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Kolmogorov invented a pair of functions to characterize the transition probabilities for a Markov process and showed that they amount to what he called an "instantaneous mean" and an "instantaneous variance".
A pair of functions is called a subsolution of (1.1)–(1.3) if (2.1).
We consider the problem of finding a pair of functions ((u r,t),f(r))).
A pair of functions ((u,v in C [0,1] times C([0,1])) is called a solution of system (1.1) if it satisfies (1.1).
The input-output method considered in this paper is the admissibility of a pair of function spaces.
Recently, Doric [10] extended the result of Zhang and Song using a pair of functions ψ and φ.
Problem 2 Determine a pair of functions ( u, λ ) defined in Ω that satisfies L u = λ ( x, φ ) (4).
Problem 1 Determine a pair of functions ( u, λ ) from the transport-like equation (2) provided that the function K is known and u satisfies conditions (3).
We say that a pair of functions ((u,v) in X) is a weak solution for problem (1.1) if identity (2.8) holds for any ((varphi,psi in X).
Similarly, a pair of functions ( u ¯ ( x, t ), v ¯ ( x, t ) ) is a weak supersolution of (1.1) if the reversed inequalities hold in Definition 2.2.
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CEO of Professional Science Editing for Scientists @ prosciediting.com