Abstract
For , we analyze a stationary superdiffusion equation in the right angle in the unknown : where is the Caputo fractional derivative. The classical solvability in the weighted fractional Hölder classes of the associated boundary problems is addressed.
1. Introduction
Fractional partial differential equations (FPDE) play a key role in the description of the so-called anomalous phenomena in nature and in the theory of complex systems (see, e.g., [1]). In particular, these equations provide a more faithful representation of the long-memory and nonlocal dependence of many anomalous processes. The signature of an anomalous diffusion species scales as a nonlinear power law in time; i.e., , . When , this is referred to as superdiffusion.
Superdiffusion is used in modelling turbulent flow [2, 3], chaotic dynamics of classical conservative systems [4], model solute transport in underground aquifers [5–7], and rivers [8–10], biophysics [11], and physical and chemical models described by the Lévy processes [12, 13]. In the present paper, we focus on the boundary value problems to the stationary superdiffusion equation.
Let be the first quarter with a boundary , , . For a fixed , we consider the linear equation in the unknown function ,subject either to the Dirichlet boundary condition (DBC),or to the Neumann boundary condition (NBC),where the functions , , , , are prescribed.
Here, the symbols and stand for the Caputo fractional derivative of order with respect to and .
Introducing the function,we define the Caputo derivative as (see, e.g., (2.4.1) in [1]) where is the ceiling function of (i.e., the smallest integer greater than or equal to ), being the Euler Gamma-function. An equivalent definition in the case of reads (see (2.4.15) in [1]) In the limit case , the Caputo fractional derivatives of boils down .
Elliptic boundary value problems, , in domain with conical and dihedral singularities have been extensively studied, starting with Kondratiev’s famous paper [14]. In this field of researches, it should be also noted the works of Kondariev and Oleinik [15], Borsuk and Kondratiev [16], Grisvard [17], and Maz’ya and Plamenevskii [18]. The results of those investigations are formulated in both and spaces and corresponding weighted classes, where the desired functions together with their derivatives are bounded with some power weights in the or norms.
Recently, a great attention in the literature has been devoted to the study of boundary value problems to the fractional Laplacian operator with order (see, e.g., [19–24] and references therein). Boundary value problems for the equation with the main part,are studied with various approaches, such as spectral technique, method of potential theory, and Fourier method (see [25–28] and references therein). We also quote the works [29, 30], where the authors presented a Galerkin finite element approximation for variational solution to the steady state fractional advection dispersion equation:where , are left and right fractional integral operators, , .
The goal of the present paper is the proof of the well-posedness and the regularity of solutions to boundary value problems (1)-(3) in weighted Hölder classes. It is worth noting that these classes allow one to control the behavior of the solution near the boundary including the corner point and at the infinity.
Outline of the Paper. In the next section, we introduce necessary functional spaces and state the main results (Theorems 3 and 4) along with the general assumptions. In Section 3, we construct the integral representation to in the case of homogenous boundary conditions. To this end, we apply Mellin transform and reduce problem (1)-(3) to the linear nonhomogenous difference equation of the first-order with variables coefficients in the two-dimensional case. Section 4 is devoted to some auxiliary results which will play a key role in the investigation. In Section 5, we estimate the seminorms of the minor derivatives , , and in Section 6 we evaluate the Hölder coefficients of the major derivatives , . In Section 7, using these estimates, we provide the proofs of Theorems 3 and 4. Moreover, in Remark 27 we show how results of Theorems 3 and 4 can be extend to the more general equation compared to (1).
2. Functional Setting and Main Results
Throughout this work, the symbol will denote a generic positive constant, depending only on the structural quantities of the problem. We will carry out our analysis in the framework of the weighted Hölder spaces. Let be arbitrary fixed. We denote by and the distance from a point to the origin and to the boundary , correspondingly. Then for every and from we define and . Note that if , then and .
For fixed , we introduce the Banach spaces and of the functions with the normswherefor .
