The main purpose of this research is to establish a new generalized ξ∗-Kannan type double controlled contraction on a sequence and obtain fixed point results for a pair of multivalued mappings in left K-sequentially complete double controlled dislocated quasi metric type spaces. New results in different setting of generalized metric spaces and ordered spaces and also new results for graphic contractions can be obtained as corollaries of our results. An example is presented to show the novelty of results. In this paper, we unify and extend some recent results in the existing literature.
Citation: Tahair Rasham, Abdullah Shoaib, Shaif Alshoraify, Choonkil Park, Jung Rye Lee. Study of multivalued fixed point problems for generalized contractions in double controlled dislocated quasi metric type spaces[J]. AIMS Mathematics, 2022, 7(1): 1058-1073. doi: 10.3934/math.2022063
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The main purpose of this research is to establish a new generalized ξ∗-Kannan type double controlled contraction on a sequence and obtain fixed point results for a pair of multivalued mappings in left K-sequentially complete double controlled dislocated quasi metric type spaces. New results in different setting of generalized metric spaces and ordered spaces and also new results for graphic contractions can be obtained as corollaries of our results. An example is presented to show the novelty of results. In this paper, we unify and extend some recent results in the existing literature.
We first assume that the readers are familiar with the notations of the Nevanlinna theory such as T(r,f),m(r,f),N(r,f) and so on, which can be found, for instant, in Hayman [12], Yang [31], Yi and Yang [32]. As is known to all, Nevanlinna theory is a powerful tool in analysing the properties of meromorphic functions including meromorphic functions in arbitrary plane regions, algebroid functions, functions of several variables, holomorphic curves, uniqueness theory of meromorphic function and complex differential equations, and so on. Moreover, it is also used in lots of areas of mathematics including potential theory, measure theory, differential geometry, topology and others.
Around 2006, Chiang and Feng [6], Halburd and Korhonen [8] established independently the difference analogues of Nevanlinna theory in C, respectively. In the past two decades, many scholars paid a lot of attention to a large number of interesting topics on complex difference of meromorphic functions by making use of the difference analogue of Nevanlinna theory, and complex difference results are in the rapid development.
In all the previous articles, Korhonen, Halburd, Chen and his students investigated the properties of solutions for a series of linear difference equations, the difference Painlevé equations, the Riccati equations, the Pielou logistic equations, and obtained the conditions of the existence and estimations of growth order for the above difference equations [1,4,5,9,27,35]; Laine, Korhonen, Yang, Chen, etc. established the difference analogue of Clunie Lemma and Mohonko Lemma, and gave the existence theorems on solutions for some complex difference equations including difference Malmquist type and some nonlinear difference equations [16,17,21]; Korhonen, Zhang, Liu, Yang paid close attention to some complex differential difference equations [11,18,19,20,24,34], in special, Liu and his co-authors [18,19,20] investigated some types of complex differential difference equations of Fermat type such as
f′(z)2+f(z+c)2=1, | (1.1) |
f′(z)2+[f(z+c)−f(z)]2=1. | (1.2) |
They proved that the transcendental entire solutions with finite order of equation (1.1) must satisfy f(z)=sin(z±Bi), where B is a constant and c=2kπ or c=(2k+1)π, k is an integer, and the transcendental entire solutions with finite order of equation (1.2) must satisfy f(z)=12sin(2z+Bi), where c=(2k+1)π, k is an integer, and B is a constant. In 2016, Gao [7] discussed the solutions for the system of complex differential-difference equations
{[f′1(z)]2+f2(z+c)2=1,[f′2(z)]2+f1(z+c)2=1, | (1.3) |
and extend the results given by Liu and Cao.
Theorem 1.1. (see [7,Theorem 1.1]). Suppose that (f1,f2) is a pair of transcendental entire solutions with finite order for the system of differential-difference equations (1.3). Then (f1,f2) satisfies
(f1(z),f2(z))=(sin(z−bi),sin(z−b1i)), |
or
(f1(z),f2(z))=(sin(z+bi),sin(z+b1i)), |
where b,b1 are constants, and c=kπ, k is a integer.
In 2012, Korhonen [14] firstly established the difference version of logarithmic derivative lemma (shortly, we may say logarithmic difference lemma) for meromorphic functions on Cm with hyper order strictly less than 23, and then used it to consider a class of partial difference equations in the same paper. Later, Cao and Korhonen [2] improved the logarithmic difference lemma to the case where the hyper order is strictly less than one. In 2019, Cao and Xu [3] further improved the logarithmic difference lemma in several variables under the condition of minimal type lim supr→∞logT(r,f)r=0. As far as we know, however, there are very little of results on solutions of complex partial difference equations by using Nevanlinna theory. By making use of the Nevanlinna theory with several complex variables (see [2,3,14]), Xu and Cao [3,29,30], Lu and Li [22] investigated the existence of the solutions for some Fermat type partial differential-difference equations with several variables, and obtained the following theorem.
