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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).

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
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References:

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