Definition 1. A function belongs to the class , for , if , , , and the norms below are finite, if , and in the case of .
Definition 2. A function belongs to the class , for , if , , , and the norms below are finite, if , and in the case of .
In a similar way we introduce the spaces and . As for and , these spaces concave with and .
and defined above are Banach spaces. Indeed, the fact that they are normed spaces is easily seen, whereas the completeness follows from Theorem 2.7 [31] together with standard arguments (see, e.g., Remark 3.1.3 in [32]).
We begin to stipulate the general assumptions.
h1 (condition on the parameters): we require
h2 (conditions on the right-hand sides of boundary conditions): we demand
h3 (conditions on the right-hand side of the equation): we suppose
We are now in the position to state our main results.
Theorem 3. Let assumptions (h1), (h2), and (17) hold and moreover for every points and andThen, (1) subject either to the DBC (2) or to the NBC (3), admits a unique classical solution on , satisfying the regularityfor . Besides, the following estimates hold:in the case of DBC (2), andif NBC (3) hold.
Theorem 4. Let assumptions (h1), (h2), and (18) hold. Then, boundary value problems (1)-(3) admit a unique classical solution on , satisfying the regularityfor . Besides, the following estimates hold:in the case of DBC (2), andif NBC (3) hold.
Remark 5. It is easily apparent that the functions,satisfy conditions (h2) and (17), (19), and (20).
The remainder of the paper is devoted to the proof of Theorems 3 and 4 in the DBC case. The proof of these theorems for NBC is almost identical and is left to the interested reader.
3. Integral Representation for in the Special Case
We first dwell on the special case where and is a finite function. Namely (2), (17), and (19) are replaced by the simpler conditionsfor some given positive and
We denote by the Mellin transform of the function . Due to conditions (30)-(32) and assumptions of Theorem 3, we can apply, at least formally, the Mellin transformation to problem (1) and (2) (see for details § 1.4 and § 2.5 in [1]). Then simple calculations lead to the equation
Introducing new variablesand new functions we rewrite the equation in the more compact formThus, we transform problem (1) and (2) to the linear nonhomogeneous difference equation of the first-order with variable coefficients. In order to solve this equation, we adapt the technique from Section 3 in [33] to our case.
Proposition 6. Let denote Euler-Mascheroni constant (see, e.g., Definition in [34]),and let and be arbitrary analytic functions such thatThen the functionsolves homogenous equation (36) (i.e., ) and does not have any poles.
If, in addition, the functions and have no zeros, then the function does not have any poles and the following estimate holds:for and for every fixed and satisfying inequalities above.
Proof. In order to verify that the function solves homogenous equation (36), it is enough to substitute in (36) and take into account the properties of the Gamma-function: . Besides, the properties of the function are simple consequences of the well-known properties of Gamma function:(i) has simple poles in the points(ii)The Stirling asymptotic formula holds: as while remains bounded
We are now in the position to construct the solution of nonhomogeneous equation (36).
Till the end of the paper, we assume that is an arbitrary fixed quantity and define the contour in the complex plane as(i) (ii)If , then consists of three parts: the half-circle , the intervals and with the small positive number , .(iii)The contour () is obtained from after its shifting to the right on .
Introducing the periodic function with period 1,we assert the following results.
Proposition 7. Let , ; conditions (32) and (31) hold, and letThen the solution of inhomogeneous equation (36) is given bywhere is defined in (39) with , .
Proof. First, we prove this statement in the case ; i.e., . To this end, we will seek the solution of (36) as a product where the unknown function solves the first-order difference inhomogeneous equation with the constant coefficientsThen adapting the technique from [35] to our case, we deduce thatif condition (45) holds.
