Picard's theorem (integral equations)

Picard's theorem (integral equations) is a theorem on the existence and uniqueness of a solution for the Fredholm integral equation of the first kind.

Fredholm integral equation of the first kind with a closed symmetric kernel $K(t,s)$ ${\ displaystyle K (t, s)}$ $K (t, s)$ kind of $\int \limits _{a}^{b}K(t,s)\phi (s)ds=f(t)$ ${\ displaystyle \ int \ limits _ {a} ^ {b} K (t, s) \ phi (s) ds = f (t)}$ $\ int \ limits _ {a} ^ {b} K (t, s) \ phi (s) ds = f (t)$ where $f(t)\in L_{2}[a,b]$ ${\ displaystyle f (t) \ in L_ {2} [a, b]}$ $f (t) \ in L _ {{2}} [a, b]$ has a unique solution in the class of functions $L_{2}[a,b]$ ${\ displaystyle L_ {2} [a, b]}$ $L _ {{2}} [a, b]$ if and only if the row $\sum _{k=1}^{\infty }\lambda _{k}^{2}f_{k}^{2}$ ${\ displaystyle \ sum _ {k = 1} ^ {\ infty} \ lambda _ {k} ^ {2} f_ {k} ^ {2}}$ $\ sum _ {{k = 1}} ^ {{\ infty}} \ lambda _ {{k}} ^ {{2}} f _ {{k}} ^ {{2}}$ converges.

Explanation

In the statement of the theorem $\lambda _{k}$ ${\ displaystyle \ lambda _ {k}}$ - characteristic numbers of the core $K(t,s)$ ${\ displaystyle K (t, s)}$ , $f_{k}=(f,\phi _{k})$ ${\ displaystyle f_ {k} = (f, \ phi _ {k})}$ - Fourier coefficients of the function $f(t)$ ${\ displaystyle f (t)}$ regarding eigenfunctions $\phi _{n}(t)$ ${\ displaystyle \ phi _ {n} (t)}$ of this kernel: $\phi _{n}(t)=\lambda _{n}\int \limits _{a}^{b}K(t,s)\phi _{n}(s)ds$ ${\ displaystyle \ phi _ {n} (t) = \ lambda _ {n} \ int \ limits _ {a} ^ {b} K (t, s) \ phi _ {n} (s) ds}$ . Symmetric core $K(t,s)$ ${\ displaystyle K (t, s)}$ called closed in $L_{2}[a,b]$ ${\ displaystyle L_ {2} [a, b]}$ if each function $\sigma (t)\in L_{2}[a,b]$ ${\ displaystyle \ sigma (t) \ in L_ {2} [a, b]}$ satisfying equality $\int \limits _{a}^{b}K(t,s)\sigma (s)ds=0$ ${\ displaystyle \ int \ limits _ {a} ^ {b} K (t, s) \ sigma (s) ds = 0}$ vanishes almost everywhere on the segment $[a,b]$ ${\ displaystyle [a, b]}$ . For a closed core, its eigenfunctions form an orthogonal complete $L_{2}[a,b]$ ${\ displaystyle L_ {2} [a, b]}$ system of functions.

Proof

Suppose there is a solution $\phi (t)\in L_{2}[a,b]$ ${\ displaystyle \ phi (t) \ in L_ {2} [a, b]}$ equations $\int \limits _{a}^{b}K(t,s)\phi (s)ds=f(t)$ ${\ displaystyle \ int \ limits _ {a} ^ {b} K (t, s) \ phi (s) ds = f (t)}$ .

Find the Fourier coefficients of the function $f(t)$ ${\ displaystyle f (t)}$ regarding eigenfunctions $\phi _{n}(t)$ ${\ displaystyle \ phi _ {n} (t)}$ of this kernel: $f_{n}=\int \limits _{a}^{b}f(t)\phi _{n}(t)dt=\int \limits _{a}^{b}\left\{\int \limits _{a}^{b}K(t,s)\phi (s)ds\right\}\phi _{n}(t)dt=\int \limits _{a}^{b}\left\{\int \limits _{a}^{b}K(t,s)\phi _{n}(t)dt\right\}\phi (s)ds={\frac {1}{\lambda _{n}}}\int \limits _{a}^{b}\phi _{n}(s)\phi (s)ds$ ${\ displaystyle f_ {n} = \ int \ limits _ {a} ^ {b} f (t) \ phi _ {n} (t) dt = \ int \ limits _ {a} ^ {b} \ left \ {\ int \ limits _ {a} ^ {b} K (t, s) \ phi (s) ds \ right \} \ phi _ {n} (t) dt = \ int \ limits _ {a} ^ { b} \ left \ {\ int \ limits _ {a} ^ {b} K (t, s) \ phi _ {n} (t) dt \ right \} \ phi (s) ds = {\ frac {1 } {\ lambda _ {n}}} \ int \ limits _ {a} ^ {b} \ phi _ {n} (s) \ phi (s) ds}$ .

