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**Existence principles for nonresonant operator and integral equations.**
*(English)*
Zbl 0999.47053

From the introduction: We establish some existence principles for nonlinear, nonresonant operator equations, and we use these results to obtain existence principles for nonlinear, nonresonant, Fredholm integral equations.

The paper is divided into two main sections. In Section 2, we discuss the operator equation \[ y(t)= (\gamma+ \tau(t)) Ly(t)+ Ny(t)\tag{1} \] defined on \([0,T]\), with \(L: L^2[0,T]\to X\) a linear operator, and \(N: X\to X\) possibly nonlinear, where \(X\subseteq L^2[0,T]\) is either \(L^p[0,T]\), \(p\geq 2\) or \(C[0,T]\). We present an existence principle which establishes the existence of a solution \(y\in X\) of (1). En route to this result, the spectral theory of \(L: L^2[0,T]\to X\) is discussed under particular hypotheses, and using the Fredholm alternative, we consider what conditions are required to ensure that the operator \(T: X\to X\) defined by \[ Ty(t):= (I- (\gamma+\tau(t)) L)y(t)\tag{2} \] on \([0,T]\), has a bounded inverse \(T^{-1}: X\to X\).

In Section 3, we apply the results of the previous section to obtain two existence principles for the Fredholm integral equation \[ y(t)= (\gamma+ \tau(t)) \int^T_0 k(t,s) y(s) ds+ \int^T_0 k(t,s)f(s,y(s)) ds\tag{3} \] on \([0,T]\). In particular, we consider what conditions are required on \(\gamma\), \(\tau\), \(k\), and \(f\) to first guarantee the existence of an \(L^p[0,T]\) \((p\geq 2)\) solution (3), and second, the existence of a \(C[0,T]\) solution of (3).

The paper is divided into two main sections. In Section 2, we discuss the operator equation \[ y(t)= (\gamma+ \tau(t)) Ly(t)+ Ny(t)\tag{1} \] defined on \([0,T]\), with \(L: L^2[0,T]\to X\) a linear operator, and \(N: X\to X\) possibly nonlinear, where \(X\subseteq L^2[0,T]\) is either \(L^p[0,T]\), \(p\geq 2\) or \(C[0,T]\). We present an existence principle which establishes the existence of a solution \(y\in X\) of (1). En route to this result, the spectral theory of \(L: L^2[0,T]\to X\) is discussed under particular hypotheses, and using the Fredholm alternative, we consider what conditions are required to ensure that the operator \(T: X\to X\) defined by \[ Ty(t):= (I- (\gamma+\tau(t)) L)y(t)\tag{2} \] on \([0,T]\), has a bounded inverse \(T^{-1}: X\to X\).

In Section 3, we apply the results of the previous section to obtain two existence principles for the Fredholm integral equation \[ y(t)= (\gamma+ \tau(t)) \int^T_0 k(t,s) y(s) ds+ \int^T_0 k(t,s)f(s,y(s)) ds\tag{3} \] on \([0,T]\). In particular, we consider what conditions are required on \(\gamma\), \(\tau\), \(k\), and \(f\) to first guarantee the existence of an \(L^p[0,T]\) \((p\geq 2)\) solution (3), and second, the existence of a \(C[0,T]\) solution of (3).

### MSC:

47J05 | Equations involving nonlinear operators (general) |

47H10 | Fixed-point theorems |

45G10 | Other nonlinear integral equations |

47N20 | Applications of operator theory to differential and integral equations |

### Keywords:

nonresonant operator equations; existence principles; nonlinear, nonresonant, Fredholm integral equations; Fredholm alternative
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\textit{M. Meehan} and \textit{D. O'Regan}, Comput. Math. Appl. 35, No. 9, 79--87 (1998; Zbl 0999.47053)

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### References:

[1] | Young, N., An introduction to Hilbert space, (1989), Cambridge University Press |

[2] | Guenther, R.B.; Lee, J.W., Partial differential equations of mathematical physics and integral equations, (1996), Dover Mineola, NY |

[3] | Kreyszig, E., Introductory functional analysis with applications, (1978), John Wiley and Sons · Zbl 0368.46014 |

[4] | M. Meehan and D. O’Regan, Existence theory for nonresonant nonlinear Fredholm integral equations and nonresonant operator equations (to appear). · Zbl 0965.45016 |

[5] | Guenther, R.B.; Lee, J.W., Some existence results for nonlinear integral equations via topological transversality, J. int. eqn. appl., 5, 195-209, (1993) · Zbl 0781.45003 |

[6] | O’Regan, D., Existence theory for nonlinear Volterra and Hammerstein integral equations, (), 601-615, River Edge, NJ · Zbl 0842.45003 |

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