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Conclusion 4. Problem (4.2) has only one solution on.
Then, (4.1) has at most one solution on.
If, then (1.2) has at least one solution on.
then problem (1.1) has at least one solution on.
Then (1.1)–(1.4) has at least one solution on.
Then BVP (4.1) has at most one solution on.
Further, one also assumes that.. Then the equation has at least one solution on.
Then boundary value problem (1.6 - 1.7 1.6 - 1.7east one solution on [ 0, T ].
Therefore, problem (5.1) has at least one solution on ([0,1]).
Therefore, according to Krasnoselskii's fixed point theorem, (2.1) has at least one solution on ((-infty,T]).
So problem (1.1) has at least one solution on [ 0, 1 ].
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