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On Coupled -Laplacian Fractional Differential Equations with Nonlinear Boundary Conditions
This paper is related to the existence and uniqueness of solutions to a coupled system of fractional differential equations (FDEs) with nonlinear -Laplacian operator by using fractional integral boundary conditions with nonlinear term and also to checking the Hyers-Ulam stability for the proposed problem. The functions involved in the proposed coupled system are continuous and satisfy certain growth conditions. By using topological degree theory some conditions are established which ensure the existence and uniqueness of solution to the proposed problem. Further, certain conditions are developed corresponding to Hyers-Ulam type stability for the positive solution of the considered coupled system of FDEs. Also, from applications point of view, we give an example.
Due to high profile accuracy and usability, FDEs become an area of interest for various fields of scientists and mathematicians. In last few years, some physical phenomenons were described through FDEs and compared with integer order differential equations which have better results, that is why researchers of different areas have paid great attention to study FDEs. The applications of FDEs can be studied in several disciplines including aerodynamics, engineering, electrical circuits, plasma physics, chemical reaction design, turbulent filtration in porous media, and signal and image processing; for further details we refer to [1–5].
Nonlinear operators have vital roles in differential equations; one of the most important operators used in FDEs is the classical nonlinear -Laplacian operator, which is defined as For further details and applications of the nonlinear -Laplacian operators, reader should study .
Researchers study different aspects of FDEs involving -Laplacian operators like the existence theory, which has been extensively investigated by using classical fixed point theory. The mentioned theory has been investigated very well for the aforesaid equations of ordinary and partial fractional differential equations. Since -Laplacian operators have been greatly applied in the mathematical modeling of large numbers of real world phenomenons devoted to physics, mechanics, dynamical systems, electrodynamics, and so forth, therefore researchers paid much attention to study such type of differential equation dealing with -Laplacian operators from different aspects including existence theory, multiplicity results, and stability analysis. For instance, Lu et al.  discussed Sturm-Liouville boundary value problems (BVP) of FDEs with -Laplacian operator for existence of two or three positive solutions by using fixed point theory. By applying Leggett-William fixed point theorem, the mentioned author studied the following problem: and also he provided proper example, where and denote standard Caputo fractional derivatives with , , , , , , , , and is continuous.
Hu and Zhang  have investigated a nonlinear FDE with -Laplacian operator for existence of solution as given by where , , and represent standard Caputo fractional derivatives, is continuous and , , , and . Zhi et al.  have investigated the existence of positive solutions for the nonlocal BVP of FDEs with -Laplacian operator and illustrated the problem with expressive example. The corresponding problem is given bywhere is -Laplacian operator and , , , , and expresses Caputo derivative of order . For further study about the existence theory and multiplicity results of -Laplacian operator involved in differential equations, see [10–14].
Using classical fixed point theory needs strong conditions to establish conditions for existence and uniqueness of solutions to FDEs and therefore restrict the applicability to certain classes of FDEs and their systems. To relax the criteria degree theory plays excellent roles for the existence of solutions to FDEs and their systems. Various degree theories including Brouwer and Leray-Schauder were established to deal with the existence theory of differential equations. An important degree theory known as topological degree theory which was introduced by Stamova  and later on extended by Isaia  has been used to establish existence theory for solutions to nonlinear differential and integral equations. The mentioned method is called prior-estimate method which needs no compactness of the operator and relaxes much the condition for existence and uniqueness of solutions to differential and integral equations. Recently, the aforesaid degree theory has been applied to investigate certain classes of FDEs with boundary conditions; see [17–19].
In recent years another aspect of FDEs which has greatly attracted the attentions of researchers is devoted to the stability analysis of the mentioned equations. Stability analysis plays significant roles in the optimization and numerical analysis of the aforesaid equations. Different kinds of stability have been studied for fractional differential equations including exponential, Mittag-Leffler, and Lyapunov stabilities; see [15, 20, 21]. An important stability was pointed out by Ulam , in 1940, which was formally introduced by Hyers , in 1941. The aforesaid stability has now been considered in many papers for classical differential equations; see [24–26]. For instance, Urs  has investigated the Hyers-Ulam stability for the following coupled periodic BVPs given as The Hyers-Ulam stability has been investigated for certain FDEs with boundary and initial conditions; see [28–30]. In many situations, Lyapunov type stability and its investigation are very difficult and time-consuming for certain nonlinear fractional differential equations. This is due to the predefined Lyapunov function which is often very difficult to construct for FDEs. Therefore, Hyers-Ulam type stability plays important roles in such a situation. Inspired from the above-mentioned work, in this paper, we study a coupled system of FDEs with nonlinear p-Laplacian operator by using topological degree theory. Further, we also investigated some conditions for the Hyers-Ulam stability of the solution to the proposed problem. The proposed problem is given by where , , , , , and and where stand for Caputo fractional derivative, is -Laplacian operator, where , denotes inverse of -Laplacian, and , , are continuous functions. Here, we remark that applying degree method to deal with existence and uniqueness and to find conditions for Hyers-Ulam stability to a coupled system of FDEs with -Laplacian operator has not been investigated properly to the best of our knowledge. Therefore thanks to the coincidence degree theory and nonlinear functional analysis greatly developed by Deimling , we establish necessary and sufficient conditions for existence and uniqueness as well as for Hyers-Ulam stability corresponding to the aforementioned problem considered by us. We also demonstrate our result through expressive example.
