In this article, we are concerned with the periodic solutions of first-order difference equation
where , is continuous with respect to , , is an integer, . We prove a result of Ambrosetti-Prodi-type for by using the method of lower and upper solutions and topological degree. We relax the coercivity assumption on in Bereanu and Mawhin  and obtain Ambrosetti-Prodi-type results.
Let be an integer, . In this article, we establish Ambrosetti-Prodi-type results of first-order difference equation
where , is continuous with respect to , , .
The Ambrosetti-Prodi problem for an equation of the form
consists of determining how varying the parameter affects the number of solutions . Usually, an Ambrosetti-Prodi-type result yields the existence of a number such that (1.2) has zero, at least one or two solutions according to , or .
The founding work is in the study by Ambrosetti and Prodi , which received immediate attention from several authors. In 1975, Fucik  was concerned with the weak solvability of the elliptic equation and obtained Ambrosetti-Prodi-type results. In 1980, Hess  studied Ambrosetti-Prodi-type results of elliptic equation, he extended the works of Ambrosetti and Prodi  and Kazdan and Warner . After that, several studies have sprung up [1,7,8, 9,10,11, 13,14,15, 16,17,18, 19,22,24].
Most of the aforementioned literature is about differential equations. Periodic problems for differential equations were studied in [12, 20, 21] Zhou [25, 26] studied periodic solutions of difference equations. Since there are many essential differences between difference equations and differential equations, such as in the continuous case, the minimum or maximum points satisfy , but in discrete case, the minimum or maximum points do not necessarily satisfy , and the definition of generalized zeros in difference is complex, and chaotic behaviors in Strogatz ; there are few researches on Ambrosetti-Prodi-type results of difference equations. Through searching for an analogue for Ambrosetti-Prodi-type results of difference equations, in 2006, Bereanu and Mawhin  were concerned with the first-order difference equation
They obtained the following:
[1, Theorem 6] Assume is continuous, with T-periodicity in the t variable, . If
Then there exists an such that
if , there is no T-periodic solution of equation (1.3),
if , there is at least one T-periodic solution of equation (1.3),
if , there are at least two T-periodic solutions of equation (1.3).
is continuous upon , .
There exist , , such that , , for all .
Assume (H1)–(H3) hold, there exists , such that
if , there is no T-periodic solution of equation (1.1),
if , there is at least one T-periodic solution of equation (1.1),
if , there are at least two T-periodic solutions of equation (1.1).
Obersnel and Omari  investigated an Ambrosetti-Prodi-type result of first-order differential equation; they studied the existence and multiplicity of solutions when the parameter exceeds a constant using normal-order upper and lower solutions and reverse-order upper and lower solutions. However, for first-order difference equations, reverse order upper and lower solutions cannot be used; in addition, lower solutions must be smaller than the upper solutions to make the method conclusive, and relevant conclusions can be found in . Hence, the multiplicity of solutions when the parameter exceeds a constant is the difficulty in this article.
In , Bereanu and Mawhin showed counterexamples when is odd, is even and , respectively. These counterexamples show that first-order difference equations have no solution when lower solutions are larger than upper solutions.
First-order difference equation
We take , , and ; hence, (H1) holds. There exist , and such that , , for all ; hence, (H2) holds. Obviously, , and hence, (H3) holds. According to Theorem 1.1, we can obtain such that
if , there is no -periodic solution of equation (1.5);
if , there is at least one -periodic solution of equation (1.5);
if , there are at least two -periodic solutions of equation (1.5).
2 Preliminary results
Let be a Banach space under norm
For convenience, we only need to consider the first-order periodic boundary value problem
The definition of the upper and lower solutions of problem (2.1) is given as follows:
is a lower solution of problem (2.1), referring to satisfies
is an upper solution of problem (2.1), referring to satisfies
is a strict lower solution of problem (2.1), referring to satisfies
is a strict upper solution of problem (2.1), referring to satisfies
Problem (2.1) has a lower solution and an upper solution , such that , , then problem (2.1) has at least one solution , such that , .
Construct auxiliary function by
Consider the modified problem
Using Brouwer fixed point theorem, at least one solution can be obtained for problem (2.2) in , whose elements can be characterized by the coordinates . Indeed, the operator is given by
which is one to one, hence invertible, and (2.2) is equivalent to the fixed point problem
in . It remains to show that if is a solution of (2.2), , then , so that is a solution of (2.1), . Suppose by contradiction that there exists a , such that , then
we can obtain
which contradicts with the definition of the lower solution.