Indeed, substituting to (48) and applying Cauchy’s residue theorem arrive at To reach these equalities, we essentially used the following properties of the functions , , and :(i) has no poles if and , satisfy (45).(ii) is periodic of period with simple poles at and multiple poles at . Besides, the following inequality holds: for (iii)There is the asymptotic representation for the bounded and and if , meet requirement (45).Note that statements (i) and (ii) are simple consequences of Proposition 6 and (32). As for assessment (iii), it can be easily drawn from (i) and (ii).
After that, we return to the representation of and obtain solution (36) asThis proves Proposition 7 in the case .
Recasting the arguments above in the case and applying Cauchy theorem allow us to prove Proposition 7 in the case of arbitrary contour , . It completes the proof of Proposition 7.
We are now in the position to obtain integral representation of the solution . Indeed, Propositions 6 and 7 providewhereNote that, conditions on hold, for example, in the domain
At last, we carry out the inverse Mellin transform to derive an explicit integral representation for ,with for and , where and , , are chosen that
In the light of (56), we can pick up and as
4. Some Technical Results
First we introduce some equivalent norms.
Proposition 8. Let and be any nonnegative numbers. Then for any functions and , , and , we have the following norm equivalence:
The proof of this statement follows with direct calculations. LetNext we represent certain estimates for the functionswhich will be frequently used to evaluate the functions and ,
Proposition 9. Let be positive number, . Then there are estimates.
Proof. For simplicity consideration, we put . Here we prove this proposition for the function . The case of is considered in the similar way. We start our consideration with the case of positive (i.e., ).
The Stirling asymptotic formula for the Gamma-function,provides that the integrand , , has the order at the infinity. Moreover, the function with has no poles in the domain Thus, by the residue theorem, we get Then, following [33], we decompose the plane into 15 subdomains, as shown in Figure 1In this decomposition is a sufficiently large number such that the terms in (65) can be neglected in all regions .
Taking into account this decomposition, we represent the function asIt is apparent that, if the function is estimated analogous to the function if . Thus, we consider here just case . To this end, we apply Stirling formula (65) and the well-known properties of the Gamma-function. Introducing the functions we arrive at the representation to the function in every domain , , and ,where the functions , , are uniformly bounded with respect to and together with .
Moreover,for belongs to one of the corresponding domains , , , It is worth mentioning that representation (70) holds for each .
If , representation (70) with aid (65) and (71) arrives at the inequalityThese relations guarantee the first estimate in Proposition 9 for .
Further, we verify the estimate if . Representation (70) and straightforward calculations lead to the inequalityTo get the same estimate in the domains , , and , it is enough to integrate by parts. Namely, let us consider the case . Based on representation (70), we can writeIt is apparent that Thus, we can enhance estimate (74), taking into account (71), so as to getRecasting the arguments above in the case of domains , , and yields the same estimate to .
After that, we combine inequalities (73) and (76) and obtain the required estimate to if .
At last, we are left to produce suitable estimate of . Standard direct calculations lead toIn conclusion, we derive thatAfter that, we remark that the terms , , and share the same properties in and , . Thus, the estimate of is very similar and we will confine ourselves the consideration of the case .
Simple conclusions draw to representations in whereTherefore, the main term in the integral isNote that the second term here is a regular function of and . In order to evaluate the first term in this representation, we apply Proposition 7.1 from [33] and obtain where and are twice continuously differentiated and bounded functions for .
In summary, we obtain the estimateFinally, this inequality together with (78) completes the proof of Proposition 9.
Recasting the proof of the previous proposition, we state the results, which will be used to evaluate , , , in Sections 5 and 6.
Proposition 10. Let , , , ; then the inequalities are fulfilled
The following results are related to the properties of the functionwhere , are some positive constant, , .
Proposition 11. Let , . Then there are the following estimates for every fixed :(i)(ii)(iii)Besides, if , then
Proof. We will carry out the detailed proof in case. The arguments for are almost identical and left to the interested reader.
Straightforward calculations lead to the representationFirst, we consider case . Asymptotic representation (70) with the decomposition in Figure 1 provides estimate in with andin . Then, this inequality together with (89) leads to statement (ii).