Here in the second equality we used that, by virtue of the hypothesis of the theorem $f(t)=\int \limits _{a}^{b}K(t,s)\phi (s)ds$ ${\ displaystyle f (t) = \ int \ limits _ {a} ^ {b} K (t, s) \ phi (s) ds}$ , in the fourth equality, which, due to the symmetry of the nucleus $\int \limits _{a}^{b}K(t,s)\phi _{n}(t)dt={\frac {1}{\lambda _{n}}}\phi _{n}(s)$ ${\ displaystyle \ int \ limits _ {a} ^ {b} K (t, s) \ phi _ {n} (t) dt = {\ frac {1} {\ lambda _ {n}}} \ phi _ {n} (s)}$ .

Equality $f_{n}={\frac {1}{\lambda _{n}}}\int \limits _{a}^{b}\phi _{n}(s)\phi (s)ds$ ${\ displaystyle f_ {n} = {\ frac {1} {\ lambda _ {n}}} \ int \ limits _ {a} ^ {b} \ phi _ {n} (s) \ phi (s) ds }$ can be rewritten as $\lambda _{n}f_{n}=\int \limits _{a}^{b}\phi _{n}(s)\phi (s)ds$ ${\ displaystyle \ lambda _ {n} f_ {n} = \ int \ limits _ {a} ^ {b} \ phi _ {n} (s) \ phi (s) ds}$ . It follows that the numbers $\lambda _{n}f_{n}$ ${\ displaystyle \ lambda _ {n} f_ {n}}$ are the Fourier coefficients of the function $\phi (t)\in L_{2}[a,b]$ ${\ displaystyle \ phi (t) \ in L_ {2} [a, b]}$ . By virtue of the well-known theorem of mathematical analysis, the series $\sum _{k=1}^{\infty }\lambda _{k}^{2}f_{k}^{2}$ ${\ displaystyle \ sum _ {k = 1} ^ {\ infty} \ lambda _ {k} ^ {2} f_ {k} ^ {2}}$ of the squares of these coefficients is convergent.

Assume, on the contrary, that the series $\sum _{k=1}^{\infty }\lambda _{k}^{2}f_{k}^{2}$ ${\ displaystyle \ sum _ {k = 1} ^ {\ infty} \ lambda _ {k} ^ {2} f_ {k} ^ {2}}$ converges. Then, by virtue of the Riesz-Fisher theorem, there exists a unique function $\phi (t)\in L_{2}[a,b]$ ${\ displaystyle \ phi (t) \ in L_ {2} [a, b]}$ for which numbers $\lambda _{n}f_{n}$ ${\ displaystyle \ lambda _ {n} f_ {n}}$ are the Fourier coefficients for the system of functions ${\mathcal {f}}\phi _{n}(t){\mathcal {g}}$ ${\ displaystyle {\ mathcal {f}} \ phi _ {n} (t) {\ mathcal {g}}}$ , i.e., the equalities hold $\lambda _{n}f_{n}=\int \limits _{a}^{b}\phi _{n}(s)\phi (s)ds$ ${\ displaystyle \ lambda _ {n} f_ {n} = \ int \ limits _ {a} ^ {b} \ phi _ {n} (s) \ phi (s) ds}$ for all $n(n=1,2,...)$ ${\ displaystyle n (n = 1,2, ...)}$ . This function $\phi (t)$ ${\ displaystyle \ phi (t)}$ satisfies the integral equation $\int \limits _{a}^{b}K(t,s)\phi (s)ds=f(t)$ ${\ displaystyle \ int \ limits _ {a} ^ {b} K (t, s) \ phi (s) ds = f (t)}$ , since by the very construction $\phi (t)$ ${\ displaystyle \ phi (t)}$ the functions $f(t)$ ${\ displaystyle f (t)}$ and $\int \limits _{a}^{b}K(t,s)\phi (s)ds$ ${\ displaystyle \ int \ limits _ {a} ^ {b} K (t, s) \ phi (s) ds}$ have the same Fourier coefficients with respect to the complete system ${\mathcal {f}}\phi _{n}(t){\mathcal {g}}$ ${\ displaystyle {\ mathcal {f}} \ phi _ {n} (t) {\ mathcal {g}}}$ native kernel functions $K(t,s)$ ${\ displaystyle K (t, s)}$ . So functions $f(t)$ ${\ displaystyle f (t)}$ and $\int \limits _{a}^{b}K(t,s)\phi (s)ds$ ${\ displaystyle \ int \ limits _ {a} ^ {b} K (t, s) \ phi (s) ds}$ identical in metric $L_{2}[a,b]$ ${\ displaystyle L_ {2} [a, b]}$ .

Literature

Krasnov M.L. Integral equations, M., Science, 1975.

Picard's theorem (integral equations)

Explanation

Proof

Literature

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