Theorem 1.2. (see [29,Theorem 1.2]). Let c=(c1,c2)∈C2. Then any transcendental entire solutions with finite order of the partial differential-difference equation
(∂f(z1,z2)∂z1)2+f(z1+c1,z2+c2)2=1 |
has the form of f(z1,z2)=sin(Az1+B), where A is a constant on C satisfying AeiAc1=1, and B is a constant on C; in the special case whenever c1=0, we have f(z1,z2)=sin(z1+B).
In 2020, Xu, Liu and Li [28] further explored the existence and forms of several systems of complex partial differential difference equations and extend the previous results given by [29,30].
Theorem 1.3. (see [28,Theorem 1.3]). Let c=(c1,c2)∈C2. Then any pair of transcendental entire solutions with finite order for the system of Fermat type partial differential-difference equations
{(∂f1(z1,z2)∂z1)2+f2(z1+c1,z2+c2)2=1,(∂f2(z1,z2)∂z1)2+f1(z1+c1,z2+c2)2=1 | (1.4) |
has the following form
(f1(z),f2(z))=(eL(z)+B1+e−(L(z)+B1)2,A21eL(z)+B1+A22e−(L(z)+B1)2), |
where L(z)=a1z1+a2z2, B1 is a constant in C, and a1,c,A21,A22 satisfy one of the following cases
(i) A21=−i, A22=i, and a1=i, L(c)=(2k+12)πi, or a1=−i, L(c)=(2k−12)πi;
(ii) A21=i, A22=−i, and a1=i, L(c)=(2k−12)πi, or a1=−i, L(c)=(2k+12)πi;
(iii) A21=1, A22=1, and a1=i, L(c)=2kπi, or a1=−i, L(c)=(2k+1)πi;
(iv) A21=−1, A22=−1, and a1=i, L(c)=(2k+1)πi, or a1=−i, L(c)=2kπi.
The above results suggest the following questions.
Question 1.1. What will happen about the solutions when f(z), f(z+c), ∂f∂z1 and ∂f∂z2 are included in the equations in Theorem B?
Question 1.2. What will happen about the solutions when the equations are turned into system under the hypothesis of Question 1.1?
In view of the above questions, this article is concerned with the description of the existence and the forms of entire solutions for several partial differential difference of Fermat type. The main tools are used in this paper are the Nevanlinna theory and difference Nevanlinna theory with several complex variables. Our main results are obtained generalized the previous theorems given by Xu and Cao, Liu, Cao and Cao [20,29]. Throughout this paper, for convenience, we assume that z+w=(z1+w1,z2+w2) for any z=(z1,z2),w=(w1,w2). The main results of this paper are stated below.
Theorem 2.1. Let c=(c1,c2)∈C2−{(0,0)}. Then any transcendental entire solution f(z1,z2) with finite order of the partial differential equation
f(z)2+[f(z+c)+∂f∂z1]2=1 | (2.1) |
must be of the form
f(z1,z2)=eL(z)+b+e−L(z)−b2, |
where L(z)=α1z1+α2z2, and L(c):=α1c1+α2c2, α1, α2,b∈C satisfy
(i) L(c)=2kπi+π2i and α1=−2i;
(ii) L(c)=2kπi+3π2i and α1=0.
The following examples show that the forms of transcendental entire solutions with finite order for equation (2.1) are precise.
Example 2.1. Let α1=−2i, α2=3π2 and b∈C, that is,
f(z1,z2)=e2z2+b+e−2z2−b2. |
Then f(z1,z2) satisfies equation (2.1) with (c1,c2)=(c1,3πi4), where c1∈C.
Example 2.2. Let α1=0, α2=2 and b=0, that is,
f(z1,z2)=e−2iz1+3π2+e2iz1−3π22. |
Then f(z1,z2) satisfies equation (2.1) with (c1,c2)=(π2,i).
If equation (2.1) contains both two partial differentials and difference, we obtain
Theorem 2.2. Let c=(c1,c2)∈C2−{(0,0)} and s=c2z1−c1z2. Then any transcendental entire solution f(z1,z2) with finite order of the partial differential equation
f(z)2+[f(z+c)+∂f∂z1+∂f∂z2]2=1 | (2.2) |
must be of the form
f(z1,z2)=eL(z)+H(s)+b+e−L(z)−H(s)−b2, |
where L(z)=α1z1+α2z2, H(s) is a polynomial in s, and L(c), α1, α2,b∈C satisfy
(i) c1=c2=kπ+34π, α1+α2=−2i and degsH≥2;
(ii) c1≠c2, H(s)≡0, and L(c)=2kπi+3π2i, α1+α2=0 or L(c)=2kπi+π2i, α1+α2=−2i.
We list the following examples to show that the forms of transcendental entire solutions with finite order for equation (2.2) are precise.
Example 2.3. Let L(z)=−3iz1+iz2, H(s)=3πi4(z1−z2)n, n∈N+ and b=0, that is,
f(z1,z2)=eL(z)+H(s)+e−L(z)−H(s)2=cos[−3z1+z2+3π4(z1−z2)n]. |
Then f(z1,z2) satisfies equation (2.2) with (c1,c2)=(3π4,3π4).