Thus, . Similarly, can be proved. Then problem (2.1) has at least one solution , such that , .□
Assume that is a strict lower solution of (2.1), is the strict upper solution of (2.1), then the problem (2.1) admits at least one solution such that . Define the open set and the open ball with the radius of . The mapping is defined by , . If is large enough, using the additivity-excision property of Brouwer degree, we have
3 Proof of the main result
Proof of Theorem 1.1
Step 1. We verify that for every , there is , such that, for all , any solution of the Cauchy problem
is a strict lower solution of the -periodic problem
Hence, by (H2), is a strict lower solution of problem (2.1).
We consider the case and prove the following claim first.
Claim For any , there is such that, for every , any solution of (3.1) satisfies .
Assume, by contradiction, that there exists such that, for every , with , there is a solution of problem (3.1) satisfying and Let be such that on , , , , and , then
For fixed , we obtain a contradiction if ; thus, our claim is proved.
In the case of , suppose that there is a sequence , with and the solution of problem (3.1) with , for any , satisfies . By the claim above, we can assume that . Thus,
We obtain the contradiction when . Hence, we have , and is a solution of (3.1).
The validity of step 1 when can be verified by a direct inspection is obtained as follows:
where is an arbitrary constant, choose
Then, we have and is a solution of (3.1).
Step 2. We show that there exists such that, for all , equation (1.1) has at least one -periodic solution. Indeed, it is easily verified that there exists such that, for all , the constant , , is a strict upper solution of problem (2.1). Furthermore, by the results proved in Step 1, problem (2.1) admits one strict lower solution satisfying for all . Therefore, equation (1.1) has at least one -periodic solution , satisfying for all , .
Step 3. We prove that the set of the parameters for which equation (1.1) has at least one -periodic solution is bounded from below. Define the set
We prove there exists , such that . Assume, by contradiction, that . Then, there exists a sequence with , and a sequence of -periodic solutions of equation (1.1) with . We claim that , otherwise, we would obtain
There would exist a function , such that
which is a contradiction. Moreover, by (H2) we have
Thus, we obtain
Let , and using (H3) yields the contradiction .
Step 4. We show the existence of at least one -periodic solution of equation (1.1) for . Let be a sequence in converging to and let be the corresponding sequence of -periodic solutions of equation (1.1) with . Let us verify that there is , such that for all . Indeed, otherwise, we can find a subsequence of , we still denote by , such that . Arguing similarly as in the proof of Step 3, thus easily leading to a contradiction as above. Therefore, is bounded in ; according to Weierstrass concentration theorem, we can obtain , . Besides, , , and when is large enough, . Sequence , i.e., , convergence to in , with , , is a -periodic solution of equation (1.1) for .
Step 5. We show that for all , equation (1.1) has at least two -periodic solutions.
Claim For any constant , there exists , such that, for all , all possible periodic solutions of equation (1.1) belong to open ball .
For every , we have
We need to show there exists a constant , such that
By (H2), we can obtain , then
Hence, all possible solutions of problem (2.1) belong to open ball .
Using the Brouwer degree theory, obviously, is a solution of problem (2.1) if and only if is a zero of , . Let , according to the claim above, we can find the corresponding such that, for all , every possible zero points of satisfy . Consequently, the Brouwer degree is well defined and does not depend upon . Using the conclusion of step 3, for , . This implies that , so that . By excision property, if .
Let be a solution of (2.1) with , then is a strict upper solution of problem (2.1) with . From Step 1, is a strict lower solution of problem (2.1). Consequently, using Remark 2.3, (2.1) with has a solution in , and
Taking sufficiently large, we deduce from the additivity property of Brouwer degree that
When , (2.1) has the second solution in .□
Funding information: This work was supported by National Natural Science Foundation of China (No. 12061064).
Author contributions: The authors claim that the research was realized in collaboration with the same responsibility. All authors read and approved the last of the manuscript.
Conflict of interest: All of the authors of this article claims that together they have no any competing interests each other.
Data availability statement: Data sharing not applicable to this article as no data sets were generated.
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