2020. "Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator" *Symmetry* 12, no. 6: 912.
https://doi.org/10.3390/sym12060912