Example 2.4. Let L(z)=−3iz1+iz2, H(s)≡0, and b=0, that is,
f(z1,z2)=eL(z)+e−L(z)2=cos(−3z1+z2). |
Then f(z1,z2) satisfies equation (2.2) with (c1,c2)=(−π2,π).
Example 2.5. Let L(z)=iz1−iz2, and b=0, that is, α1+α2=0 and
f(z1,z2)=eL(z)+e−L(z)2=cos(z1−z2). |
Then f(z1,z2) satisfies equation (2.2) with (c1,c2)=(2π,π2).
When equation (2.1) is turned to system of functional equations, we have
Theorem 2.3. Let c=(c1,c2)∈C2−{(0,0)}. Then any pair of finite order transcendental entire solution (f1,f2) for the system of the partial differential equations
{f1(z)2+[f2(z+c)+∂f1∂z1]2=1,f2(z)2+[f1(z+c)+∂f2∂z1]2=1 | (2.3) |
must be of the form
(f1(z),f2(z))=(eL(z)+b+e−L(z)−b2,A1eL(z)+b+A2e−L(z)−b2), |
where L(z)=α1z1+α2z2, and α1,α2,b,A1,A2∈C satisfy one of the following cases:
(i) if α1=0, then α2c2=2kπi+π2i, A1=A2=−1 or α2c2=2kπi+3π2i, A1=A2=1 or α2c2=2kπi, A1=−i,A2=i or α2c2=(2k+1)πi, A1=i,A2=−i, k∈Z;
(ii) if α1=−2i, then α1c1+α2c2=2kπi+π2i, A1=A2=1 or α1c1+α2c2=2kπi+3π2i, A1=A2=−1, k∈Z.
Some examples show the existence of transcendental entire solutions with finite order for system (2.3).
Example 2.6. Let L(z)=iz2 and b=0, that is, α1=0, α2=i and
(f1,f2)=(eiz2+e−iz22,−eiz2+e−iz22)=(cosz2,−cosz2). |
Then (f1,f2) satisfies system (2.3) with (c1,c2)=(c1,π2), where c1∈C.
Example 2.7. Let L(z)=iz2 and b=0, that is, α1=0, α2=i and
(f1,f2)=(eiz2+e−iz22,−ieiz2+ie−iz22)=(cosz2,sinz2). |
Then (f1,f2) satisfies system (2.3) with (c1,c2)=(c1,2π), where c1∈C.
Example 2.8. Let L(z)=−2iz1+iz2 and b=0, that is, α1=−2i, α2=i and
(f1,f2)=(eL(z)+e−L(z)2,−eL(z)+e−L(z)2)=(cos(−2z1+z2),−cos(−2z1+z2)). |
Then (f1,f2) satisfies system (2.3) with (c1,c2)=(−π2,π2).
Similar to the argument as in the proof of Theorem 2.3, we can get the following result easily.
Theorem 2.4. Let c=(c1,c2)∈C2 and c1≠c2. Then any pair of transcendental entire solution (f1,f2) with finite order for the system of the partial differential equations
{f1(z)2+[f2(z+c)+∂f1∂z1+∂f1∂z2]2=1,f2(z)2+[f1(z+c)+∂f2∂z1+∂f2∂z2]2=1 | (2.4) |
must be of the form
(f1(z),f2(z))=(eL(z)+b+e−L(z)−b2,A1eL(z)+b+A2e−L(z)−b2), |
where L(z)=α1z1+α2z2, and α1,α2,b,A1,A2∈C satisfy one of the following cases:
(i) if α1+α2=0, then α1c1+α2c2=2kπi+π2i, A1=A2=−1 or α1c1+α2c2=2kπi+3π2i, A1=A2=1 or α1c1+α2c2=2kπi, A1=−i,A2=i or α1c1+α2c2=(2k+1)πi, A1=i,A2=−i, k∈Z;
(ii) if α1+α2=−2i, then α1c1+α2c2=2kπi+π2i, A1=A2=1 or α1c1+α2c2=2kπi+3π2i, A1=A2=−1, k∈Z.
The following example shows that the condition c1≠c2 in Theorem 2.4 can not be removed.
Example 2.9. Let c1≠0 and
(f1,f2)=(eg(z1,z2)+e−g(z1,z2)2,−ieg(z1,z2)+ie−g(z1,z2)2), |
where g(z1,z2)=z1−z2+β(z1−z2)n, β≠0 and n∈N+. Then (f1,f2) is a pair of transcendental entire solutions with finite order for system (2.4) with c=(c1,c1).
The following lemmas play the key roles in proving our results.
Lemma 3.1. ([25,26]). For an entire function F on Cn, F(0)≠0 and put ρ(nF)=ρ<∞. Then there exist a canonical function fF and a function gF∈Cn such that F(z)=fF(z)egF(z). For the special case n=1, fF is the canonical product of Weierstrass.