Let us verify point (ii). One can easily check that After that, recasting the proof of Proposition 9 with aid the last inequality arrives at the estimateThen we are left to evaluate the terms , We restrict ourselves the estimate of , the remaining terms are evaluated the same way. Asymptotic (70) with and the change of variables, , provide the representationAfter that, following the proof of Proposition 9, we integrate by parts and deduce The standard calculations lead toIn conclusion, we reach the estimateCollecting this inequality with (93), we deduce the first estimate in statement (ii) of Proposition 11.
Further, to improve the estimate for , we consider two different cases:
(i) (ii) It is apparent that, in the first case there isand, therefore, keeping in mind the previous estimate of , we deduce thatComing to the second case (i.e., ), the change of the variable, , leads toThe straightforward calculations ensure the uniformly boundedness of the first term in (102); moreover, this estimate is independent of , , and .
Further, we treat the second term in (102). Integrating by parts leads to After that, the properties of functions , , and allow us to extend this estimate and conclude Hence, this estimate completes the proof of statement (ii).
Finally, we are left to verify statement (iii). As for the first equality in (iii), it is a simple consequence of Lemma 5.3 [36]. Let us check the second inequality in (iii). To this end, we rewrite as where we set Then representation (70) with and direct calculations providewhich impliesThus, the claim is proven.
Finally, we complete this preliminary section with two estimates that will be frequently used in the following sections.
Proposition 12. Let , , , , . Then the following inequalities are fulfilled: (i) for every fixed . Besides,(ii)where the positive constant is independent of , , , , , and .
Proof. To show the validity of statement (i), it is enough to recast the arguments in Propositions 11 and 10 if .
Next, we obtain the second inequality from this proposition. For the sake of clarity, we consider case . In the opposite case, we exchange the function by and repeat the arguments below.
The direct calculations together with the change of variables, , reduce the function to the formAt this point, we consider the two different cases for :
(i) (ii) In the first case, we have Note that we apply integrating by parts in order to reach the last estimate.
Further, we obtain the same estimate in the second case. To this end, we use the easily verified inequality,and deduce the boundIn summary, we complete the proof of Proposition 12.
5. Estimates on Minor Derivatives of
To evaluate the function , we use representation (57) with and , . The change of variablesin the integrals in (57) yieldswhere
5.1. Estimate on Maximum of
Before evaluating the functions and , , , we describe the suitable properties of the kernel .
Lemma 13. Let and let , . Then there are estimates. (i)(ii)If or , the following representations hold: where is a Dirac delta function and (iii)Here the constant is independent on , and .
Proof. First of all, we verify statement (i). It is easy to see that inequality (51) guarantees for Then the results of Proposition 9 with aid of (125) provide statement (i).
As for statement (ii), we confirm ourselves the case of , due to the arguments in the case are similar.
First, we represent , , as Here we use the same reasons as in the proof of Proposition 9.
Then, as mentioned in the proof of Proposition 7 (see (ii) there) the function has the simple poles if . Thus, following Cauchy’s residue theorem, we rewrite aswhere we set Concerning the estimate of , we apply Lemma 5.3 from [36] and deduceThen the asymptotic representation of Gamma-function (see, e.g., in [1]),provides the estimateHence, we are left to evaluate the function . To this end, we rewrite the term in the formwhere we put Treating the first term in (132) via Proposition 9 and inequalities (125), we arrive atConcerning , we getand thereforeAfter that, statement (ii) of Proposition 12 (with ) providesIn summary, we can conclude thatFinally, coming to , we first rewrite it asThen, the tedious calculations with Propositions 9 and 10 and estimates (125) entailHence, representations (127) and (132) together with estimates (134), (138), and (140) provide statement (ii) for the function whereAt last, the proof of statement (iii) is simple consequences of statements (i) and (ii) from this proposition. Thus, the claim is proven.