2. Axillary Results
Definition 1. The integral with fractional order of Riemann-Liouville type is defined for the function as provided that the integral on the right converges pointwise on .
Definition 2. The derivative with fractional order of Caputo type is defined for the function as where , is the integer part of such that the integral on the right converges pointwise on .
Lemma 3. Let and , and then the general solution of FDE is given by for some , and is smallest integer such that .
Let be the space of all continuous functions endowed with a norm which is obviously a Banach space. Then the product space denoted by under the norms is also a Banach space which will be used throughout this paper. For the coincidence degree theory and nonlinear functional analysis, we recall the following definitions which can be traced in [15, 16, 31] as follows.
Definition 4. Let the class of all bounded set of be denoted by Then the mapping for Kuratowski measure of noncompactness is defined as where .
Proposition 5. The following are the characteristics of Kuratowski measure : (1)For relative compact , then Kuratowski measure .(2)For seminorm , , , and .(3) yields ; .(4).(5)
Definition 6. Assume that is bounded and continuous mapping such that . Then is a -Lipschitz, where such that Then is called strict -contraction under the condition
Definition 7. The function is -condensing if Therefore yields
Further we have which is Lipschitz for , such that The condition yields that is a strict contraction.
Proposition 8. The mapping is -Lipschitz with constant if and only if , which is said to be compact.
Proposition 9. The operator is -Lipschitz for some constant if and only if , which is Lipschitz with constant .
Theorem 10. Let be a -contraction andUnder the conditions is bounded for and , with degree Then, has at least one fixed point.
Lemma 11 (see ). Let be a -Laplacian operator. Then (i)if , , and , then (ii)if and , then
Definition 12. Let . Then the operator equation given by is called Hyers-Ulam stable if, for any , the inequality given as has a unique fixed point say with constant such that , for all holds.
In view of Definition 12, we give the following definition.
Definition 13. The system of Hammerstein type integral equationsis called Hyers-Ulam stable such that, for () and for all and for every solution to the systemthere exists a unique solution of (21) satisfying the following system of inequalities:
3. Some Data Dependence Assumptions
To proceed further, let the following hypothesis hold: ()The nonlocal functions and where satisfy the following: where .()To satisfy the following growth conditions by the constants and for with the nonlocal functions and we have ()The functions and satisfy the following growth conditions under the constants , : ()There exist real valued constants and , and for all ,
4. Main Results
Theorem 14. Let be integrable function for FDEs and with integral boundary conditions; then the solution ofis provided bywhere is Green’s function, given by
Proof. Applying the operator on (28) and using Lemma 3, we get from problem (28) the following equivalent integral form asBy using conditions , we get . From (31), we haveApplying the operator on (32) and using Lemma 3 again, we get from problem (32) the following equivalent integral form given byBy using the condition in (33), we obtain Also in view of condition in (33), we getBy substituting the values of , , and in (33), we get the following integral equation:
According to Theorem 14, the equivalent system of Hammerstein type integral equations corresponding to coupled system (6) is given bywhere is defined asFrom and clearly,Further, we define the operators and byand for byHence we have , , and . Therefore the operator equation of Hammerstein type integral equations (36) is given byThus the solution of Hammerstein type equation (36) is the fixed points of operator equation (41).
Theorem 15. In view of hypotheses and , the operator is -Lipschitz and satisfies the growth condition given by
Proof. From condition and using , we getwhich implies that , where . To obtain the growth condition we haveUpon simplification, we get from (44) Similarly, we getNow where
Theorem 16. In view of hypothesis , the operator is continuous and satisfies the growth condition given bywhere , , and for each .
Proof. Consider bounded set with sequence converging to in . We have to show that as . Therefore, we have to considerDue to continuity of , one has as . Thus in view of Lebesgue dominated convergent theorem, we have as . Thus . Hence is continuous. Now for growth condition (49), we haveFrom, (51), we obtain the following result
Theorem 17. The operator is compact and -Lipschitz with constant zero.
Proof. Consider a sequence in bounded set such that is subset of ; in view of (49), we seeThus is bounded for Let us, for any , considerBy simplification, we get the following result:And also in same fashion, one hasTherefore if , then the right hand side of both sides of (55) and (56) tends to zero. Thus are equicontinuous, and therefore is equicontinuous on . Hence, thanks to the Arzelá-Ascoli theorem, is compact. Also, by proposition is -Lipschitz with constant zero.
Theorem 19. In view of assumptions – such that , then the coupled system (6) has at least one solution together with condition that the set of the solutions is bounded in .