Remark 3.1. Here, denote ρ(nF) to be the order of the counting function of zeros of F.
Lemma 3.2. ([23]). If g and h are entire functions on the complex plane C and g(h) is an entire function of finite order, then there are only two possible cases: either
(a) the internal function h is a polynomial and the external function g is of finite order; or else
(b) the internal function h is not a polynomial but a function of finite order, and the external function g is of zero order.
Lemma 3.3. ([13,Lemma 3.1]). Let fj(≢0),j=1,2,3, be meromorphic functions on Cm such that f1 is not constant, and f1+f2+f3=1, and such that
3∑j=1{N2(r,1fj)+2¯N(r,fj)}<λT(r,f1)+O(log+T(r,f1)), |
for all r outside possibly a set with finite logarithmic measure, where λ<1 is a positive number. Then either f2=1 or f3=1, where N2(r,1f) is the counting function of the zeros of f in |z|≤r, where the simple zero is counted once, and the multiple zero is counted twice.
Suppose that f(z) is a transcendental entire solution with finite order of equation (2.1). Thus, we rewrite (2.1) as the following form
[f(z)+i(f(z+c)+∂f∂z1)][f(z)−i(f(z+c)+∂f∂z1)]=1, | (3.1) |
which implies that both f(z)+i(f(z+c)+∂f∂z1) and f(z)−i(f(z+c)+∂f∂z1) have no poles and zeros. Thus, by Lemmas 3.1 and 3.2, there thus exists a polynomial p(z) such that
f(z)+i(f(z+c)+∂f∂z1)=ep(z),f(z)−i(f(z+c)+∂f∂z1)=e−p(z). |
This leads to
f(z)=ep(z)+e−p(z)2, | (3.2) |
f(z+c)+∂f(z)∂z1=ep(z)−e−p(z)2i. | (3.3) |
Substituting (3.2) into (3.3), it yields that
−(∂p∂z1+i)ep(z+c)+p(z)+(∂p∂z1+i)ep(z+c)−p(z)−e2p(z+c)=1. | (3.4) |
Obviously, ∂p∂z1+i≢0. Otherwise, e2p(z+c)≡−1, thus p(z) is a constant. Then f(z) is a constant, this a contradiction. By Lemma 3.3, it follows from (3.4) that
(∂p∂z1+i)ep(z+c)−p(z)=1. | (3.5) |
Thus, in view of (3.4) and (3.5), we conclude that
−(∂p∂z1+i)ep(z)−p(z+c)=1. | (3.6) |
Equations (3.5) and (3.6) mean that e2[p(z+c)−p(z)]=−1. Thus, it yields that p(z)=L(z)+H(s)+b, where L is a linear function as the form L(z)=α1z1+α2z2, e2L(c)=−1, and H(s) is a polynomial in s:=c2z1−c1z2, α1,α2,b∈C.
On the other hand, it follows from (3.5) and (3.6) that (∂p∂z1+i)2=−1, that is, ∂p∂z1=0 or ∂p∂z1=−2i. In view of (3.5) or (3.6), it yields that eL(c)=i and α1=−2i or eL(c)=−i and α1=0, that is, L(c)=2kπi+π2i and α1=−2i or L(c)=2kπi+3π2i and α1=0. Moreover, by combining with p(z)=L(z)+H(s)+b, we have that H′c2=η, where η is a constant. If c2=0, then H(s)=H(−c1z2), and combining with e2(p(z+c)−p(z))=−1, then it follows that degsH≤1. If c2≠0, it is easy to get that H(s) is a polynomial in s with degsH≤1. Hence, we can conclude that L(z)+H(s)+b is a linear form in z1,z2, w.l.o.g, we denote that the form is α1z1+α2z2+b, where α1,α2,b∈C. Thus, by combining with (3.5) and (3.6), we have
f(z1,z2)=eL(z)+b+e−L(z)−b2, |
where L(z)=α1z1+α2z2, and L(c), α1, α2,b∈C satisfy L(c)=2kπi+π2i and α1=−2i or L(c)=2kπi+3π2i and α1=0, k∈Z.
Therefore, this completes the proof of Theorem 2.1.
Suppose that f(z) is a transcendental entire solution with finite order of equation (2.2). Similar to the argument as in the proof of Theorem 2.1, there exists a nonconstant polynomial p(z) such that (3.2) and
(∂p∂z1+∂p∂z2+i)ep(z+c)−p(z)=1, | (3.7) |
−(∂p∂z1+∂p∂z2+i)ep(z)−p(z+c)=1. | (3.8) |
In view of (3.7) and (3.8), we have e2(p(z+c)−p(z))=−1. Thus, it yields that p(z)=L(z)+H(s)+b, where L is a linear function as the form L(z)=α1z1+α2z2, H(s) is a polynomial in s, and α1,α2,b∈C, and e2L(c)=−1.
