In this paper the existence and nonexistence results of positive solutions are obtained for SturmLiouville boundary value problema, b, c, d 0 are constants and satisfy (a + b)(c + d) > 0. The discussion is based on the positivity estimation for the Green's function of associated linear boundary value problem and the fixed point index theory in cones.
In this article, we are concerned with the existence of mild solutions as well as approximate controllability for a class of fractional evolution equations with nonlocal conditions in Banach spaces. Sufficient conditions of existence of mild solutions and approximate controllability for the desired problem are presented by introducing a new Green’s function and constructing a control function involving Gramian controllability operator. The discussions are based on Schauder’s fixed point theorem as well as the theory of α-order solution operator and α-order resolvent operator. An example is given to illustrate the feasibility of our theoretical results.
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