• Bulletin of the Korean Mathematical Society
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• v.27 no.2
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• pp.151-155
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• 1990

Recently, H.Z.Ming [7] obtained a necessary and sufficient condition for the existence of a solution to a general operator equation. In the present paper, we obtain such conditions in general forms and give some examples. We begin with the well-known Fan-Browder fixed point theorem, from which we deduce two general theorems on such necessary and sufficient conditiions. We give some examples of such conditions, which are improved versions of fixed point theorems of Halpern-Bergman [5], Ky Fan [3], [4], Kaczynski [6], Reich [9], Schauder [10], Tychonoff [11], and Ming [7]. In fact, we restate Ming's result in its correct form. The following is known as the Fan-Browder fixed point theorem [1], [2].

• Korean Journal of Mathematics
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• v.27 no.1
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• pp.119-129
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• 2019

Balance function is one of the joint factors to determine fall in risk theory. It helps to moderate the progression and riskiness of falls for detecting balance and fall risk factors. Nevertheless, the objective measures for balance function require expensive equipment with the assessment of any expertise. We establish the existence and uniqueness of a multi-order fractional differential equations based on ${\psi}$-Hilfer operator on time scales with balance function. This class describes the dynamic of time scales derivative. Our tool is based on the Schauder fixed point theorem. Here, sufficient conditions for Ulam-stability are given.