On the other hand, from (3.7) and (3.8), we have (∂p∂z1+∂p∂z2+i)2=−1. Noting that p(z)=L(z)+H(s)+b, it follow that
[α1+α2+H′(c2−c1)+i]2=−1. | (3.9) |
If c1=c2, then H′(c2−c1)≡0. This shows that H(s) can be any degree polynomial in s. In fact, we can assume that degsH≥2. If degsH≤1, then L(z)+H(s)+b is still a linear function of the form a1z1+a2z2, this means that H(s)≡0. In view of (3.9), it yields that α1+α2=0 or α1+α2=−2i. If α1+α2=0, that is, α1=−α2, combining with c1=c2, then it leads to L(c)=α1c1−α1c1=0, that is, e2L(c)=1, this is a contradiction with e2L(c)=−1. If α1+α2=−2i, then L(c)=(α1+α2)c1=−2ic1. By combining with eL(c)=−i, we have c1=c2=kπ+34π.
If c1≠c2, then H′ must be a constant, that is, degsH≤1. Hence, we can conclude that L(z)+H(s)+b is a linear form in z1,z2. Thus, this means that H(s)≡0 and H′(c2−c1)≡0. Hence, we have p(z)=α1z1+α2z2+b, where α1,α2,b∈C. So, in view of (3.9), we also have α1+α2=0 or α1+α2=−2i. Further, In view of (3.7) or (3.8), we conclude that L(c)=2kπi+π2i and α1+α2=−2i or L(c)=2kπi+3π2i and α1+α2=0, k∈Z.
Therefore, this completes the proof of Theorem 2.2.
Suppose that (f1(z),f2(z)) is a pair of transcendental entire solutions with finite order of (2.3). By using the same argument as in the proof of Theorem 2.1, there exist two polynomials p(z),q(z) in C2 such that
{f1(z)=ep(z)+e−p(z)2,f2(z+c)+∂f1∂z1=ep(z)−e−p(z)2i,f2(z)=eq(z)+e−q(z)2,f1(z+c)+∂f2∂z1=eq(z)−e−q(z)2i. | (4.1) |
In view of (4.1), it follows that
−(i+∂p∂z1)ep(z)+q(z+c)+(i+∂p∂z1)e−p(z)+q(z+c)−e2q(z+c)≡1, | (4.2) |
and
−(i+∂q∂z1)eq(z)+p(z+c)+(i+∂q∂z1)e−q(z)+p(z+c)−e2p(z+c)≡1. | (4.3) |
Thus, by applying Lemma 3.3, we can deduce from (4.2) and (4.3) that
(i+∂p∂z1)e−p(z)+q(z+c)=1,or−(i+∂p∂z1)ep(z)+q(z+c)=1 |
and
(i+∂q∂z1)e−q(z)+p(z+c)=1,or−(i+∂q∂z1)eq(z)+p(z+c)=1. |
Now, we will discuss four cases below.
Case 1.
{(i+∂p∂z1)e−p(z)+q(z+c)=1,(i+∂q∂z1)e−q(z)+p(z+c)=1. | (4.4) |
Thus, it follows that −p(z)+q(z+c)=η1 and −q(z)+p(z+c)=η2, that is, p(z+2c)−p(z)=η2+η1 and q(z+2c)−q(z)=η2+η1, here and below η1,η2 are constants in C, which each occurrence can be inconsistent. So, we can deduce that p(z)=L(z)+H(s)+B1 and q(z)=L(z)+H(s)+B2, where L(z) is a linear function of the form L(z)=α1z1+α2z2, H(s) is a polynomial in s:=c1z2−c2z1, α1,α2,B1,B2 are constants in C. Noting that (i+∂p∂z1) and (i+∂q∂z1) are constants, it follows that p(z)=L(z)+B1 and q(z)=L(z)+B2. Thus, by combining with (4.2)-(4.4), it yields that
{(i+α1)eL(c)+B2−B1=1,(i+α1)eL(c)+B1−B2=1,−(i+α1)e−L(c)+B1−B2=1,−(i+α1)e−L(c)+B2−B1=1, | (4.5) |
where L(c):=α1c1+α2c2, this thus leads to
(i+α1)2=−1,e2L(c)=−1,e2(B1−B2)=1. | (4.6) |
Then we have α1=0 or α1=−2i.
