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Section 4 gives the unique solution of (1.1) and an example is introduced.
In Section 4 we established an existence theorem for the solution of an initial value problem, which not only gives the unique solution but also locates the domain for the solution.
Proof Let the mapping f a ( t ) = t a e − t k k be defined on ( 0, ∞ ), then f a ′ ( t ) = t a − 1 e − t k k [ a − t k ] ; f a ′ ( t ) = 0 gives the unique solution t 0 = a 1 k, which implies that f a is an increasing function on ( 0, t 0 ) and decreasing on ( t 0, ∞ ).
For a given the unique solution of the boundary value problem (2.4). is given by (2.5).
Here, we give the unique solution of BVP (1.1) under the conditions that and is mixed nonmonotone in and does not need to be separable by using the cone theory and the Banach contraction mapping principle.
Therefore, (18) gives the unique optimal solution for the powers.
Therefore, ({ lim_{krightarrow infty }r^{k} t,theta)}) gives the unique local solution of the Cauchy problem (2.5).
Given and, the unique solution of (2.5). is given by (2.6).
Proposition 2.3 Let x α be given as the unique solution of (2.2).
Theorem 2.4 Let w α be given as the unique solution of (2.2).
Proposition 2.3 Let w α be given as the unique solution of (2.2).
