Abstract
In this paper, by using the partial order method, the existence and uniqueness of a solution for systems of a class of abstract operator equations in Banach spaces are discussed. The result obtained in this paper improves and unifies many recent results. Two applications to the system of nonlinear differential equations and the systems of nonlinear differential equations in Banach spaces are given, and the unique solution and interactive sequences which converge the unique solution and the error estimation are obtained.
1. Introduction
Guo and Lakshmikantham [1] introduced the definition of the mixed monotone operator and the coupled fixed point, and there are many good results (see [2–23]). Recently, from paper [6], using the monotone iterative techniques, the iterative unique solution of the following nonlinear mixed monotone Fredholm-type integral equations in Banach spaces is obtained:where and .
In this paper, the following nonlinear abstract operator equations in Banach spaces are discussed:where and is a partial interval in which is denoted as the following:
For convenience, the following assumptions are made: There exist positive bounded operators which satisfy. , and for any , the following is obtained: . There exists a positive bounded operator , and for any , the following is obtained: in which the spectral radius satisfies
In this paper, firstly, by using the partial order method, the existence and uniqueness of a solution for systems of a class of abstract operator equations in Banach spaces are discussed. And next, two applications to the system of nonlinear integral equations and the system of nonlinear differential equations in Banach spaces are given, and the unique solution and interactive sequences which converge a unique solution and the error estimation are obtained.
2. The Interactive Solution of Abstract Operator Equations
Let be a cone in , i.e., a closed convex subset, such that for any and . A partial order in is defined as . A cone is said to be normal if there exists a constant which satisfies , implying , where denotes the zero element of . And, the smallest number is called as the normal constant of and denoted as . The cone is normal iff every ordered interval is bounded.
The following theorem is the main results in this section.
Theorem 1. Let be a cone in , . Suppose that satisfies conditions . Then,(i)There exists a unique solution of equation (2) in , and for any solutions of equation (2) , one has .(ii)For any initial value , the following iterative sequences are constructed:which satisfy , and for any ,there exists a natural number which satisfies as , the following is obtained:
Proof. By , it is known that the operator is reversible. And, from condition , is the positive operator. LetThen, equation (7) can be substituted by the following:By conditions , it is easy to obtain that operators satisfy the following:(1)(2) are the mixed monotone operator(3), where Letting , the following two results are obtained by mathematical induction:In fact, from (1) and (3), one hasSuppose that for , one has (12) and (13). Then, as , by (2) and (3), the following is obtained:Then, it is known thatThen, for any natural number , (12) and (13) are obtained by mathematical induction.
Next, it is proved that is Cauchy sequences. From condition , it is known thatthen by ([14], V 3.9), .
Thus, for any the following is obtained:Then, there exists a natural number which satisfiesAnd, by (12) and (13), it is obtained thatSo, by (13), it is known thatThen, by the normality of and (19), it is known thatThus, the following is obtained:i.e., is Cauchy sequences. So, there exists (D is bounded), such that .
And, by , the normality of , and (19), one obtainsthereforeThus, , , andso by (2), (3), and (11), it is also obtained thatLetting and by (27), .
Then, by the definition of , one obtains , i.e., is a solution of equation (2).
Lastly, it is proven that the solution is unique. Supposing that also satisfies equation (2), then by (11) and mathematical induction, the following is obtained:Thus, .
And, letting in (24), as , the following is obtained:Similarly, as , the following is obtained:The proof is complete.
Remark 1. In Theorem 1, it is only supposed that operators satisfy the partial condition, and the unique solution and interactive sequences which converge a unique solution are obtained.
3. The Application of Nonlinear Integral Equations
In this section, the following nonlinear integral equations are considered:where (here, the continuity of is not assumed) and , , , and is a real Banach space with norm .
In this section, the iterative solution of a nonlinear integral equation (33) is discussed. For convenience, the following assumptions are made: For the nonnegative bounded continuous function , and nonnegative integrable , , one has There exists a constant , for any , which satisfies For any , the following is satisfied: .
In this section, the following main theorem is obtained.
Theorem 2. Let be a normal cone in . Suppose conditions hold. Then, there exists a unique solution of equation (2) , and there are iterative sequences converging to the unique solution, and corresponding error estimates are given.
Proof. Let . Then, is a cone. Thus, by the normal of , is also normal.
The following operator is considered:where for any ,Then, . It is easy to know that is a solution of (33) if and only if is a solution of the following integral equations:Next, from conditions , it is obtained that the operators satisfy the whole condition of Theorem 1.
In fact, :(i)Let Then, and . Thus, Therefore, for the equation , there exists a unique solution . Then, by , for any , the following is obtained: Obviously, .(ii)By , the following is obtained: Similarly, .(iii)From , the following is obtained: Then, by (41), it is known that Therefore, from (i), (ii), and (iii), letting in Theorem 1, it is easy to know that the condition holds. Finally, for any initial value , by constructing the iterative sequences one has , and for any , there exists a natural number which satisfies as , the following is obtained: This completes the proof of Theorem 2.
4. The Application of Nonlinear Differential Equations
In this section, the following nonlinear initial value problems of the differential equation are considered:where , , , and is a real Banach space with norm .
For convenience, the following assumptions are made: There exists the nonnegative bounded integrable functions which satisfy There exists constant , for any , which satisfies For any , the following is satisfied: .
Then, the following theorem is obtained.
Theorem 3. Let be a normal cone in . Suppose that conditions hold. Then, there exists a unique solution of equation (48) , and there are iterative sequences converging to the unique solution, and corresponding error estimates are given.
Proof. Firstly, differential equation (48) is turned into integral equations. For any fixed , the following one-order linear ordinary differential initial value problems in Banach spaces are investigated:It is easy to know that is a solution of (52) if and only if is a solution of the following integral equations:Next, the operator is defined as the following:Obviously, is a solution of (48) if and only ifNext, similar to the proof of Theorem 2, it is tested whether the operators satisfy the whole condition of Theorem 1 from conditions . Therefore, the result of Theorem 3 is obtained from Theorem 1.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
The author read and approved the final manuscript.
Acknowledgments
The author was supported by the Project of National Social Science Fund of China (NSSF) (18BTY015) and the Shandong Province Higher Educational Science and Technology Program (J16LI01).