Assume that α1=0. If eL(c)=i, that is, L(c)=α2c2=2kπi+π2i, then it follows from (4.5) that eB2−B1=−1. Noting that system (4.1), we have
f1(z)=ep(z)+e−p(z)2=eL(z)+B1+e−L(z)−B12, |
and
f2(z)=eL(z)+B2+e−L(z)−B22=eL(z)+B1+B2−B1+e−L(z)−B1+B1−B22=−eL(z)+B1+e−L(z)−B12=−f1(z). |
If eL(c)=−i, that is, L(c)=α2c2=2kπi+3π2i, then it follows from (4.5) that eB2−B1=1. Thus, it yields from (4.1) that
f2(z)=eL(z)+B2+e−L(z)−B22=eL(z)+B1+B2−B1+e−L(z)−B1+B1−B22=eL(z)+B1+e−L(z)−B12=f1(z). |
Assume that α1=−2i. If eL(c)=i, that is, L(c)=−2ic1+α2c2=2kπi+π2i, then it follows from (4.5) that eB2−B1=1. Noting that system (4.1), we have
f2(z)=eL(z)+B2+e−L(z)−B22=eL(z)+B1+B2−B1+e−L(z)−B1+B1−B22=eL(z)+B1+e−L(z)−B12=f1(z). |
If eL(c)=−i, that is, L(c)=−2ic1+α2c2=2kπi+3π2i, then it follows from (4.5) that eB2−B1=−1. Thus, it yields from (4.1) that
f2(z)=eL(z)+B2+e−L(z)−B22=eL(z)+B1+B2−B1+e−L(z)−B1+B1−B22=−eL(z)+B1+e−L(z)−B12=−f1(z). |
Case 2.
{(i+∂p∂z1)e−p(z)+q(z+c)=1,−(i+∂q∂z1)eq(z)+p(z+c)=1. |
Thus, it follows that −p(z)+q(z+c)=η1 and q(z)+p(z+c)=η2, that is, q(z+2c)+q(z)=η1−η2. Noting that q(z) is a nonconstant polynomial in C2, we can get a contradiction.
Case 3.
{−(i+∂p∂z1)ep(z)+q(z+c)=1,(i+∂q∂z1)e−q(z)+p(z+c)=1. |
Thus, it follows that p(z)+q(z+c)=η1 and −q(z)+p(z+c)=η2, that is, p(z+2c)+p(z)=η2+η1. Noting that p(z) is a nonconstant polynomial in C2, we can get a contradiction.
Case 4.
{−(i+∂p∂z1)ep(z)+q(z+c)=1,−(i+∂q∂z1)eq(z)+p(z+c)=1. | (4.7) |
Thus, it follows that p(z)+q(z+c)=η1 and q(z)+p(z+c)=η2, that is, p(z+2c)−p(z)=η2−η1 and q(z+2c)−q(z)=η1−η2. Similar to the same argument as in Case 1, we get that p(z)=L(z)+B1 and q(z)=−L(z)+B2, where L(z)=α1z1+α2z2, α1,α2,B1,B2 are constants in C. Thus, by combining with (4.2)-(4.3) and (4.7), it yields that
{(i+α1)eL(c)−B1−B2=1,(i−α1)e−L(c)−B1−B2=1,−(i+α1)e−L(c)+B1+B2=1,−(i−α1)eL(c)+B1+B2=1, | (4.8) |
which leads to (i−α1)2=(i+α1)2=−1, that is, α1=0. Then, it leads to
e2L(c)=1,e2(B1+B2)=−1. | (4.9) |
If eL(c)=1, that is, L(c)=2kπi, then it follows from (4.8) that eB1+B2=i. Noting that system (4.1), we have
f1(z)=ep(z)+e−p(z)2=eL(z)+B1+e−L(z)−B12, |
and
f2(z)=e−L(z)+B2+eL(z)−B22=eL(z)+B1−B1−B2+e−L(z)−B1+B1+B22=−ieL(z)+B1+ie−L(z)−B12. |
If eL(c)=−1, that is, L(c)=(2k+1)πi, then it follows from (4.8) that eB1+B2=−i. Noting that system (4.1), we have
f2(z)=e−L(z)+B2+eL(z)−B22=eL(z)+B1−B1−B2+e−L(z)−B1+B1+B22=ieL(z)+B1−ie−L(z)−B12. |
Therefore, this completes the proof of Theorem 2.3.
By observing the argument as in Theorems 2.1 and 2.2, when f(z) is a function with one complex variable, we can obtain the following conclusions easily.
Theorem 5.1. Then the following complex differential equation
f(z)2+[f(z)+f′(z)]2=1 | (5.1) |
has no any transcendental entire solutions with finite order.
Theorem 5.2. Let c∈C−{0}. Suppose that the complex differential difference equation
f(z)2+[f(z+c)+f′(z)]2=1 | (5.2) |
admits a transcendental entire solution with finite order. Then f must be of the form
f(z)=e2iz+b+e−2iz−b2, |
and c=−π4±kπ, k∈Z, where b∈C.
Theorem 5.3. The following system of complex differential equations
{f1(z)2+[f2(z)+f′2(z)]2=1,f2(z)2+[f1(z)+f′1(z)]2=1 | (5.3) |
has no any transcendental entire solutions with finite order.
For further studying the solutions of (2.1)-(2.4), we raise the following question which can not be solved now.
Question 5.1. How to describe the transcendental meromorphic solutions of (2.1)–(2.4)?
We thank the referee(s) for reading the manuscript very carefully and making a number of valuable and kind comments which improved the presentation.
This work was supported by the National Natural Science Foundation of China (11561033), the Natural Science Foundation of Jiangxi Province in China (20181BAB201001), and the Foundation of Education Department of Jiangxi (GJJ190876, GJJ191042, GJJ190895) of China.