We are ready now to state estimates of the function .
Lemma 14. Let assumptions (h1), (31), and (32) hold. Then the function represented with (119) satisfies inequalities
Proof. At the beginning, we verify the first inequality in this lemma. To this end, we evaluate the term . Putting in (119), we concludehere we use the simple inequality Concerning , we apply statement (i) from Lemma 13 with to deduce with the constant is independent of , , and . Coming to the term , applying Proposition 11 to the functionand keeping in mind Proposition 9 with inequality (125), we getIn conclusion, we haveif
Then, collecting this inequality with (143) and (145) yieldsFurther, we estimate . Assumption (31) provides thatwhere the function satisfies conditions (31) and (32). Thus, substituting this representation of in (143) and recasting arguments above, we deduceIt is easy to see that the second estimate of Lemma 14 is proved with the similar arguments.
Finally, we are left to estimate , For simplicity consideration, we evaluate . Another case is studied with the same arguments. Using statement (ii) of Lemma 13 with , we rewrite as After that, statements (ii) and (iii) of Lemma 13 and Proposition 10 yield inequality This completes the proof of Lemma 14.
5.2. Estimate on ,
First, based on representation (57) and formula (2.1.17) [1], we obtainwhere we set
After that, recasting the arguments of Lemma 14 with aid of Propositions 9–11, we can state the following results.
Lemma 15. Let and , and let the nonnegative numbers and satisfy . Then the kernel possesses the following properties: (i)(ii)(iii)(iv)If either and , or and , , then where the functions and satisfy statements (i)-(iii) of this lemma and
Now we are ready to describe properties of low derivatives to the function .
Lemma 16. Let assumptions of Lemma 14 hold, and . Then there are the estimates.(i)(ii)
Proof. Taking into account (154), we will carry out the detailed proof of the inequality The remaining terms in (i) and (ii) are estimated in the same way and with the recasting of the corresponding arguments of Lemma 14.
We begin to evaluate the first term in (163). To this end, we represent aswhere we put Statement (ii) of Lemma 15 providesAs for terms , , , and , we apply statement (iii) of Lemma 15 and deduce Finally, statement (i) of Lemma 16 guarantees the estimate to Hence, representation (164) and inequalities for entailIn order to complete the proof, we need a similar estimate to the term . We exchange in (154) by the function from (iv) in Lemma 15. After that, recasting the arguments above together with statement (iv) in Lemma 15 derivesThis finishes the proof of the lemma.
The following result is a simple consequence of Lemmas 14 and 16, interpolation inequalities, and Proposition 8.
Proposition 17. Let assumptions of Lemma 14 hold. Then
6. Estimates on ,
We begin our consideration with the estimates of the function . Here, we use representation (119) to the function with ; i.e., .
Recasting the arguments leading to (127) arrives at where we putAfter that, straightforward calculations arrive at
Thus, we can rewrite in the form
We are now in the position to calculate . Indeed, formula (2.1.17) [1] and direct calculations provide the representation where we put with After that, introducing new variablesand new functions we rewrite the functions , , and in the more comfortable forms
In order to reach the last equality in (181), we recast the arguments of Lemma 13 which lead to representations (132), (134), and (138). Indeed,andwhere the positive constant is independent of and .
Hence, we obtain
6.1. Estimates on Maximum
Further, in virtue of representations (176) and (181), estimates of and follow from the corresponding estimates of the functions .
First, we describe the properties of the kernels and .
Lemma 18. Let , , . Then there are the following estimates. (i)(ii)(iii)where the positive constant is independent of , , and .
Proof. First, applying Proposition 11 with and and using definition (85), we havewhereThen, applying statement (iii) in Propositions 11 and 12 with aid (125) arrives at the first inequality of this proposition.
Further, we proceed with a detailed proof of statement (ii). The proof of statement (iii) is almost identical and is left to the interested reader.
