On the absence of global solutions for some q-difference inequalities
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In this paper, we obtain sufficient conditions for the nonexistence of global solutions for some classes of q-difference inequalities. Our approach is based on the weak formulation of the problem, a particular choice of the test function, and some q-integral inequalities.
KeywordsNonexistence Global solution q-difference inequalities
MSC39A13 35A23 35B44
The study of sufficient conditions for the nonexistence of global solutions to differential equations or inequalities provides important information in theory as in applications. First, sufficient conditions for the absence of solutions provide necessary conditions for the existence of solutions. Second, useful information on limiting behaviors of many physical systems can be obtained via the nonexistence criteria. Indeed, having an information on the blowing-up of solutions can help in preventing accidents and malfunction in industry. It helps also in improving the performance of machines and extending their lifespan.
There are several works in the literature concerning the nonexistence of solutions for different classes of differential equations or inequalities involving nonstandard derivatives. In particular, the study of the absence of solutions for different types of fractional differential problems has received a great attention from many researchers. In this direction, we refer the reader to [15, 16, 18, 19, 20, 21] and the references therein. However, to the best of our knowledge, there are no investigations on the nonexistence of solutions in quantum calculus.
The q-difference calculus or quantum calculus is an old subject, which is rich in history and in applications. It was initiated by Jackson [11, 12] and developed by many researchers (see, e.g., [1, 6, 8]). We can find in the literature several papers dealing with the existence of solutions for different kinds of q-difference equations; see, for example, [3, 4, 5, 9, 10, 13, 17, 24] and the references therein.
In this paper, we obtain sufficient criteria for the absence of global solutions to problems (1)–(2) and (3)–(4). The proofs are based on an extension of the test function method due to Mitidieri and Pohozaev  to quantum calculus.
The paper is organized as follows. In Sect. 2, we recall some basic concepts on q-calculus and present some properties and lemmas that will be used in the proofs of our results. Section 3 is devoted to study the nonexistence of global solutions for problem (1)–(2). In Sect. 4, we establish a nonexistence result for problem (3)–(4).
2 Preliminaries on quantum calculus
In this section, we recall some basic concepts on quantum calculus and provide some useful properties.
Next, we recall the following q-integration-by-parts rule.
B. Samet extends his appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
The authors declare that they carried out all the work in this manuscript and read and approved the final manuscript.
This work is supported by the Distinguished Scientist Fellowship Program (DSFP) at King Saud University.
The authors declare that they have no competing interests.
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