The authors declare that none of the authors have any competing interests in the manuscript.
[1] |
T. Abdeljawad, N. Mlaiki, H. Aydi, N. Souayah, Double controlled metric type spaces and some fixed point results, Mathematics, 6 (2018), 320. doi: 10.3390/math6120320. doi: 10.3390/math6120320
![]() |
[2] | T. Abdeljawad, J. O. Alzabut, A. Mukheimer, Y. Zaidan, Best proximity points for cyclical contraction mappings with 0-boundedly compact decompositions, J. Comput. Anal. Appl., 15 (2013), 678–685. |
[3] |
T. Abdeljawad, J. Alzabut, A. Mukheimer, Y. Zaidan, Banach contraction principle for cyclical mappings on partial metric spaces, Fixed Point Theory Appl., 2012 (2012), 154. doi: 10.1186/1687-1812-2012-154. doi: 10.1186/1687-1812-2012-154
![]() |
[4] |
S. S. Alshoraify, A. Shoaib, M. Arshad, New types of F-contraction for multivalued mappings and related fixed point results in abstract spaces, J. Funct. Spaces, 2019 (2019), 1812461. doi: 10.1155/2019/1812461. doi: 10.1155/2019/1812461
![]() |
[5] |
I. Altun, N. A. Arifi, M. Jleli, A. Lashin, B. Samet, A new approach for the approximations of solutions to a common fixed point problem in metric fixed point theory, J. Funct. Spaces, 2016 (2016), 6759320. doi: 10.1155/2016/6759320. doi: 10.1155/2016/6759320
![]() |
[6] |
M. Arshad, Z. Kadelburg, S. Radenović, A. Shoaib, S. Shukla, Fixed points of α-dominated mappings on dislocated quasi metric spaces, Filomat, 31 (2017), 3041–3056. doi: 10.2298/FIL1711041A. doi: 10.2298/FIL1711041A
![]() |
[7] | J. P. Aubin, Mathematical methods of games and economic theory, North-Holland, Amsterdam, 1979. Available from: https://doi.org/10.1002/zamm.19810611016. |
[8] |
H. Aydi, M. F. Bota, E. Karapinar, S. Mitrović, Fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl., 2012 (2012), 88. doi: 10.1186/1687-1812-2012-88. doi: 10.1186/1687-1812-2012-88
![]() |
[9] |
T. G. Bhaskar, V. Laskhmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379–1393. doi: 10.1016/j.na.2005.10.017. doi: 10.1016/j.na.2005.10.017
![]() |
[10] | S. Bohnenblust, S. Karlin, Contributions to the theory of games, Princeton University Press, Princeton, 1950. |
[11] | S. Carl, S. Heikkila, Fixed point theory in ordered sets and applications; from differential and integral equations to game theory, New York: Springer, 2010. doi: 10.1007/978-1-4419-7585-0. |
[12] |
F. Gu, L. Wang, Some coupled fixed-point theorems in two quasi-partial metric spaces, Fixed Point Theory Appl., 2014 (2014), 19. doi: 10.1186/1687-1812-2014-19. doi: 10.1186/1687-1812-2014-19
![]() |
[13] |
J. Harjani, K. Sadarangani, Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations, Nonlinear Anal., 72 (2010), 1188–1197. doi: 10.1016/j.na.2009.08.003. doi: 10.1016/j.na.2009.08.003
![]() |
[14] |
T. Kamran, M. Samreen, O. U. A. Ain, A generalization of b-metric space and some fixed point theorems, Mathematics, 5 (2017), 19. doi: 10.3390/math5020019. doi: 10.3390/math5020019
![]() |
[15] |
E. Karapınar, İ. M. Erhan, A. Öztürk, Fixed point theorems on quasi-partial metric spaces, Math. Comput. Model., 57 (2013), 2442–2448. doi: 10.1016/j.mcm.2012.06.036. doi: 10.1016/j.mcm.2012.06.036
![]() |
[16] |
J. L. Li, Several extensions of the Abian-Brown fixed point theorem and their applications to extended and generalized Nash equalibria on chain-complete posets, J. Math. Anal. Appl., 409 (2014), 1084–1094. doi: 10.1016/j.jmaa.2013.07.070. doi: 10.1016/j.jmaa.2013.07.070
![]() |
[17] |
N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Mathematics, 6 (2018), 194. doi: 10.3390/math6100194. doi: 10.3390/math6100194
![]() |
[18] |
S. B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math., 30 (1969), 475–488. doi: 10.2140/pjm.1969.30.475. doi: 10.2140/pjm.1969.30.475
![]() |
[19] |
M. Nazam, A. Arif, C. Park, H. Mahmood, Some results in cone metric spaces with applications in homotopy theory, Open Math., 18 (2020), 295–306. doi: 10.1515/math-2020-0025. doi: 10.1515/math-2020-0025