At the beginning, we evaluate the first term in the left-hand side of the inequality in (ii)Simple calculations lead to where we set At this point, we estimate each term separately.
By Proposition 9 and estimates (125),where the positive constant is independent of and .
Concerning , keeping in mind Propositions 9-10 and estimates (125), we deducewhere the positive constant is independent of and .
In summary, we can concludeIn order to obtain the last inequality, we use the condition .
Then, we are left to evaluate the second term in inequality (ii)To this end, coming to representation (191), we deducewithProposition 9, statement (ii) in Proposition 11 (with , ), and estimate (125) lead toThus, we haveFinally, Propositions 9 and 10 provide inequalitywhich guarantees thatThen, collecting this inequality with the estimate for yieldswith the positive constant independent of and .
The last inequality together with the analogous estimate for completes the proof of statement (ii). This finishes the proof of Lemma 18.
Now we estimate the function .
Lemma 19. Let assumptions of Lemma 14 hold. Then
Proof. Here we prove in detail the estimateThe rest inequalities are obtained with the same techniques and with the recasting the arguments in Lemma 14.
By representations (176) and (181), we haveFurther, we estimate separately.
By statement (i) in Lemma 18,Concerning , we rewrite this function in the form analogous to (164) where we put It is apparent that the estimates of each are simple consequence of Lemma 18. Thus, we deduceThe last inequality with (206) and (207) providesThis finishes the proof of Lemma 19.
6.2. Estimates on Hölder Seminorms of
First of all, we return to representations (176) and (181) and evaluate each function separately.
Proposition 20. Let assumptions of Lemma 14 hold. Then
Proof. It is apparent that (181) and properties of the function ensure estimates of .
Next we evaluate the term . For simplicity consideration, we assumeand putIn virtue of representation (181) and Lemma 18, we have Then we are left to tackle the second term in the right-hand side of the last inequality. Simple calculations lead to At this point, recasting the proof of statement (i) in Lemma 18 arrives at In conclusion, we obtain the estimate In order to reach the last inequality we use property (32) to the function .
Summarizing, we have inequalityThe same arguments in the case of the difference guarantee estimateThus, collecting estimates for we reach the conclusionThis completes the proof of Proposition 20.
Next we obtain the same results to the function . For each point , , , : and , we putFurther, we describe the properties of the kernel .
Lemma 21. Let and , , be some positive numbers, , and letThen the kernel possesses the following:(i)(ii) (iii)(iv)where the positive constant is independent of , , .
Since the proof of this result is technically tedious and repeating certain steps in the proof of Lemma 18, we provide one in the Appendix.
Then, based on Lemma 21, we can conclude the following.
Lemma 22. Let assumptions of Lemma 14 hold. Then there is the estimate
Proof. We provide here the estimate of . The case of is treated in similar arguments.
It should be noted that it is enough to evaluate in case of (223). Indeed, in the opposite case, its estimate follows from Lemma 19. Thus, we assume that (223) hold. Introducing the domainwe rewrite the function in a more suitable form: Then we consider the differencewhere we set Then puttingand making change of variables (179), we reduce the domain to either in the case of or if : At this point, we estimate each term separately.
Change of variables (179) leads to Applying decomposition like (164) to each term in the representation above, then statements (i) and (ii) in Lemma 21 provide estimateIndeed, let us verify this with the example of the first term in the representation to . Standard calculations lead to Then, properties of the function and Lemma 21 with , , yield To reach the last inequality we apply (223) to (233). The remaining terms in the representation of are evaluated in the same way.
Concerning , we repeat the arguments above with changing by and obtain the estimate like (236) to the function .
Coming to , change of variables (179), Lemma 21, and condition (223) arrive atFinally, we are left to estimate . Change of variables (179) and statement (iv) in Lemma 21 lead toCollecting this inequality with (236) and (239), we get the boundThis finishes the proof of Lemma 22.
