Abstract
In this paper, existence criteria for single and multiple positive solutions of periodic boundary value problems for first order difference equations of the form � 4 x(k) + f(k,x(k + 1)) = 0, k 2 [0,T], x(0) = x(T + 1), are established by using the fixed point theorem in cones. An example is also given to illustrate the main results.
Highlights
Due to the wide application in many fields such as science, economics, neural network, ecology, cybernetics, etc., the theory of nonlinear difference equations has been widely studied since 70’s of last century, see, for example, [1, 2, 19, 20]
We are concerned with the existence of single and multiple positive solutions of periodic boundary value problems (PBVPs) for first order difference equation
In [24], Wang obtained the existence of multiple positive solutions of PBVP (1.1) by using the Leggett-Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression when condition (A) holds
Summary
Due to the wide application in many fields such as science, economics, neural network, ecology, cybernetics, etc., the theory of nonlinear difference equations has been widely studied since 70’s of last century, see, for example, [1, 2, 19, 20]. We are concerned with the existence of single and multiple positive solutions of PBVP for first order difference equation. In [22], by using a fixed point theorem, Sun considered the existence of one positive solution of the PBVP (1.1) when the following condition holds:. In [24], Wang obtained the existence of multiple positive solutions of PBVP (1.1) by using the Leggett-Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression when condition (A) holds. Motivated by the results mentioned above, in this paper, we shall obtain existence criteria for single and multiple positive solutions to the PBVP (1.1) by means of a fixed point theorem in cones.
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