![]() |
[20] |
J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223–239. doi: 10.1007/s11083-005-9018-5. doi: 10.1007/s11083-005-9018-5
![]() |
[21] |
A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to metrix equations, Proc. Am. Math. Soc., 132 (2004), 1435–1443. doi: 10.1090/S0002-9939-03-07220-4. doi: 10.1090/S0002-9939-03-07220-4
![]() |
[22] | T. Rasham, A. Shoaib, C. Park, M. Arshad, Fixed point results for a pair of multi dominated mappings on a smallest subset in K -sequentially dislocated quasi metric space with application, J. Comput. Anal. Appl., 25 (2018), 975–986. |
[23] | T. Rasham, A. Shoaib, N. Hussain, M. Arshad, Fixed point results for a pair of α∗-dominated multivalued mappings with applications, UPB Sci. Bull., Ser. A, Appl. Math. Phys., 81 (2019), 3–12. |
[24] |
A. Shazad, T. Rasham, G. Marino, A. Shoaib, On fixed point results for α∗-ψ-dominated fuzzy contractive mappings with graph, J. Intell. Fuzzy Syst., 38 (2020), 3093–3103. doi: 10.3233/JIFS-191020. doi: 10.3233/JIFS-191020
![]() |
[25] |
T. Rasham, Q. Mahmood, A. Shahzad, A. Shoaib, A. Azam, Some fixed point results for two families of fuzzy A-dominated contractive mappings on closed ball, J. Intell. Fuzzy Syst., 36 (2019) 3413–3422. doi: 10.3233/JIFS-181153. doi: 10.3233/JIFS-181153
![]() |
[26] |
T. Rasham, A. Shoaib, B. A. S. Alamri, M. Arshad, Multivalued fixed point results for new generalized F-dominated contractive mappings on dislocated metric space with application, J. Funct. Spaces, 2018 (2018), 4808764. doi: 10.1155/2018/4808764. doi: 10.1155/2018/4808764
![]() |
[27] |
T. Rasham, A. Shoaib, N. Hussain, B. A. S. Alamri, M. Arshad, Multivalued fixed point results in dislocated b-metric spaces with application to the system of nonlinear integral equations, Symmetry, 11 (2019), 40. doi: 10.3390/sym11010040. doi: 10.3390/sym11010040
![]() |
[28] |
B. Samet, C. Vetro, P. Vetro, Fixed point theorems for (α,ψ)-contractive type mappings, Nonlinear Anal., 75 (2012), 2154–2165. doi: 10.1016/j.na.2011.10.014. doi: 10.1016/j.na.2011.10.014
![]() |
[29] | M. Samreen, T. Kamran, M. Postolache, Extended b-metric space, extended b-comparison function and nonlinear contractions, UPB Sci. Bull., Ser. A, Appl. Math. Phys., 80 (2018), 21–28. |
[30] |
M. Sarwar, M. Rahman, G. Ali, Some fixed point results in dislocated quasi metric (dq-metric) spaces, J. Inequal. Appl., 2014 (2014), 278. doi: 10.1186/1029-242X-2014-278. doi: 10.1186/1029-242X-2014-278
![]() |
[31] |
M. Sgroi, C. Vetro, Multi-valued F-contractions and the solution of certain functional and integral equations, Filomat, 27 (2013), 1259–1268. doi: 10.2298/FIL1307259S. doi: 10.2298/FIL1307259S
![]() |
[32] | M. Nazam, C. Park, A. Hussain, M. Arshad, J. R. Lee, Fixed point theorems for F-contractions on closed ball in partial metric spaces, J. Comput. Anal. Appl., 27 (2019), 759–769. |
[33] | A. Shoaib, Fixed point results for α∗-ψ -multivalued mappings, Bull. Math. Anal. Appl., 8 (2016), 43–55. |
[34] |
A. Shoaib, S. S. Alshoraify, M. Arshad, Double controlled dislocated quasi-metric type spaces and some results, J. Math., 2020 (2020), 3734126. doi: 10.1155/2020/3734126. doi: 10.1155/2020/3734126
![]() |
[35] |
A. Shoaib, S. Kazi, A. Tassaddiq, S. S. Alshoraify, T. Rasham, Double controlled quasi-metric type spaces and some results, Complexity, 2020 (2020), 3460938. doi: 10.1155/2020/3460938. doi: 10.1155/2020/3460938
![]() |
[36] |
A. Shoaib, P. Kumam, S. S. Alshoraify, M. Arshad, Fixed point results in double controlled dislocated quasi metric type spaces, AIMS Math., 6 (2021), 1851–1864. doi: 10.3934/math.2021112. doi: 10.3934/math.2021112
![]() |
[37] |
L. Zhu, C. Zhu, C. Chen, Z. Stojanović, Multidimensional fixed points for generalized Ψ-quasi-contractions in quasi-metric-like spaces, J. Inequal. Appl., 2014 (2014), 27. doi: 10.1186/1029-242X-2014-27. doi: 10.1186/1029-242X-2014-27
![]() |
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