At this point, we come back to . Keeping in mind Proposition 20, Lemma 22, and representations (176) and (181), we deduce the inequality
Further, recasting arguments in Proposition 20 and Lemmas 22 and 14 and applying Proposition 8, we assert the result.
Lemma 23. Let assumption of Lemma 14 hold. Then
In order to obtain the same results to the functions and , , it is enough to repeat the arguments from this section. Thus, we have.
Remark 24. Under conditions of Lemma 23, the functions , , , satisfy inequalities (23) and (27) and
7. Proofs of Theorems 3 and 4
Under restrictions (30) and (31), the proof of Theorems 3 and 4 follows from the arguments of Sections 3–6. Indeed, Proposition 17 and Lemmas 16, 19, and 23 with Remark 24 provide existence of a solution which satisfies estimates (22), (23) and (26), (27). Next, direct calculations with aid of representation (176) allow one to verify that the constructed solution with (57) satisfies (1) and the corresponding boundary conditions. Further, the uniqueness of the solution follows from the coercive estimates. This completes the proof of Theorems 3 and 4 if (30) and (31) hold.
At this point, we remove restriction (30). To this end, we introduce the new functionThe following properties of the function could be easily checked with aid in [37] and properties of the functions and (see (h2), (19)): if , thenAfter that, we look for the solution to (1), (2) in the formwhere the unknown function solves the boundary value problem with homogenous boundary conditionwithIt is apparent that the function satisfies conditions (h3), (19). Thus, arguments of Sections 3–6 guarantee the one-valued solvability to (248) and, besides, the function satisfies properties (22), (23), and (27) and (26). Then, we return the function and obtain the one-to-one solvability of the original problem, where the solution possesses the same properties as . Thus, Theorems 3 and 4 are proven in the absence of (30).
Finally, we are left to remove restriction (31). To this end, it is worth to remark that a function which satisfies condition (31) belongs to with any positive and hence also with , . Further, it is enough to repeat the arguments from Sections 3–7 in order to prove Theorem 3, if satisfies (17). If meets requirement (18), the same reasons hold. This finishes the proof of Theorems 3 and 4.
Remark 25. Requirement (30) could be removed without introducing the function . To this end, it is enough to recast the arguments of Sections 3–6 to the problems with the homogenous equation and nonhomogeneous boundary conditions subject either to the Dirichlet boundary condition (DBC) or to the Neumann boundary condition (NBC)
Remark 26. Arguments from Sections 3–6 guarantee that the requirement on the weight in the assumption (h1) can be changed by
Remark 27. Actually, with inessential modifications in the proofs, the very same results hold for the more general equationwhere . The details are left to the interested reader.
Appendix
Proof of Lemma 21
To prove statement (i), we estimate the first term in this inequality. The second one is evaluated with the same arguments. Let us rewrite the first term in the formwhere is defined in (191). After that, (191), (193), and (199) arrive at with the positive constant being independent of , , .
Note that to reach the last inequality, we use the fact that . Thus, claim (i) is proven.
Concerning statement (ii), we apply inequalities analogous to (199) and (201) and obtainThen, keeping in mind this inequality, we deduce Next, recasting this argument in the case of the term provides the analogous estimate to the second term in the left-hand side of (ii). This completes the proof of (ii).
It is apparent that statement (iii) follows from direct calculations and representation (A.1) and Propositions 11 and 12.
As for (iv), we prove the first inequality; the second one is obtained with the same arguments. Let us consider the differencewhere we putfor
Introducing new functionswe rewrite the functions in the suitable forms Recasting the arguments of Propositions 9 and 10 arrives at By this inequality together with Propositions 9 and 10, we haveIn conclusion, we obtain
Finally, we estimate the term . To this end, we again represent via and then using Propositions 11 and 12 deduce Collecting this inequality with (A.11), we obtain the first inequality in (iv). This completes the proof of Lemma 21.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.